Question 1

A small wheel with 6 cogs is meshed with a larger wheel that has 14 cogs. If the smaller wheel makes 21 complete revolutions, how many revolutions will the larger wheel make while the teeth remain in continuous mesh?

Accepted Answer

Introduction / Context: This is a classic gear and cogs question which appears frequently in aptitude tests. Two wheels with different numbers of teeth mesh together, and the number of revolutions of one wheel is related to the number of revolutions of the other wheel. The idea is based on the fact that the number of tooth engagements must be the same for both wheels, so the total teeth passing a contact point must match on each wheel. Given Data / Assumptions: - Smaller wheel has 6 cogs or teeth. - Larger wheel has 14 cogs or teeth. - The smaller wheel makes 21 revolutions. - Wheels are meshed without slipping, so tooth engagements match exactly. - We need the number of revolutions made by the larger wheel. Concept / Approach: When two gears mesh, the total number of teeth that pass through the point of contact must be equal for both gears. If the smaller wheel has fewer teeth, it will make more revolutions than the larger wheel to keep the teeth count matched. Mathematically, we equate teeth passed by each wheel: teeth per revolution multiplied by number of revolutions for one wheel equals the same product for the other wheel. Step-by-Step Solution: Step 1: Teeth passed by the smaller wheel in one revolution = 6. Step 2: Teeth passed by the larger wheel in one revolution = 14. Step 3: If the smaller wheel makes 21 revolutions, total teeth passed by it = 6 * 21. Step 4: Let the revolutions of the larger wheel be R. Then total teeth passed by the larger wheel = 14 * R. Step 5: Since teeth engagements must match, set 6 * 21 = 14 * R. Step 6: Calculate 6 * 21 = 126, so 126 = 14 * R. Step 7: Solve for R to get R = 126 / 14 = 9. Verification / Alternative check: As another check, we can also consider the inverse proportionality between revolutions and number of teeth when tooth flow is equal. The ratio of revolutions is the inverse of the ratio of teeth: revolutions of smaller : revolutions of larger = 14 : 6. Given that revolutions of smaller are 21, we can set 21 : R = 14 : 6 and solve for R. This cross multiplication also leads to R = 9, confirming the initial calculation. Why Other Options Are Wrong: 7 revolutions would result in only 14 * 7 = 98 teeth passing on the larger wheel, which does not match 6 * 21 = 126 teeth from the smaller wheel. 11 or 13 revolutions produce even more mismatched total teeth counts. Only 9 revolutions gives 14 * 9 = 126, which matches exactly the small wheel tooth count and is therefore the only consistent answer. Common Pitfalls: Students sometimes confuse the direct and inverse ratios, mistakenly thinking that the wheel with more teeth must make more revolutions. In reality, for the same tooth flow, the wheel with more teeth makes fewer revolutions. Another pitfall is to divide the number of teeth incorrectly or forget that both wheels must pass the same total number of teeth at the contact point. Final Answer: The larger wheel will make 9 complete revolutions while the smaller wheel makes 21 revolutions.

Question 2

The incomes of two persons A and B are in the ratio 3 : 4. Each of them saves Rs 100 per month. If the ratio of their monthly expenditures is 1 : 2, what are the monthly incomes of A and B respectively?

Accepted Answer

Introduction / Context: This is a typical ratio based income and expenditure question. Two persons A and B have incomes in a given ratio and both save the same fixed amount every month. The ratio of their expenditures is also given. Using these relationships, we are asked to determine their actual monthly incomes in rupees. This tests the ability to translate verbal conditions into algebraic equations and then solve for the unknowns. Given Data / Assumptions: - Income ratio of A to B is 3 : 4. - Each person saves Rs 100 per month. - Expenditure ratio of A to B is 1 : 2. - Expenditure of a person = income minus savings. - We need the actual incomes of A and B in rupees. Concept / Approach: Let the common factor for incomes be k. Then the income of A is 3k and the income of B is 4k. Since both save Rs 100, the expenditures are 3k minus 100 and 4k minus 100. The ratio of these expenditures is 1 : 2. We set up the equation (3k minus 100) divided by (4k minus 100) equals 1 divided by 2 and solve for k. After determining k, we compute actual incomes 3k and 4k and match them with the options. Step-by-Step Solution: Step 1: Let income of A = 3k and income of B = 4k. Step 2: Since each saves Rs 100, expenditure of A = 3k - 100 and expenditure of B = 4k - 100. Step 3: Given that expenditure ratio A : B = 1 : 2, write (3k - 100) / (4k - 100) = 1 / 2. Step 4: Cross multiply to get 2(3k - 100) = 4k - 100. Step 5: Simplify 6k - 200 = 4k - 100 which leads to 2k = 100. Step 6: Solve for k to get k = 50. Step 7: Income of A = 3k = 3 * 50 = Rs 150, income of B = 4k = 4 * 50 = Rs 200. Verification / Alternative check: Check the savings and expenditures with these values. A saves Rs 100, so expenditure of A = 150 - 100 = Rs 50. B saves Rs 100, so expenditure of B = 200 - 100 = Rs 100. The expenditure ratio is 50 : 100 which simplifies to 1 : 2, matching the condition. The income ratio 150 : 200 simplifies to 3 : 4, which also matches. Hence, the derived incomes are fully consistent with the given data. Why Other Options Are Wrong: Rs 200 and Rs 400 do not satisfy the condition that both save Rs 100 and have expenditure ratio 1 : 2. Rs 100 and Rs 300 give an income ratio of 1 : 3, not 3 : 4. Rs 100 and Rs 200 lead to a ratio of 1 : 2 which again does not match. Only Rs 150 and Rs 200 give the required income ratio of 3 : 4 along with a savings of Rs 100 each and an expenditure ratio of 1 : 2. Common Pitfalls: Students often mix up the ratio of income with the ratio of expenditure and may try to directly impose 3 : 4 on expenditures. Another common mistake is to forget that savings are the same fixed amount for both, not in ratio. This can lead to setting up incorrect equations. Careless algebra, especially while cross multiplying and simplifying, can also cause wrong values of k and therefore wrong income figures. Final Answer: The monthly incomes are Rs 150 for A and Rs 200 for B.

Question 3

If A : B = 1 : 4 and B : C = 2 : 3, then what is the combined ratio A : B : C for the three quantities?

Accepted Answer

Introduction / Context: This question is a straightforward ratio chaining problem. We are given two separate ratios involving three variables A, B and C. To find the combined three term ratio A : B : C, we need to make the common term B consistent in both given ratios and then combine the information. This type of question checks understanding of how to merge partial ratio information into a single comprehensive ratio. Given Data / Assumptions: - A : B = 1 : 4. - B : C = 2 : 3. - B is the common term in both ratios. - We are required to find the three term ratio A : B : C. Concept / Approach: When two ratios share a common term, we first rewrite them so that the value of the common term is represented by the same number in both ratios. Then, we can directly read off the corresponding values of A and C. This method leverages the idea that ratios may be multiplied by the same factor on both sides without changing their meaning. The key is to align the second term of the first ratio with the first term of the second ratio since both represent B. Step-by-Step Solution: Step 1: Write the first ratio: A : B = 1 : 4. Step 2: Write the second ratio: B : C = 2 : 3. Step 3: The common term B is 4 units in the first ratio and 2 units in the second ratio. Step 4: To make B equal in both ratios, find a common multiple of 4 and 2 which is 4. Step 5: Keep A : B = 1 : 4 as it is. Multiply B : C = 2 : 3 by 2 to get 4 : 6. Step 6: Now B is 4 in both, so A : B : C = 1 : 4 : 6. Verification / Alternative check: You can check by assuming an actual value for B. Suppose B corresponds to 4 units. From A : B = 1 : 4, A would be 1 unit. From B : C = 2 : 3, if B is 4 then 2 units map to 4, so 1 unit is 2, and 3 units is 6. Therefore, C is 6. This directly produces the ratio A : B : C = 1 : 4 : 6, confirming the earlier ratio alignment method is correct. Why Other Options Are Wrong: 2 : 8 : 7 would imply C is 7 when B is 8, but that does not match the scaled ratio B : C = 2 : 3. Similarly, 2 : 8 : 5 and 1 : 4 : 5 misrepresent the relationship between B and C. Only 1 : 4 : 6 maintains both original conditions correctly when broken back into A : B and B : C. Common Pitfalls: Students sometimes attempt to directly join the ratios as 1 : 4 : 3 or 1 : 2 : 3, which is incorrect because it ignores the need to equalize the common term B. Another frequent error is multiplying the wrong side of one ratio or failing to scale both parts of the ratio by the same factor. Careful alignment of the common variable is critical in such problems. Final Answer: The combined ratio of the three quantities is 1 : 4 : 6.

Question 4

If y divided by (x minus z) equals (y plus x) divided by z, and also equals x divided by y, that is y/(x - z) = (y + x)/z = x/y, then what is the ratio z : y : x?

Accepted Answer

Introduction / Context: This algebraic ratio question involves three expressions all equal to the same common value. The given condition y/(x - z) = (y + x)/z = x/y links the three variables x, y and z. The aim is to determine the ratio z : y : x. This is more challenging than direct ratio problems because it requires using algebraic manipulation and reasoning with a common parameter to express each variable in terms of another. Given Data / Assumptions: - y/(x - z) = (y + x)/z = x/y. - All denominators are assumed to be non zero so the expressions are valid. - The equality holds for some positive common value. - We are asked to find the ratio z : y : x in simplest integer form. Concept / Approach: Let the common value of all three expressions be k. Then we have three equations: x / y = k, so x = k y. Also, (y + x) / z = k so y + x = k z, and y / (x - z) = k so y = k (x - z). Substituting x in terms of y into the other equations allows us to solve for z in terms of y, and then we can express the ratio z : y : x in whole numbers by eliminating k. Step-by-Step Solution: Step 1: Let x / y = k, so x = k y. Step 2: From (y + x) / z = k, we have y + x = k z. Step 3: Substitute x = k y into Step 2 to get y + k y = k z, so y(1 + k) = k z. Step 4: Hence z = y(1 + k) / k. Step 5: From y / (x - z) = k, we have y = k (x - z). Step 6: Substitute x = k y and z from Step 4 into y = k (x - z). Step 7: This gives y = k (k y - y(1 + k) / k) which simplifies to y = k^2 y - (1 + k) y. Step 8: Rearranging gives y(k^2 - 1) = y(1 + k), so k^2 - 1 = 1 + k. Step 9: Solve k^2 - k - 2 = 0, which factors as (k - 2)(k + 1) = 0, so k = 2 (positive solution). Step 10: With k = 2, x = 2y and z = y(1 + 2) / 2 = 3y/2. Step 11: Write z : y : x = (3y/2) : y : 2y = 3 : 2 : 4. Verification / Alternative check: Take a convenient value such as y = 2 units. Then x = 2y = 4, and z = 3y/2 = 3. Substitute into the original expressions: y/(x - z) = 2/(4 - 3) = 2, (y + x)/z = (2 + 4)/3 = 6/3 = 2, and x/y = 4/2 = 2. All three equal 2, confirming that the ratio z : y : x = 3 : 2 : 4 satisfies the given condition. Why Other Options Are Wrong: 1 : 2 : 3, 4 : 3 : 2 and 2 : 3 : 4 do not satisfy the system when substituted back into the expressions. For each of these, if you pick a value for y and compute x and z according to the ratio, at least one of the three fractions y/(x - z), (y + x)/z and x/y will differ from the others. Therefore, they cannot represent a consistent solution to the given equation. Common Pitfalls: A common mistake is to assume that each ratio can be treated independently without considering the shared common value. Another pitfall is algebraic manipulation errors when substituting and simplifying expressions involving k, especially when clearing denominators. Learners may also incorrectly cancel terms or forget that y, x and z must remain positive to interpret a meaningful ratio. Final Answer: The required ratio of the three variables is z : y : x = 3 : 2 : 4.

Question 5

At a university entrance exam, 40 percent of the candidates are actually capable and 60 percent are incapable. Among the capable candidates, 80 percent pass the test, and among the incapable candidates, 30 percent pass the test. What is the ratio of…

Accepted Answer

Introduction / Context: This problem is a probability and ratio question framed in an examination setting. We are given information about how many candidates are genuinely capable versus incapable and the probabilities of passing for both groups. The objective is to find the fraction of passing candidates who actually belong to the capable group. This is conceptually similar to a Bayes theorem style reasoning, although we can handle it with direct percentage arithmetic and careful counting. Given Data / Assumptions: - 40 percent of candidates are capable. - 60 percent of candidates are incapable. - 80 percent of capable candidates pass the test. - 30 percent of incapable candidates pass the test. - We are interested in the ratio of capable candidates who pass to the total number of passing candidates. Concept / Approach: Assume a convenient total number of candidates, such as 100, to convert percentages into actual counts. Then compute how many of these 100 are capable and how many are incapable. Next, find how many pass in each group using the given pass percentages. Adding the passes from both groups gives the total number of students who pass the test. Finally, the ratio asked for is the number of capable students who pass divided by the total number of students who pass, which can be simplified to lowest terms. Step-by-Step Solution: Step 1: Assume total candidates = 100 for easy calculation. Step 2: Capable candidates = 40 percent of 100 = 40. Step 3: Incapable candidates = 60 percent of 100 = 60. Step 4: Passing capable candidates = 80 percent of 40 = 0.8 * 40 = 32. Step 5: Passing incapable candidates = 30 percent of 60 = 0.3 * 60 = 18. Step 6: Total passing candidates = 32 + 18 = 50. Step 7: Ratio of capable passing candidates to total passing candidates = 32 / 50. Step 8: Simplify 32 / 50 by dividing numerator and denominator by 2 to get 16 / 25. Verification / Alternative check: We can compute the same ratio using decimal probabilities. The proportion of capable passers is 0.4 * 0.8 = 0.32, and the proportion of incapable passers is 0.6 * 0.3 = 0.18. Total pass proportion is 0.32 + 0.18 = 0.50. The required ratio is 0.32 / 0.50 = 0.64, which is identical to 16 / 25 when written as a fraction. This confirms the consistency of our earlier calculation with percentage based reasoning. Why Other Options Are Wrong: 11/24 is approximately 0.458 which is much smaller than 0.64 and does not match the computed ratio. 1/5 equals 0.2 and is far too small. 6/5 equals 1.2, which is impossible because it implies more capable passers than total passers. Only 16/25 matches the correct fraction 32 divided by 50 obtained from the step by step calculations. Common Pitfalls: A common mistake is to average the pass percentages directly or to confuse the proportion of capable candidates with the proportion of passing candidates. Some students mistakenly treat 80 percent and 30 percent as if they were applied to the same base group. Another pitfall is failing to choose a convenient base total, like 100, which makes calculations much easier and clearer. Final Answer: The proportion of capable students among all students who pass the test is 16/25.

Question 6

Raju invests Rs 6,000 in a scheme. Santhosh invests 30 percent less than Raju, and Rajitha invests 25 percent more than Santhosh. What is the ratio of the amount invested by Rajitha to the total amount invested by all three of them together?

Accepted Answer

Introduction / Context: This investment based ratio question is about three people, Raju, Santhosh and Rajitha, investing different amounts in relation to each other. One amount is given explicitly, and the other two are described using percentage increases and decreases. Once we convert all these amounts into actual rupee values, we can compute the ratio of one person investment to the total combined investment. Given Data / Assumptions: - Raju invests Rs 6,000. - Santhosh invests 30 percent less than Raju. - Rajitha invests 25 percent more than Santhosh. - We need the ratio of Rajitha investment to the total investment of all three combined. Concept / Approach: The approach involves sequential percentage calculations. First, we compute Santhosh investment by reducing Raju amount by 30 percent. Then we use Santhosh amount to calculate Rajitha investment by increasing it by 25 percent. After obtaining all three absolute amounts, we add them to get the total investment. Finally, we express Rajitha amount over the total amount as a simplified ratio with whole number terms. Step-by-Step Solution: Step 1: Raju investment = Rs 6,000. Step 2: Santhosh invests 30 percent less than Raju, so Santhosh amount = 6,000 * (1 - 0.30) = 6,000 * 0.70 = Rs 4,200. Step 3: Rajitha invests 25 percent more than Santhosh, so Rajitha amount = 4,200 * (1 + 0.25) = 4,200 * 1.25. Step 4: Compute 4,200 * 1.25 = 4,200 * (5/4) = 4,200 * 5 / 4 = 21,000 / 4 = Rs 5,250. Step 5: Total investment of all three = 6,000 + 4,200 + 5,250 = Rs 15,450. Step 6: Required ratio = Rajitha investment : total investment = 5,250 : 15,450. Step 7: Simplify the ratio by dividing both terms by 150 to get 35 : 103. Verification / Alternative check: Alternatively, we can work entirely in relative terms first. Let Raju amount be 6,000 units. Santhosh amount is 70 percent of that, and Rajitha amount is 125 percent of Santhosh amount. That means Rajitha amount is 6,000 * 0.70 * 1.25 which also simplifies to 5,250. This reconfirms the earlier calculation. Re adding the three amounts again gives 15,450, so the ratio 5,250 : 15,450 is consistent. Why Other Options Are Wrong: 25 : 114, 15 : 108 and 41 : 94 do not reduce to the fraction 5,250 divided by 15,450. When simplified, none of these options represents 35 : 103. Only 35 : 103 maintains the correct proportional relationship between Rajitha share and the total investment after simplification. Common Pitfalls: A typical mistake is using the 25 percent increase directly on Raju amount instead of on Santhosh amount. Another common error is miscalculating 30 percent less by subtracting 30 rupees per 1000 instead of 30 percent of the total. There can also be simplification errors when reducing 5,250 : 15,450 to lowest terms. Carefully dividing by a common factor such as 150 helps avoid such issues. Final Answer: The ratio of Rajitha investment to the total investment of all three is 35 : 103.

Question 7

A store has cans of orange, pineapple and mixed fruit juices in the ratio 8 : 9 : 15. The store sells 25 percent of the orange, 33.33 percent of the pineapple and 20 percent of the mixed fruit juice cans. What is the ratio of the remaining cans of o…

Accepted Answer

Introduction / Context: This ratio and percentage question is based on stock management in a store. Initially, the store has three types of juice cans in a known ratio. A certain percentage of each type is sold, which reduces the quantities. We are asked to determine the new ratio of the remaining cans on the shelves. This problem tests the ability to work with ratios, percentages and proportional reasoning all together. Given Data / Assumptions: - Initial ratio of orange : pineapple : mixed fruit juice cans is 8 : 9 : 15. - 25 percent of orange cans are sold, so 75 percent remain. - 33.33 percent (approximately one third) of pineapple cans are sold, so two thirds remain. - 20 percent of mixed fruit cans are sold, so 80 percent remain. - We need the ratio of remaining orange, pineapple and mixed fruit cans. Concept / Approach: We can treat the original ratio as actual counts by assuming a convenient total unit, say 8 orange cans, 9 pineapple cans and 15 mixed fruit cans. Then we apply the given percentages to find remaining cans of each type. After computing the remaining quantities, we express them as a ratio and simplify to obtain the final ratio in lowest terms. The absolute total is not important; only the relative values matter. Step-by-Step Solution: Step 1: Assume initial stock: orange = 8 units, pineapple = 9 units, mixed fruit = 15 units. Step 2: Orange sold = 25 percent of 8 = 0.25 * 8 = 2 units, so remaining orange = 8 minus 2 = 6 units. Step 3: Pineapple sold = 33.33 percent of 9 which is about 1/3 of 9 = 3 units, so remaining pineapple = 9 minus 3 = 6 units. Step 4: Mixed fruit sold = 20 percent of 15 = 0.20 * 15 = 3 units, so remaining mixed fruit = 15 minus 3 = 12 units. Step 5: Remaining stock ratio = 6 : 6 : 12. Step 6: Simplify 6 : 6 : 12 by dividing each term by 6 to get 1 : 1 : 2. Verification / Alternative check: Instead of using exact counts, we could directly multiply each original ratio term by the remaining fraction. For orange, remaining fraction is 75 percent or 3/4, so 8 * 3/4 = 6. For pineapple, remaining fraction is about 2/3, so 9 * 2/3 = 6. For mixed fruit, remaining fraction is 80 percent or 4/5, so 15 * 4/5 = 12. These values match the earlier calculation, confirming that the final ratio is correctly simplified to 1 : 1 : 2. Why Other Options Are Wrong: 6 : 6 : 13 would be correct only if mixed fruit remaining were 13 units, which is not supported by the 20 percent sale of 15. 12 : 15 : 19 and 4 : 9 : 13 do not reflect the percentages of sold stock correctly and would imply inconsistent or incorrect percentage reductions. The only ratio that aligns with the computed remaining quantities of 6, 6 and 12 is 1 : 1 : 2. Common Pitfalls: Some students treat percentages as absolute quantities instead of fractions of the current count. Another common mistake is incorrectly approximating 33.33 percent and not recognizing it as one third. Others may forget to simplify the final ratio, leaving it as 6 : 6 : 12 instead of reducing to 1 : 1 : 2. Consistent use of fractional or percentage arithmetic avoids these errors. Final Answer: The ratio of remaining cans of orange, pineapple and mixed fruit juices is 1 : 1 : 2.

Question 8

Six Kirloskar pumps can fill a tank in 7 days, while two similar USHA pumps can fill the same tank in 18 days. What is the ratio of the efficiency (work rate) of one Kirloskar pump to one USHA pump?

Accepted Answer

Introduction / Context: This is a work and time problem involving two different brands of pumps, Kirloskar and USHA. Multiple pumps of each type can fill the same tank in different times. We are asked to compare the efficiency of one pump of each type. Efficiency here means the rate at which a single pump can fill the tank, measured as a fraction of the tank per day. This type of question is common in aptitude tests because it combines proportional reasoning and work rate calculations. Given Data / Assumptions: - Six Kirloskar pumps fill the tank in 7 days. - Two USHA pumps fill the same tank in 18 days. - All pumps of the same brand have equal efficiency. - The tank size is the same in both cases. - We need the ratio of efficiency of one Kirloskar pump to one USHA pump. Concept / Approach: Work rate is defined as work done per unit time. If six Kirloskar pumps together fill one full tank in 7 days, their combined rate is 1 divided by 7 tanks per day. Therefore, a single Kirloskar pump does one sixth of that. Similarly, if two USHA pumps fill the tank in 18 days, their combined rate is 1 divided by 18 tanks per day. So a single USHA pump does half of that. Once we have individual rates, we can form their ratio and simplify. Step-by-Step Solution: Step 1: Combined rate of six Kirloskar pumps = 1 tank / 7 days = 1/7 tank per day. Step 2: Rate of one Kirloskar pump = (1/7) / 6 = 1 / 42 tank per day. Step 3: Combined rate of two USHA pumps = 1 tank / 18 days = 1/18 tank per day. Step 4: Rate of one USHA pump = (1/18) / 2 = 1 / 36 tank per day. Step 5: Efficiency ratio of one Kirloskar pump to one USHA pump = (1/42) : (1/36). Step 6: Simplify the ratio by taking reciprocals: (1/42) : (1/36) = 36 : 42. Step 7: Divide both terms by 6 to get 6 : 7. Verification / Alternative check: We can check the reasonableness of the ratio. Since a USHA pump alone fills the tank in 36 days and a Kirloskar pump in 42 days, a USHA pump is slightly faster. That means its efficiency is slightly higher, so the ratio of Kirloskar to USHA should be somewhat less than 1. Indeed, 6 : 7 corresponds to 6/7 which is less than 1, confirming that the direction and magnitude of the ratio make sense. Why Other Options Are Wrong: 7 : 6 would suggest that one Kirloskar pump is more efficient than one USHA pump, which contradicts the filling times derived from the data. 7 : 54 is not consistent with any simple fraction obtained from the work rates 1/42 and 1/36. The option “None of these” is incorrect because 6 : 7 matches the correct simplified ratio of the two efficiencies. Common Pitfalls: Students sometimes mistakenly compare the total days taken by multiple pumps directly without converting to single pump rates. Another frequent error is to set the ratio using 42 : 36 instead of 36 : 42 or to forget to simplify the final ratio. Care must be taken when dividing the combined rate to obtain single pump efficiency and when simplifying the fractional ratio. Final Answer: The ratio of the efficiency of a Kirloskar pump to a USHA pump is 6 : 7.

Question 9

If (2a + 3b) divided by (5a - 3b) equals 23/5, that is (2a + 3b)/(5a - 3b) = 23/5, then what is the value of the ratio b/a?

Accepted Answer

Introduction / Context: This is an algebraic equation in which a linear expression in a and b divided by another linear expression in a and b equals a given fraction. We are asked to find the ratio b/a. Such questions are common in aptitude tests where the goal is to practice manipulating algebraic expressions and solving for ratios rather than exact values of the variables. Given Data / Assumptions: - The equation is (2a + 3b)/(5a - 3b) = 23/5. - Variables a and b are non zero so that the denominator 5a - 3b is not zero. - We must find the value of the ratio b/a. Concept / Approach: We start by cross multiplying the given fraction equality to remove the denominators. This yields a linear relation between a and b. Then we rearrange the resulting equation so that the ratio b/a appears explicitly. By factoring out common terms, we can solve for b/a as a simple fraction. The individual values of a and b are not required, only their ratio matters. Step-by-Step Solution: Step 1: Start from (2a + 3b)/(5a - 3b) = 23/5. Step 2: Cross multiply to clear denominators: 5(2a + 3b) = 23(5a - 3b). Step 3: Expand both sides: left side = 10a + 15b, right side = 115a - 69b. Step 4: Bring all terms to one side: 10a + 15b - 115a + 69b = 0. Step 5: Combine like terms: (10a - 115a) + (15b + 69b) = -105a + 84b = 0. Step 6: Rearrange to get 84b = 105a. Step 7: Divide both sides by 21 to get 4b = 5a. Step 8: Therefore b/a = 5/4. Verification / Alternative check: To verify, choose convenient values satisfying b/a = 5/4, for example a = 4 and b = 5. Substitute into the original expression: 2a + 3b = 2*4 + 3*5 = 8 + 15 = 23, and 5a - 3b = 5*4 - 3*5 = 20 - 15 = 5. Then (2a + 3b)/(5a - 3b) = 23/5 which matches the given fraction exactly. This confirms that the ratio b/a = 5/4 is correct. Why Other Options Are Wrong: 4/5 is the reciprocal of the correct ratio and would give different values when substituted back into the expression. 20/23 and 23/20 bear no direct relationship to the simplified form of 84/105 and will not satisfy the given equation upon substitution. Only 5/4 preserves the equality (2a + 3b)/(5a - 3b) = 23/5 for suitable non zero values of a and b. Common Pitfalls: One common mistake is incorrect expansion or sign errors when moving terms from one side of the equation to the other. Another pitfall is attempting to isolate a or b directly instead of focusing on the ratio b/a. Also, some students may mistakenly simplify 84/105 as 4/5 without checking that it represents b/a, not a/b, which reverses the final answer. Final Answer: The required value of the ratio is b/a = 5/4.

Question 10

What is the third proportional to the numbers 9 and 45, that is, the value x such that 9 : 45 = 45 : x?

Accepted Answer

Introduction / Context: This question asks for the third proportional to two given numbers. If a, b and c are in proportion such that a : b = b : c, then c is called the third proportional to a and b. This is a standard concept in ratio and proportion that often appears in aptitude questions. Here we are given a = 9 and b = 45 and asked to find c. Given Data / Assumptions: - First number a = 9. - Second number b = 45. - We want c such that 9 : 45 = 45 : c. - All numbers are positive real numbers. Concept / Approach: From the definition, for a, b, c to be in continued proportion we must have a/b = b/c. This can be rearranged to c = b^2 / a. Therefore, once a and b are known, we compute c by squaring the second number and dividing by the first. This method is a quick way to obtain the third proportional without explicitly dealing with the full ratio fraction. Step-by-Step Solution: Step 1: Use the proportion 9 : 45 = 45 : x where x is the third proportional. Step 2: Write it as a fraction: 9 / 45 = 45 / x. Step 3: Cross multiply to get 9x = 45 * 45. Step 4: Compute 45 * 45 = 2,025. Step 5: Solve 9x = 2,025 to get x = 2,025 / 9. Step 6: Compute 2,025 / 9 = 225. Step 7: Therefore, the third proportional to 9 and 45 is 225. Verification / Alternative check: As a check, verify that the ratio 9 : 45 equals the ratio 45 : 225. Simplify 9 : 45 to 1 : 5 and 45 : 225 also simplifies to 1 : 5 since both numerator and denominator are divided by 45. Because the ratios match, we confirm that 225 maintains the correct proportional relationship and is indeed the third proportional. Why Other Options Are Wrong: 405 and 81 do not satisfy the condition 9 : 45 = 45 : x when substituted for x. For example, 45 : 405 simplifies to 1 : 9, not 1 : 5. The value 5 is far too small and completely breaks the proportional sequence. Only 225 makes both ratios equal, so it is the unique correct answer by the definition of third proportional. Common Pitfalls: A common error is to confuse third proportional with mean proportional, which is related to geometric mean, or to improperly square the wrong number. Some learners also forget to simplify the final ratio when checking their result, leading them to think a wrong number might also work. Remembering the formula c = b^2 / a is a fast and reliable way to handle such questions. Final Answer: The third proportional to 9 and 45 is 225.

Question 11

Two numbers are in the ratio 2 : 3 and their sum is 75. What is the larger of the two numbers?

Accepted Answer

Introduction / Context: This is a straightforward question on numbers in a given ratio with their sum known. The two numbers share a fixed proportional relationship, and the total of the two is given. These two pieces of information are enough to determine each number exactly. Such questions train the ability to convert ratio based descriptions into actual values using the concept of total parts. Given Data / Assumptions: - Ratio of the numbers is 2 : 3. - Sum of the two numbers is 75. - We need to find the larger of the two numbers. Concept / Approach: When two numbers are in the ratio m : n, we can write them as 2k and 3k for some positive value k. Their sum then becomes 2k + 3k which is 5k. If the sum is known, we solve 5k equal to that sum to get k, and then each number is found by substituting k back into 2k and 3k. The larger number corresponds to the bigger coefficient in the original ratio, which here is 3k. Step-by-Step Solution: Step 1: Let the two numbers be 2k and 3k, following the ratio 2 : 3. Step 2: Their sum is 2k + 3k = 5k. Step 3: Given that 2k + 3k = 75, so 5k = 75. Step 4: Solve for k to get k = 75 / 5 = 15. Step 5: The first number is 2k = 2 * 15 = 30. Step 6: The second number is 3k = 3 * 15 = 45. Step 7: The larger of the two numbers is 45. Verification / Alternative check: Check that the numbers satisfy the original conditions. The ratio 30 : 45 simplifies to 2 : 3 when both terms are divided by 15. Their sum is 30 + 45 = 75, which matches the given total. Since both conditions are satisfied, the calculation is correct. Therefore 45 is confirmed as the larger number. Why Other Options Are Wrong: 30 is the smaller number, not the larger, although it is part of the correct pair. 50 and 48 would make the sum or the ratio inconsistent with the given data. For instance, pairing 25 and 50 would give a sum of 75 but a ratio of 1 : 2, not 2 : 3. Thus only 45 fits as the larger number in the correct ratio based pair whose sum is 75. Common Pitfalls: Some students mistakenly divide the sum directly by one of the ratio terms or miscalculate the total parts. For example, they might divide 75 by 3 only, ignoring the fact that 2 parts and 3 parts together make 5 parts. Another pitfall is to swap which term of the ratio represents which number, but in this question we only need the larger number, which is always linked to the bigger ratio term 3k. Final Answer: The larger of the two numbers is 45.

Question 12

Visitors to a show are charged Rs 15 per person on the first day, Rs 7.50 per person on the second day and Rs 2.50 per person on the third day. The attendance on these three days is in the ratio 2 : 5 : 13 respectively. What is the average charge pe…

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Introduction / Context: This problem is an application of weighted averages. The ticket prices differ on three different days, and the number of visitors also differs each day. We are asked to calculate the overall average charge per visitor for the entire three day show. Since the attendance is not the same every day, a simple arithmetic mean of the three prices would be incorrect. Instead, we must weight each price by the corresponding attendance ratio. Given Data / Assumptions: - Ticket price on day one is Rs 15 per visitor. - Ticket price on day two is Rs 7.50 per visitor. - Ticket price on day three is Rs 2.50 per visitor. - Attendance ratio for the three days is 2 : 5 : 13. - We need the overall average ticket charge per person for all three days combined. Concept / Approach: To compute the weighted average price, assume a convenient total attendance that respects the given ratio. For example, let the number of visitors on day one be 2x, on day two be 5x and on day three be 13x. Then compute the total revenue from each day (price multiplied by visitors for that day). Add the revenues to get total revenue and add the visitors to get total number of persons. The average price is then total revenue divided by total attendance. The factor x cancels out in the ratio calculation. Step-by-Step Solution: Step 1: Let attendance on day one = 2 units, day two = 5 units, day three = 13 units. Step 2: Revenue from day one = 2 * Rs 15 = Rs 30. Step 3: Revenue from day two = 5 * Rs 7.50 = Rs 37.50. Step 4: Revenue from day three = 13 * Rs 2.50 = Rs 32.50. Step 5: Total revenue for three days = 30 + 37.50 + 32.50 = Rs 100. Step 6: Total attendance = 2 + 5 + 13 = 20 units. Step 7: Average charge per visitor = total revenue / total attendance = 100 / 20 = Rs 5. Verification / Alternative check: We can verify this by choosing an actual number of visitors, for example multiply each attendance unit by 10 so that day one has 20 visitors, day two has 50 visitors and day three has 130 visitors. Compute revenue with these numbers and divide by total visitors. The revenue values will scale by 10, as will total visitors, and the average will still be Rs 5. This confirms that using ratio units was sufficient and correct. Why Other Options Are Wrong: Rs 5.50, Rs 6 and Rs 7 all exceed the computed weighted average of Rs 5. Simply averaging the three ticket prices (15, 7.50 and 2.50) would give (15 + 7.50 + 2.50)/3 = 25/3 which is about 8.33, but that ignores the attendance differences and clearly does not match any option. Only Rs 5 reflects the correct total revenue divided by the correct total number of visitors based on the attendance ratio. Common Pitfalls: A very common mistake is to average the three prices directly without accounting for the number of visitors each day. Another pitfall is mixing up the ratio of attendance and mistakenly treating it as money distribution rather than head count. Precise multiplication of price and attendance units before summing ensures the correct weighted average is obtained. Final Answer: The overall average charge per visitor over the three days is Rs 5.

Question 13

Which of the following sets of three side lengths cannot represent the sides of a right angled triangle?

Accepted Answer

Introduction / Context: This is a geometry and Pythagoras theorem question. In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Many common sets of integers that satisfy this relationship are called Pythagorean triples. The question asks us to identify which of the given sets of three side lengths does not satisfy the relationship and therefore cannot form a right angled triangle. Given Data / Assumptions: - Four sets of side lengths are given: (3, 4, 5), (5, 12, 13), (8, 15, 17), and (12, 15, 18). - All lengths are positive numbers representing the sides of a potential triangle. - We must check whether each set can satisfy Pythagoras theorem in some ordering. Concept / Approach: For any given triple of side lengths, we first identify the largest side and treat it as the possible hypotenuse. Then we compute the square of that side and compare it with the sum of the squares of the other two sides. If the equality holds, the triple can represent a right angled triangle. If it fails, the triple is not a Pythagorean triple and cannot represent a right angled triangle with those side lengths. Step-by-Step Solution: Step 1: For 3, 4, 5 the largest side is 5. Check 3^2 + 4^2 = 9 + 16 = 25 and 5^2 = 25, so it is a right angled triangle. Step 2: For 5, 12, 13 the largest side is 13. Check 5^2 + 12^2 = 25 + 144 = 169 and 13^2 = 169, so it is a right angled triangle. Step 3: For 8, 15, 17 the largest side is 17. Check 8^2 + 15^2 = 64 + 225 = 289 and 17^2 = 289, so it is also a right angled triangle. Step 4: For 12, 15, 18 the largest side is 18. Compute 12^2 + 15^2 = 144 + 225 = 369. Step 5: Compute 18^2 = 324. Step 6: Since 369 is not equal to 324, 12, 15 and 18 do not satisfy Pythagoras theorem and therefore cannot form a right angled triangle. Verification / Alternative check: We can also verify by checking if any side ordering for 12, 15, 18 works, but the hypotenuse must always be the largest side. Swapping labels among the smaller sides does not change the sum of their squares, so 144 + 225 remains 369 regardless of order. Since the square of 18 is fixed as 324, no rearrangement will satisfy the equality. This confirms that 12, 15, 18 cannot represent a right angled triangle. Why Other Options Are Wrong: 3, 4, 5 is the most well known Pythagorean triple. 5, 12, 13 and 8, 15, 17 are also standard triples used frequently in geometry and aptitude problems. All of these satisfy a^2 + b^2 = c^2 for suitable ordering of sides. Only 12, 15, 18 fails that condition and is therefore the only incorrect candidate for a right angled triangle. Common Pitfalls: A common error is to miscalculate one of the squares or the sum, for example computing 15^2 incorrectly. Another mistake is to assume any triple that looks similar to a known Pythagorean triple must also work without checking the arithmetic. Always square the sides carefully and verify the equality rather than relying purely on pattern recognition. Final Answer: The set of lengths that cannot be the sides of a right angled triangle is 12, 15, 18.

Question 14

In a hospital there are 209 doctors and nurses together. If the ratio of the number of doctors to the number of nurses is 11 : 8, how many nurses are there in the hospital?

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Introduction / Context: This is a simple ratio and total problem involving staff in a hospital. We are told how many doctors and nurses there are in total and given the ratio of doctors to nurses. From this information, we can determine the exact number of doctors and nurses separately. Such questions are common in aptitude tests and help reinforce the idea that ratios can be converted to actual counts when the total is known. Given Data / Assumptions: - Total number of doctors and nurses together is 209. - Ratio of doctors to nurses is 11 : 8. - Let doctors be the first term and nurses be the second term of the ratio. - We need the number of nurses only. Concept / Approach: If the ratio of doctors to nurses is 11 : 8, we can assume there are 11k doctors and 8k nurses for some positive number k. Their sum is 11k + 8k which is 19k. Since the total is given as 209, we can solve 19k = 209. Once k is found, we multiply it by 8 to get the number of nurses and by 11 to get the number of doctors if needed. Step-by-Step Solution: Step 1: Let number of doctors = 11k and number of nurses = 8k. Step 2: Total staff = 11k + 8k = 19k. Step 3: Given total staff = 209, so 19k = 209. Step 4: Solve for k: k = 209 / 19. Step 5: Compute 209 / 19 = 11. Step 6: Number of nurses = 8k = 8 * 11 = 88. Verification / Alternative check: To verify, calculate the number of doctors as 11k = 11 * 11 = 121. Add doctors and nurses to get 121 + 88 = 209, which matches the given total. Also, the ratio 121 : 88 simplifies to 11 : 8 when both terms are divided by 11, confirming that the allocation is correct. Why Other Options Are Wrong: 96, 108 and 121 do not satisfy both conditions simultaneously. For example, if nurses were 96 and doctors were 209 minus 96 which is 113, the ratio 113 : 96 does not reduce to 11 : 8. Similarly, other assumed nurse counts fail to yield the target ratio and total. Only 88 nurses with 121 doctors perfectly fit the ratio and sum conditions. Common Pitfalls: One common mistake is to reverse the ratio and treat nurses as the first term and doctors as the second term, which leads to wrong numbers. Another error is miscalculating k by dividing 209 by something other than the sum of the ratio terms 19. Careful identification of which group corresponds to which part of the ratio is essential. Final Answer: The number of nurses in the hospital is 88.

Question 15

What is the fourth proportional to the numbers 2, 3 and 6, that is, the value x such that 2 : 3 = 6 : x?

Accepted Answer

Introduction / Context: This problem asks for the fourth proportional to three given numbers. If a, b, c and d are in proportion such that a : b = c : d, then d is called the fourth proportional to a, b and c. These problems are fundamental in ratio and proportion and help reinforce cross multiplication techniques and the underlying structure of proportional relationships. Given Data / Assumptions: - First number a = 2. - Second number b = 3. - Third number c = 6. - We seek d such that 2 : 3 = 6 : d. Concept / Approach: From the definition of proportion, a/b = c/d. Therefore, d can be found by rearranging the equation to d = (b * c) / a. Once we know a, b and c, we simply apply this formula. Alternatively, we can express the ratio as a fraction and use cross multiplication to solve for d directly. Both methods are equivalent and give the same final result. Step-by-Step Solution: Step 1: Write the proportion 2 : 3 = 6 : x, where x is the fourth proportional. Step 2: Express this as a fraction: 2 / 3 = 6 / x. Step 3: Cross multiply: 2x = 3 * 6. Step 4: Compute 3 * 6 = 18. Step 5: Solve for x: 2x = 18, so x = 18 / 2 = 9. Step 6: Therefore, the fourth proportional to 2, 3 and 6 is 9. Verification / Alternative check: As a quick check, substitute x = 9 back into the ratio and verify equality. The ratio 2 : 3 is equal to 2/3, and 6 : 9 simplifies to 2 : 3 as well since both 6 and 9 can be divided by 3. Therefore 6/9 also equals 2/3. Since both ratios match, x = 9 is the correct fourth proportional. Why Other Options Are Wrong: 10, 7 and 8 do not maintain the proportion. For instance, 6 : 10 simplifies to 3 : 5, which is not equal to 2 : 3. Similarly, 6 : 7 and 6 : 8 do not match the original 2 : 3 ratio. Only 9 ensures that 2 : 3 = 6 : 9 when simplified. Common Pitfalls: Some learners confuse third and fourth proportionals and may incorrectly use c^2/a instead of b*c/a. Others may perform cross multiplication incorrectly, perhaps writing 2/3 = x/6 or swapping numerator and denominator. Writing out the proportion clearly and cross checking the simplified ratios helps avoid these mistakes. Final Answer: The fourth proportional to 2, 3 and 6 is 9.

Question 16

A truck covers 550 metres in one minute, while a bus covers 33 kilometres in three quarters of an hour. What is the ratio of the speed of the truck to the speed of the bus?

Accepted Answer

Introduction / Context: This is a speed comparison problem involving a truck and a bus travelling different distances in different times. We are asked to compute the ratio of their speeds. To compare speeds correctly, they must be expressed in the same units, for example kilometres per hour. Once both speeds are converted to a common unit, forming the ratio is straightforward. Given Data / Assumptions: - The truck covers 550 metres in one minute. - The bus covers 33 kilometres in three quarters of an hour which is 45 minutes. - We need the ratio speed of truck : speed of bus. Concept / Approach: Speed is distance divided by time. The truck speed is given in metres per minute and the bus speed in kilometres per hour, so we convert both to kilometres per hour for an easy comparison. After conversion, we will have speeds S_truck and S_bus in the same units. The required ratio is then S_truck : S_bus, which we simplify to smallest integer terms. Step-by-Step Solution: Step 1: Truck distance = 550 metres in 1 minute. Step 2: Convert 550 metres to kilometres: 550 metres = 0.55 kilometres. Step 3: Time for the truck is 1 minute, which is 1/60 hour. Step 4: Truck speed = distance / time = 0.55 / (1/60) = 0.55 * 60 = 33 km per hour. Step 5: Bus distance = 33 kilometres in three quarters of an hour, which is 0.75 hour. Step 6: Bus speed = 33 / 0.75 = 44 km per hour. Step 7: Speed ratio truck : bus = 33 : 44. Step 8: Simplify 33 : 44 by dividing both terms by 11 to get 3 : 4. Verification / Alternative check: We can also work in metres per minute for both vehicles and confirm the same ratio. The truck speed is 550 metres per minute. The bus travels 33,000 metres in 45 minutes, so its speed is 33,000 / 45 = 733.33 metres per minute approximately. Then the ratio 550 : 733.33 approximately simplifies to 3 : 4, consistent with the km per hour method. This double check confirms the correctness of our ratio. Why Other Options Are Wrong: 1 : 3, 2 : 3 and 1 : 4 represent much smaller truck speed relative to the bus than what the calculations show. Since the truck speed is 33 km per hour and the bus speed is 44 km per hour, the fraction is 33/44 which equals 3/4, not any of the other options. Therefore the only correct ratio is 3 : 4. Common Pitfalls: A common error is mixing units, for example comparing metres per minute directly with kilometres per hour without conversion. Another mistake is to mis convert three quarters of an hour as 3/4 minutes instead of 45 minutes or 0.75 hour. Careful unit conversion and consistent use of either kilometres per hour or metres per minute avoid these issues. Final Answer: The ratio of the speed of the truck to the speed of the bus is 3 : 4.

Question 17

A shopkeeper sells a book at a discount of 20 percent on its printed price but still makes a profit of 15 percent on the cost price. What is the ratio of the cost price to the printed price of the book?

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Introduction / Context: This question combines the concepts of marked price (printed price), cost price and selling price with both discount and profit present. The shopkeeper offers a 20 percent discount on the printed price to customers but still earns a 15 percent profit on the cost price. We are asked to find the ratio of cost price to printed price. This tests understanding of how discounts and profits relate to each other through the selling price. Given Data / Assumptions: - Discount offered on the printed price is 20 percent. - Profit made on the cost price is 15 percent. - Selling price is therefore both 80 percent of the printed price and 115 percent of the cost price. - We need the ratio cost price : printed price. Concept / Approach: Let the printed price be M and the cost price be C. The selling price S is 80 percent of M because of the 20 percent discount, so S = 0.80M. The same selling price is 115 percent of C because of the 15 percent profit, so S = 1.15C. By equating these two expressions for S, we can express C in terms of M and then write the ratio C : M. Finally, we simplify the ratio into integers by clearing decimals and reducing to lowest terms. Step-by-Step Solution: Step 1: Let printed price = M and cost price = C. Step 2: Discount 20 percent means selling price S = 80 percent of printed price = 0.80M. Step 3: Profit 15 percent on cost means S = 115 percent of cost price = 1.15C. Step 4: Since both expressions equal the selling price, set 0.80M = 1.15C. Step 5: Rearrange to express C in terms of M: C = (0.80 / 1.15) * M. Step 6: Write 0.80 as 80/100 and 1.15 as 115/100, so C = (80/100) / (115/100) * M = (80/115) * M. Step 7: Therefore, the ratio C : M = 80/115 : 1 which simplifies to 80 : 115. Step 8: Divide both numbers by 5 to simplify: 80 : 115 becomes 16 : 23. Verification / Alternative check: Pick actual numbers using the ratio 16 : 23. Suppose cost price C = Rs 16 and printed price M = Rs 23. Selling price after discount is 80 percent of 23 which is 0.80 * 23 = Rs 18.40. Profit on cost is then 18.40 minus 16 = Rs 2.40 which is 15 percent of 16. Both the discount and profit conditions are satisfied, confirming that the ratio 16 : 23 is consistent with the given information. Why Other Options Are Wrong: 20 : 23 and 23 : 20 correspond to different relationships between cost and printed price that do not maintain both a 20 percent discount and a 15 percent profit when tested with actual values. 23 : 16 would imply cost price greater than printed price, which is unrealistic in this context. Only 16 : 23 gives a selling price that is exactly 80 percent of printed price and 115 percent of cost price at the same time. Common Pitfalls: A common mistake is to confuse discount and profit bases, for instance treating the 20 percent as a reduction on cost price or the 15 percent as based on printed price. Another error is to add or subtract the percentages directly without writing proper equations. Ensuring that discount is applied to printed price and profit to cost price, and equating both expressions for selling price, keeps the reasoning correct. Final Answer: The ratio of cost price to printed price of the book is 16 : 23.

Question 18

Two numbers x and y are such that their mean proportional is 8 and the third proportional to them is 512. What are the values of x and y?

Accepted Answer

Introduction / Context:This aptitude question tests the concept of mean proportional (geometric mean) and third proportional between two numbers. Such questions are common in ratio and proportion chapters in competitive exams. Understanding how to convert verbal statements like “mean proportional is 8” and “third proportional is 512” into algebraic equations is the key. Once the equations are formed, the problem reduces to solving a simple system in x and y. Given Data / Assumptions: x and y are two positive real numbers. Their mean proportional (geometric mean) is 8. The third proportional to x and y is 512. We assume the usual definitions of mean and third proportional from ratio and proportion. Concept / Approach:The mean proportional of two numbers x and y is defined as a number m such that x : m = m : y, which gives m^2 = x * y. The third proportional to x and y is a number T such that x : y = y : T, which leads to T = y^2 / x. We will use these definitions to form two equations: one from the mean proportional and one from the third proportional. Solving these equations together gives the required values of x and y. Step-by-Step Solution:Mean proportional m = 8 implies m^2 = x * y, so x * y = 8^2 = 64.Third proportional T = 512 means x : y = y : 512, so y^2 / x = 512.From y^2 / x = 512, we get y^2 = 512 * x.We also know x * y = 64, so y = 64 / x.Substitute y = 64 / x into y^2 = 512 * x.(64 / x)^2 = 512 * x.4096 / x^2 = 512 * x.4096 = 512 * x^3.x^3 = 4096 / 512 = 8, so x = 2.Using x * y = 64 gives 2 * y = 64, so y = 32. Verification / Alternative check:Check mean proportional: x * y = 2 * 32 = 64, so geometric mean = sqrt(64) = 8, which matches the given condition.Check third proportional: x : y = 2 : 32 = 1 : 16 and y : 512 = 32 : 512 = 1 : 16, so the ratio is preserved. Thus 512 is indeed the third proportional to 2 and 32. Why Other Options Are Wrong:Options like “2 and 16”, “4 and 32”, “4 and 16”, or “8 and 8” do not satisfy both conditions simultaneously. For example, if x = 4 and y = 32, the product is 128, whose square root is not 8, so the mean proportional condition fails. Similarly, other pairs do not give y^2 / x equal to 512. Only the pair (2, 32) works for both equations. Common Pitfalls:Students sometimes confuse arithmetic mean with geometric mean and incorrectly use (x + y) / 2 = 8 instead of sqrt(x * y) = 8. Another common mistake is misinterpreting the definition of third proportional and writing x^2 / y instead of y^2 / x. Also, some candidates forget to verify both conditions and stop after satisfying only one relation, leading to wrong answers. Final Answer:The required numbers are 2 and 32, which satisfy both the mean proportional and the third proportional conditions.

Question 19

If 3A = 7B = 13C, what is the ratio A : B : C based on this proportional relationship?

Accepted Answer

Introduction / Context:This problem checks understanding of proportional equations of the form 3A = 7B = 13C. Such expressions are very common in ratio and proportion questions in aptitude exams. The goal is to convert a chained equality into a clean ratio A : B : C by expressing each variable in terms of a common constant and then simplifying the resulting fractions. Given Data / Assumptions: We are given 3A = 7B = 13C. We assume A, B and C are positive real quantities. We need to find the ratio A : B : C. Concept / Approach:Whenever we see an expression like 3A = 7B = 13C, we can assume that each of these is equal to some common constant k. That is, 3A = k, 7B = k and 13C = k. This allows us to write A, B and C individually in terms of k. Once they are written in fractional form, we can remove the denominator by multiplying all terms by the least common multiple and then read off the final ratio. Step-by-Step Solution:Assume 3A = 7B = 13C = k.Then A = k / 3, B = k / 7 and C = k / 13.So A : B : C = (k / 3) : (k / 7) : (k / 13).Cancel the common factor k to get 1 / 3 : 1 / 7 : 1 / 13.To remove denominators, multiply each term by the least common multiple of 3, 7 and 13, which is 273.(1 / 3) * 273 = 91, (1 / 7) * 273 = 39, (1 / 13) * 273 = 21.Therefore A : B : C = 91 : 39 : 21. Verification / Alternative check:Take A = 91, B = 39 and C = 21. Then 3A = 273, 7B = 273 and 13C = 273, which confirms that 3A = 7B = 13C.Since the condition is satisfied, the ratio 91 : 39 : 21 is correct. Why Other Options Are Wrong:Options like 21 : 39 : 91 or 39 : 91 : 21 reverse or permute the terms incorrectly, so they do not satisfy 3A = 7B = 13C when we test them. The option 13 : 7 : 3 directly copies coefficients and ignores the reciprocal nature of ratios when moving from 3A = 7B = 13C to A : B : C. Thus, only 91 : 39 : 21 works properly. Common Pitfalls:A very common mistake is to take the ratio as 3 : 7 : 13 or its simple variations. Candidates sometimes forget that the coefficients 3, 7 and 13 are multiplied with A, B and C, so when we isolate A, B and C, we actually get reciprocals 1 / 3, 1 / 7 and 1 / 13. Not handling this reciprocal step correctly leads to an incorrect ratio. Always express each term in terms of a common constant first and then simplify. Final Answer:The required ratio is 91 : 39 : 21.

Question 20

What smallest number must be added to each of the numbers 94, 24, 100 and 26 so that the four resulting numbers are in proportion?

Accepted Answer

Introduction / Context:This question checks the concept of forming a proportion by adding the same number to several given quantities. The goal is to find a constant k such that the modified numbers are in proportion. Here, four numbers are given, and we require that after adding k to each, the first two and the last two form equal ratios. Problems like this are standard in the ratio and proportion chapter of quantitative aptitude. Given Data / Assumptions: Original numbers are 94, 24, 100 and 26. The same number k is added to each of them. The resulting numbers 94 + k, 24 + k, 100 + k and 26 + k must be in proportion in the sense that 94 + k : 24 + k = 100 + k : 26 + k. We assume all numbers are real and that k is chosen so the ratios are well defined. Concept / Approach:If four numbers a, b, c and d are in proportion, then a : b = c : d. This can be converted into the equation a * d = b * c. In this question, after adding k, our four numbers become a = 94 + k, b = 24 + k, c = 100 + k and d = 26 + k. Setting a : b = c : d and cross multiplying gives us one algebraic equation in k, which we solve to obtain the required constant. Once k is found, we can quickly check if the condition holds. Step-by-Step Solution:Let k be the number added to each given number.Then the new numbers are 94 + k, 24 + k, 100 + k and 26 + k.For them to be in proportion, (94 + k) : (24 + k) = (100 + k) : (26 + k).Cross multiply: (94 + k) * (26 + k) = (24 + k) * (100 + k).Expand the left side: (94 + k) * (26 + k) = 94 * 26 + 94k + 26k + k^2.This is 2444 + 120k + k^2.Expand the right side: (24 + k) * (100 + k) = 24 * 100 + 24k + 100k + k^2.This is 2400 + 124k + k^2.Set them equal: 2444 + 120k + k^2 = 2400 + 124k + k^2.Cancel k^2 from both sides and simplify: 2444 + 120k = 2400 + 124k.2444 - 2400 = 124k - 120k gives 44 = 4k, so k = 11. Verification / Alternative check:Add k = 11 to each number: 94 + 11 = 105, 24 + 11 = 35, 100 + 11 = 111, 26 + 11 = 37.Check the ratios: 105 : 35 = 3 : 1 and 111 : 37 = 3 : 1 as well, since 105 / 35 = 3 and 111 / 37 = 3.Therefore, the four new numbers are indeed in proportion, confirming that k = 11 is correct. Why Other Options Are Wrong:If we try k = 10, 9, 8 or 12, the resulting ratios do not match exactly on both sides. For example, with k = 10, we would get 104 : 34 and 110 : 36; these fractions are not equal when simplified. Only k = 11 gives equal ratios, so other options are incorrect. Common Pitfalls:Some students mistakenly form the equation using only three numbers or treat “continued proportion” as requiring (24 + k)^2 = (94 + k)(100 + k), which is a different condition. Others may make algebraic errors while expanding products or forget to cancel k^2 correctly. Always write the proportion clearly, cross multiply carefully, and simplify the resulting linear equation step by step. Final Answer:The required number to be added to each term is 11.

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