Question 1

For the numbers 653, 749, 872, 933, and 541, if in each number the first digit is replaced by the third digit, the second digit is replaced by the first digit, and the third digit is replaced by the second digit, which resulting number is the greate…

Accepted Answer

Introduction / Context: This problem tests your ability to manipulate digits of three digit numbers according to a specific rule and then compare the transformed results. Such digit manipulation and comparison tasks are common in arithmetic reasoning and help assess attention to detail as well as understanding of place value. Given Data / Assumptions: - The original numbers are 653, 749, 872, 933, and 541.- For each number, a digit transformation rule is applied.- The rule is: first digit is replaced by the third digit, second digit is replaced by the first digit, and third digit is replaced by the second digit.- We then compare the resulting transformed numbers to find the largest. Concept / Approach: Let a generic three digit number have digits A (hundreds), B (tens), and C (units). The rule says that A is replaced by C, B is replaced by A, and C is replaced by B. In other words, the new number has digits C, A, B in that order. Therefore, for any given three digit number, we can rewrite it in this permuted form and then evaluate the resulting numbers. After computing all the new numbers, we simply compare them numerically to find the greatest one. Step-by-Step Solution: - Take 653: here A = 6, B = 5, C = 3. The new number is C A B, that is 3 6 5 or 365.- Take 749: A = 7, B = 4, C = 9. The new number is 9 7 4, which is 974.- Take 872: A = 8, B = 7, C = 2. The new number is 2 8 7, which is 287.- Take 933: A = 9, B = 3, C = 3. The new number is 3 9 3, which is 393.- Take 541: A = 5, B = 4, C = 1. The new number is 1 5 4, which is 154.- Now compare the transformed values: 365, 974, 287, 393, and 154.- Among these, 974 is the largest.- 974 came from the original number 749. Verification / Alternative check: One quick check is to observe that the new hundreds digit is the original units digit. So any original number with the largest units digit is a strong candidate to give the largest transformed number. Among the units digits 3, 9, 2, 3, and 1, the largest is 9 from 749. That means the transformed number from 749 will have 9 in the hundreds place and is likely to be the greatest. The full numerical calculation confirms this intuition. Why Other Options Are Wrong: - 933 gives 393 after transformation, which is far smaller than 974.- 872 becomes 287, which has hundreds digit 2 and is clearly smaller.- 653 becomes 365, still below 974.- 541 becomes 154, which is the smallest among the transformed numbers. Common Pitfalls: A common mistake is to misapply the digit swapping rule, for example exchanging only two digits or misreading the order C, A, B. Another issue is forgetting that the largest hundreds digit dominates the comparison of three digit numbers. Always carefully write down the mapping from old to new digits and compute each transformed number before comparing them. Doing quick checks on place values helps verify that your answers are reasonable. Final Answer: The original number that becomes the greatest after the given transformation is 749.

Question 2

In elementary mathematics, how is the expression 0 divided by 0 classified?

Accepted Answer

Introduction / Context: This conceptual question asks about the nature of the expression 0 divided by 0, which often appears in algebraic manipulations and calculus. Understanding the correct classification of this expression helps prevent serious mistakes in problem solving, especially when simplifying fractions or dealing with limits. Given Data / Assumptions: - We are dealing with the expression 0 divided by 0.- Division is considered in the context of real numbers.- We must decide whether the result should be a specific number, undefined, infinite, or indeterminate. Concept / Approach: Division a / b in arithmetic means the number which when multiplied by b gives a. For example, 6 / 3 is 2 because 2 * 3 = 6. For 0 / 0, we are looking for a number x such that x * 0 = 0. However, any real number multiplied by 0 gives 0. This means there are infinitely many possible values of x that satisfy x * 0 = 0. Because there is no unique answer, the expression 0 / 0 does not have a well defined numerical value. In mathematics, such expressions are typically called indeterminate forms rather than simply undefined. The term indeterminate emphasizes that more information is required, often in the form of a limit or additional conditions. Step-by-Step Solution: - Consider the definition of division: for nonzero b, a / b is the unique number x such that x * b = a.- For 0 / 0, we would need to find x such that x * 0 = 0.- But for any real x, x * 0 = 0, so there is no unique solution.- Because there is no single value that 0 / 0 must take, it cannot be assigned a definite numerical value.- In many algebraic contexts, this is called an indeterminate form, indicating that the form by itself gives no information about the limit or value of a function. Verification / Alternative check: Imagine trying to simplify different fractions that look like 0 / 0. For example, in limits, the expression (x - x) / (x - x) simplifies to 0 / 0, but depending on the original function, the limit might be 1, 2, 5, or any other number. Another function like (x^2 - x^2) / (x - x) also gives 0 / 0 superficially but can lead to a different limiting value. Because the same symbolic pattern 0 / 0 can arise from many different underlying expressions with different behaviors, it is considered indeterminate. Additional analysis is always needed. Why Other Options Are Wrong: - Option 0: Choosing 0 as the value would ignore the fact that many other values also satisfy x * 0 = 0.- Option 1: There is no reason to pick 1, and 1 * 0 = 0 does not uniquely distinguish it.- Option Undefined: In basic arithmetic, division by zero such as 5 / 0 is undefined, but 0 / 0 is a special case categorized as an indeterminate form because of its many possible interpretations.- Option Infinity: Assigning infinity would be arbitrary and not consistent with the requirement of a unique real number result. Common Pitfalls: Many learners treat all division by zero expressions as the same and call them undefined without distinguishing 0 / 0 from other cases like 5 / 0. While both cannot be assigned an ordinary real value, 0 / 0 is special in algebra and calculus because it can correspond to many different limiting values depending on context. Always treat 0 / 0 as an indeterminate form that signals the need for deeper analysis rather than as a number you can plug into formulas. Final Answer: The expression 0 divided by 0 is classified as indeterminate.

Question 3

Rajitha has a calculator on which the digit keys have been interchanged in three disjoint pairs, and from the given correct outputs she deduces which digit keys are unchanged. Which set of three digit keys remains in its original position?

Accepted Answer

Introduction / Context: This is a logical puzzle based on a faulty calculator whose digit keys have been swapped in pairs. The puzzle provides a few sample calculations where the inputs are sequences of keys pressed and the outputs are numerically correct. From this information, we must infer which digit keys were never moved and therefore still produce their original digits. Such questions test pattern recognition, logical reasoning, and comfort with number operations. Given Data / Assumptions: - The calculator has digit keys 0 through 9.- Exactly three disjoint pairs of digit keys have swapped positions.- When Rajitha presses a key labelled d, the calculator internally reads some other digit f(d) because of the swap.- The calculator still performs arithmetic correctly on the internal digits and displays correct numerical results.- The following keypresses and displayed outputs are known: 1. Pressed 95 + 84, displayed 128. 2. Pressed 128 + 64, displayed 195. 3. Pressed 912 + 43, displayed 942. 4. Pressed 178 + 19, displayed 192. Concept / Approach: Let f(d) denote the actual digit value read when the key labelled d is pressed. Because only pairs were swapped, f is a permutation made up of three transpositions and four fixed points. For each equation, the sum of the numbers built from f applied to the pressed digits must equal the displayed result. By writing each pressed number digit by digit in terms of f and forming equations, we can deduce the mapping. Once the mapping is known, the fixed points give us the digits that were not moved. Step-by-Step Solution: - From 95 + 84 = 128, form 10 * f(9) + f(5) plus 10 * f(8) + f(4) equals 128.- From 128 + 64 = 195, form 100 * f(1) + 10 * f(2) + f(8) plus 10 * f(6) + f(4) equals 195.- From 912 + 43 = 942, form 100 * f(9) + 10 * f(1) + f(2) plus 10 * f(4) + f(3) equals 942.- From 178 + 19 = 192, form 100 * f(1) + 10 * f(7) + f(8) plus 10 * f(1) + f(9) equals 192.- Solving this system carefully, while remembering that each digit 0 through 9 appears exactly once as a value of f(d), yields the unique mapping: f(2) = 4, f(4) = 2, f(3) = 8, f(8) = 3, f(5) = 6, f(6) = 5.- All other digits are fixed: f(0) = 0, f(1) = 1, f(7) = 7, f(9) = 9.- Therefore, the digits 0, 1, 7, and 9 are unchanged, and any set containing only these digits is a correct set of unchanged keys.- Among the answer choices, the triple 7, 0, 1 consists entirely of unchanged digits. Verification / Alternative check: Check one of the equations using the mapping. For 95 + 84, we have f(9) = 9 and f(5) = 6, so the first number is 96. Also f(8) = 3 and f(4) = 2, so the second number is 32. Their sum is 96 + 32 = 128, which matches the displayed result. Similar checks on the other three sums confirm the mapping is consistent. This validates the conclusion about which digits are unchanged. Why Other Options Are Wrong: - 9, 0, 3: Digit 3 is involved in a swap with 8, so it is not unchanged.- 1, 3, 8: Both 3 and 8 are swapped, so this set cannot represent unchanged keys.- 2, 4, 1: Digits 2 and 4 are swapped with each other, so they are not fixed.- 5, 6, 9: Digits 5 and 6 form another swapped pair and are not unchanged. Common Pitfalls: Learners may try to guess the answer from only one equation or misunderstand that the keys were swapped in pairs, not arbitrarily. Ignoring the permutation structure leads to contradictions. A systematic approach that uses all the given equations and the condition that each digit is used exactly once resolves the puzzle uniquely. Always keep track of which digits have already been assigned in the mapping to avoid conflicts. Final Answer: The set of digit keys that remain in their original positions is 7, 0, 1.

Question 4

In arithmetic, when a negative number is divided by another negative number, what is the sign of the resulting quotient?

Accepted Answer

Introduction / Context: This is a basic sign rule question from arithmetic and algebra. Questions about the signs of results when multiplying or dividing positive and negative numbers commonly appear in aptitude tests. Knowing these rules saves time and helps prevent errors in longer calculations. Given Data / Assumptions: - We consider division of one negative number by another negative number.- Both numerator and denominator are nonzero real numbers.- We are asked only about the sign of the quotient, not the exact value. Concept / Approach: The sign rules for multiplication and division are closely related. When multiplying, positive times positive is positive, positive times negative is negative, and negative times negative is positive. Division is the inverse operation of multiplication, so the same sign rules carry over. Specifically, dividing a negative number by another negative number yields a positive result because it corresponds to answering the question: which number multiplied by a negative gives another negative? The answer must be positive. Step-by-Step Solution: - Let the two negative numbers be -a and -b, where a and b are positive real numbers.- The quotient is (-a) / (-b).- We can rewrite this as (-1 * a) / (-1 * b).- Group the sign factors: (-1 / -1) * (a / b).- The ratio (-1 / -1) is equal to 1, which is positive.- The ratio a / b is a positive number because a and b are positive.- Therefore, the product of these two positive quantities is positive.- Hence (-a) / (-b) is a positive number. Verification / Alternative check: Take a concrete example: (-10) divided by (-2). If the result were x, then x * (-2) should equal -10. The only real number x that satisfies x * (-2) = -10 is x = 5, which is positive. Similar examples such as (-9) / (-3) = 3 also demonstrate that the quotient is positive. This matches the general rule that a negative divided by a negative gives a positive result. Why Other Options Are Wrong: - Option -ve: A negative result would correspond to dividing numbers with opposite signs, such as negative by positive or positive by negative, which is not the case here.- Option 0: The quotient is zero only when the numerator is zero, not just because both numbers are negative.- Option Cannot be determined: The sign is completely determined by the signs of numerator and denominator, so there is no ambiguity.- Option Depends on numbers: The magnitude may depend on the numbers, but the sign does not as long as both are negative. Common Pitfalls: A common source of mistakes is mixing up the rules for adding negative numbers with those for multiplying or dividing them. While adding two negatives makes the result more negative, multiplying or dividing two negatives gives a positive result. Always remember that for multiplication and division, like signs give positive and unlike signs give negative. Final Answer: When a negative number is divided by another negative number, the quotient is positive (+ve).

Question 5

If the positions of the first and fifth digits of the number 14278359 are interchanged, the second and sixth digits are interchanged, the third and seventh digits are interchanged, and the fourth and eighth digits are interchanged, which digit will …

Accepted Answer

Introduction / Context: This question is a digit position puzzle that tests your ability to track multiple swaps within a single number. Such problems are designed to check attention to detail and understanding of positional notation in numbers. Correctly following each interchange step is essential to identify the digit at a specified final position. Given Data / Assumptions: - Original number: 14278359.- Positions are counted from left to right as first through eighth digit.- Swaps to be performed are: 1. First digit with fifth digit. 2. Second digit with sixth digit. 3. Third digit with seventh digit. 4. Fourth digit with eighth digit.- After all swaps, we must find the second digit from the right end. Concept / Approach: Label each digit of the original number by its position so that you can clearly follow the swaps. It helps to write the digits in a sequence and then systematically perform each interchange. Because every swap is between symmetric positions (first with fifth, second with sixth, and so on), the operation is straightforward if done methodically. After performing all swaps, count digits from the right to identify the second from the right. Step-by-Step Solution: - Write the original number with positions: 1 4 2 7 8 3 5 9.- Positions: 1st digit 1, 2nd 4, 3rd 2, 4th 7, 5th 8, 6th 3, 7th 5, 8th 9.- Swap positions 1 and 5: new 1st digit becomes 8, new 5th digit becomes 1.- Swap positions 2 and 6: new 2nd digit becomes 3, new 6th digit becomes 4.- Swap positions 3 and 7: new 3rd digit becomes 5, new 7th digit becomes 2.- Swap positions 4 and 8: new 4th digit becomes 9, new 8th digit becomes 7.- After all swaps, the digits from position 1 to 8 are: 8, 3, 5, 9, 1, 4, 2, 7.- The number after rearrangement is 83591427.- Counting from the right end, the rightmost digit is 7 and the second from the right is 2. Verification / Alternative check: A quick check is to verify that each original digit still appears exactly once after the swaps and that no digit was lost or duplicated. The final sequence 8, 3, 5, 9, 1, 4, 2, 7 contains exactly the same digits as 1, 4, 2, 7, 8, 3, 5, 9. This suggests the swaps were performed correctly. Rechecking the second from the right in 83591427 confirms that it is indeed 2. Why Other Options Are Wrong: - 5: This occurs as the third digit from the left, not the second from the right.- 7: This becomes the rightmost digit, not the second from the right.- 4: In the final number, 4 is in the sixth position from the left, which is third from the right, not second.- 8: This is the leftmost digit, far from the right end. Common Pitfalls: Many learners confuse the left and right positions or forget that all the swaps are to be applied to the original number, not step by step incorrectly. Another common mistake is to perform only some of the swaps or to miscount the final position when reading from the right. Writing the digits clearly and updating them carefully after each swap greatly reduces the chance of errors. Final Answer: After the specified rearrangement, the second digit from the right end is 2.

Question 6

Out of 6561 identical looking balls, exactly one ball is heavier than all the others. Using a balance scale that can compare the weights of two groups of balls, what is the minimum number of weighings needed to be certain of finding the heavy ball?

Accepted Answer

Introduction / Context: This is a classic puzzle about finding one heavier ball among many identical looking balls using a balance scale. It tests your ability to use divide and conquer strategies and to think in terms of base representations, specifically base 3. Such problems often appear in logical reasoning and competitive exams, especially in sections involving puzzles and optimization. Given Data / Assumptions: - There are 6561 balls in total.- All balls look identical, but exactly one ball is heavier than the others.- A balance scale is available which compares the total weight on its left pan to the total weight on its right pan.- Each weighing tells us whether the left side is heavier, the right side is heavier, or both sides are equal.- We want the minimum number of weighings that guarantees we can identify the heavy ball. Concept / Approach: The optimal strategy uses a ternary or base three approach. In each weighing, you can split the remaining candidate balls into three groups as equal in size as possible: left pan, right pan, and one group set aside. Depending on the outcome of the weighing, you can discard two thirds of the balls and keep one third for further investigation. This is because there are three possible outcomes for each weighing: left heavier, right heavier, or balanced. The idea is similar to a ternary search where the search space reduces by a factor of three each time. The number of weighings required is the smallest integer k such that 3^k is at least the total number of balls. Step-by-Step Solution: - Determine how many balls can be resolved with k weighings using the base three idea.- With 1 weighing, you can distinguish at most 3 possibilities (3^1).- With 2 weighings, at most 3^2 = 9 balls can be resolved.- With 3 weighings, at most 3^3 = 27 balls can be handled.- With 4 weighings, 3^4 = 81 balls.- With 5 weighings, 3^5 = 243 balls.- With 6 weighings, 3^6 = 729 balls.- With 7 weighings, 3^7 = 2187 balls.- With 8 weighings, 3^8 = 6561 balls.- Our problem has exactly 6561 balls, which equals 3^8.- Therefore, 8 weighings are sufficient to guarantee finding the heavier ball. Verification / Alternative check: You can imagine the process: initially divide the 6561 balls into three groups of 2187 each. Weigh two of these groups. If one side is heavier, the heavy ball is among those 2187 balls. If the scale balances, the heavy ball is among the remaining 2187 not weighed. This process can be repeated, always dividing the current candidate group into three equal parts. After each weighing, the candidate set shrinks by a factor of three. After 8 such steps, you will be left with exactly one ball that must be heavier, confirming that 8 weighings are enough. Why Other Options Are Wrong: - 2414 and 204: These values are far larger than the optimal and do not represent a minimal strategy.- 87: This is also much larger than needed and indicates using a naive process such as comparing very small groups repeatedly.- 10: While 10 weighings are more than enough, they are not minimal because 8 already suffice. Common Pitfalls: Many learners attempt a binary split strategy that divides the balls into two groups each time, which is less efficient with a three outcome scale. Another common error is to underestimate the power of exponential growth and assume that many more weighings are needed. Remember to match the number of outcomes per weighing with the base of the representation used for counting possibilities. A balance scale with three possible outcomes naturally suggests a base three strategy. Final Answer: The minimum number of weighings required to be certain of finding the heavy ball is 8.

Question 7

In the three digit number 103, what is the place value of the digit 0?

Accepted Answer

Introduction / Context: This question tests basic understanding of place value in the decimal number system, specifically the role of the digit zero in a number like 103. Place value is a foundation for all further work in arithmetic and number operations, so it is important to distinguish between position (place) and actual contribution to the value of the number. Given Data / Assumptions: - The number given is 103.- It is a three digit number with digits 1, 0, and 3.- The digit 1 is in the hundreds place, 0 is in the tens place, and 3 is in the ones place.- We are asked for the place value of the digit 0. Concept / Approach: In the decimal place value system, the value of a digit depends on its position. The digit in the ones place is multiplied by 1, the digit in the tens place is multiplied by 10, and the digit in the hundreds place is multiplied by 100. More generally, a digit in the k th place from the right (starting with ones as first) has a place value equal to digit times 10^(k - 1). When the digit is zero, its place value is zero regardless of the place, because zero multiplied by any power of ten is still zero. Nevertheless, the zero still serves an important role as a placeholder to indicate that there are no tens in the number. Step-by-Step Solution: - Write the number 103 in expanded form.- 103 = 1 * 100 + 0 * 10 + 3 * 1.- The digit 1 in the hundreds place contributes 1 * 100 = 100 to the total.- The digit 0 in the tens place contributes 0 * 10 = 0 to the total.- The digit 3 in the ones place contributes 3 * 1 = 3 to the total.- Therefore, the place value of the digit 0 is 0. Verification / Alternative check: If the tens digit were not zero but some other digit, say 5, the number would be 153, and the tens digit would contribute 5 * 10 = 50. In 103, the absence of any tens is represented by 0 in the tens place, whose place value is 0. This shows that zero in a place always contributes zero to the total value but still influences how we interpret the other digits by shifting them into higher place positions. Why Other Options Are Wrong: - 100: This is the place value of the digit 1 in the hundreds place, not the digit 0.- 10: This is the place value factor for the tens place, but the digit itself is zero, so the product is 0.- 103: This is the value of the entire number, not the place value of a single digit.- 1: This is the place value of a digit in the ones place if that digit is 1, not the tens digit 0. Common Pitfalls: Some learners confuse the name of the place with the place value. The tens place always uses the factor 10, but if the digit there is 0, then the place value is 0. Another mistake is to think that zero can be ignored entirely, when in fact it is essential for correctly representing numbers like 103, where it ensures that the 1 remains in the hundreds place rather than shifting into the tens place. Final Answer: The place value of the digit 0 in 103 is 0.

Question 8

In the context of psychological and employment testing, aptitude tests are primarily designed to measure which of the following aspects of human ability?

Accepted Answer

Introduction / Context: This verbal reasoning question focuses on the purpose of aptitude tests in academic, psychological, and employment settings. Aptitude tests are commonly used in recruitment processes, entrance examinations, and career guidance to assess a range of cognitive abilities rather than knowledge of specific subjects. Understanding what they measure helps candidates prepare effectively and interpret test results correctly. Given Data / Assumptions: - The options mention general reasoning, perceptual speed, verbal comprehension, and a combined alternative.- We are asked to identify what aptitude tests are designed to measure.- Physical strength is also listed as an option, which suggests a contrast between cognitive and physical attributes. Concept / Approach: Aptitude tests are standardized assessments designed to evaluate potential to learn, solve problems, and adapt to new tasks. They usually focus on mental abilities, such as logical reasoning, numerical ability, spatial visualization, verbal comprehension, and speed of perception or decision making. These tests aim to predict how well a person will perform in training or in a particular job role. Physical strength is typically measured by other types of tests, such as physical fitness or performance tests, not by aptitude tests. Step-by-Step Solution: - Consider general reasoning: this includes logical thinking, pattern recognition, and problem solving. Aptitude tests almost always have reasoning components.- Perceptual speed refers to how quickly and accurately a person can notice details, compare information, and make simple decisions under time pressure. Many aptitude tests include sections that measure this ability, for example, symbol matching tasks.- Verbal comprehension involves understanding written or spoken language, interpreting meanings, and drawing inferences from text. Verbal ability sections in aptitude tests specifically target this skill.- Since aptitude tests are designed to measure a wide range of these mental abilities together, they do not focus solely on one of them but on a combination.- Physical strength is not a primary target of aptitude tests; it is evaluated through entirely different types of assessments.- Therefore, the best description among the options is that aptitude tests measure all of the listed cognitive abilities. Verification / Alternative check: Look at typical sections of well known aptitude test batteries used by employers and educational institutions. They often include verbal reasoning, numerical reasoning, abstract or logical reasoning, clerical speed and accuracy, and sometimes spatial ability. These correspond closely to general reasoning, perceptual speed, and verbal comprehension. This confirms that aptitude tests cover more than a single narrow skill and aim at a broad profile of mental abilities. Why Other Options Are Wrong: - General reasoning alone is too narrow because aptitude tests usually assess multiple skill areas.- Perceptual speed alone does not capture verbal or numerical thinking.- Verbal comprehension alone ignores important non verbal factors such as numerical and spatial skills.- Physical strength is generally outside the scope of aptitude testing and is not the main objective of these tests. Common Pitfalls: Some candidates think aptitude tests are just reasoning tests or just verbal tests because of their experience with a particular examination. In reality, aptitude batteries can be broad and multi dimensional. Another misconception is to confuse aptitude tests with achievement tests, which measure what a person has already learned in a subject like mathematics or history. Remember that aptitude tests are about potential and cognitive abilities, not about specific knowledge or physical characteristics. Final Answer: Aptitude tests are designed to measure all of the above cognitive abilities, including general reasoning, perceptual speed, and verbal comprehension.

Question 9

In binary representation used in digital electronics and computer systems, which digit represents the OFF state?

Accepted Answer

Introduction / Context: This question checks your understanding of the basic binary system, which is fundamental in computer science and digital electronics. Binary uses only two digits, and these digits correspond closely to physical states such as ON and OFF, or high and low voltage, in digital circuits. Knowing which digit corresponds to the OFF state is essential for interpreting logic diagrams and truth tables. Given Data / Assumptions: - Binary representation uses two digits: 0 and 1.- These digits are mapped to logical and physical states in digital systems.- We are asked specifically which digit represents the OFF state. Concept / Approach: In digital electronics and binary logic, the digit 1 generally represents the presence of a signal, a high voltage level, logical true, or ON. In contrast, the digit 0 represents the absence of a signal, a low voltage level, logical false, or OFF. This mapping is consistent across most digital systems and programming languages. The binary system is thus a two state encoding of physical ON and OFF conditions. Step-by-Step Solution: - Recall that binary digits are 0 and 1.- In logical terms, 1 corresponds to true or ON, and 0 corresponds to false or OFF.- In hardware, 1 generally corresponds to a higher voltage level, while 0 corresponds to a lower or zero voltage level.- Because OFF means the signal or power is not present, this aligns with the digit 0.- Therefore, in binary representation the OFF state is represented by digit 0. Verification / Alternative check: Look at common logic gates such as AND, OR, and NOT in digital electronics. Their truth tables always use 0 to represent the false or OFF condition and 1 for the true or ON condition. For example, in an AND gate, input 0 means the line is OFF and cannot contribute to turning the output ON. These standard conventions confirm that 0 is associated with OFF in binary systems. Why Other Options Are Wrong: - Option 1: This digit corresponds to ON, true, or high state in binary logic.- Options 2, 3, and 4: These digits are not even valid binary digits and are not used to represent basic ON or OFF states in binary coded systems. Common Pitfalls: Beginners sometimes confuse the mapping when they see diagrams where a line labelled 1 is drawn low and 0 is drawn high for stylistic reasons. The correct interpretation always follows the underlying convention: 1 is a logically active or ON state, and 0 is logically inactive or OFF. Remember that the binary system is strictly limited to digits 0 and 1, so any value outside this set cannot represent a basic binary state. Final Answer: In binary representation, the digit representing the OFF state is 0.

Question 10

Approximately how much does one US gallon of water weigh in kilograms at normal room temperature?

Accepted Answer

Introduction / Context: This general knowledge and practical reasoning question asks about the approximate weight of one US gallon of water in kilograms. Such conversions are useful in everyday life, science, and engineering, especially when working with recipes, fuel calculations, or laboratory measurements that switch between volume and mass units. Given Data / Assumptions: - The question refers to a US gallon, not an imperial gallon.- We assume normal room temperature conditions where the density of water is close to 1 kilogram per liter.- We must choose the closest value in kilograms from the options provided. Concept / Approach: The key relationship is between volume in liters and mass in kilograms for water. At standard conditions, the density of water is approximately 1 kilogram per liter, meaning 1 liter of water has a mass close to 1 kilogram. One US gallon is defined as approximately 3.785 liters. Therefore, the mass of one US gallon of water is approximately 3.785 kilograms. The options show several values near this figure, and we must pick the one that matches this standard conversion. Step-by-Step Solution: - Recall the conversion between US gallons and liters: 1 US gallon is approximately 3.785 liters.- Assume the density of water is about 1 kilogram per liter under normal conditions.- Therefore, the mass of 1 US gallon of water is roughly equal to 3.785 liters multiplied by 1 kilogram per liter.- This gives an approximate mass of 3.785 kilograms.- Compare this value with the answer options: 2.741, 2.956, 3.418, 3.785, and 4.500 kilograms.- The option 3.785 kilograms matches the standard conversion and is closest to the true value. Verification / Alternative check: If necessary, you can verify the reasoning by remembering that 1 liter is exactly 1000 cubic centimeters, and the density of pure water at about 4 degrees Celsius is very close to 1 gram per cubic centimeter, leading to 1 kilogram per liter. Even at room temperature, this approximation remains good for everyday purposes. Multiplying 1 kilogram per liter by 3.785 liters confirms a mass of around 3.785 kilograms for 1 US gallon of water. Why Other Options Are Wrong: - 2.741 kgs and 2.956 kgs are significantly below the expected 3.785 kilograms and would correspond to a lower density than water actually has.- 3.418 kgs is closer but still underestimates the standard conversion.- 4.500 kgs is higher than the expected value and does not match the standard gallon to liter conversion. Common Pitfalls: One common confusion is between US gallons and imperial gallons. An imperial gallon is about 4.546 liters, which would give a mass closer to 4.546 kilograms of water, not 3.785 kilograms. The question explicitly refers to a US gallon, so using the imperial conversion leads to the wrong answer. Another mistake is to forget that the density of water is approximately 1 kilogram per liter, making the calculation more complicated than necessary. Final Answer: One US gallon of water weighs approximately 3.785 kilograms at normal room temperature.

Question 11

In arithmetic reasoning, what is the smallest possible number of ducks that can swim in the described formation where there are two ducks in front of a duck, two ducks behind a duck, and one duck between two ducks?

Accepted Answer

Introduction / Context: This puzzle style question about ducks is a classic example of arithmetic reasoning and logical arrangement. The wording sounds as if many ducks are needed, but with careful visualization we can see that a much smaller number satisfies all conditions. Questions like this test the ability to translate a verbal description into a mental picture and to avoid being misled by the apparent complexity of the language. Given Data / Assumptions: - Ducks are assumed to stand or swim in a straight line for simplicity. - The formation must satisfy all three conditions at the same time. - The conditions are: two ducks in front of a duck, two ducks behind a duck, and one duck between two ducks. - We are asked for the smallest possible number of ducks that fits all conditions together, not each condition separately with different groups. Concept / Approach: The key idea is to identify whether one small group of ducks can simultaneously satisfy all three positional conditions. Instead of first assuming a large number of ducks, we test the smallest possible arrangements and see if there is a way to interpret the phrases from the point of view of different ducks in the line. The phrase two ducks in front of a duck does not require that every duck has two ducks in front, only that there exists at least one duck for which this is true. The same logic applies to two ducks behind a duck and one duck between two ducks. Step-by-Step Solution: Step 1: Consider three ducks in a straight line and label them A, B, and C from left to right. Step 2: For duck C, there are ducks A and B in front of it, so we have two ducks in front of a duck satisfied by looking from the position of duck C. Step 3: For duck A, there are ducks B and C behind it, so we have two ducks behind a duck satisfied by looking from the position of duck A. Step 4: For duck B, duck A is in front and duck C is behind, so there is exactly one duck between two ducks, which satisfies the third condition. Step 5: All three conditions are met simultaneously by the same arrangement of only three ducks, so there is no need to add more ducks. Verification / Alternative check: We can verify by trying to use fewer than three ducks. With only one duck, none of the conditions about ducks in front or behind can be met. With two ducks, there can be one duck in front of another and one behind another, but we can never have two in front or two behind. For four or more ducks, many arrangements will also satisfy the conditions, but the question specifically asks for the smallest number. Since three ducks already satisfy all statements and fewer than three cannot, three is the minimal valid answer. Why Other Options Are Wrong: Option 5: With five ducks the conditions can be satisfied, but this is not the smallest possible number, so it does not answer the question correctly. Option 7: Seven ducks are more than enough, but again this ignores the requirement to find the minimum number of ducks that works. Option 9: Nine ducks also can be arranged to meet the description, yet the question clearly emphasises the smallest possible number, which nine is not. Option None of these: This would mean that no listed number works, but we have already demonstrated that three ducks do work, so this option is incorrect. Common Pitfalls: A frequent mistake is to assume that every duck in the group must simultaneously have two ducks in front and two behind, which forces students to imagine long lines of ducks. Another common error is to forget that the conditions can be satisfied by different ducks in the same formation. Learners may also overlook the minimality requirement and choose a larger number that works without checking whether a smaller arrangement is possible. Visualizing the positions or drawing a simple sketch usually reduces these errors. Final Answer: The smallest possible number of ducks in the formation is 3.

Question 12

Two friends Jack and Bill discuss the ages of Bills three children; the product of their ages is 72 and the sum of their ages equals Jacks birth date, and using the extra clue that there is a unique eldest child who has just started piano lessons, f…

Accepted Answer

Introduction / Context: This well known puzzle about the ages of Bills three children combines number factorization, logical reasoning, and the use of indirect information. The story mentions the product of the ages, the sum matching a birth date, and an extra clue involving the eldest child. Such questions appear in logical reasoning and arithmetic reasoning sections to test systematic thinking rather than pure calculation speed. Given Data / Assumptions: - Bill has exactly three children. - The product of their ages is 72. - The sum of their ages equals Jack's birth date, which must be a possible calendar day number between 1 and 31 inclusive. - Even after knowing the product and the sum, Jack initially remains unsure about the ages. - The final clue that there is a unique eldest child who just started piano lessons allows Jack to determine the correct ages. Concept / Approach: The main technique is to list all possible triples of positive integer ages whose product is 72. For each triple, we compute the sum and interpret it as a possible birth date. The statement that Jack still does not know the ages after hearing that the sum equals his birth date implies that more than one triple corresponds to the same sum, otherwise he would have identified the ages immediately. Therefore we look for sums that appear more than once among the candidate triples. Finally, the information that there is a unique eldest child eliminates any triple where the two oldest ages are equal, since that would imply joint eldest children rather than a single eldest child. Step-by-Step Solution: Step 1: List positive integer triples whose product is 72, such as (1, 1, 72), (1, 2, 36), (1, 3, 24), (1, 4, 18), (1, 6, 12), (1, 8, 9), (2, 2, 18), (2, 3, 12), (2, 4, 9), (2, 6, 6), (3, 3, 8), and (3, 4, 6). Step 2: Compute sums: examples include 1 + 6 + 12 = 19, 1 + 8 + 9 = 18, 2 + 6 + 6 = 14, 3 + 3 + 8 = 14, and so on. Step 3: Identify sums that repeat. The sum 14 occurs for two different triples: (2, 6, 6) and (3, 3, 8). This means that if Jack's birth date is 14, he would still be unsure which triple is correct after hearing the product and the sum. Step 4: The story says Jack remains uncertain until Bill adds that his eldest child has started piano lessons. This implies there is a unique eldest child. Step 5: In the triple (2, 6, 6) there are two children aged 6, so there is no single eldest child. In (3, 3, 8) the child aged 8 is uniquely oldest. Therefore (3, 3, 8) is the only triple consistent with all clues. Verification / Alternative check: We can quickly verify that 3 * 3 * 8 = 72 and that the sum 3 + 3 + 8 = 14, which is indeed a valid day number for a birth date. Checking the rejected triple (2, 6, 6) also gives a product of 72 and sum 14, confirming why Jack remained uncertain before the final clue. The last statement about the eldest child is precisely what disambiguates the two possibilities. No other triple offers this exact combination of repeated sum, matching product, and existence of a unique eldest child. Why Other Options Are Wrong: Option 2, 6, 6: This triple has a correct product and sum, but it has two eldest children tied at age 6, so there is no unique eldest child. Option 1, 6, 12: The product is 72 but the sum is 19, which would have uniquely identified the ages immediately, contradicting the story of Jack's temporary confusion. Option 1, 8, 9: Here the product is 72 and the sum is 18, again unique as a sum, so Jack would not have remained unsure after hearing the sum. Option 2, 3, 12: This also has product 72 but a sum of 17, which is a distinct birth date and gives no ambiguity. Common Pitfalls: Many learners focus only on the original product and ignore the narrative clue about Jack's uncertainty. Others may stop after finding any triple that multiplies to 72 rather than systematically listing all possibilities. Another common error is forgetting that a calendar birth date must be between 1 and 31, which restricts the possible sums. A final pitfall is misinterpreting the phrase eldest child and not realizing that any triple with two equal largest ages is ruled out. Final Answer: The ages of Bills three children are 3 years, 3 years, and 8 years.

Question 13

The difference between two numbers is 5 and six times the smaller number is 6 less than four times the greater number; find the values of the two numbers.

Accepted Answer

Introduction / Context: This problem is a standard example of forming and solving simultaneous linear equations from a word description. Arithmetic reasoning questions like this test the ability to translate relationships stated in words into algebraic equations and then solve them in a structured way. The information about a difference and a comparison of multiples of the numbers is sufficient to determine both values exactly. Given Data / Assumptions: - There are two real numbers, one greater and one smaller. - The difference between the greater number and the smaller number is 5. - Six times the smaller number is 6 less than four times the greater number. - We assume the numbers are positive, consistent with typical aptitude questions, but the algebra works for general real numbers. Concept / Approach: We introduce variables to represent the greater and the smaller number, then carefully convert each English statement into an algebraic equation. The phrase difference between two numbers is 5 becomes one equation, and the phrase six times the smaller lacks by 6 from four times the greater becomes the second equation. Solving this pair of simultaneous equations gives unique values for the two numbers. This is a systematic approach widely applicable to many linear word problems. Step-by-Step Solution: Step 1: Let the greater number be x and the smaller number be y. Step 2: The difference condition gives x - y = 5. Step 3: The statement six times the smaller lacks by 6 from four times the greater means that four times the greater is six more than six times the smaller. So we write 4 * x = 6 * y + 6. Step 4: Rearrange the second equation as 4x - 6y = 6. Step 5: From x - y = 5, express x in terms of y as x = y + 5. Step 6: Substitute x = y + 5 into 4x - 6y = 6 to get 4(y + 5) - 6y = 6. Step 7: Expand and simplify: 4y + 20 - 6y = 6 gives -2y + 20 = 6, so -2y = -14 and therefore y = 7. Step 8: Substitute y = 7 back into x = y + 5 to obtain x = 12. Step 9: Thus the greater number is 12 and the smaller number is 7. Verification / Alternative check: Check the difference: 12 - 7 = 5, which matches the first condition. Check the second condition: six times the smaller is 6 * 7 = 42 and four times the greater is 4 * 12 = 48. The phrase lacks by 6 means 42 is 6 less than 48, which is correct. Therefore both conditions are satisfied exactly by the pair (12, 7), confirming our solution. No other option pair given in the choices simultaneously satisfies both equations. Why Other Options Are Wrong: Option 13 and 8: The difference is 5, but 6 * 8 = 48 and 4 * 13 = 52, and 48 is not exactly 6 less than 52. Option 15 and 10: The difference is 5, but 6 * 10 = 60 and 4 * 15 = 60, so there is no difference of 6 between these multiples. Option 14 and 9: The difference is again 5, but 6 * 9 = 54 and 4 * 14 = 56, which differ by only 2, not 6. Option 17 and 12: These differ by 5, but 6 * 12 = 72 and 4 * 17 = 68, making the smaller multiple greater than the larger multiple, contrary to the condition given. Common Pitfalls: One common mistake is misreading the phrase lacks by 6 and writing 6y = 4x + 6 instead of 4x = 6y + 6. Another error is reversing the difference equation as y - x = 5, which changes the sign and leads to incorrect values. Some learners also try to guess and check options without forming equations, which can be time consuming and error prone for more complex problems. Writing variables, forming equations carefully, and then solving them systematically is the most reliable method. Final Answer: The two numbers are 12 and 7.

Question 14

Express seventeen tens in standard unit form by writing the equivalent number in terms of hundreds, tens, and ones.

Accepted Answer

Introduction / Context: This question belongs to basic place value and unit form concepts in arithmetic. Understanding how tens, hundreds, and ones relate is fundamental in primary mathematics and is often revisited in aptitude tests to ensure that candidates have a clear mental model of our base ten number system. Here we are asked to convert seventeen tens into a standard expanded form that explicitly shows hundreds, tens, and ones. Given Data / Assumptions: - We are working in the base ten number system. - One ten is equal to ten ones. - Ten tens equal one hundred. - The phrase seventeen tens means 17 groups of ten, which must be rewritten in standard unit form as hundreds, tens, and ones. Concept / Approach: The base ten system organizes numbers into powers of ten: ones, tens, hundreds, thousands, and so on. Each place is ten times the value of the place to its right. To convert seventeen tens into standard form, we first convert tens into an equivalent number of ones, obtain the overall number, and then express that number back into hundreds, tens, and ones. Recognizing when a collection of tens can be regrouped into hundreds is the main idea tested here. Step-by-Step Solution: Step 1: Interpret the phrase seventeen tens as 17 multiplied by 10 because each ten is a group of 10 ones. Step 2: Compute 17 * 10 = 170. This means seventeen tens are equal to 170 ones. Step 3: Now we express 170 in standard place value form. The number 170 has digits 1, 7, and 0, reading from left to right as hundreds, tens, and ones. Step 4: The digit 1 in the hundreds place represents 1 hundred, which equals 100. Step 5: The digit 7 in the tens place represents 7 tens, which equals 70. Step 6: The digit 0 in the ones place represents 0 ones, so there are no single units left over. Step 7: Therefore, seventeen tens can be written as 1 hundred, 7 tens, and 0 ones in unit form. Verification / Alternative check: We can verify by expanding the standard unit form back into a single number. One hundred plus seventy plus zero equals 100 + 70 + 0 = 170. This matches the earlier calculation of 17 * 10. Another way to check is to imagine grouping ten tens into a hundred: from seventeen tens, ten tens form one hundred, and seven tens remain. That gives 1 hundred and 7 tens, exactly the structure in the correct option. Since both methods give the same result, we can be confident in the answer. Why Other Options Are Wrong: Option 17 ones: This corresponds to the number 17, which would be 1 ten and 7 ones, not seventeen tens. Option 170 ones: This describes the total number of ones correctly but does not express the number in hundreds, tens, and ones unit form as required by the question. Option 1 ten and 7 ones: This is the expanded form of 17, not of 170, so it represents seventeen ones, not seventeen tens. Option 7 tens and 10 ones: Ten ones make 1 ten, so this combines to 8 tens or 80, which is far less than 170. Common Pitfalls: Learners sometimes confuse the phrases seventeen and seventeen tens, treating them as the same. Another common mistake is to stop at 170 ones without translating that into hundreds, tens, and ones. Some students may also mis-handle regrouping, for example thinking that 10 ones always stay as 10 ones instead of regrouping into 1 ten. Practising conversions in both directions, from unit form to standard form and back, helps reinforce these place value relationships. Final Answer: Seventeen tens in unit form is 1 hundred, 7 tens, and 0 ones.

Question 15

In basic geometry, a chord that passes through the center of a circle is called what type of line segment?

Accepted Answer

Introduction / Context: This question belongs to elementary Euclidean geometry. It checks familiarity with basic terms related to circles, such as chord, radius, diameter, secant, and tangent. Knowing the definitions and how they are related is essential for solving more advanced problems involving arcs, angles, and circle theorems. Given Data / Assumptions: - We are dealing with a circle in a plane, which has a fixed center point and all points on the circle at a constant distance from the center. - A chord is defined as a line segment whose endpoints both lie on the circle. - The chord in the question specifically passes through the center of the circle. - We need to identify the standard geometric name for such a chord. Concept / Approach: In circle geometry, the radius is the line segment from the center to any point on the circle. A diameter is a chord that passes through the center and whose length is twice the radius. A secant is a line that cuts the circle at two points, extending beyond the circle on both sides, while a tangent touches the circle at exactly one point. Recognizing that a chord passing through the center is the longest possible chord leads directly to the concept of diameter. Step-by-Step Solution: Step 1: Recall that a chord is any line segment with both endpoints on the circle. Step 2: The special case where such a chord passes through the center of the circle has a standard name in geometry. Step 3: By definition, a diameter is a chord that passes through the center and connects two points on the circle, with length equal to twice the radius. Step 4: A tangent touches the circle at only one point and does not pass through the center. Step 5: A secant is a full line that intersects the circle in two points, not merely a segment inside it. Step 6: The word perigram is not a standard geometric term associated with circles in basic geometry. Step 7: Therefore the only correct term for a chord that passes through the center is diameter. Verification / Alternative check: To verify, imagine a circle with center O and radius r. Any radius has length r, while a diameter is formed by combining two opposite radii along a straight line, giving length 2 * r. This diameter clearly passes through the center O and has endpoints on the circle, so it meets the definition of a chord that passes through the center. No other named object in the basic list matches this description exactly, which confirms the identification. Why Other Options Are Wrong: Option secant: A secant is a line that intersects the circle at two points and continues outside the circle. It is not limited to the segment inside the circle and does not have to pass through the center. Option tangent: A tangent only touches the circle at one point and never passes through the interior of the circle, so it cannot be the chord described. Option perigram: This is not a standard name used for chords or lines in classical school level circle geometry. Option radius: A radius connects the center to a point on the circle, but it has only one endpoint on the circle, so it is not a chord with two endpoints on the circle. Common Pitfalls: Students sometimes confuse radius and diameter, thinking that any segment through the center is called a radius. Another confusion arises between secant and chord, since both relate to two points on the circle, but a secant is an infinite line rather than a finite segment. Remembering that a diameter is both a chord and the longest chord in a circle helps solidify the correct concept. Final Answer: The chord that passes through the center of a circle is called the diameter.

Question 16

Identify which of the following given numbers is a composite number, that is, a positive integer with more than two distinct positive divisors.

Accepted Answer

Introduction / Context: This question tests the understanding of prime and composite numbers, which is a fundamental topic in number theory and arithmetic reasoning. Composite numbers have more than two distinct positive divisors, while prime numbers have exactly two distinct positive divisors, namely 1 and the number itself. The special cases 0 and 1 are neither prime nor composite and often appear in aptitude questions to check conceptual clarity. Given Data / Assumptions: - The options shown are 0, 19, 29, 91, and 1. - We use the standard definitions of prime and composite numbers for positive integers greater than 1. - The task is to pick a number that has at least one non trivial factorization apart from 1 and itself. Concept / Approach: A composite number n greater than 1 has at least one divisor d such that 1 < d < n. Primes such as 19 and 29 do not have such divisors. The numbers 0 and 1 are handled separately: 1 has only one positive divisor, and 0 is divisible by every nonzero integer but is not classified as prime or composite in usual definitions. Therefore, we check each option for factorization. If a number can be expressed as a product of two smaller positive integers (both greater than 1), it is composite. Step-by-Step Solution: Step 1: Examine 19. It is not divisible by 2, 3, 5, or any integer smaller than its square root except 1. Therefore 19 is prime. Step 2: Examine 29. Similar checks show that 29 is not divisible by 2, 3, 5, or any integer smaller than its square root except 1, so 29 is also prime. Step 3: Examine 0. Zero is divisible by every nonzero integer but by convention is not considered either prime or composite, so it is not the desired answer. Step 4: Examine 1. The integer 1 has only one positive divisor, which is itself, and therefore fails the requirement of having more than two divisors. It is neither prime nor composite. Step 5: Examine 91. Try simple factors: 7 * 13 = 91. Both 7 and 13 are greater than 1 and less than 91, so 91 has at least two proper divisors beyond 1 and 91. Step 6: Since 91 can be written as a product of 7 and 13, it is a composite number. Verification / Alternative check: To verify, list the positive divisors of 91: 1, 7, 13, and 91. Because there are more than two distinct positive divisors, 91 fits the definition of a composite number. For comparison, the divisors of 19 are just 1 and 19, and similarly for 29. The integer 1 has only the divisor 1, and 0 does not meet the standard conditions for primality or compositeness. Therefore only 91 satisfies the criterion given in the question. Why Other Options Are Wrong: Option 0: Although 0 is divisible by all nonzero integers, it is not classified as composite in basic number theory and is treated as a special case. Option 19: This number has exactly two divisors, 1 and 19, and is therefore prime, not composite. Option 29: Just like 19, the number 29 has only 1 and 29 as divisors, so it is also prime. Option 1: The number 1 has only one positive divisor and fails the requirement for either prime or composite classification. Common Pitfalls: A frequent misunderstanding is to treat 1 or 0 as composite simply because they do not seem prime. Another error is not testing divisibility properly and assuming that larger numbers like 91 must be prime. Remember that checking divisibility by small primes (2, 3, 5, 7, and so on) up to the square root of the number is usually sufficient to decide primality for test level questions. Final Answer: The composite number among the given options is 91.

Question 17

Identify the correct set of multiples of 18 from the given options, noting that a multiple of 18 must be exactly divisible by 18 without leaving any remainder.

Accepted Answer

Introduction / Context: This question checks the understanding of multiples and divisibility, which is a core arithmetic skill. Recognizing whether a number is a multiple of a given integer is important in topics such as least common multiples, factors, and divisibility rules. Here we are asked to determine which of the listed numbers are multiples of 18 and then choose the correct option summarizing that information. Given Data / Assumptions: - The base number is 18. - The individual numbers presented are 144, 72, and 306. - A multiple of 18 is any number that can be written as 18 * n for some integer n. - We assume all numbers are in base ten and are positive integers. Concept / Approach: To determine whether a number is a multiple of 18, we can test divisibility by 18 directly or break 18 into its prime factors 2 and 9, then check divisibility by both 2 and 9. A number is divisible by 2 if it ends in an even digit and divisible by 9 if the sum of its digits is a multiple of 9. Once we confirm that each number fits these conditions, we can decide whether all, some, or none of the numbers are multiples of 18 and choose the appropriate option. Step-by-Step Solution: Step 1: Check 144. It is even, so divisible by 2. The sum of its digits is 1 + 4 + 4 = 9, which is a multiple of 9. Therefore 144 is divisible by both 2 and 9, hence by 18. Step 2: Check 72. It is also even. The sum of digits is 7 + 2 = 9, a multiple of 9, so 72 is divisible by 18 as well. Step 3: Check 306. It is even, ending in 6. The sum of digits is 3 + 0 + 6 = 9, which again is a multiple of 9, so 306 is divisible by both 2 and 9 and therefore by 18. Step 4: Alternatively, we can divide each number by 18 directly: 144 / 18 = 8, 72 / 18 = 4, and 306 / 18 = 17, each yielding an integer quotient. Step 5: Since all three numbers are multiples of 18, the correct statement is that all of the listed numbers are multiples of 18. Verification / Alternative check: We can rewrite each number explicitly as a product of 18 and an integer: 144 = 18 * 8, 72 = 18 * 4, and 306 = 18 * 17. Because all three can be expressed in this way, they all qualify as multiples of 18. This matches the earlier divisibility tests and confirms that the correct option is the one stating that all of the above numbers satisfy the condition. Why Other Options Are Wrong: Option 144: Selecting this alone would ignore the fact that 72 and 306 are also multiples of 18. Option 72: Similarly, choosing only 72 fails to recognize that 144 and 306 also meet the divisibility requirement. Option 306: Picking only 306 is incorrect because 144 and 72 have already been shown to be multiples of 18. Option None of these: This would mean none of the numbers are multiples of 18, which contradicts our calculations. Common Pitfalls: A common mistake is to check only divisibility by 2, forgetting the factor of 9, or vice versa. Some students also miscalculate the digit sum or misperform the division by 18. Using both the factor method (2 and 9) and direct division can help confirm the result. Another pitfall is to assume that only one option can be correct in questions where sometimes all given values satisfy the condition. Final Answer: All of the given numbers 144, 72, and 306 are multiples of 18, so the correct option is All of the above.

Question 18

A girl counts on the fingers of her left hand by calling the thumb 1, index finger 2, middle finger 3, ring finger 4, little finger 5, and then reversing direction (ring finger 6, middle finger 7, index finger 8, thumb 9, and so on); if she continue…

Accepted Answer

Introduction / Context: This question is a classic arithmetic reasoning puzzle based on repeating patterns and modular arithmetic. The girl counts on her fingers forward and backward, creating a cyclical sequence of fingers that repeats after a fixed number of counts. We are asked to determine which finger corresponds to a large count, 1995, without listing all intermediate steps one by one. Recognizing the repeating cycle and using division with remainders is the key to efficient solving. Given Data / Assumptions: - The girl uses five fingers of her left hand: thumb, index, middle, ring, and little finger. - She labels them in order as 1 thumb, 2 index, 3 middle, 4 ring, 5 little. - After reaching the little finger, she reverses direction and continues 6 ring, 7 middle, 8 index, 9 thumb, then repeats the pattern. - We assume she continues indefinitely in this forward backward manner until reaching 1995. Concept / Approach: The sequence of fingers follows a repeating pattern. Instead of tracking all 1995 counts, we identify the length of one full cycle and then compute the position of 1995 within that cycle using modular arithmetic. Once we know the remainder when 1995 is divided by the cycle length, we match that remainder to the corresponding finger in one cycle. This is a standard approach to problems involving periodic behaviour. Step-by-Step Solution: Step 1: Write down the sequence of fingers for the first few counts: 1 thumb, 2 index, 3 middle, 4 ring, 5 little, 6 ring, 7 middle, 8 index, 9 thumb, and then the pattern repeats. Step 2: Observe that the distinct finger sequence over one full cycle of counts is: thumb, index, middle, ring, little, ring, middle, index. This cycle has length 8. Step 3: Represent the cycle as positions 1 to 8: 1 thumb, 2 index, 3 middle, 4 ring, 5 little, 6 ring, 7 middle, 8 index. Step 4: To find the finger for count 1995, divide 1995 by 8, the cycle length, and find the remainder. Step 5: Perform the division: 1995 / 8 gives a quotient of 249 with a remainder of 3, since 8 * 249 = 1992 and 1995 - 1992 = 3. Step 6: A remainder of 3 means that the 1995th count falls on the same position in the cycle as count 3. Step 7: From the cycle, position 3 corresponds to the middle finger. Therefore the girl stops at the middle finger. Verification / Alternative check: We can verify the logic by checking a smaller example. For instance, count 11 also gives remainder 3 when divided by 8 (since 11 - 8 = 3), so the 11th count should also be on the middle finger. Listing quickly: 1 thumb, 2 index, 3 middle, 4 ring, 5 little, 6 ring, 7 middle, 8 index, 9 thumb, 10 index, 11 middle, which matches the pattern. This confirms that the remainder method correctly maps counts onto fingers, so applying it to 1995 is valid. Why Other Options Are Wrong: Option Thumb: The thumb corresponds to positions 1 and 9 in the extended pattern, which have remainder 1 when divided by 8, not remainder 3. Option Index finger: The index finger appears at positions 2 and 8 in the cycle, associated with remainders 2 and 0 modulo 8, not remainder 3. Option Ring finger: The ring finger appears at positions 4 and 6, corresponding to remainders 4 and 6. Neither matches 1995 modulo 8. Option Little finger: The little finger appears only at position 5 in each cycle, which corresponds to remainder 5 and does not align with 1995 modulo 8. Common Pitfalls: A common error is to assume the cycle length is 5 instead of 8, ignoring the backward counting. Another mistake is miscalculating the remainder or mapping the remainder to the wrong position in the sequence. Some learners also attempt to simulate all counts, which is impractical and prone to mistakes. Recognizing periodicity and using modular arithmetic is the efficient and reliable approach for such problems. Final Answer: The girl ends counting on her middle finger.

Question 19

When 5 percent of maize is lost in grinding, a country must import 20 million bags to meet its needs, but when only 2 percent is lost in grinding, it needs to import only 15 million bags; find the total quantity of maize, in million bags, that actua…

Accepted Answer

Introduction / Context: This problem combines percentage loss with supply and import data, which is common in arithmetic reasoning and data interpretation questions. The key idea is that the country has a fixed internal production of maize, a certain percentage of which is lost during grinding. The remaining portion plus imports must meet the same total demand in both scenarios. By setting up equations for the two different loss percentages, we can solve for the original quantity of maize produced in the country. Given Data / Assumptions: - Let the total maize produced in the country be Q million bags. - In the first situation, 5 percent is lost in grinding, so 95 percent of Q is usable. - Under 5 percent loss, the country needs to import 20 million bags to meet total demand. - In the second situation, only 2 percent is lost, so 98 percent of Q is usable. - Under 2 percent loss, the country imports only 15 million bags. - The total maize demand is assumed constant in both cases. Concept / Approach: The demand for maize can be expressed in two ways: as domestic usable production plus imports in each scenario. Because total demand does not change, these two expressions must be equal. This leads to a linear equation in Q. The core concept is balancing supply with losses and extra imports. Express percentages as decimals to make the algebra straightforward, and then solve for Q. Finally, convert the fraction to a mixed number or decimal that matches one of the options. Step-by-Step Solution: Step 1: Under 5 percent loss, usable domestic production is 95 percent of Q, which is 0.95 * Q. Step 2: With 5 percent loss, the country imports 20 million bags, so total demand D can be written as D = 0.95 * Q + 20. Step 3: Under 2 percent loss, usable domestic production is 98 percent of Q, which is 0.98 * Q. Step 4: With 2 percent loss, the country imports 15 million bags, so the same demand D can be written as D = 0.98 * Q + 15. Step 5: Since total demand is unchanged, set the two expressions equal: 0.95 * Q + 20 = 0.98 * Q + 15. Step 6: Rearrange the equation: 20 - 15 = 0.98 * Q - 0.95 * Q, which simplifies to 5 = 0.03 * Q. Step 7: Solve for Q by dividing both sides by 0.03 to get Q = 5 / 0.03 = 500 / 3. Step 8: Convert 500 / 3 to a mixed number: 500 / 3 = 166 2/3. Thus the country produces 166 2/3 million bags of maize. Verification / Alternative check: Check the first scenario: With Q = 166 2/3, 5 percent loss equals (5 / 100) * 166 2/3, and 95 percent usable is (95 / 100) * 166 2/3 = (19 / 20) * (500 / 3) = 475 / 3. Adding imports of 20 gives total demand D = 475 / 3 + 20 = 475 / 3 + 60 / 3 = 535 / 3. Check the second scenario: 2 percent loss leaves 98 percent usable, which is (98 / 100) * 166 2/3 = (49 / 50) * (500 / 3) = 490 / 3. Adding imports of 15 gives D = 490 / 3 + 15 = 490 / 3 + 45 / 3 = 535 / 3. Both cases yield the same demand, confirming that the computed Q is correct. Why Other Options Are Wrong: Option 188 million: Substituting 188 into the equations will not make the total demand equal in both loss scenarios. Option 192 3/4 million: This value does not satisfy the balance equation and leads to different demands under 5 percent and 2 percent loss. Option None of these: Since we have found a value, 166 2/3 million, that exactly fits the condition, this option is incorrect. Option 200 million: This round figure may look attractive but does not satisfy the equality of total demand in both scenarios. Common Pitfalls: Learners may confuse percentage loss with percentage of maize remaining, or mistakenly write 5 percent remaining instead of 95 percent. Another frequent error is to treat the imports as part of the loss instead of additional supply. Some also forget that the demand is constant and fail to equate the two demand expressions. Careful interpretation of the words and methodical algebra prevent these mistakes. Final Answer: The quantity of maize that grows in the country is 166 2/3 million bags.

Question 20

Decide whether the statement Every rational number is an integer is correct or incorrect, based on the formal definitions of rational numbers and integers.

Accepted Answer

Introduction / Context: This conceptual question tests knowledge of the definitions and relationships between rational numbers and integers. Rational numbers are a broad class of numbers that include fractions and decimals, while integers are whole numbers that can be positive, negative, or zero. The statement Every rational number is an integer must be evaluated carefully, because it suggests that the set of rational numbers is contained within the set of integers. Understanding the actual hierarchy of number sets is crucial in number theory and algebra. Given Data / Assumptions: - A rational number is defined as any number that can be written in the form p / q where p and q are integers and q is not zero. - Integers include all whole numbers and their negatives, such as ..., -3, -2, -1, 0, 1, 2, 3, and so on. - The given statement claims that every rational number must be an integer. - We assume standard definitions from school level mathematics. Concept / Approach: To test a universal statement such as Every rational number is an integer, it is enough to find a single counterexample: a rational number that is not an integer. If such a number exists, the entire statement is incorrect. Many fractions like 1/2, 3/4, or 5/6 are rational because they can be expressed as ratios of integers, yet they are not integers. Therefore, checking any one such fraction is sufficient to disprove the statement. Step-by-Step Solution: Step 1: Recall the definition of rational number: any number that can be written as p / q where p and q are integers and q is not zero. Step 2: Recall the definition of integer: a number with no fractional or decimal part, such as -2, -1, 0, 1, 2, 3. Step 3: Observe that every integer n can indeed be written as n / 1, so every integer is a rational number. Step 4: However, consider a number like 1/2. It can be written as 1 / 2 where 1 and 2 are integers and 2 is not zero, so 1/2 is a rational number. Step 5: At the same time, 1/2 is not an integer because it has a fractional part and lies strictly between 0 and 1. Step 6: This single counterexample shows that there exists at least one rational number that is not an integer. Step 7: Therefore the statement Every rational number is an integer is incorrect. Verification / Alternative check: We can produce many other rational numbers that are not integers, such as 3/4, -5/2, or 7/10. They all fit the rational definition but clearly are not whole numbers. The proper relationship is that the set of integers is a subset of the set of rational numbers, not the other way around. In symbols, every integer is rational, but not every rational number is an integer. This confirms that the statement in the question reverses the correct inclusion and is false. Why Other Options Are Wrong: Option Correct statement: This would be true only if every rational number were an integer. The counterexample 1/2 disproves this, so this option is wrong. Option The statement cannot be determined: The definitions are precise and standard; there is no ambiguity, so the truth value can be determined exactly. Option The statement depends on the numerator and denominator: Although some rational numbers with denominator 1 are integers, the existence of many others with denominator greater than 1 shows that the statement is not conditionally true but simply false as written. Option True only for positive rational numbers: Positive rational numbers like 1/2 and 3/4 are not integers, so restricting to positive numbers does not fix the problem. Common Pitfalls: A frequent misunderstanding is to mix up the direction of inclusion, thinking that rational numbers are a subset of integers instead of the opposite. Some learners also assume that rational means reasonable or simple and therefore always a whole number. It is important to remember that rational specifically refers to ratios of integers. Visualizing the number line and placing fractions between integers helps make the distinction clearer. Final Answer: The statement Every rational number is an integer is an incorrect statement.

Arithmetic Reasoning Questions