Question 1

Five farmers have 7, 9, 11, 13, and 14 apple trees respectively in their orchards. Each farmer finds that every tree in his own orchard yields exactly the same number of apples. It is further observed that if the 3rd farmer gives one apple to the 1s…

Accepted Answer

Introduction / Context: This arithmetic reasoning question involves equal yields per tree and redistribution of apples so that all farmers finally have the same number of apples. Each farmer has a different number of apple trees, but within each orchard every tree yields the same number of apples. After certain transfers of apples among farmers, all of them end up with equal totals. The aim is to determine the per tree yields of the 3rd and 4th farmers. Given Data / Assumptions: Farmer 1 has 7 trees, farmer 2 has 9 trees, farmer 3 has 11 trees, farmer 4 has 13 trees, and farmer 5 has 14 trees. Within each orchard, every tree yields the same number of apples, so totals are 7y1, 9y2, 11y3, 13y4, and 14y5. The 3rd farmer gives 1 apple to the 1st farmer. The 5th farmer gives 3 apples to the 2nd farmer and 3 apples to the 4th farmer, a total of 6 apples. After these transfers, all farmers have the same total number of apples. We must find the yields per tree of the 3rd farmer (y3) and the 4th farmer (y4). Concept / Approach: Let A be the final common number of apples each farmer holds after all transfers. Express the final total for each farmer in terms of the original totals and the number of apples given or received. Equate these expressions to A to form linear equations involving y1, y2, y3, y4, and y5. The crucial relation needed to answer the question involves y3 and y4, which we can obtain by focusing on the equations for the 3rd and 4th farmers. Step-by-Step Solution: Step 1: Initial totals: farmer 1 has 7y1 apples, farmer 2 has 9y2, farmer 3 has 11y3, farmer 4 has 13y4, and farmer 5 has 14y5. Step 2: After transfers: farmer 1 receives 1 apple from farmer 3, so total becomes 7y1 + 1. Step 3: Farmer 2 receives 3 apples from farmer 5, so total becomes 9y2 + 3. Step 4: Farmer 3 gives 1 apple, so total becomes 11y3 - 1. Step 5: Farmer 4 receives 3 apples from farmer 5, so total becomes 13y4 + 3. Step 6: Farmer 5 gives away 6 apples in total, so total becomes 14y5 - 6. Step 7: All final totals are equal to A: 7y1 + 1 = 9y2 + 3 = 11y3 - 1 = 13y4 + 3 = 14y5 - 6 = A. Step 8: Equate the expressions for farmers 3 and 4: 11y3 - 1 = 13y4 + 3. Step 9: Rearranging gives 13y4 - 11y3 = -4. Step 10: Check the given option pairs for (y3, y4). For (11, 9), compute 13*9 - 11*11 = 117 - 121 = -4, which satisfies the equation. Verification / Alternative check: The other option pairs do not satisfy 13y4 - 11y3 = -4. For example, for (17, 9): 13*9 - 11*17 = 117 - 187 = -70; for (9, 11): 13*11 - 11*9 = 143 - 99 = 44; for (9, 9): 117 - 99 = 18. Only (11, 9) yields -4. Because the condition of equal final totals constrains y3 and y4 through this equation, the only consistent pair is y3 = 11 and y4 = 9. Why Other Options Are Wrong: Each alternative pair fails to satisfy the key relationship 13y4 - 11y3 = -4 that comes from equating the final totals of the 3rd and 4th farmers. As a result, those pairs would lead to unequal final totals and cannot describe a situation where all farmers end with the same number of apples. Common Pitfalls: This type of problem can be confusing because there are many variables and totals to track. A common mistake is to try to solve for all yields explicitly rather than focusing on the essential equations that involve the unknowns asked in the question. Another pitfall is miscounting the apples transferred, especially from the 5th farmer, who gives to two different farmers. Carefully constructing equations and then checking only the necessary pairs simplifies the reasoning. Final Answer: The yields per tree for the 3rd and 4th farmers are 11 and 9 apples respectively.

Question 2

In a family, each daughter has the same number of brothers as she has sisters, and each son has twice as many sisters as he has brothers. Based on this information, how many sons are there in the family?

Accepted Answer

Introduction / Context: This classic family puzzle is an arithmetic reasoning and basic algebra question. It describes relationships between numbers of sons and daughters using counts of brothers and sisters from each child’s perspective. By carefully translating the statements into equations and solving them, we can determine the number of sons in the family. Given Data / Assumptions: Let the family have s sons and d daughters. Each daughter has the same number of brothers as sisters. Each son has twice as many sisters as he has brothers. We assume that all children are either sons or daughters and that there is at least one of each. Concept / Approach: From the point of view of any daughter, the number of brothers is s and the number of sisters is d - 1, because she does not count herself. The statement that these two numbers are equal leads to the equation s = d - 1. From the point of view of any son, the number of brothers is s - 1 and the number of sisters is d. The statement that each son has twice as many sisters as brothers gives d = 2(s - 1). Solving this small system of two equations in two unknowns yields the number of sons and daughters. Step-by-Step Solution: Step 1: From each daughter: number of brothers = s, number of sisters = d - 1, and these are equal. So s = d - 1. Step 2: Rearrange s = d - 1 to get d = s + 1. Step 3: From each son: number of brothers = s - 1, number of sisters = d, and each son has twice as many sisters as brothers. So d = 2(s - 1). Step 4: Substitute d = s + 1 into d = 2(s - 1): s + 1 = 2(s - 1). Step 5: Expand the right side: s + 1 = 2s - 2. Step 6: Rearrange: 1 + 2 = 2s - s, so 3 = s. Step 7: Therefore, the number of sons s is 3. The number of daughters d = s + 1 = 4, although the question asks only for the number of sons. Verification / Alternative check: Check from a daughter’s perspective: with s = 3 and d = 4, each daughter has 3 brothers and 4 - 1 = 3 sisters, so that condition holds. From a son’s perspective, each son has 3 - 1 = 2 brothers and 4 sisters, and 4 is twice 2, so that condition also holds. Both statements are satisfied, confirming that 3 sons is correct. Why Other Options Are Wrong: If s = 1 or s = 2, the equations s = d - 1 and d = 2(s - 1) do not yield consistent integer values for d that satisfy both conditions simultaneously. For s = 4, you get d = 5 from the first equation but d = 6 from the second, which is contradictory. Thus options A, B, and D lead to inconsistent family structures. Common Pitfalls: A frequent error is to count the child himself or herself when listing brothers or sisters, which changes the counts by one and leads to incorrect equations. Another mistake is to assume that the number of sons and daughters must be the same. Writing the relationships algebraically avoids both issues and makes the solution straightforward. Final Answer: The family has 3 sons.

Question 3

A bird lover is asked how many birds he has in his basket. He replies: "All are parrots but six, all are pigeons but six, and all are koels but six." Based on this information, how many birds are there in the basket in total?

Accepted Answer

Introduction / Context: This is a well known logical puzzle involving three types of birds in a basket. The owner describes the collection using three similar statements, each mentioning that all birds are of one type except six. The key is to interpret these sentences correctly and turn them into a simple system of equations for the numbers of parrots, pigeons, and koels. Once interpreted properly, the arithmetic is straightforward. Given Data / Assumptions: Let the basket contain parrots, pigeons, and koels only. Statement 1: all are parrots but six, so all birds except six are parrots. Statement 2: all are pigeons but six, so all birds except six are pigeons. Statement 3: all are koels but six, so all birds except six are koels. Let the numbers of parrots, pigeons, and koels be p, q, and r respectively. Total number of birds is N = p + q + r. Concept / Approach: The phrase “all are parrots but six” means that the birds that are not parrots number six. That is, the non parrots are the pigeons and koels, whose total is six. Similarly, “all are pigeons but six” implies that non pigeons (parrots plus koels) total six. And “all are koels but six” means non koels (parrots plus pigeons) total six. These three equations can be solved simultaneously to obtain p, q, r, and thus the total N. Step-by-Step Solution: Step 1: From “all are parrots but six,” non parrots are pigeons and koels: q + r = 6. Step 2: From “all are pigeons but six,” non pigeons are parrots and koels: p + r = 6. Step 3: From “all are koels but six,” non koels are parrots and pigeons: p + q = 6. Step 4: Add all three equations: (q + r) + (p + r) + (p + q) = 6 + 6 + 6 = 18. Step 5: The left side simplifies to 2(p + q + r) = 2N, so 2N = 18. Step 6: Therefore N = 18 / 2 = 9. Step 7: Hence the total number of birds in the basket is 9. Verification / Alternative check: We can also find individual counts: from p + q = 6 and q + r = 6, subtracting gives p - r = 0, so p = r. Similarly, from p + r = 6 and p + q = 6 you get r = q, so p = q = r. Since p + q + r = 9 and all three are equal, each type must have 3 birds. Then non parrots (pigeons plus koels) are 3 + 3 = 6, and similarly for the other two statements. This matches the puzzle description perfectly. Why Other Options Are Wrong: Options 8, 10, or 18 do not allow the birds to be split into three equal groups that satisfy all three conditions simultaneously. Trying to assign integer counts with those totals soon leads to inconsistencies between the equations q + r = 6, p + r = 6, and p + q = 6. Common Pitfalls: Many people initially misread the statements as referring to how many of each type there are, rather than to how many are not of a certain type. Another pitfall is to attempt trial and error with random numbers of birds instead of setting up simple equations. Interpreting “all are type X but six” as “the birds that are not type X total six” is the crucial insight. Final Answer: The total number of birds in the basket is 9.

Question 4

It takes five minutes for one person to pass a rumour to two other persons. Each newly informed person then continues the process in the same way, so that every informed person passes the rumour on to two more people in the next five minutes. How ma…

Accepted Answer

Introduction / Context: This arithmetic reasoning question models the spread of a rumour using a branching pattern. It is effectively a problem of geometric progression. The rumour starts with one person, and in fixed time intervals, each informed person passes the rumour to two new people. You must determine how many such five minute intervals are needed for the total number of people who know the rumour to reach 729. Given Data / Assumptions: Initially, 1 person knows the rumour. Every five minutes, each informed person tells the rumour to 2 new people. Once a person knows the rumour, he or she continues to spread it to new people in the next step. The process repeats identically each five minute period. The target is a total of 729 informed persons. Concept / Approach: At each stage, the number of people who know the rumour multiplies by 3. This happens because each current knower remains informed and also informs two new people. Thus, if the total number of informed people after n steps is Tn, we get Tn = 3^n, assuming we start counting steps after the first five minute period. The problem becomes finding n such that 3^n = 729 and then multiplying n by 5 minutes to get the total time. Step-by-Step Solution: Step 1: After the first 5 minutes, the original person tells 2 people, so the total who know the rumour is 3 (1 original + 2 new). Step 2: After the second 5 minute period, these 3 people each tell 2 new people, adding 6 more, so the total becomes 9. This is 3^2. Step 3: Similarly, after the third 5 minute period, the total becomes 27 (3^3); after the fourth, 81 (3^4); after the fifth, 243 (3^5). Step 4: After the sixth 5 minute period, the total becomes 729 (3^6). Step 5: Therefore, it takes 6 such periods to reach 729 people. Each period is 5 minutes long. Step 6: Total time = 6 * 5 minutes = 30 minutes. Verification / Alternative check: Check the powers: 3^1 = 3, 3^2 = 9, 3^3 = 27, 3^4 = 81, 3^5 = 243, and 3^6 = 729. This confirms that after six rounds the total informed population is exactly 729. No smaller exponent of 3 gives 729, so the number of steps cannot be fewer than six. Why Other Options Are Wrong: Option B 25 minutes and option D 36 minutes correspond to non integer multiples of 5 or to wrong numbers of steps. Option A 45 minutes would correspond to nine steps (9 * 5), which would give 3^9 = 19,683 informed people, far more than required. Only 30 minutes matches the step count obtained from 3^6. Common Pitfalls: Some learners mistakenly assume that the number of informed people doubles instead of tripling, ignoring that each knower remains informed while also informing two more people. Others miscount the number of rounds needed to reach 729, often stopping at 243 or 81. Carefully tracking the total after each step and recognizing the geometric progression pattern 3^n helps avoid these mistakes. Final Answer: It takes 30 minutes for the rumour to reach 729 persons.

Question 5

In the number 94257636, if 2 is subtracted from each odd digit and 3 is added to each even digit, a new number is formed. In this new number, how many distinct digits appear exactly twice?

Accepted Answer

Introduction / Context: This arithmetic reasoning question tests your ability to work carefully with digits and simple transformation rules. Starting from an eight digit number, you are asked to apply different operations to odd and even digits and then count how many distinct digits appear exactly twice in the transformed number. It is a good exercise in attention to detail and counting. Given Data / Assumptions: The original number is 94257636. For each odd digit, subtract 2 to get the new digit. For each even digit, add 3 to get the new digit. The digits are transformed independently, position by position. We must examine the resulting sequence of digits and count how many different digit values occur exactly twice. Concept / Approach: The method is straightforward: process the number digit by digit. Identify whether each digit is odd or even, apply the required arithmetic, and write down the new digit. After obtaining the new eight digit sequence, count how many times each distinct digit appears. The question is not asking how many times digits repeat in total, but how many distinct digit values have a frequency of exactly two in the new number. Step-by-Step Solution: Step 1: Write the digits of 94257636: 9, 4, 2, 5, 7, 6, 3, 6. Step 2: Apply the rule: odd digits → subtract 2; even digits → add 3. Step 3: 9 (odd) becomes 9 - 2 = 7. Step 4: 4 (even) becomes 4 + 3 = 7. Step 5: 2 (even) becomes 2 + 3 = 5. Step 6: 5 (odd) becomes 5 - 2 = 3. Step 7: 7 (odd) becomes 7 - 2 = 5. Step 8: 6 (even) becomes 6 + 3 = 9. Step 9: 3 (odd) becomes 3 - 2 = 1. Step 10: 6 (even) becomes 6 + 3 = 9. Step 11: The new sequence is 7, 7, 5, 3, 5, 9, 1, 9. Verification / Alternative check: Count the frequency of each digit in the new number. Digit 7 appears twice, digit 5 appears twice, digit 9 appears twice, digit 3 appears once, and digit 1 appears once. Therefore, three distinct digits (7, 5, and 9) each occur exactly twice. The correct count is therefore 3. Why Other Options Are Wrong: Option A 0 would mean no digit appears exactly twice, which is not true. Option B 1 and option C 2 both underestimate the actual number of distinct digits that appear two times. They could result from miscalculating one or two of the transformations or miscounting frequencies. Common Pitfalls: A typical mistake is to misclassify a digit as odd or even, especially 0, 2, 4, 6, and 8, or to perform subtraction and addition incorrectly when doing the transformations mentally. Another error occurs when counting frequencies quickly without writing them down. Listing the new digits clearly and tallying each distinct value prevents these errors. Final Answer: In the new number, 3 distinct digits appear exactly twice.

Question 6

In arithmetic conversion, how many feet are there in exactly one metre (1 m), to five decimal places?

Accepted Answer

Introduction / Context: This arithmetic reasoning question checks knowledge of a standard unit conversion between the metric system and the imperial system. The conversion between metres and feet is often needed in quantitative aptitude, physics, and engineering problems. Here, you are asked specifically how many feet correspond to one metre, given to five decimal places. Given Data / Assumptions: The question uses the standard international definition of the metre and the foot. We are asked for the number of feet in exactly one metre. The answer is to be selected from the given decimal approximations. Concept / Approach: The internationally accepted relationship is that 1 inch is exactly 2.54 centimetres, and 1 foot is 12 inches. Consequently, 1 foot equals 30.48 centimetres. A metre is 100 centimetres. Therefore, to find how many feet are in a metre, we divide 100 centimetres by 30.48 centimetres per foot. The result is approximately 3.28084 feet per metre. Recognizing or recalling this standard factor is extremely useful for quick conversions in many aptitude questions. Step-by-Step Solution: Step 1: Use the standard conversion 1 inch = 2.54 centimetres. Step 2: Since 1 foot = 12 inches, 1 foot = 12 * 2.54 = 30.48 centimetres. Step 3: 1 metre = 100 centimetres by definition. Step 4: Number of feet in 1 metre = total centimetres in 1 metre divided by centimetres per foot. Step 5: Compute 100 / 30.48, which is approximately 3.28084. Step 6: Compare this with the options; option B gives 3.28084, matching the correct conversion factor. Verification / Alternative check: As a check, reverse the conversion. If 1 metre equals 3.28084 feet, then 3.28084 feet multiplied by 30.48 centimetres per foot equals approximately 100 centimetres. This round trip consistency confirms that 3.28084 is accurate to five decimal places and that the other decimals are not correct representations of 1 metre in feet. Why Other Options Are Wrong: Options A 3.214, C 3.2516, and D 3.21456 are close in magnitude but do not yield exactly 100 centimetres when converted back. They correspond to incorrect or approximate conversion factors. Only 3.28084 matches the accepted international standard for metre to feet conversion. Common Pitfalls: Learners sometimes memorize approximate values like 3.3 feet for 1 metre and then misinterpret them as exact. Others confuse the reciprocal conversion 1 foot ≈ 0.3048 metre with the direction from metre to feet. Remembering the more precise value 3.28084 for competitive exams allows for more accurate calculations in related problems. Final Answer: Exactly one metre is equal to approximately 3.28084 feet.

Question 7

Consider the number 43403. The digit 3 appears twice in this number, in different places. What is the difference between the local (place) values of the two 3s in 43403?

Accepted Answer

Introduction / Context: This arithmetic reasoning question tests understanding of the concept of local or place value of digits in a number. The digit 3 appears in two different positions in the number 43403, and each position corresponds to a different place value. The problem asks for the numerical difference between these two place values. Given Data / Assumptions: The number under consideration is 43403. The digits from left to right are 4, 3, 4, 0, and 3. The left 3 is in the thousands place, and the right 3 is in the ones place. Local value means digit multiplied by the value of its place (ones, tens, hundreds, thousands, and so on). Concept / Approach: In the decimal place value system, the rightmost position is the ones place, then tens, hundreds, thousands, and higher powers of 10 moving left. The place value of a digit equals that digit multiplied by 10 raised to the power corresponding to its position. The difference between the local values of the two 3s is simply the larger place value minus the smaller place value. Step-by-Step Solution: Step 1: Write the number 43403 and identify the positions of the digit 3. One 3 is the second digit from the left, the other 3 is the last digit on the right. Step 2: Determine the place of the left 3. From right to left, the positions are ones (3), tens (0), hundreds (4), thousands (3), ten thousands (4). So the left 3 is in the thousands place. Step 3: The local value of the left 3 is 3 * 1000 = 3000. Step 4: Determine the place of the right 3. It is in the ones place. Step 5: The local value of the right 3 is 3 * 1 = 3. Step 6: The difference between the local values is 3000 - 3 = 2997. Verification / Alternative check: You can rewrite 43403 as 4 * 10^4 + 3 * 10^3 + 4 * 10^2 + 0 * 10^1 + 3 * 10^0. The two 3s appear as 3 * 10^3 and 3 * 10^0. Their values are 3,000 and 3 respectively. Subtracting gives 3,000 - 3 = 2,997, reinforcing the earlier calculation. Why Other Options Are Wrong: Option A 0 would mean the two place values are equal, which is clearly not true. Option B 3,030 and option C 3,003 may come from adding or otherwise confusing the positions, but they do not represent the difference 3000 - 3. Only 2,997 equals the correct subtraction. Common Pitfalls: A common mistake is to confuse digit value with place value. The digit value of both 3s is 3, but their local values differ because of their positions. Another error is misidentifying the positions in the number, especially when counting from the wrong side. Always count places from the rightmost digit as ones, then tens, hundreds, and so on. Final Answer: The difference between the local values of the two 3s is 2,997.

Question 8

If the least common multiple (LCM) and the highest common factor (HCF) of two numbers are equal to each other, what must be true about the two numbers?

Accepted Answer

Introduction / Context: This arithmetic reasoning question focuses on properties of the least common multiple (LCM) and the highest common factor (HCF), also called the greatest common divisor (GCD). It asks what can be concluded about two numbers if their LCM and HCF happen to be the same value. Understanding how LCM and HCF relate to the numbers helps in many number theory and divisibility problems. Given Data / Assumptions: There are two positive integers, say a and b. The LCM of a and b equals their HCF. We need to determine what relationship between a and b must hold. Concept / Approach: In general, for any two positive integers a and b, the product of their LCM and HCF is equal to the product of the numbers: LCM(a, b) * HCF(a, b) = a * b. If LCM and HCF are equal, say both equal to k, then k * k = a * b. This means a * b = k^2. When the LCM and HCF are equal, the only way this equality holds consistently is when both numbers themselves equal k, so that a = k and b = k. Step-by-Step Solution: Step 1: Let the two numbers be a and b. Let their HCF be h and their LCM be l. Step 2: Use the standard identity: h * l = a * b. Step 3: Given that h = l, denote this common value by k. Then the identity becomes k * k = a * b. Step 4: So a * b = k^2. But the HCF of a and b is k, which means k divides both a and b. Step 5: Write a = k * m and b = k * n, where m and n are integers with HCF 1 (since k is the highest common factor). Step 6: Then a * b = (k * m) * (k * n) = k^2 * m * n. Step 7: From k^2 * m * n = k^2, it follows that m * n = 1. Step 8: The only positive integers whose product is 1 are m = 1 and n = 1. Step 9: Therefore a = k and b = k, so the two numbers are equal to each other. Verification / Alternative check: Consider a simple numerical example. Let both numbers be 12. Then HCF(12, 12) = 12 and LCM(12, 12) = 12. They are equal and equal to the numbers themselves. If you try different numbers like 6 and 18, HCF is 6 and LCM is 18; these are not equal. This supports the theoretical conclusion that equality of LCM and HCF can only occur when the two numbers themselves are equal. Why Other Options Are Wrong: Option A “Prime Numbers” is not necessarily true because two numbers that are equal could be composite, such as 12 and 12. Option B “Composite Numbers” is not always true because the equal numbers could be prime, such as 7 and 7. Option D “Co-prime Numbers” is incorrect, since co-prime numbers have HCF 1 and LCM equal to their product, which is usually larger than 1, so the two are not equal. Common Pitfalls: A common confusion is to assume that prime numbers must be involved whenever HCF and LCM are mentioned. Another error is to misapply the identity between LCM, HCF, and the product of the numbers. Remember that the special equality case for LCM and HCF simply forces the two numbers to coincide. Final Answer: If the LCM and HCF of two numbers are equal, then the numbers themselves must be equal.

Question 9

Find the least number by which 175760 must be divided in order to make the quotient a perfect cube.

Accepted Answer

Introduction / Context: This arithmetic reasoning question involves prime factorization and properties of perfect cubes. You are given a number, 175760, and asked to find the smallest integer by which it should be divided so that the resulting quotient is a perfect cube. Understanding how exponents of prime factors must behave for a number to be a perfect cube is the key to solving such problems efficiently. Given Data / Assumptions: The original number is 175760. We seek the least positive integer k such that 175760 / k is a perfect cube. A perfect cube is a number whose prime factor exponents are all multiples of 3. We may use prime factorization to understand the structure of 175760. Concept / Approach: First, factor 175760 into prime factors and examine the exponents of these primes. For a number to be a cube, each exponent in its prime factorization must be divisible by 3. Since we are dividing by k, we are reducing some of these exponents. Our goal is to remove just enough prime factors so that the remaining exponents become multiples of 3, and the product of the removed factors is as small as possible. We do this by choosing the smallest exponents to subtract that achieve multiples of 3. Step-by-Step Solution: Step 1: Prime factorize 175760. It factors as 175760 = 2^4 * 5^1 * 13^3. Step 2: For a perfect cube, all exponents of primes must be multiples of 3. Step 3: Currently, the exponents are: for 2 it is 4, for 5 it is 1, and for 13 it is 3. Step 4: The exponent 3 for 13 is already a multiple of 3, so we do not need to remove any factor of 13. Step 5: For the prime 2, the exponent is 4. To reach the nearest lower multiple of 3, we reduce it to 3 by dividing by 2^1. Step 6: For the prime 5, the exponent is 1. The nearest lower multiple of 3 is 0, so we must remove 5^1 by dividing by 5. Step 7: The smallest number we must divide by is therefore 2^1 * 5^1 = 2 * 5 = 10. Step 8: After division, the quotient is 175760 / 10 = 2^3 * 13^3, which is (2 * 13)^3 = 26^3, a perfect cube. Verification / Alternative check: Check that dividing by any smaller factor will not produce a cube. For example, dividing only by 2 gives exponents 3 for 2, 1 for 5, and 3 for 13, so the exponent of 5 is still 1, not a multiple of 3. Dividing only by 5 leaves the exponent of 2 at 4, which is still not a multiple of 3. Thus, you need to remove at least one factor of 2 and one factor of 5, giving the product 10 as the minimal divisor. Why Other Options Are Wrong: Option A 6 removes 2 and 3, but 3 is not even a prime factor of 175760, and the exponent of 5 remains unchanged, so the result is not a cube. Option C 9 removes 3^2, again not helpful because 3 is not present in the prime factorization. Option D 30 removes 2, 3, and 5, which is more than necessary and not the least number that works. Common Pitfalls: Students sometimes confuse the process for making a number a perfect square with that for a perfect cube and try to pair exponents instead of grouping them in triples. Others attempt to multiply by a number to get a cube instead of dividing, which is a different type of question. Always remember that to make a quotient a perfect cube by division, you must remove primes until all exponents are multiples of 3. Final Answer: The least number by which 175760 must be divided to make it a perfect cube is 10.

Question 10

A child has 1,200 five rupee coins and wants to arrange them in a rectangular pattern such that the number of rows equals the number of columns, that is, in a perfect square array. What is the minimum number of additional coins he needs so that this…

Accepted Answer

Introduction / Context: This arithmetic reasoning question is about perfect squares. The child has 1,200 identical coins and wants to arrange them in a square array such that the number of rows equals the number of columns. Because 1,200 is not a perfect square, he must add some more coins. You are asked to find the smallest number of additional coins required to make the total coin count a perfect square. Given Data / Assumptions: The child initially has 1,200 coins. An arrangement in which rows equal columns requires a perfect square number of coins. We seek the least non negative integer x such that 1,200 + x is a perfect square. The answer must be chosen from the given options. Concept / Approach: The idea is to find the smallest perfect square greater than 1,200. The additional coins needed will be the difference between that perfect square and 1,200. To do this efficiently, compute the integer square root of 1,200, find the next integer, and square it. The result is the next perfect square after 1,200. This is a standard trick in such questions and avoids listing squares one by one. Step-by-Step Solution: Step 1: Find the largest integer whose square is less than or equal to 1,200. Step 2: 34^2 = 1,156 and 35^2 = 1,225. So 34^2 is less than 1,200 and 35^2 is greater than 1,200. Step 3: Therefore, the next perfect square after 1,200 is 35^2 = 1,225. Step 4: The number of additional coins required is 1,225 - 1,200. Step 5: Compute the difference: 1,225 - 1,200 = 25. Step 6: Hence, the child must add 25 more coins to have a total that is a perfect square. Verification / Alternative check: Check that 1,225 coins can indeed be arranged as a 35 by 35 square, because 35 rows of 35 coins each gives exactly 35 * 35 = 1,225 coins. Also verify that 1,200 itself is not a perfect square by noting that 34^2 = 1,156 and 35^2 = 1,225, so there is no integer whose square is 1,200. Thus, 25 is the smallest addition required. Why Other Options Are Wrong: Option A 125 would take the total to 1,325, which is not a perfect square. Option B 96 would give a total of 1,296, which is 36^2, but it is more than the minimum 25 required to reach the smaller perfect square 1,225. Option C 35 raises the total to 1,235, which is not a perfect square. The problem asks for the minimum, so 25 is the only correct answer. Common Pitfalls: Some students do not look for the nearest perfect square and instead jump to a larger square like 36^2, thereby adding more coins than necessary. Others may confuse the process and try to subtract coins to reach a lower perfect square, which is not allowed in this problem. Remember that you must add the smallest amount to reach the next perfect square above the current total. Final Answer: The minimum number of additional coins needed is 25.

Question 11

In a certain coded operation, the symbols are redefined as follows: "x" means "-", "÷" means "+", "-" means "÷", and "+" means "x". Using these new meanings, evaluate the expression 24 - 4 ÷ 6 x 3 + 4.

Accepted Answer

Introduction / Context: This arithmetic reasoning question involves operator coding and order of operations. The usual meanings of the symbols "+", "-", "x", and "÷" are changed to new meanings. You must first translate the given expression using the new definitions and then evaluate it using standard arithmetic precedence rules. This tests both symbol substitution skills and understanding of the correct order of operations. Given Data / Assumptions: Symbol "x" means subtraction "-". Symbol "÷" means addition "+". Symbol "-" means division "÷". Symbol "+" means multiplication "x". The original expression is 24 - 4 ÷ 6 x 3 + 4. After substitution, we evaluate using normal precedence: division and multiplication before addition and subtraction, working from left to right. Concept / Approach: The key steps are: (1) carefully substitute each operator in the expression with its new meaning; (2) obtain a new expression involving the usual operators +, -, x, and ÷; and (3) evaluate this new expression using standard arithmetic rules. It is important not to change the order of the numbers, only the meanings of the operators. This type of problem punishes careless substitution or ignoring operator precedence. Step-by-Step Solution: Step 1: Start with the original coded expression: 24 - 4 ÷ 6 x 3 + 4. Step 2: Replace each operator with its true meaning: "-" means "÷", so 24 - 4 becomes 24 ÷ 4 as the first operation symbol. "÷" means "+", so 4 ÷ 6 becomes 4 + 6. "x" means "-", so 6 x 3 becomes 6 - 3. "+" means "x", so 3 + 4 becomes 3 x 4. Step 3: After substitution, the entire expression becomes: 24 ÷ 4 + 6 - 3 x 4. Step 4: Apply normal precedence rules. First perform division and multiplication from left to right, then addition and subtraction. Step 5: Compute 24 ÷ 4 = 6. The expression now is 6 + 6 - 3 x 4. Step 6: Next do multiplication: 3 x 4 = 12. The expression becomes 6 + 6 - 12. Step 7: Finally, perform addition and subtraction from left to right: 6 + 6 = 12, and 12 - 12 = 0. Verification / Alternative check: If you prefer, you can recompute the entire process carefully to confirm. Rewriting the sequence of operations and recalculating yields the same final expression 6 + 6 - 12 and the same result 0. Since both passes lead to 0, the answer is consistent and correct. Why Other Options Are Wrong: Options 1, 2, and 3 would arise only if the substitution of operators or the order of operations were handled incorrectly, such as evaluating operations strictly left to right without honoring precedence, or forgetting that "+" has been recoded to mean multiplication. None of these options matches the properly evaluated result. Common Pitfalls: The most common mistakes are to substitute operators incorrectly, to change numbers by accident when recoding, or to evaluate the transformed expression strictly from left to right, ignoring the standard rule that division and multiplication must be handled before addition and subtraction. Writing the substituted expression clearly and then applying arithmetic rules step by step helps avoid these errors. Final Answer: The value of the coded expression is 0.

Question 12

In a parking lot there are only two types of vehicles: two-wheelers and four-wheelers. The total number of vehicles is 200, and the total number of wheels counted is 580. How many four-wheeler vehicles are there in the parking lot?

Accepted Answer

Introduction / Context: This arithmetic reasoning question uses a simple system of linear equations to model a parking lot with two-wheelers and four-wheelers. You are given the total number of vehicles and the total number of wheels and must determine how many vehicles are four-wheelers. This type of problem frequently appears in aptitude tests to check understanding of simultaneous equations and logical setup. Given Data / Assumptions: Total number of vehicles (two-wheelers plus four-wheelers) is 200. Each two-wheeler has 2 wheels. Each four-wheeler has 4 wheels. Total number of wheels is 580. There are no other types of vehicles besides two-wheelers and four-wheelers. Concept / Approach: Let the number of two-wheelers be T and the number of four-wheelers be F. The total vehicle count gives one equation, T + F = 200. The total wheel count gives another equation, 2T + 4F = 580. Solving this pair of linear equations will yield the values of T and F. Once F is known, we have the required number of four-wheelers. Step-by-Step Solution: Step 1: Let T be the number of two-wheelers and F be the number of four-wheelers. Step 2: From the total number of vehicles, T + F = 200. Step 3: From the total number of wheels, 2T + 4F = 580. Step 4: Simplify the wheels equation by dividing by 2: T + 2F = 290. Step 5: Now we have two equations: T + F = 200 and T + 2F = 290. Step 6: Subtract the first equation from the second: (T + 2F) - (T + F) = 290 - 200. Step 7: This gives F = 90. Step 8: Therefore, there are 90 four-wheelers in the parking lot. Verification / Alternative check: Using F = 90 in the first equation T + F = 200, we get T = 200 - 90 = 110 two-wheelers. Check the wheels: 110 two-wheelers contribute 110 * 2 = 220 wheels, and 90 four-wheelers contribute 90 * 4 = 360 wheels. Total wheels = 220 + 360 = 580, which matches the given data, confirming the solution is correct. Why Other Options Are Wrong: Option A 110 is actually the number of two-wheelers, not four-wheelers. Option C 100 and option D 180 do not satisfy both the vehicle count and the wheel count simultaneously when used in the equations, and so they are inconsistent with the given information. Common Pitfalls: A common error is to confuse the counts of two-wheelers and four-wheelers or to miswrite the wheel equation as 2T + 4F = 200 instead of equaling 580. Another pitfall is to try guessing values instead of solving the equations systematically, which can waste time and lead to mistakes. Writing and solving the equations clearly is the most reliable method. Final Answer: There are 90 four-wheelers in the parking lot.

Question 13

If the two straight lines p x + y = 2 and 3 x + y = 4 are parallel to each other, what is the value of p?

Accepted Answer

Introduction / Context:This arithmetic reasoning question tests your understanding of the condition for two straight lines to be parallel in coordinate geometry. When two linear equations in x and y represent parallel lines, their slopes must be equal. By carefully comparing the coefficients of x and y in the given equations, we can find the required value of p that makes the two lines parallel. Given Data / Assumptions:- First line: p x + y = 2.- Second line: 3 x + y = 4.- Both equations are in the standard linear form a x + b y = c.- We assume x and y are real variables and p is a real constant to be determined. Concept / Approach:For a straight line written as a x + b y = c, the slope m is given by m = -a / b, provided b is not zero. Two lines are parallel if and only if they have the same slope but different intercepts. Therefore, if line 1 has slope m1 and line 2 has slope m2, the condition for parallelism is m1 = m2. We will compute the slopes of both lines and equate them to solve for p. The constant terms on the right side do not affect the slopes, only the intercepts. Step-by-Step Solution:- Rewrite the first line p x + y = 2. Here a1 = p and b1 = 1.- The slope of the first line is m1 = -a1 / b1 = -p / 1 = -p.- Rewrite the second line 3 x + y = 4. Here a2 = 3 and b2 = 1.- The slope of the second line is m2 = -a2 / b2 = -3 / 1 = -3.- For the lines to be parallel, set m1 = m2, so -p = -3.- Multiply both sides by -1 to get p = 3. Verification / Alternative check:If p = 3, both equations become 3 x + y = 2 and 3 x + y = 4. These clearly have the same left side 3 x + y and different constants, so they represent two distinct parallel lines with equal slope and different intercepts. This confirms that p = 3 is consistent with the geometric meaning of parallel lines. Why Other Options Are Wrong:- Option 1: If p = 1, the first line slope becomes -1, which does not match slope -3 of the second line.- Option 2: If p = 2, the first line slope is -2, still not equal to -3.- Option 4: If p = 4, the first line slope is -4, again different from -3.- Option 5: p = 5 gives slope -5, which also fails the parallelism condition. Common Pitfalls:Many learners incorrectly compare constant terms or think that for parallel lines all coefficients must be proportional, including the constant term. For parallel but distinct lines, only the ratios of the coefficients of x and y matter, not the constant term. Another common mistake is to forget that the slope for a x + b y = c is -a / b and to reverse a and b. Moving terms to slope intercept form y = m x + c helps clarify this. Always check that you are equating slopes, not intercepts, when working with parallel lines. Final Answer:The value of p that makes the two lines parallel is 3.

Question 14

To measure exactly 31 kg of rice on a balance scale, including a 1 kg standard weight you already have, what is the minimum number of distinct weights required?

Accepted Answer

Introduction / Context:This arithmetic reasoning puzzle is based on the idea of representing a given weight using a set of standard weights on a balance scale. The goal is to find the minimum number of distinct weights required, given that we already have a 1 kg weight, so that we can measure exactly 31 kg of rice. This relates to representing numbers in terms of powers of 2 on a balance scale in a straightforward way where all weights are placed on one side and the object is on the other side. Given Data / Assumptions:- We need to measure exactly 31 kg of rice.- We have a balance scale and standard weights.- One 1 kg weight is already available.- We assume we place all standard weights on one pan and the rice on the other pan, not allowing weights on both sides.- We want to minimize the total number of distinct standard weights including the existing 1 kg weight. Concept / Approach:To measure any integer weight up to some maximum using weights all placed on one side, a very efficient system is based on powers of 2. Any positive integer can be written as a sum of distinct powers of 2, which is the idea behind binary representation. To express 31 as a sum of powers of 2, we write 31 = 16 + 8 + 4 + 2 + 1. Thus, the required weights are 1 kg, 2 kg, 4 kg, 8 kg, and 16 kg. Since 1 kg is already available, we only need to ensure the complete set of these five distinct weights. Step-by-Step Solution:- Express 31 in binary or as a sum of powers of 2.- 31 = 16 + 8 + 4 + 2 + 1.- The powers of 2 involved are 1, 2, 4, 8, and 16.- Therefore, you need weights of 1 kg, 2 kg, 4 kg, 8 kg, and 16 kg.- The total number of distinct weights is 5.- With these weights on one pan, you can place the rice on the other pan and combine the weights to get 31 kg exactly. Verification / Alternative check:Check that no smaller set works. If you try to remove any one of the powers of 2, you can no longer construct 31 using only sums of the remaining weights. For example, without the 16 kg weight, the largest sum you can make from 1, 2, 4, and 8 kg is 15 kg, which is not enough. Similarly, you cannot reach 31 kg without all five powers of 2. Hence 5 is minimal. Why Other Options Are Wrong:- Option 3: With only three weights you cannot represent 31 kg as a sum of distinct weights including 1 kg.- Option 7: Seven weights are more than necessary and not minimal.- Option 9: Nine weights are far too many for such a simple target.- Option 6: Six weights would include redundant values and are not needed because five already achieve the goal. Common Pitfalls:A common mistake is to think in terms of repeated uses of a single weight, for example imagining weighing 1 kg thirty one times. In such puzzles, each standard weight is used at most once in a given weighing. Another error is to forget the binary idea and choose arbitrary weights that do not combine efficiently. Remember that powers of 2 are optimal for covering a range of integer values on a balance with all weights on one side. Final Answer:The minimum number of distinct weights needed, including the 1 kg weight, is 5.

Question 15

In a class, if when 1 student is absent the remaining students can be divided into 6 equal groups, and when 2 students are absent the remaining students can be divided into 7 equal groups, what is the total strength of the class?

Accepted Answer

Introduction / Context:This arithmetic reasoning question checks your understanding of divisibility, remainders, and how to translate conditions about grouping students into a mathematical problem. The class strength must satisfy two divisibility conditions based on how many students are absent. By converting the statements into congruence relations, we can solve for the total number of students using simple modular arithmetic. Given Data / Assumptions:- Let the total number of students in the class be N.- When 1 student is absent, N - 1 is divisible by 6.- When 2 students are absent, N - 2 is divisible by 7.- We seek a positive integer N that satisfies both conditions simultaneously. Concept / Approach:The key idea is to model the conditions using congruences. The statement that N - 1 is divisible by 6 means N - 1 is a multiple of 6, or N is 1 more than a multiple of 6. In modular arithmetic, this is written as N ≡ 1 (mod 6). Similarly, N - 2 divisible by 7 means N ≡ 2 (mod 7). We then need a number that satisfies both congruences at the same time. Because the moduli 6 and 7 are relatively prime, we can use simple trial or the Chinese remainder idea on numbers within a reasonable range, or we can reason directly by listing candidates. Step-by-Step Solution:- Condition 1: N ≡ 1 (mod 6). So N can be 1, 7, 13, 19, 25, 31, 37, and so on, each obtained by adding 6.- Condition 2: N ≡ 2 (mod 7). So N can be 2, 9, 16, 23, 30, 37, 44, and so on, each obtained by adding 7.- Find the common value in both sequences.- Looking at the lists, the number 37 appears in both sequences.- Therefore N = 37 satisfies both conditions.- Check: N - 1 = 36, which is divisible by 6, and N - 2 = 35, which is divisible by 7. Verification / Alternative check:We can verify quickly by substituting the answer into the original statement. With 37 total students, if 1 is absent, there are 36 present, which can be divided into 6 equal groups of 6 students each. If 2 students are absent, there are 35 present, which can be divided into 7 equal groups of 5 students each. Both conditions are fully satisfied without contradiction, confirming that 37 is correct. Why Other Options Are Wrong:- Option 28: 28 - 1 = 27 does not divide evenly by 6, so the first condition fails.- Option 19: 19 - 1 = 18 works for divisibility by 6, but 19 - 2 = 17 is not divisible by 7.- Option 57: 57 - 1 = 56 is not a multiple of 6, so the first condition fails.- Option 49: 49 - 2 = 47 is not divisible by 7, so the second condition fails. Common Pitfalls:A common mistake is to misread the problem and treat 6 and 7 as the group sizes when students are present rather than understanding that these refer to equal grouping possibilities after some are absent. Another pitfall is to ignore one of the divisibility conditions, solving only for one congruence and not checking the other. Always translate such word problems into clear congruence statements and then systematically search for a value that satisfies all of them. Final Answer:The total number of students in the class is 37.

Question 16

The sum of the interior angles of a polygon is twice the sum of its exterior angles taken one at each vertex. How many sides does the polygon have?

Accepted Answer

Introduction / Context:This question tests concepts from basic geometry about polygons, specifically the relationship between interior angles, exterior angles, and the number of sides. Understanding the standard formulas for the sum of interior angles and the sum of exterior angles of a polygon is crucial for solving many geometry problems that appear in aptitude and competitive exams. Given Data / Assumptions:- Let the polygon have n sides.- The sum of its interior angles is twice the sum of its exterior angles taken one at each vertex.- The polygon is assumed to be simple and convex so that the usual angle sum formulas apply. Concept / Approach:Two key geometric facts are used here. First, for an n sided polygon, the sum of interior angles is given by (n - 2) * 180 degrees. This arises by dividing a polygon into (n - 2) triangles. Second, for any convex polygon, the sum of the exterior angles, taking one exterior at each vertex, is always 360 degrees, regardless of the number of sides. The problem states that the sum of the interior angles is twice the sum of the exterior angles, so we can set up an equation using these formulas and solve for n. Step-by-Step Solution:- Let n be the number of sides of the polygon.- Sum of interior angles = (n - 2) * 180 degrees.- Sum of exterior angles (one at each vertex) = 360 degrees.- Given condition: sum of interior angles = 2 * sum of exterior angles.- Substitute the formulas: (n - 2) * 180 = 2 * 360.- Simplify the right side: 2 * 360 = 720.- So we have (n - 2) * 180 = 720.- Divide both sides by 180: n - 2 = 720 / 180.- Compute the division: 720 / 180 = 4.- Hence n - 2 = 4, so n = 4 + 2 = 6. Verification / Alternative check:Check the result for n = 6. A hexagon has interior sum (6 - 2) * 180 = 4 * 180 = 720 degrees. The sum of exterior angles is 360 degrees. Twice the sum of exterior angles is 2 * 360 = 720 degrees, which matches the interior sum. Therefore, the condition is satisfied exactly when n = 6, confirming the correctness of the answer. Why Other Options Are Wrong:- Option 4: For n = 4, the interior sum is (4 - 2) * 180 = 360 degrees, which equals the exterior sum, not twice.- Option 5: For n = 5, interior sum is (5 - 2) * 180 = 540 degrees, which is not equal to 720 degrees.- Option 7: For n = 7, interior sum is (7 - 2) * 180 = 900 degrees, which is more than twice 360 degrees.- Option 8: For n = 8, interior sum is (8 - 2) * 180 = 1080 degrees, again not twice the exterior sum. Common Pitfalls:Learners sometimes mistakenly think that the sum of the exterior angles equals the sum of the interior angles, or they confuse interior and exterior measures at each vertex. It is important to remember that the total of all exterior angles is fixed at 360 degrees for any convex polygon, while the interior sum grows linearly with the number of sides. Whenever a question relates interior and exterior sums, carefully substitute the correct formulas rather than guessing. Writing down the equation step by step helps avoid accidental algebra errors. Final Answer:The polygon has 6 sides.

Question 17

Thrice the square of a natural number decreased by four times the number equals 50 more than the number. What is that natural number?

Accepted Answer

Introduction / Context:This question is an example of forming and solving a quadratic equation from a verbal statement about a natural number. Such problems are very common in aptitude tests as they assess your ability to translate words into algebra and then solve the resulting equation. Here the relationships involve the square of a number, multiples of the number, and a constant difference of 50. Given Data / Assumptions:- Let the natural number be n.- Thrice the square of the number means 3 * n^2.- Four times the number means 4 * n.- The expression thrice the square decreased by four times the number equals 50 more than the number. Concept / Approach:The phrase thrice the square of a natural number decreased by 4 times the number describes an algebraic expression that we can write as 3 * n^2 - 4 * n. The phrase 50 more than the number describes n + 50. The problem states that these two quantities are equal. Therefore, we set up the equation 3 * n^2 - 4 * n = n + 50 and then bring all terms to one side to obtain a standard quadratic equation of the form a n^2 + b n + c = 0. We then solve this quadratic and interpret which solution is acceptable as a natural number. Step-by-Step Solution:- Start from the verbal condition: thrice the square decreased by four times the number equals 50 more than the number.- Write it algebraically: 3 * n^2 - 4 * n = n + 50.- Move all terms to the left side: 3 * n^2 - 4 * n - n - 50 = 0.- Simplify the like terms: 3 * n^2 - 5 * n - 50 = 0.- This is a quadratic equation with a = 3, b = -5, c = -50.- Compute the discriminant D = b^2 - 4 * a * c = (-5)^2 - 4 * 3 * (-50).- D = 25 + 600 = 625.- The square root of 625 is 25.- The solutions are n = (5 ± 25) / (2 * 3) = (5 ± 25) / 6.- So n = 30 / 6 = 5 or n = -20 / 6 which simplifies to -10 / 3.- Since n is a natural number, discard the negative non integer value.- Therefore n = 5. Verification / Alternative check:Substitute n = 5 back into the original statement. Thrice the square is 3 * 5^2 = 3 * 25 = 75. Four times the number is 4 * 5 = 20. So thrice the square decreased by four times the number is 75 - 20 = 55. The right side is 50 more than the number, which is 5 + 50 = 55. Both sides match, so n = 5 satisfies the condition perfectly. Why Other Options Are Wrong:- Option 4: If n = 4, 3 * 4^2 - 4 * 4 = 48 - 16 = 32, but 50 more than 4 is 54, so it fails.- Option 7: If n = 7, 3 * 7^2 - 4 * 7 = 147 - 28 = 119, while 7 + 50 = 57, not 119.- Option 6: If n = 6, 3 * 6^2 - 4 * 6 = 108 - 24 = 84, but 6 + 50 = 56.- Option 8: If n = 8, 3 * 8^2 - 4 * 8 = 192 - 32 = 160, while 8 + 50 = 58. Common Pitfalls:Candidates sometimes misinterpret the phrase decreased by and write 4 * n - 3 * n^2 instead of 3 * n^2 - 4 * n. Another frequent mistake is to forget to move all terms to one side of the equation, leading to incorrect quadratic coefficients. Always rewrite verbal expressions carefully, verify the signs, and then solve the quadratic using the discriminant or factorization when possible. Finally, remember to check which solutions satisfy the requirement of being a natural number. Final Answer:The required natural number is 5.

Question 18

If you have 18 tens and 20 ones, what is the total number they represent?

Accepted Answer

Introduction / Context:This question checks your understanding of place value in the base ten number system. Tens and ones are fundamental concepts in early arithmetic and appear frequently in reasoning questions that test basic number sense. Converting a description such as 18 tens and 20 ones into a single number requires careful handling of place values and simple addition. Given Data / Assumptions:- There are 18 tens.- There are 20 ones.- We work in the standard base ten system, where 1 ten equals 10 ones.- We need the total value when tens and ones are combined. Concept / Approach:In the decimal system, each ten represents a value of 10. So 18 tens represent 18 multiplied by 10. Ones represent single units. Therefore, the total number is the sum of the value of all tens and all ones. The key is to convert the tens into their equivalent number of ones or directly into their numeric value, and then add the ones. This reinforces the idea that the digit in the tens place has a value ten times the digit itself. Step-by-Step Solution:- Calculate the value of 18 tens.- Each ten is equal to 10, so 18 tens is 18 * 10.- 18 * 10 = 180.- Next, calculate the value of 20 ones.- Each one is simply 1, so 20 ones is 20.- Add the value of the tens and ones: 180 + 20.- 180 + 20 = 200.- Therefore, 18 tens and 20 ones together represent the number 200. Verification / Alternative check:Consider grouping the 20 ones into tens. Since 10 ones make 1 ten, 20 ones are equal to 2 tens. So in total there are 18 tens plus 2 tens, that is 20 tens. Now 20 tens are equal to 20 * 10 = 200. This alternate viewpoint also leads to the same answer and confirms that the reasoning is correct. Why Other Options Are Wrong:- Option 1820: This looks like a concatenation of 18 and 20, but that is not how place value works.- Option 180: This represents only 18 tens and ignores the value of the 20 ones.- Option 160: This would imply that the ones somehow reduced the total rather than increasing it.- Option 220: This overcounts the contribution of tens and ones and does not match either decomposition method. Common Pitfalls:A common mistake is to treat the expression 18 tens 20 ones as the four digit number 1820 without converting tens and ones to their actual values. Another issue is failing to regroup the ones into tens when that makes the calculation clearer. Always remember that place value is multiplicative: the tens digit has a value equal to digit times 10, and the ones digit has value equal to digit times 1. Final Answer:The total number represented by 18 tens and 20 ones is 200.

Question 19

In engineering mechanics, Lami theorem is applicable only to which type of system of forces in equilibrium?

Accepted Answer

Introduction / Context:This question belongs to the topic of engineering mechanics and deals with Lami theorem, a very useful relation for solving problems involving three forces in equilibrium. Understanding the exact conditions under which Lami theorem can be applied is important for correctly choosing it as a tool in statics problems rather than misusing it in more general force systems. Given Data / Assumptions:- The question asks about the type of force system for which Lami theorem is applicable.- We assume the forces are acting on a single rigid body in equilibrium.- The options mention different combinations of concurrent and coplanar forces. Concept / Approach:Lami theorem states that if a body is in equilibrium under the action of three forces, then each force is proportional to the sine of the angle between the other two forces. Symbolically, if the three forces are F1, F2, and F3, making angles opposite to them α, β, and γ respectively, then F1 / sin α = F2 / sin β = F3 / sin γ. This theorem is valid only under specific conditions: the three forces must be concurrent, meaning they meet at a single point, and coplanar, meaning they lie in the same plane. Also, the body must be in equilibrium so that the vector sum of the forces is zero. Step-by-Step Solution:- Lami theorem deals explicitly with three forces acting on a point in equilibrium.- For three forces to be in equilibrium, their vector sum must be zero, and they must act at the same point or be concurrent.- The theorem uses angles between forces, so all three forces must be in a single plane, that is, be coplanar.- Therefore, the correct condition is that the three forces are coplanar and concurrent.- From the options, this corresponds to coplanar and concurrent forces. Verification / Alternative check:Consider a standard example: three strings meeting at a point supporting a mass, each string making a different angle with the vertical. The tensions in the three strings are the three concurrent forces. All forces lie in a single vertical plane. Lami theorem is used to relate these three tensions and solve for magnitudes. If the forces were not concurrent or not coplanar, the standard Lami relation would not hold and we would need more general vector methods. Why Other Options Are Wrong:- Non concurrent forces: Lami theorem cannot be used when forces do not meet at one point because the equilibrium condition then involves moments and distributed effects.- Coplanar forces only: Coplanarity alone is not enough; the forces must also be concurrent.- Any type of forces: This is too broad and incorrect because Lami theorem is not a universal equilibrium condition.- Non coplanar concurrent forces: For such three dimensional systems, Lami theorem in its standard planar form does not apply. Common Pitfalls:A frequent error is to try to apply Lami theorem to more than three forces or to forces that are not concurrent. Some students also forget about the requirement of coplanarity and attempt to use it in three dimensional configurations. When solving statics problems, always check that you have exactly three forces, that they intersect at a point, and that they lie in a plane before using this theorem. Final Answer:Lami theorem is applicable only for coplanar and concurrent forces in equilibrium.

Question 20

Seven ninths of the people present in a hall are sitting on nine thirteenths of the chairs available, and the rest of the people are standing. If there are 28 empty chairs, how many chairs would still be empty if everyone in the hall were sitting?

Accepted Answer

Introduction / Context:This is a classic arithmetic reasoning problem based on fractions and word interpretation. It involves people, chairs, and the concept of empty seats. By carefully translating the statements into algebraic equations involving fractions, you can find both the total number of chairs and the total number of people, and then answer the question about how many chairs remain empty if every person sits down. Given Data / Assumptions:- Let the number of people in the hall be P.- Let the total number of chairs be C.- Seven ninths of the people are sitting.- These seated people occupy nine thirteenths of the chairs.- There are 28 empty chairs under this seating arrangement.- We assume all chairs are identical and each chair can hold one person. Concept / Approach:Translate the fractions into equations. If seven ninths of the people are sitting, then the number of seated people is (7 / 9) * P. The problem states that those seated people occupy nine thirteenths of the chairs, so the number of occupied chairs is (9 / 13) * C. Since each seated person uses exactly one chair, these two quantities must be equal. At the same time, we know the number of empty chairs is C minus the occupied chairs, and this is given as 28. Using these two relations, we can solve for C and then find P. Finally, if every person sits, the number of empty chairs will be C - P. Step-by-Step Solution:- Number of seated people = (7 / 9) * P.- Number of occupied chairs = (9 / 13) * C.- Equate seated people and occupied chairs: (7 / 9) * P = (9 / 13) * C.- The number of empty chairs is C - (9 / 13) * C = (4 / 13) * C.- We are told that empty chairs are 28, so (4 / 13) * C = 28.- Solve for C: C = 28 * 13 / 4.- Compute 28 * 13 = 364, and 364 / 4 = 91, so C = 91.- Now find P using (7 / 9) * P = (9 / 13) * 91.- Compute (9 / 13) * 91 = 9 * 7 = 63.- So (7 / 9) * P = 63, which gives P = 63 * 9 / 7 = 81.- If everyone sits, the number of empty chairs becomes C - P = 91 - 81 = 10. Verification / Alternative check:Check the original situation using C = 91 and P = 81. Seated people are (7 / 9) * 81 = 63. Chairs used are (9 / 13) * 91 = 63. Empty chairs are 91 - 63 = 28, which matches the given information. Under full seating, 81 chairs are used and 10 remain empty. This confirms the solution. Why Other Options Are Wrong:- Option 15: Would correspond to P = 76 given C = 91, which does not respect the initial fraction relations.- Option 12 and option 18: Both contradict either the fraction equalities or the given 28 empty chair condition.- Option 8: Similarly fails to satisfy the equations when you recompute the number of people and chairs. Common Pitfalls:Students often misinterpret which fraction applies to people and which to chairs, or they forget to equate the number of seated people with the number of occupied chairs. Another common error is to ignore the condition about empty chairs, which is essential to compute the total number of chairs. Work step by step: write precise equations, solve for one variable at a time, and then answer the final question about empty chairs when everyone sits. Final Answer:If everyone in the hall were sitting, there would still be 10 chairs empty.

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