Question 1

For real numbers a, b, and c, the expression a^2 + b^2 + c^2 - ab - bc - ca equals zero if and only if which of the following conditions holds?

Accepted Answer

Introduction / Context: This algebra question tests understanding of symmetric expressions in three variables and the use of identities to factor or simplify them. The expression a^2 + b^2 + c^2 - ab - bc - ca may look complicated, but with some algebraic manipulation it can be rewritten in a way that reveals when it becomes zero. Recognizing such patterns is very helpful in inequality problems and in simplifying expressions in algebra and coordinate geometry. Given Data / Assumptions: - We consider real numbers a, b, and c. - The expression in question is E = a^2 + b^2 + c^2 - ab - bc - ca. - We want to know under what condition E = 0 holds. - Options involving products like abc = 1 or abc = 0 are also given, but we must check them carefully rather than assume multiplicative relationships. Concept / Approach: There is a well-known identity that rewrites a^2 + b^2 + c^2 - ab - bc - ca as a sum of squares involving pairwise differences. Specifically, E can be expressed as ((a - b)^2 + (b - c)^2 + (c - a)^2) / 2. Since squares of real numbers are always non-negative, E is always greater than or equal to zero. The only way E can be zero is if each squared term is zero, i.e., all pairwise differences vanish and a, b, c are all equal. This directly leads to the required condition. Step-by-Step Solution: Step 1: Start from the identity for pairwise differences: (a - b)^2 = a^2 + b^2 - 2ab, (b - c)^2 = b^2 + c^2 - 2bc, (c - a)^2 = c^2 + a^2 - 2ca. Step 2: Add these three expressions: (a - b)^2 + (b - c)^2 + (c - a)^2 = (a^2 + b^2 - 2ab) + (b^2 + c^2 - 2bc) + (c^2 + a^2 - 2ca). Step 3: Combine like terms: we get 2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca. Step 4: Factor out 2: this becomes 2(a^2 + b^2 + c^2 - ab - bc - ca). Step 5: Therefore a^2 + b^2 + c^2 - ab - bc - ca = ((a - b)^2 + (b - c)^2 + (c - a)^2) / 2. Step 6: A sum of squares equals zero if and only if each square is zero. So (a - b)^2 = 0, (b - c)^2 = 0, and (c - a)^2 = 0, meaning a = b, b = c, and c = a. Step 7: These equalities together imply a = b = c. Verification / Alternative check: If a = b = c = k, then E = k^2 + k^2 + k^2 - k^2 - k^2 - k^2 = 0, confirming that equality of the three numbers makes the expression zero. On the other hand, if a, b, c are not all equal, at least one difference such as a - b or b - c is non-zero, making at least one squared term positive. Then the sum of squares is positive and E is greater than zero. Thus E = 0 occurs if and only if a = b = c. Why Other Options Are Wrong: Option abc = 1 or abc = -1: These conditions relate to the product of a, b, and c and say nothing about pairwise equality. There are many triples with product 1 or -1 for which a^2 + b^2 + c^2 - ab - bc - ca is not zero. Option abc = 0: This means at least one of the numbers is zero, but the others need not match. E is generally positive for such choices unless they all coincide, which is a special case of a = b = c. Option None of these: Incorrect because the condition a = b = c is both necessary and sufficient for the expression to be zero and is listed explicitly. Common Pitfalls: Students often try to factor the expression incorrectly or look for multiplicative conditions like abc = 1 without checking them. Others might test only a few numerical examples and jump to conclusions. Using the identity that expresses the expression as a sum of squared differences is the most reliable way to understand its behaviour and to see that equality of all three variables is the exact condition for it to vanish. Final Answer: The expression equals zero if and only if a, b, and c are all equal, that is, a = b = c.

Question 2

Evaluate the numerical expression 3 × 3 × 3 + 3 + 3 ÷ 3 using the standard order of operations.

Accepted Answer

Introduction / Context: This question tests the correct use of the order of operations (often remembered as BODMAS or PEMDAS). The expression 3 × 3 × 3 + 3 + 3 ÷ 3 contains multiplication, addition, and division. To evaluate it correctly, we must perform multiplications and divisions from left to right before performing additions. Misordering the operations or grouping terms incorrectly can lead to wrong answers. Given Data / Assumptions: - The expression is 3 × 3 × 3 + 3 + 3 ÷ 3. - Multiplication and division have the same precedence and are carried out from left to right. - Addition is performed after multiplication and division. Concept / Approach: According to the order of operations, we first calculate all products and quotients in the order they appear from left to right, and then we calculate sums. There are no brackets or powers written explicitly here, so the only precedence issue is between ×, ÷, and +. Evaluating step by step avoids accidental grouping such as (3 × 3 × 3 + 3 + 3) ÷ 3, which would not follow the standard rules. Step-by-Step Solution: Step 1: Start with the expression: 3 × 3 × 3 + 3 + 3 ÷ 3. Step 2: Evaluate the multiplications from left to right. First compute 3 × 3 = 9. Step 3: Multiply that result by the next 3: 9 × 3 = 27. So the expression becomes 27 + 3 + 3 ÷ 3. Step 4: Next handle the division 3 ÷ 3 = 1. After this step, the expression becomes 27 + 3 + 1. Step 5: Now perform the additions from left to right: 27 + 3 = 30, then 30 + 1 = 31. Step 6: Therefore, the value of the expression is 31. Verification / Alternative check: We can check by grouping the multiplication as 3^3 = 27 (since 3 × 3 × 3 is 3 cubed). Then the expression is 27 + 3 + 1 = 31, which matches the previously obtained result. Any rearrangement that changes the order of operations, such as dividing the entire sum by 3, leads to a different answer and is not allowed under standard rules. Why Other Options Are Wrong: Option 11 or 3.33 or 9: These results arise from misapplying the order of operations, such as incorrectly dividing the whole expression by 3 or performing addition before handling all multiplications and divisions. Option None of the above: Incorrect because 31 is listed among the options and is the correct value. Common Pitfalls: Common mistakes include evaluating the expression left to right without regard to operator precedence, or interpreting the last 3 ÷ 3 as applied to the entire preceding sum. Remember that multiplication and division must be completed first, in sequence, and only then can addition and subtraction be carried out. Keeping to this discipline ensures consistent and correct evaluation of mixed operations. Final Answer: The value of 3 × 3 × 3 + 3 + 3 ÷ 3 is 31.

Question 3

How many distinct rectangles with positive integer side lengths (in centimetres) can be drawn having a perimeter of 36 cm?

Accepted Answer

Introduction / Context: This question involves counting the number of distinct rectangles with a fixed perimeter and integer side lengths. A rectangle is determined by its length and breadth, and a given perimeter imposes a linear relationship between these two side lengths. We must find all possible integer pairs that satisfy this relationship, while treating rectangles with interchanged side lengths as the same shape (since rotating a rectangle does not produce a new rectangle). Given Data / Assumptions: - Let the side lengths of the rectangle be a and b (in cm), both positive integers. - The perimeter P is given by P = 2(a + b) and equals 36 cm. - Therefore a + b = 18. - Rectangles with sides (a, b) and (b, a) are considered the same rectangle because the shape is unchanged by swapping length and breadth. Concept / Approach: We need to find positive integer solutions to the equation a + b = 18. There are many ordered pairs (a, b), but we must identify distinct rectangles up to swapping sides. To do this, we can list all ordered solutions with a Step-by-Step Solution: Step 1: Start with a + b = 18, where a and b are positive integers. Step 2: To avoid double-counting, impose a Step 3: List possible values of a and the corresponding b values: a = 1 ⇒ b = 17, a = 2 ⇒ b = 16, a = 3 ⇒ b = 15, a = 4 ⇒ b = 14, a = 5 ⇒ b = 13, a = 6 ⇒ b = 12, a = 7 ⇒ b = 11, a = 8 ⇒ b = 10, a = 9 ⇒ b = 9. Step 4: For a > 9, we would get a > b, which we have already counted by symmetry when a Step 5: This gives us 9 distinct pairs (a, b), hence 9 distinct rectangles. Verification / Alternative check: We can double-check by counting all ordered positive integer solutions to a + b = 18, which are 17 pairs (since a can be 1 through 17). Because each unordered pair (a, b) with a ≠ b appears twice in this list (once as (a, b) and once as (b, a)), and the pair (9, 9) appears only once, we can count: there are 8 pairs with unequal sides that correspond to 8 rectangles, plus one square with sides (9, 9). Thus 8 + 1 = 9 rectangles, matching our earlier list. Why Other Options Are Wrong: Options 8, 10, 11: These counts either undercount or overcount the actual number of distinct integer side pairs that satisfy the perimeter condition. Careful enumeration shows exactly 9 possibilities. Option None of these: Incorrect because 9 is a listed option and has been established as the correct count. Common Pitfalls: A common mistake is to count ordered pairs without adjusting for symmetry, leading to an answer of 17 instead of 9. Another error is to stop listing too early and miss some valid pairs. Enforcing a Final Answer: There are 9 distinct rectangles with integer side lengths and perimeter 36 cm.

Question 4

Which of the following numbers is divisible by 24?

Accepted Answer

Introduction / Context: This question tests divisibility rules, specifically for determining quickly whether a number is divisible by 24. Since 24 factors as 8 × 3, a number is divisible by 24 exactly when it is divisible by both 8 and 3. Using divisibility tests for 8 and 3 allows us to check each option without performing full long division, which saves time in competitive exams. Given Data / Assumptions: - We need to find which one of the numbers 37804, 2015736, 36718, and 53810 is divisible by 24. - 24 = 8 × 3, so divisibility by 24 requires divisibility by 8 and by 3 simultaneously. - We will apply standard divisibility tests: for 8, check the last three digits; for 3, check the sum of digits. Concept / Approach: A number is divisible by 8 if its last three digits form a number divisible by 8. A number is divisible by 3 if the sum of its digits is a multiple of 3. We apply these tests to each candidate. Only numbers that pass both tests are divisible by 24. We then match the correct candidate with the answer choices. Step-by-Step Solution: Step 1: Check 37804. Last three digits are 804. 804 ÷ 8 = 100.5, so it is not divisible by 8. Therefore 37804 is not divisible by 24. Step 2: Check 2015736. Last three digits are 736. 736 ÷ 8 = 92 exactly, so it is divisible by 8. Now sum its digits: 2 + 0 + 1 + 5 + 7 + 3 + 6 = 24. Since 24 is divisible by 3, 2015736 is divisible by 3. Hence 2015736 is divisible by both 8 and 3, so it is divisible by 24. Step 3: Check 36718. Last three digits are 718. 718 ÷ 8 = 89.75, so 36718 is not divisible by 8 and therefore not by 24. Step 4: Check 53810. Last three digits are 810. 810 ÷ 8 = 101.25, so not divisible by 8. Hence 53810 is not divisible by 24. Step 5: Only 2015736 passes both divisibility tests. Verification / Alternative check: We can confirm by direct division for 2015736: dividing by 24 stepwise or using a calculator gives an integer quotient with no remainder. Since the other numbers already fail one of the necessary tests (for 8), they cannot be divisible by 24. Therefore 2015736 is the only correct choice. Why Other Options Are Wrong: Options 37804, 36718, 53810: Each of these fails the divisibility test for 8. Even if some might be divisible by 3, failing divisibility by 8 means they are not divisible by 24. Option None of these: Incorrect because 2015736 is in fact divisible by 24 and is one of the listed options. Common Pitfalls: A typical error is checking divisibility by 3 alone and ignoring divisibility by 8, or vice versa. Another mistake is checking all digits for divisibility by 8 instead of just the last three. Remembering that 24 factors into 8 and 3, and applying both tests properly, is the key to solving such problems quickly and correctly. Final Answer: The number 2015736 is divisible by 24.

Question 5

Express 5008 kilometres in scientific notation, using a coefficient between 1 and 10 multiplied by an appropriate power of 10.

Accepted Answer

Introduction / Context: Scientific notation is a standard way of writing very large or very small numbers as a product of a number between 1 and 10 and an integer power of 10. It is widely used in science and engineering for clear and compact representation. Here we need to express 5008 kilometres in proper scientific notation, ensuring the coefficient is between 1 and 10 and the power of 10 is chosen correctly. Given Data / Assumptions: - The number to be converted is 5008 km. - Scientific notation has the form c × 10^n where 1 ≤ c < 10 and n is an integer. - We must keep the units in kilometres, not convert to metres or other units. Concept / Approach: To write 5008 in scientific notation, we move the decimal point so that there is exactly one non-zero digit to the left of the decimal point. We count how many places the decimal point moves to determine the exponent of 10. Moving the decimal to the left increases the exponent; moving it to the right would decrease it. We then attach the unit km back to the final expression. Step-by-Step Solution: Step 1: Write 5008 as a decimal number: 5008.0. Step 2: Move the decimal point to obtain a number between 1 and 10. Placing the decimal after the first digit 5 gives 5.008. Step 3: Count how many places the decimal moved: from 5008.0 to 5.008, the decimal moves 3 places to the left. Step 4: Moving 3 places to the left corresponds to multiplying by 10^3. So 5008 = 5.008 × 10^3. Step 5: Attach the unit: 5008 km = 5.008 × 10^3 km. Verification / Alternative check: We can verify by reversing the process: 5.008 × 10^3 = 5.008 × 1000 = 5008, because 5.008 × 1000 shifts the decimal three places to the right. The value is restored exactly to 5008, so the scientific notation is correct. The coefficient 5.008 satisfies 1 ≤ 5.008 < 10, so the form is valid. Why Other Options Are Wrong: Option 50.08 × 10^3 km: Here the coefficient 50.08 is not between 1 and 10, so this is not proper scientific notation. Option 5.008 × 10^-3 km: This represents 0.005008 km, which is much smaller than 5008 km and therefore incorrect as a representation of 5008 km. Option 500.8 × 10^3 km: This equals 500800 km, which is different from 5008 km by a factor of 100, so it is wrong. Option None of these: Incorrect because 5.008 × 10^3 km is a valid option and correctly represents 5008 km in scientific notation. Common Pitfalls: One frequent mistake is to stop after writing 50.08 × 10^2, which is numerically correct but not in proper scientific notation because the coefficient is greater than 10. Another error is to miscount the number of decimal places moved, resulting in the wrong power of 10. Always ensure that the coefficient is between 1 and 10 and then verify by reversing the conversion. Final Answer: 5008 km written in scientific notation is 5.008 × 10^3 km.

Question 6

A two-digit number has digits that satisfy three conditions: (A) the sum of the digits is 15, (B) the difference of the squares of the digits is 45 (larger digit squared minus smaller digit squared), and (C) the difference of the digits (tens digit …

Accepted Answer

Introduction / Context: This is a data sufficiency style question about a two-digit number whose tens and units digits must satisfy several conditions. Rather than asking directly for the number, the problem asks which combination of statements provides enough information to determine the number uniquely. Understanding how to translate each condition into an equation and how they interact is key to solving data sufficiency questions correctly. Given Data / Assumptions: - Let the two-digit number have tens digit t and units digit u, with t between 1 and 9 and u between 0 and 9. - Statement A: t + u = 15 (sum of the digits is 15). - Statement B: t^2 - u^2 = 45 (difference of squares of the digits, larger digit squared minus smaller digit squared, equals 45). - Statement C: t - u = 3 (difference of their digits, interpreted as tens digit minus units digit, is 3). - The goal is to decide which pairs of statements are sufficient to find the unique number, not necessarily to compute it in every case, though computing it once can help verify sufficiency. Concept / Approach: We treat each statement as an equation and examine combinations. From B, we can factor t^2 - u^2 as (t - u)(t + u) = 45. Using integer factor pairs of 45, together with digit constraints, gives possible (t, u) pairs. We then see how combining A, B, and C in different pairs restricts the possibilities to a single number. If any pair of statements always leads to one unique (t, u) pair, that pair is sufficient. Step-by-Step Solution: Step 1: Use statement B alone: (t - u)(t + u) = 45. Possible positive factor pairs of 45 are (1, 45), (3, 15), and (5, 9). They give potential (t - u, t + u) pairs. Solving: For (3, 15): t = (15 + 3) / 2 = 9, u = (15 - 3) / 2 = 6. For (5, 9): t = (9 + 5) / 2 = 7, u = (9 - 5) / 2 = 2. The pair (1, 45) would give t = 23, u = 22, which is invalid for digits. Step 2: From B alone we have two digit pairs: (9, 6) and (7, 2), corresponding to numbers 96 and 72, so B alone is not sufficient. Step 3: Combine A and B. A says t + u = 15. Among the pairs from B, only (9, 6) has sum 15 (7 + 2 = 9). Thus A and B together uniquely give t = 9 and u = 6, so the number is 96. Hence A and B together are sufficient. Step 4: Combine B and C. C says t - u = 3. Of the B-pairs (9, 6) and (7, 2), only (9, 6) has t - u = 3 (7 - 2 = 5). Therefore B and C together again uniquely give 96 and are sufficient. Step 5: Combine A and C. A gives t + u = 15, and C gives t - u = 3. Solving these simultaneously: adding the equations gives 2t = 18, so t = 9 and u = 6. This is a unique solution, so A and C together are also sufficient. Step 6: Since any one of the three pairs (A, B), (B, C), or (C, A) is sufficient by itself, the correct description is that any one of these pairs is enough. Verification / Alternative check: We can verify by checking that no other digit pair meets all conditions simultaneously. The pair (9, 6) satisfies t + u = 15, t^2 - u^2 = 81 - 36 = 45, and t - u = 3. The pair (7, 2) satisfies B but violates both A and C. Thus, each combination A & B, B & C, and A & C isolates the correct pair (9, 6) uniquely, confirming the sufficiency of each pair. Why Other Options Are Wrong: Option B and C together are sufficient: This is true but incomplete, because A and B together, and A and C together are also sufficient; the option fails to state the broader sufficiency pattern. Option C and A together are sufficient: This is true but again does not acknowledge that B can pair with either A or C to yield sufficiency. Option A and B together are sufficient: Also true but not the only sufficient pair. Option None of these: Incorrect because a precise description exists, namely that any one of the three pairs is sufficient. Common Pitfalls: A common mistake is to treat data sufficiency as if only one specific pair must work, overlooking that other combinations may also be enough. Another pitfall is misinterpreting difference of the digits and difference of squares as absolute differences rather than oriented differences, which can cause ambiguity in solving. Clearly specifying t - u and t^2 - u^2 as oriented differences ensures consistent equations and a unique solution in each pair case. Final Answer: Any one pair of A and B, B and C, or C and A is sufficient to determine the number uniquely.

Question 7

Hemavathi cuts a cake into two equal halves and then cuts one half into smaller pieces of equal size. Each of these small pieces weighs 15 grams. If she now has a total of 9 pieces of cake, how heavy was the original cake?

Accepted Answer

Introduction / Context: This word problem is a simple application of fractions and multiplication in a real-life context. A cake is cut into two halves, one half is further cut into equal small pieces, and we are told how many pieces there are in total and the weight of each small piece. The task is to determine the total weight of the original cake from this information. Given Data / Assumptions: - The original cake is cut into two equal halves, so each half weighs half of the original weight. - One half is cut into smaller equal pieces; each small piece weighs 15 grams. - The other half remains as one large piece. - In total, Hemavathi now has 9 pieces of cake: 1 large half and some number of small pieces. - All pieces are assumed to have precise weights as stated (no wastage or crumbs lost). Concept / Approach: We first determine how many of the 9 pieces are small pieces and then use the weight of each small piece to find the weight of the half that was cut. Since that half is exactly half the original cake, doubling its weight gives the total weight of the cake. The reasoning is straightforward, but we must correctly interpret that only one half is cut into small pieces and count the number of small pieces accordingly. Step-by-Step Solution: Step 1: Let the weight of the original cake be W grams. Then each half weighs W/2 grams. Step 2: One half remains uncut as a single large piece of weight W/2 grams. Step 3: The other half is divided into equal small pieces, each weighing 15 grams. Step 4: Hemavathi has 9 pieces in total. One of these is the uncut half; therefore, the number of small pieces is 9 - 1 = 8. Step 5: The total weight of the small pieces is 8 × 15 grams = 120 grams. Step 6: This total 120 grams comes from the half that was cut, so W/2 = 120 grams. Step 7: Multiply both sides by 2 to obtain W = 240 grams. Verification / Alternative check: We can verify by reconstructing the situation with W = 240 grams. Each half weighs 240/2 = 120 grams. One half remains as a single piece of 120 grams. The other half is cut into 120/15 = 8 pieces, each of 15 grams. Thus there are 8 small pieces plus 1 large half, making 9 pieces in total. All details of the story are satisfied, so the solution is consistent. Why Other Options Are Wrong: Option 280 g or 180 g or 170 g: For any of these values, the weight of a half would not be a multiple of 15 that allows exactly 8 small pieces and a total of 9 pieces. For example, if the cake weighed 180 g, half would be 90 g and cutting that into 15 g pieces would yield 6 pieces, not 8, contradicting the total piece count. Option None of these: Incorrect because 240 g is one of the options and matches the scenario precisely. Common Pitfalls: A common misunderstanding is to assume both halves are cut into small pieces, leading to the wrong number of pieces and an incorrect total weight. Another mistake is to divide the total number of pieces incorrectly between large and small pieces or to forget that the large half is one whole piece. Carefully distinguishing the uncut half from the cut half and counting pieces accurately leads to the correct answer. Final Answer: The original cake weighed 240 grams.

Question 8

In an arithmetic reasoning puzzle, an intelligent boy is assembling a jigsaw with 275 pieces. Each day that he fits pieces together there are fewer pieces left, and it is reasonable to assume that he fits more pieces each day because the number left…

Accepted Answer

Introduction / Context:This arithmetic reasoning question describes a boy who is completing a jigsaw puzzle by fitting more pieces each day than on the previous day. The situation naturally leads to an increasing sequence of numbers and tests understanding of arithmetic progressions and summations. The puzzle asks how many complete days are required if the boy starts by fitting 20 pieces and then increases his daily output by exactly one extra piece every new day until all 275 pieces are used. Given Data / Assumptions:Total number of pieces in the jigsaw puzzle is 275.On the first day, the boy fits 20 pieces.Each subsequent day he fits exactly one more piece than on the previous day.He continues this pattern without any break until the puzzle is fully completed.We assume that he finishes exactly on a day when the total fitted pieces reaches 275 without exceeding it. Concept / Approach:The number of pieces fitted per day forms an arithmetic progression. The first term of this sequence is 20, and the common difference is 1 because he adds one extra piece each day. If he works for n days, then the total number of pieces fitted is the sum of the first n terms of this arithmetic progression. The sum of n terms of an arithmetic progression with first term a and common difference d is S = n/2 * [2a + (n - 1) * d]. We equate this total to 275 and solve for n. Step-by-Step Solution:Step 1: Let n be the total number of days he works on the puzzle.Step 2: The first term a is 20, and the common difference d is 1, so the daily pieces are 20, 21, 22, and so on.Step 3: Use the arithmetic progression sum formula: S = n/2 * [2a + (n - 1) * d].Step 4: Substitute S = 275, a = 20 and d = 1: 275 = n/2 * [40 + (n - 1) * 1].Step 5: Simplify inside the brackets: 275 = n/2 * (n + 39).Step 6: Multiply both sides by 2 to clear the denominator: 550 = n * (n + 39).Step 7: Rearrange to form a quadratic equation: n^2 + 39n - 550 = 0.Step 8: Solve the quadratic equation. The discriminant is 39^2 + 4 * 550 = 1521 + 2200 = 3721, and the square root of 3721 is 61.Step 9: Use the quadratic formula n = [-39 ± 61] / 2. The positive solution is n = (61 - 39) / 2 = 22 / 2 = 11.Step 10: Therefore the boy completes the puzzle in exactly 11 days. Verification / Alternative check:We can verify by directly summing the terms. The daily fitted pieces are 20, 21, 22, and so on up to the 11th term. The 11th term is 20 + (11 - 1) * 1 = 30. The average of the first and last terms is (20 + 30) / 2 = 25. There are 11 terms, so the total is 25 * 11 = 275, which matches the required number of pieces. This confirms that 11 days is correct. Why Other Options Are Wrong:The option 15 days would give far more than 275 pieces because the sequence would continue increasing for too many days. The option 13 days also overshoots the required sum. The option 16 days is even larger and clearly does not match the total pieces. None of these alternatives produce a sum of exactly 275 using the given pattern, so they are incorrect. Common Pitfalls:A very common mistake is to divide 275 by 20 or by some average without recognizing that the number of pieces fitted is increasing every day. Another error is to treat the situation as if the number of pieces fitted per day is constant. Some learners might also misapply the arithmetic progression formula by forgetting the factor of one half or misplacing the values of a and n. Careful algebra is required to obtain the correct value of n. Final Answer:The boy completes the entire 275 piece jigsaw puzzle after exactly 11 days.

Question 9

In three coloured boxes, Red, Green and Blue, a total of 108 balls are placed. There are twice as many balls in the green and red boxes combined as there are in the blue box, and there are twice as many balls in the blue box as there are in the red …

Accepted Answer

Introduction / Context:This quantitative reasoning question involves three coloured boxes, each containing a certain number of balls. The question gives relationships between the numbers of balls in the red, green and blue boxes and the overall total. It tests the ability to translate verbal conditions into algebraic equations and then solve them systematically to find the exact count in one particular box, namely the green box. Given Data / Assumptions:There are three boxes: Red, Green and Blue.The total number of balls in all three boxes together is 108.The combined number of balls in the green and red boxes is twice the number of balls in the blue box.The number of balls in the blue box is twice the number of balls in the red box.All ball counts are non negative integers. Concept / Approach:The most efficient approach is to introduce algebraic variables for the number of balls in each box and then convert the relationships into linear equations. This leads to a system of simultaneous equations that can be solved step by step. Once the value for one box is known, substitution allows us to compute the values for the other boxes and verify that their sum matches the total of 108 balls. Step-by-Step Solution:Step 1: Let R be the number of balls in the red box, G be the number of balls in the green box and B be the number of balls in the blue box.Step 2: The statement "twice as many balls in the green and red boxes combined as in the blue box" gives the equation R + G = 2B.Step 3: The statement "twice as many in the blue box as in the red box" gives the equation B = 2R.Step 4: The total number of balls is 108, so R + G + B = 108.Step 5: Substitute B = 2R into the equation R + G = 2B to get R + G = 2 * 2R = 4R, so G = 4R - R = 3R.Step 6: Now substitute B = 2R and G = 3R into the total equation: R + 3R + 2R = 108.Step 7: Simplify: 6R = 108, so R = 108 / 6 = 18.Step 8: From G = 3R, we get G = 3 * 18 = 54. So the green box contains 54 balls.Step 9: As a check, B = 2R = 36, and indeed 18 + 54 + 36 = 108. Verification / Alternative check:We can verify the relationships directly with the calculated values. The blue box has 36 balls, and the red plus green boxes together have 18 + 54 = 72 balls. This is exactly twice 36, so the first condition holds. The blue box has 36 balls, which is twice the 18 balls in the red box, so the second condition holds. The sum 18 + 54 + 36 is 108, so the total is also correct. This confirms that the solution is consistent. Why Other Options Are Wrong:If the green box had 76 balls, the total would exceed 108 when the other constraints are applied. If it had 64 balls or 48 balls, the relationships R + G = 2B and B = 2R cannot be satisfied simultaneously with an integer value for R that also makes the total equal 108. Only 54 balls in the green box satisfy all given conditions. Common Pitfalls:One common mistake is to misinterpret "twice as many in the green and red boxes combined as in the blue box" and instead write G = 2B or R = 2B. Another pitfall is to forget to include the total sum equation, which is essential for determining the exact values rather than just ratios. Care must also be taken not to mix up which box is twice which. Working systematically with variables and equations avoids these errors. Final Answer:The number of balls in the green box is 54.

Question 10

In Euclidean geometry, a plane is considered an undefined term because it:

Accepted Answer

Introduction / Context:This conceptual geometry question asks why a plane in Euclidean geometry is called an undefined term. Many learners encounter the words point, line and plane at the very beginning of geometry, yet they are not defined in the usual way using previously known concepts. Understanding the role of undefined terms is essential, because they form the foundation on which all other geometric definitions and theorems are built. Given Data / Assumptions:We are working in the context of Euclidean geometry, the standard geometry taught in most schools.Three basic geometric concepts are traditionally treated as undefined: point, line and plane.The question focuses specifically on the concept of a plane.We are asked to choose the statement that correctly explains why a plane is called an undefined term.We assume standard textbook conventions for definitions and axiomatic structure. Concept / Approach:In an axiomatic system like Euclidean geometry, some basic objects are not defined in terms of simpler objects. Instead, they are described informally and understood through their properties and relationships. These are called undefined or primitive terms. A plane is one such primitive concept. It is not given a formal definition using previously introduced shapes; it is described as a flat surface extending without end, but that description itself relies on intuition. The correct answer must capture this idea that the plane is described informally rather than formally defined. Step-by-Step Solution:Step 1: Review the role of undefined terms in geometry. They serve as starting points and are accepted without formal definitions.Step 2: Consider the common informal description of a plane as a flat, two dimensional surface that extends infinitely in all directions.Step 3: Recognize that such a description is not a rigorous definition. It uses intuitive ideas like flatness and infinity that themselves are not defined in elementary geometry.Step 4: Analyze the options. Option a gives a descriptive property, but it acts more like a definition, not an explanation of why the term is undefined.Step 5: Option b talks about where shapes can be constructed, which is a consequence of the existence of a plane but not a reason for being undefined.Step 6: Option c clearly states that the plane is described generally and not using a formal definition, which matches the idea of an undefined term.Step 7: Option d mentions naming a plane using three noncollinear points, which is a naming convention, not an explanation of why it is undefined.Step 8: Therefore, option c is the best explanation. Verification / Alternative check:If we look at any axiomatic presentation of Euclidean geometry, such as in many school textbooks, we see that point, line and plane are introduced as basic undefined terms. Their properties are captured through postulates and theorems. For example, there is a postulate that through any three noncollinear points there is exactly one plane. This postulate uses the term plane without defining it, confirming that the concept is primitive. This matches the idea expressed in option c that the plane is described in a general way and not formally defined. Why Other Options Are Wrong:Option a describes a property of a plane but does not explain its status as an undefined term. Option b refers to a use of planes in constructing other shapes, but many objects could serve as a setting for construction without being undefined. Option d is about one way to name or specify a plane using three noncollinear points, which again does not address the reason for its undefined status. Thus, these options are partial properties and not the core conceptual reason. Common Pitfalls:Students often confuse a property or description with the logical reason for calling something undefined. Another common mistake is to assume that every mathematical object must have a formal definition directly in terms of earlier ones, without recognizing that some basic concepts must be taken as primitive. Remembering that undefined terms are the foundation of the system helps avoid these misunderstandings. Final Answer:The plane is considered an undefined term because it is described generally, not using a formal definition.

Question 11

In basic arithmetic, what is the product of a nonzero number and its reciprocal?

Accepted Answer

Introduction / Context:This question tests understanding of the concept of a reciprocal, also called the multiplicative inverse, and how it behaves under multiplication. Knowing what happens when a number is multiplied by its reciprocal is fundamental to algebra, fractions and many higher level topics. The focus is on nonzero numbers, since zero does not have a reciprocal. Given Data / Assumptions:We are considering a nonzero real number, which we can call a.The reciprocal or multiplicative inverse of a is defined as 1 divided by a.The question asks for the product a * (1 / a).Zero is excluded, because division by zero is undefined and zero has no reciprocal.Standard rules of multiplication and division of real numbers apply. Concept / Approach:The reciprocal of a nonzero number a is the number that, when multiplied by a, gives 1. This is the defining property of the multiplicative inverse. In symbolic form, if a is not zero, then a * (1 / a) = 1. So the correct choice must reflect this fundamental identity. The approach here is to apply this definition directly rather than trying to plug in specific numerical examples only. Step-by-Step Solution:Step 1: Let a be an arbitrary nonzero real number.Step 2: By definition, the reciprocal or multiplicative inverse of a is 1 / a.Step 3: Consider the product of a and its reciprocal: a * (1 / a).Step 4: Use the rule of multiplication and division: a * (1 / a) can be viewed as a divided by a.Step 5: For any nonzero number a, a / a = 1, because a is exactly one times itself.Step 6: Therefore the product of a nonzero number and its reciprocal is always equal to 1. Verification / Alternative check:We can confirm this by trying some concrete examples. If the number is 2, its reciprocal is 1 / 2, and 2 * (1 / 2) = 1. If the number is 5, its reciprocal is 1 / 5, and 5 * (1 / 5) = 1. If the number is -3, the reciprocal is 1 / -3, and -3 * (1 / -3) = 1 as well. These examples from both positive and negative numbers show that the result is consistently 1 whenever the number is nonzero. Why Other Options Are Wrong:Option a claims the product is 0, which would be true only if one of the factors were zero, but that contradicts the condition that the number is nonzero. Option c suggests that the product is the negative of the number, which does not match the definition of a reciprocal and fails in any simple test, since 2 * (1 / 2) is not equal to -2. Option d suggests that the product equals the number itself, which is also not correct; for example, 3 * (1 / 3) equals 1, not 3. Only 1 is consistent with the definition for every nonzero number. Common Pitfalls:Some learners confuse additive inverse (which gives a sum of 0) with multiplicative inverse (which gives a product of 1). Another error is to forget that the reciprocal is always 1 divided by the number, not the number itself or its negative. Clear separation of these concepts is very important for success in algebra. Final Answer:The product of a nonzero number and its reciprocal is always 1.

Question 12

What is the multiplicative inverse (reciprocal) of the integer 7?

Accepted Answer

Introduction / Context:This question focuses on the idea of a multiplicative inverse, also known as a reciprocal, for a specific integer. Understanding inverses is essential in algebra because they are used to solve equations, simplify expressions and understand the structure of number systems. Here the integer in question is 7, and we must identify which option represents its multiplicative inverse. Given Data / Assumptions:The integer given is 7, which is nonzero.The multiplicative inverse of a nonzero number a is the number that satisfies a * (its inverse) = 1.We are working in the real numbers.Division by zero is not allowed, so zero cannot be a reciprocal of any number.The goal is to find a number x such that 7 * x = 1. Concept / Approach:By definition, the multiplicative inverse of a number a is 1 / a. This definition directly gives the inverse once we know a. Therefore the multiplicative inverse of 7 is 1 / 7. Another way to think about it is to ask which number x satisfies 7 * x = 1. Solving for x gives x = 1 / 7, which matches the reciprocal definition. Any option that does not satisfy this equation or involves zero is incorrect. Step-by-Step Solution:Step 1: Let a be 7. The multiplicative inverse of a is defined as 1 / a.Step 2: Substitute a = 7 into this definition to get the candidate inverse 1 / 7.Step 3: Check the defining property of a multiplicative inverse: a * (1 / a) must equal 1 for nonzero a.Step 4: Compute 7 * (1 / 7). This is equal to 7 / 7, which simplifies to 1.Step 5: Therefore 1 / 7 satisfies the requirement and is the multiplicative inverse of 7.Step 6: Compare with other options to ensure they do not satisfy the same property. Verification / Alternative check:We can quickly test each option by multiplying it by 7. Option a is 1, and 7 * 1 = 7, not 1, so this fails. Option b is -1 / 7, and 7 * (-1 / 7) = -1, which is the negative of 1, so this also fails. Option c is 0, and 7 * 0 = 0, so it clearly fails and zero can never be a multiplicative inverse. Option d is 1 / 7, and as already shown, 7 * (1 / 7) = 1, so this passes the test. This confirms that 1 / 7 is the correct reciprocal. Why Other Options Are Wrong:The number 1 is the multiplicative identity, not the inverse of 7. It leaves 7 unchanged when multiplied, giving 7 instead of 1. The number -1 / 7 produces a product of -1, which fails the requirement. Zero is never a multiplicative inverse because any number multiplied by zero is zero, not 1. These observations rule out all the incorrect options. Common Pitfalls:Students sometimes confuse additive inverses and multiplicative inverses. The additive inverse of 7 is -7, since 7 + (-7) = 0, but that is unrelated to the multiplicative inverse, which focuses on products equal to 1. Another pitfall is thinking that the inverse of a whole number must also be a whole number, which is not true; many inverses are fractions. Final Answer:The multiplicative inverse of 7 is 1/7.

Question 13

Two angles whose measures add up to exactly 90 degrees are called what type of angles?

Accepted Answer

Introduction / Context:This geometry vocabulary question asks for the special name given to two angles whose measures sum to 90 degrees. Knowing names for pairs of angles is important in geometry because these relationships often appear in problems involving triangles, parallel lines and right angles. The focus here is on the pair that forms a right angle when combined. Given Data / Assumptions:There are two angles being considered.The sum of their measures is exactly 90 degrees.We are asked for the standard geometric term used for such a pair of angles.The available answer choices include several different types of angle relationships.We assume standard terminology used in school level Euclidean geometry. Concept / Approach:There are two key words that often cause confusion: complementary and supplementary. Complementary angles are two angles whose measures add up to 90 degrees, while supplementary angles are two angles whose measures add up to 180 degrees. The other terms in the options, such as corresponding and alternate angles, describe positional relationships when lines are cut by a transversal, not simple sums of measures. Therefore the correct answer must match the 90 degree sum, which is complementary. Step-by-Step Solution:Step 1: Recall the definition of complementary angles: two angles are complementary if the sum of their measures is 90 degrees.Step 2: Recall the definition of supplementary angles: two angles are supplementary if the sum of their measures is 180 degrees.Step 3: Compare the condition in the question with these definitions. The question states that the two angles have a sum of 90 degrees.Step 4: This matches exactly the definition of complementary angles.Step 5: Therefore the correct classification is complementary angles. Verification / Alternative check:We can use a simple numeric example. Suppose one angle measures 30 degrees and the other measures 60 degrees. Together they sum to 30 + 60 = 90 degrees. This is a classic example of complementary angles. In contrast, a pair of 100 degrees and 80 degrees would sum to 180 degrees and be called supplementary angles. This comparison confirms that the 90 degree sum corresponds to complementary, not supplementary. Why Other Options Are Wrong:Supplementary angles, listed in option a, have a sum of 180 degrees, not 90 degrees. Corresponding angles, in option b, are formed when a transversal cuts two lines and refer to matching positions, not to a particular sum. Alternate angles, in option d, also describe positions relative to a transversal and lines, such as alternate interior angles, again not a specific total measure. None of these match the condition of adding up to 90 degrees. Common Pitfalls:A very common memory mistake is confusing complementary with supplementary, especially because both words sound somewhat similar. A helpful trick is to associate the letter C in complementary with the number 90 by thinking of a right angle corner. Another idea is to remember that S in supplementary can be linked with straight angle, which has 180 degrees. Final Answer:Two angles whose measures add up to 90 degrees are called Complementary angles.

Question 14

Which of the following sets of three positive integers represents a Pythagorean triple, that is, the side lengths of a right angled triangle?

Accepted Answer

Introduction / Context:This question checks understanding of Pythagorean triples, which are sets of three positive integers that can be the side lengths of a right angled triangle. Such triples satisfy the Pythagorean theorem, which is fundamental in geometry. The task is to look at each proposed triple and decide which one satisfies the condition a^2 + b^2 = c^2 for some ordering of the numbers where c is the largest. Given Data / Assumptions:Each option contains three positive integers.We assume that if they form a right triangle, the largest of the three integers will be the hypotenuse.The Pythagorean theorem states that in a right triangle with legs a and b and hypotenuse c, we have a^2 + b^2 = c^2.Only one option is intended to be a valid Pythagorean triple.We must compute the squares and compare sums carefully. Concept / Approach:To identify a Pythagorean triple, first identify the largest number in the set as the potential hypotenuse. Then compute the squares of the other two numbers, add them and compare the result with the square of the largest number. If and only if a^2 + b^2 equals c^2, the triple is valid. We repeat this check for each option until we find one that satisfies the condition. Step-by-Step Solution:Step 1: For option a, the numbers are 68, 72 and 81, with 81 as the largest. Compute 68^2 = 4624 and 72^2 = 5184. The sum is 4624 + 5184 = 9808, while 81^2 = 6561, so 9808 is not equal to 6561. Option a is not a Pythagorean triple.Step 2: For option b, the numbers are 35, 38 and 42, with 42 as the largest. Compute 35^2 = 1225 and 38^2 = 1444. The sum is 2669, while 42^2 = 1764, so 2669 is not equal to 1764. Option b is not valid.Step 3: For option c, the numbers are 27, 38 and 42, with 42 again the largest. Compute 27^2 = 729 and 38^2 = 1444. The sum is 2173, but 42^2 is 1764, so there is no equality and option c is not a Pythagorean triple.Step 4: For option d, the numbers are 33, 44 and 55, with 55 as the largest. Compute 33^2 = 1089 and 44^2 = 1936. The sum is 1089 + 1936 = 3025. Now compute 55^2, which is also 3025. Since the sums match, option d is a Pythagorean triple. Verification / Alternative check:Once we have identified 33, 44 and 55 as a Pythagorean triple, we can note that this triple is actually a scaled version of the well known 3, 4, 5 triple. If we multiply 3, 4 and 5 by 11, we get 33, 44 and 55. Because scaling all sides of a right triangle by the same factor preserves the right angle, 33, 44 and 55 must also satisfy the Pythagorean relation. This provides a fast confirmation. Why Other Options Are Wrong:In each of the other options, when we square the two smaller numbers and add the results, the sum does not equal the square of the largest number. This means that they cannot be the sides of a right angled triangle. They might be sides of some triangle, but they do not satisfy the Pythagorean theorem, so they are not Pythagorean triples. Common Pitfalls:A common mistake is to assume that numbers that look close to each other or that differ by similar gaps automatically form a Pythagorean triple. Another pitfall is computing squares incorrectly or forgetting to identify the largest number as the candidate hypotenuse. Careful arithmetic and systematic checking prevent these errors. Final Answer:The only set that forms a Pythagorean triple is 33, 44, 55.

Question 15

Which of the following capital letters of the English alphabet has more than one line of symmetry?

Accepted Answer

Introduction / Context:This question tests knowledge of symmetry, specifically lines of symmetry in capital letters of the English alphabet. A line of symmetry divides a shape into two mirror image halves. Some letters have no line of symmetry, some have exactly one, and a few have more than one line of symmetry. The problem asks which letter among the given options has more than one line of symmetry. Given Data / Assumptions:We are dealing with capital letters A, C and O.We consider the standard block style of these letters as used in basic geometry discussions.A line of symmetry is a straight line such that reflecting the letter across that line leaves the letter unchanged.The question focuses on letters with more than one line of symmetry, not just at least one.We assume an idealized geometric representation of the letters. Concept / Approach:To solve this, we must visualize each letter and identify possible reflection lines. A has one vertical line of symmetry in most standard fonts. C, drawn as an open curve, usually does not have a vertical line of symmetry and does not have more than one clear line of symmetry. The letter O, when drawn as a perfect circle or an ideal ellipse, has multiple lines of symmetry: in the circular case, infinitely many, and even in an ideal symmetric oval, at least two distinct lines, one vertical and one horizontal. Therefore O clearly has more than one line of symmetry. Step-by-Step Solution:Step 1: Consider the letter A. It is typically drawn as two slanting sides joined by a horizontal bar. It has a vertical line of symmetry down the middle, but no horizontal line of symmetry because the top and bottom are different shapes.Step 2: Since A has exactly one line of symmetry, it does not meet the condition of having more than one line of symmetry.Step 3: Now consider the letter C. It looks like a curved arc open on one side. In a standard block style, it has no vertical line of symmetry because the open side is on one side only. It also has no horizontal symmetry, since the top and bottom curves are not mirror images about a horizontal axis.Step 4: Because C has no clear line of symmetry in the idealized sense, it certainly does not have more than one line of symmetry.Step 5: Finally, consider the letter O. If we model O as a perfect circle, it has infinitely many lines of symmetry through its center. Even if we model it as a symmetric ellipse, it still has at least two symmetry lines, one vertical and one horizontal.Step 6: Thus, among A, C and O, only O has more than one line of symmetry. Verification / Alternative check:An easy check is to imagine folding a printed letter on paper. For A, folding along a vertical line through the middle aligns the two halves, but any other fold will not. For C, no straight fold will give perfect overlap. For O, one can fold vertically, horizontally and along many diagonal lines through the center, and the two halves will match exactly. This simple folding test confirms that O has more than one line of symmetry. Why Other Options Are Wrong:Option a, A, has only one vertical line of symmetry, so it does not satisfy the requirement of more than one. Option b, C, has at most one approximate line of symmetry in certain stylized fonts and often none in an ideal geometric interpretation, but definitely not more than one. Option d, All of the above, is incorrect because A and C do not both have more than one line of symmetry. Only O clearly meets the stated condition. Common Pitfalls:Students may misread the phrase and focus on letters that have at least one line of symmetry instead of more than one. Another source of confusion is relying on a particular stylized font rather than the standard geometric form. It is important to apply the formal idea of reflection symmetry rather than rough visual guesses. Final Answer:The capital letter that has more than one line of symmetry among the given options is O.

Question 16

A tangent to a circle touches the circle at exactly one point. A tangent line contains at most how many chords of that circle?

Accepted Answer

Introduction / Context:This question tests understanding of the concepts of a tangent and a chord in circle geometry. A tangent is a line that touches a circle at exactly one point, while a chord is a line segment whose endpoints both lie on the circle. The problem asks how many chords a tangent line can contain at most. This requires careful distinction between a full line and a line segment that lies inside the circle. Given Data / Assumptions:We have a fixed circle in the plane.A tangent to this circle is a straight line that touches the circle at exactly one point.A chord is defined as a segment that has both endpoints on the circle.We are asked for the maximum possible number of distinct chords that lie entirely on the tangent line.We assume standard Euclidean definitions without degeneracy. Concept / Approach:To determine how many chords can lie on a tangent, we must recall that for a chord to exist, both its endpoints must be distinct points on the circle, and the interior of the segment lies inside the circle. A tangent line intersects the circle at a single point, not two. Therefore it cannot provide two distinct points on the circle for a chord. At best, it has one common point with the circle, and a chord requires two. Thus the number of chords on a tangent line is zero. Step-by-Step Solution:Step 1: Recall the definition of a chord. A chord is any line segment with both endpoints on the circle.Step 2: Recall the definition of a tangent. A tangent to a circle touches the circle at exactly one point.Step 3: For a line to contain a chord of the circle, it must intersect the circle at two distinct points, which will be the endpoints of the chord.Step 4: A tangent line, by definition, intersects the circle at only one point, so there is no second distinct point of intersection.Step 5: Since there is no pair of distinct intersection points, there is no possible chord whose entire segment lies on the tangent line.Step 6: Therefore the maximum number of chords of the circle that the tangent can contain is zero. Verification / Alternative check:We can compare this with a secant line, which cuts the circle at exactly two points. A secant line does contain one chord: the segment joining those two intersection points. In contrast, a tangent line is the limiting case where the two intersection points merge into one. As the secant line rotates to become tangent, the chord shrinks to a single point and is no longer considered a chord. This visualization supports the conclusion that a true tangent line does not contain any nondegenerate chord. Why Other Options Are Wrong:Option a suggests that there could be exactly one chord on the tangent line, but that would require two distinct intersection points, which contradicts the definition of a tangent. Option b and option c suggest even more chords, which is impossible since even one chord cannot exist on a line that touches at only one point. Thus all these options conflict with the fundamental geometric definitions. Common Pitfalls:Some learners misinterpret the single point of contact as a very short chord. However, a chord must have length and two distinct endpoints. A single point has no length and does not qualify. Another pitfall is confusing the behavior of secants with tangents. Remember that tangents are special precisely because they intersect the circle at only one point. Final Answer:A tangent to a circle contains at most 0 chords of that circle.

Question 17

The weight of Sona is four times that of Mona. The weight of Mona is 2.5 times that of Hona. The weight of Hona is 1.5 times that of Dhona. The weight of Dhona is 2.5 times that of Rona. Who is the heaviest among Sona, Mona, Hona, Dhona and Rona?

Accepted Answer

Introduction / Context:This question presents a chain of comparative statements about the weights of five people: Sona, Mona, Hona, Dhona and Rona. Instead of giving exact weights, it gives ratios that relate one person to another. The task is to reason through these relationships and identify who is the heaviest. This type of problem is common in quantitative aptitude tests and helps develop comfort with ratios and proportional reasoning. Given Data / Assumptions:The weight of Sona is four times the weight of Mona.The weight of Mona is 2.5 times the weight of Hona.The weight of Hona is 1.5 times the weight of Dhona.The weight of Dhona is 2.5 times the weight of Rona.All weights are positive real numbers.We want to know which person has the greatest weight. Concept / Approach:A convenient strategy is to express every person’s weight in terms of a single basic variable, for example the weight of Rona. If the relationships are consistent, we can then see the relative sizes of each person’s weight by comparing the coefficients in front of that variable. The one with the largest coefficient will be the heaviest. Step-by-Step Solution:Step 1: Let the weight of Rona be x kilograms. We do not need the exact value of x, only the ratios.Step 2: Dhona’s weight is 2.5 times Rona’s weight, so Dhona weighs 2.5x kilograms.Step 3: Hona’s weight is 1.5 times Dhona’s weight, so Hona weighs 1.5 * 2.5x = 3.75x kilograms.Step 4: Mona’s weight is 2.5 times Hona’s weight, so Mona weighs 2.5 * 3.75x = 9.375x kilograms.Step 5: Sona’s weight is four times Mona’s weight, so Sona weighs 4 * 9.375x = 37.5x kilograms.Step 6: Now compare the coefficients: Rona has x, Dhona has 2.5x, Hona has 3.75x, Mona has 9.375x and Sona has 37.5x.Step 7: Since 37.5x is the largest of these values, Sona is clearly the heaviest. Verification / Alternative check:We can also take a simple concrete choice for Rona’s weight, such as x = 2 kilograms, just to check the ordering. Then Dhona weighs 5 kilograms, Hona weighs 7.5 kilograms, Mona weighs 18.75 kilograms and Sona weighs 75 kilograms. The exact numbers do not matter, but the ordering clearly shows that Sona is heavier than Mona, Mona is heavier than Hona, and so on. This numerical example matches the symbolic reasoning and confirms that Sona is the heaviest. Why Other Options Are Wrong:Mona is heavy but still lighter than Sona, because Sona is four times Mona. Hona is lighter than Mona because Mona is 2.5 times Hona. Dhona is lighter than Hona since Hona is 1.5 times Dhona. Rona is the lightest because everyone else’s weight is a multiple of Rona’s weight that is greater than 1. Therefore none of these names except Sona can correspond to the heaviest person. Common Pitfalls:One common error is to misread the chain and reverse the ratios, for example treating “Mona is 2.5 times Hona” as “Hona is 2.5 times Mona.” Another pitfall is trying to guess the answer without working through all the relationships. Expressing every weight in terms of one base variable keeps the relationships clear and prevents confusion. Final Answer:The heaviest person among the five is Sona.

Question 18

In the following arithmetic equation, correct the equality by interchanging the positions of exactly two operation signs: 4 – 10 × 5 + 9 ÷ 3 = 51. Which pair of signs must be interchanged to make the equation true under normal BODMAS rules?

Accepted Answer

Introduction / Context:This question involves an arithmetic expression with multiple operations: subtraction, multiplication, addition and division. The expression is currently incorrect, and the task is to make it correct by interchanging exactly two operation signs. This tests understanding of both operator precedence rules, often remembered as BODMAS, and careful evaluation of expressions. Given Data / Assumptions:The original equation is 4 – 10 × 5 + 9 ÷ 3 = 51.We must interchange two operation signs from among –, ×, + and ÷.Only one interchange is allowed, and the new expression must be evaluated using normal precedence rules: multiplication and division before addition and subtraction, and left to right at the same level.The right hand side is fixed at 51.We assume usual arithmetic on real numbers. Concept / Approach:An efficient approach is to compute the value of the original expression using BODMAS, then systematically test each possible swap suggested in the options. Each time, we recompute the value and check whether it equals 51. Care must be taken to apply precedence correctly. Because the expression is not very long, this trial and check method is straightforward and reliable. Step-by-Step Solution:Step 1: Evaluate the original expression 4 – 10 × 5 + 9 ÷ 3.Step 2: Apply multiplication and division first: 10 × 5 = 50 and 9 ÷ 3 = 3, so the expression becomes 4 – 50 + 3.Step 3: Now evaluate from left to right: 4 – 50 = -46, then -46 + 3 = -43. So the left hand side is -43, not 51.Step 4: Consider option a, swapping × and –. The new expression becomes 4 × 10 – 5 + 9 ÷ 3. Evaluate: 4 × 10 = 40, 9 ÷ 3 = 3, so 40 – 5 + 3 = 35 + 3 = 38, not 51.Step 5: Consider option b, swapping ÷ and ×. The new expression becomes 4 – 10 ÷ 5 + 9 × 3. Evaluate: 10 ÷ 5 = 2 and 9 × 3 = 27, so 4 – 2 + 27 = 2 + 27 = 29, not 51.Step 6: Consider option c, swapping + and –. The expression becomes 4 + 10 × 5 – 9 ÷ 3. Evaluate: 10 × 5 = 50 and 9 ÷ 3 = 3, so 4 + 50 – 3 = 54 – 3 = 51, which matches the right hand side.Step 7: Consider option d, swapping – and ÷. The expression becomes 4 ÷ 10 × 5 + 9 – 3. Evaluate: 4 ÷ 10 = 0.4, 0.4 × 5 = 2, so 2 + 9 – 3 = 11 – 3 = 8, not 51.Step 8: Only option c produces 51, so it is the correct choice. Verification / Alternative check:We can confirm the correctness of the modified expression for option c separately. With plus and minus swapped we have 4 + 10 × 5 – 9 ÷ 3. Using BODMAS, 10 × 5 = 50 and 9 ÷ 3 = 3. The expression becomes 4 + 50 – 3. Compute 4 + 50 = 54, and then 54 – 3 = 51. This equals the required right hand side, confirming that the equation becomes valid under this interchange. Why Other Options Are Wrong:For option a, the expression becomes 4 × 10 – 5 + 9 ÷ 3, which simplifies to 4 × 10 = 40 and 9 ÷ 3 = 3, giving 40 – 5 + 3 = 38, not 51. For option b, the expression becomes 4 – 10 ÷ 5 + 9 × 3, which leads to 4 – 2 + 27 = 29. For option d, the expression becomes 4 ÷ 10 × 5 + 9 – 3, which yields 8. None of these match 51, so they are incorrect. Common Pitfalls:A frequent mistake is to ignore operator precedence and evaluate operations strictly from left to right. This gives incorrect intermediate results and can lead to wrong conclusions about which swap works. Another pitfall is performing arithmetic errors in multiplication or division. Working carefully and applying BODMAS consistently is essential. Final Answer:The equation becomes correct when we interchange the signs + and –.

Question 19

The weights of four boxes are 30, 70, 60 and 20 kilograms. Which of the following cannot be the total weight, in kilograms, of any combination of these boxes if in a combination each box can be used at most once?

Accepted Answer

Introduction / Context:This problem is about subset sums. We are given the weights of four different boxes and asked which given total weight cannot be made by selecting some or all of the boxes, with the restriction that each box can be used at most once in a combination. This is a common type of question in aptitude tests that checks systematic reasoning and combinatorial thinking. Given Data / Assumptions:The four box weights are 30 kilograms, 70 kilograms, 60 kilograms and 20 kilograms.Each combination can use any subset of these boxes, including possibly only one box or all four boxes.No box may be used more than once in any combination.We are given four candidate totals: 180, 190, 120 and 150 kilograms.We must find which total cannot be formed by adding the weights of some subset of the four boxes. Concept / Approach:The straightforward method is to list possible sums of subsets of the four box weights. Since there are only four boxes, the number of subsets is small. We can compute all distinct sums that can arise and then compare them with the candidate totals. The total that does not appear in our list of achievable sums is the required answer. Step-by-Step Solution:Step 1: Let the weights be 30, 70, 60 and 20 kilograms.Step 2: First consider combinations of one box: the possible totals are 30, 70, 60 and 20.Step 3: Consider combinations of two boxes: 30 + 70 = 100, 30 + 60 = 90, 30 + 20 = 50, 70 + 60 = 130, 70 + 20 = 90, and 60 + 20 = 80. The distinct two box sums are 50, 80, 90, 100 and 130.Step 4: Consider combinations of three boxes: 30 + 70 + 60 = 160, 30 + 70 + 20 = 120, and 30 + 60 + 20 = 110, while 70 + 60 + 20 = 150. The three box sums are 160, 120, 110 and 150.Step 5: Consider the combination of all four boxes: 30 + 70 + 60 + 20 = 180.Step 6: Collect all distinct possible totals: 20, 30, 50, 60, 70, 80, 90, 100, 110, 120, 130, 150, 160 and 180.Step 7: Compare these with the options: 180 appears, 120 appears, 150 appears, but 190 does not appear in the list.Step 8: Therefore 190 kilograms cannot be formed with any combination of the given boxes. Verification / Alternative check:We can also reason more directly about the maximum possible weight and near maximum weights. The sum of all four boxes is 180 kilograms, so no combination can exceed 180. Therefore any option greater than 180, such as 190, is immediately impossible. This gives a very fast check. The other totals 120 and 150 are less than 180, so they could potentially be sums. From the earlier subset calculations, we see that 120 and 150 are indeed achievable, while 190 is not. This cross checks our conclusion. Why Other Options Are Wrong:The total 180 is obtained by using all four boxes: 30 + 70 + 60 + 20 = 180. The total 120 is obtained by the combination 30 + 70 + 20. The total 150 is obtained by the combination 70 + 60 + 20. Because each of these totals matches at least one valid subset of boxes, they can be formed and therefore are not correct answers to the question. Common Pitfalls:One common mistake is to overlook the possibility of using all four boxes and conclude that a large total like 180 or 150 cannot be made. Another mistake is careless arithmetic that causes either double counting or missing some subset. A calm, systematic listing of subsets, or a quick look at the maximum possible sum, helps avoid such errors. Final Answer:The only total that cannot be obtained from any combination of the four boxes is 190 kilograms.

Question 20

In a row of 15 children, when Raju is shifted three places towards the right, he becomes 8th from the right end. What was his original position from the left end of the row before he moved?

Accepted Answer

Introduction / Context:This is a classic ranking and arrangement problem that often appears in reasoning tests. The problem describes a row of children and gives Raju’s position from the right after he has moved three places to the right. The task is to determine his original position from the left. Solving such problems requires understanding how left and right positions are related when the total number in the row is known. Given Data / Assumptions:The total number of children in the row is 15.After moving three places to the right, Raju becomes 8th from the right end.Before moving, he was three places to the left of his new position.Positions from the left and positions from the right are counted starting at 1.We assume that no other child changes position during Raju’s movement. Concept / Approach:There are two key ideas. First, for a line of N people, the relationship between a person’s position from the left and from the right is: left position + right position - 1 = N. Second, when Raju moves three places to the right, his position from the left increases by 3. If we find his new position from the left and subtract 3, we obtain his original position from the left. Step-by-Step Solution:Step 1: Use the relationship between left and right positions. For a total of 15 children, left position + right position - 1 = 15.Step 2: After the move, Raju is 8th from the right. Let his new position from the left be L_new.Step 3: Apply the formula: L_new + 8 - 1 = 15.Step 4: Simplify: L_new + 7 = 15, so L_new = 15 - 7 = 8.Step 5: So after the movement, Raju is 8th from the left as well as 8th from the right.Step 6: Raju originally was three places to the left of this new position, so his original left position L_original = L_new - 3.Step 7: Compute L_original = 8 - 3 = 5.Step 8: Therefore Raju was originally 5th from the left end of the row. Verification / Alternative check:We can visualize the row briefly. With 15 children, the positions from the left are numbered 1 through 15, and from the right they are 15 through 1. If Raju is now 8th from the right, he is at the position that matches left position 8. If he moved three places to the right to reach position 8, then he must have been at position 5 earlier. This simple visual line check is consistent with the algebraic solution. Why Other Options Are Wrong:If Raju had originally been at position 14 or 12 from the left, moving three places to the right would place him beyond the end of the row, which is impossible. If he had been 6th from the left, moving three places to the right would make him 9th from the left, which would correspond to 7th from the right in a row of 15, not 8th. Therefore these options do not match the given information. Common Pitfalls:Some learners mistakenly add or subtract positions from the wrong side or forget to use the total number in the row when relating left and right positions. Others forget that moving to the right increases the left position number. Keeping the relationship left position + right position - 1 = total can help avoid these errors. Final Answer:Raju’s original position from the left end of the row was 5.

Arithmetic Reasoning Questions

Questions

Practice smarter