Question 1

Two alloys P and Q are prepared by mixing tin and lead in the ratios 12 : 5 and 4 : 3 respectively. If equal quantities (by weight) of alloys P and Q are melted together to form a new alloy R, what is the ratio of tin to lead in alloy R?

Accepted Answer

Introduction / Context:Alligation and mixture problems involving alloys are standard in aptitude exams. Here, we are given two alloys P and Q made of tin and lead in different ratios. Equal masses of these alloys are melted together to create a third alloy R. The question asks for the resulting ratio of tin to lead in R. This tests understanding of how to convert ratios into fractional compositions and how to combine them correctly when mixing equal quantities. Given Data / Assumptions: Alloy P has tin : lead = 12 : 5. Alloy Q has tin : lead = 4 : 3. Equal weights of alloy P and alloy Q are mixed. Alloys are homogeneous, so the ratio applies uniformly to each unit mass. Concept / Approach:The key idea is to express the composition of each alloy in terms of fractional parts of tin and lead. Since equal weights of P and Q are mixed, we can assume 1 unit weight of each alloy for simplicity. We then calculate how much tin and how much lead comes from each alloy, add them, and finally express the combined tin and lead as a simplified ratio. This is a direct application of ratio and fraction manipulation. Step-by-Step Solution: Step 1: For alloy P, tin : lead = 12 : 5. Total parts = 12 + 5 = 17. So, in 1 unit of P, tin = 12/17 and lead = 5/17. Step 2: For alloy Q, tin : lead = 4 : 3. Total parts = 4 + 3 = 7. So, in 1 unit of Q, tin = 4/7 and lead = 3/7. Step 3: Equal weights of P and Q are mixed, so total tin in R = 12/17 + 4/7. Compute tin: 12/17 + 4/7 = (12 * 7 + 4 * 17) / 119 = (84 + 68) / 119 = 152 / 119. Step 4: Total lead in R = 5/17 + 3/7. Compute lead: 5/17 + 3/7 = (5 * 7 + 3 * 17) / 119 = (35 + 51) / 119 = 86 / 119. Step 5: Ratio tin : lead in R = 152 : 86. Divide both numbers by 2 to simplify: 152 : 86 = 76 : 43. Verification / Alternative check:To check, ensure the ratio is in lowest terms. The greatest common divisor of 76 and 43 is 1 (43 is a prime and does not divide 76), so 76 : 43 is already the simplest form. Also, the tin fraction 152/238 and lead fraction 86/238 simplify to 76/119 and 43/119, reinforcing the same ratio 76 : 43. Why Other Options Are Wrong: 84 : 65, 57 : 35 and 78 : 47 do not match the computed combined tin and lead amounts and do not come from correctly summing the fractional contributions. They are simply distractors and do not satisfy the arithmetic derived from the given ratios. Common Pitfalls:Students may incorrectly add ratios directly like 12 : 5 with 4 : 3, which is not valid. Another mistake is to mix up numerator and denominator when converting ratios into fractions. Some learners also forget that equal weights are mixed and instead assume equal volumes without standardizing, though here equal quantities by weight are explicitly mentioned, making it straightforward. Final Answer:The ratio of tin to lead in alloy R is 76 : 43.

Question 2

The ratio of three numbers is 2 : 3 : 5. If the sum of the three numbers is 275, then what is the largest among these three numbers?

Accepted Answer

Introduction / Context:This is a direct ratio problem involving three numbers. We know their ratio and their total sum, and we need to find the largest number. Such questions test whether a student can convert a ratio into actual values when the sum is given. It is a standard topic in ratio and proportion in quantitative aptitude sections. Given Data / Assumptions: Three numbers are in the ratio 2 : 3 : 5. The sum of the three numbers is 275. We need to find the largest of the three numbers. The numbers can be fractional because the total is not necessarily a multiple of the total ratio sum. Concept / Approach:When numbers are in a ratio, we can assume them to be 2k, 3k and 5k for some common multiplier k. The sum then becomes 2k + 3k + 5k, which must equal the given total 275. Solving this simple linear equation gives k, and then each number is calculated by multiplying the ratio part by k. The largest number corresponds to the largest ratio term. Step-by-Step Solution: Let the three numbers be 2k, 3k and 5k. Step 1: Sum of the numbers = 2k + 3k + 5k = 10k. Step 2: We are told that the sum is 275, so 10k = 275. Step 3: Solve for k: k = 275 / 10 = 27.5. Step 4: First number = 2k = 2 * 27.5 = 55. Step 5: Second number = 3k = 3 * 27.5 = 82.5. Step 6: Third number (largest) = 5k = 5 * 27.5 = 137.5. Verification / Alternative check:Add the three computed numbers: 55 + 82.5 + 137.5 = 275, which matches the given total. Also, the ratio 55 : 82.5 : 137.5 simplifies to 2 : 3 : 5 if we divide each term by 27.5. This double check confirms that the calculations and the largest number are correct. Why Other Options Are Wrong: 142, 110.5 and 82.5 do not correspond to 5k when k = 27.5. 82.5 is actually the middle number, not the largest one. Values like 142 and 110.5 break the ratio 2 : 3 : 5 when combined with the other two implied numbers. Common Pitfalls:A typical mistake is to think that all three numbers must be integers and reject fractional values, which is not necessary in ratio problems unless the question explicitly restricts them. Another error is to miscalculate the sum of ratio terms or to assign k incorrectly, leading to wrong final values. Careful arithmetic in forming and solving the equation 10k = 275 avoids these issues. Final Answer:The largest of the three numbers is 137.5.

Question 3

If a : b is 32 : 35 and b : c is 21 : 32, then what is the simplified ratio a : c?

Accepted Answer

Introduction / Context:This question checks understanding of how to combine two given ratios involving three variables a, b and c. Such problems are common in ratio and proportion chapters of aptitude tests. We are given a : b and b : c, and we need to find the relationship between a and c directly as a : c. Given Data / Assumptions: a : b = 32 : 35. b : c = 21 : 32. All quantities are positive and we want the simplified ratio a : c. Concept / Approach:From the two ratios, we can write a/b and b/c as fractions. The ratio a : c is obtained by multiplying a/b with b/c, because a/c = (a/b) * (b/c). After we compute this product, we simplify the resulting fraction to lowest terms to get the final ratio in simplest form. Step-by-Step Solution: Given a : b = 32 : 35, we have a/b = 32/35. Given b : c = 21 : 32, we have b/c = 21/32. To find a : c, compute a/c = (a/b) * (b/c). So, a/c = (32/35) * (21/32). Cancel 32 in numerator and denominator: a/c = 21/35. Simplify 21/35 by dividing numerator and denominator by 7. This gives a/c = 3/5. Therefore, a : c = 3 : 5. Verification / Alternative check:We can construct actual numbers that satisfy both ratios. Let a = 32k and b = 35k from the first ratio. From the second ratio, let b = 21m and c = 32m. Equate b: 35k = 21m. Choose k = 3 and m = 5 so that 35 * 3 = 105 and 21 * 5 = 105. Then a = 32 * 3 = 96 and c = 32 * 5 = 160. The ratio 96 : 160 simplifies by dividing both parts by 32, giving 3 : 5, which matches our earlier result. Why Other Options Are Wrong: 5 : 3 would correspond to a/c = 5/3, which is the reciprocal of the correct ratio. 1 : 1 would only occur if a and c were equal, which they are not. 5 : 7 does not agree with the product of the given ratios and is simply a distractor. Common Pitfalls:Some students mistakenly add the ratios or average them instead of multiplying a/b by b/c. Others attempt to directly compare 32 : 35 and 21 : 32 without forming a fraction, leading them away from the correct method. Remember that to link a and c when b is common, we rely on multiplication of the corresponding fractions and then simplify. Final Answer:The required ratio is 3 : 5.

Question 4

In an army selection process, the ratio of selected to unselected candidates was 6 : 1. If 30 fewer candidates had applied and 10 fewer had been selected, then the ratio of selected to unselected candidates would have become 7 : 1. How many candidat…

Accepted Answer

Introduction / Context:This is a classic selection and ratio problem involving total candidates, selected candidates and unselected candidates. It checks the ability to translate ratio statements into algebraic equations and then solve for the total number of participants. This type of question appears frequently in competitive exams to test reasoning with ratios, totals and hypothetical changes. Given Data / Assumptions: Initial ratio of selected : unselected candidates = 6 : 1. If 30 fewer candidates had applied, the total would reduce by 30. If 10 fewer had been selected, the new selected count becomes original selected minus 10. Under these new conditions, the ratio selected : unselected becomes 7 : 1. We are asked to find the original number of candidates who applied. Concept / Approach:First, we express the original numbers of selected and unselected candidates in terms of a common variable using the ratio 6 : 1. That also gives us the original total number of candidates. Next, we adjust these numbers according to the hypothetical situation: subtract 30 from the total and 10 from the selected, and deduce the new unselected count from the new total. Using the new ratio 7 : 1, we get an equation in the variable. Solving this equation yields the original total number of applicants. Step-by-Step Solution: Step 1: Let originally selected candidates = 6k and unselected candidates = 1k. Then original total candidates = 6k + k = 7k. Step 2: In the new scenario, 30 fewer candidates applied. So new total = 7k - 30. Step 3: Also, 10 fewer candidates are selected. So new selected = 6k - 10. Step 4: New unselected = new total - new selected = (7k - 30) - (6k - 10) = k - 20. Step 5: Given that new ratio selected : unselected = 7 : 1. So (6k - 10) / (k - 20) = 7 / 1. Step 6: Cross-multiply: 6k - 10 = 7(k - 20). This gives 6k - 10 = 7k - 140. Rearrange: 7k - 6k = 140 - 10. So k = 130. Step 7: Original total candidates = 7k = 7 * 130 = 910. Verification / Alternative check:Originally, selected = 6 * 130 = 780, unselected = 130. Ratio = 780 : 130 = 6 : 1 (correct). In new scenario: total = 910 - 30 = 880, selected = 780 - 10 = 770, unselected = 880 - 770 = 110. New ratio = 770 : 110 = 7 : 1, which matches the condition. So the answer 910 is confirmed. Why Other Options Are Wrong: 1820, 455, and 2730 do not satisfy both ratio conditions when substituted. For example, if total = 455, selected and unselected would not yield the required new ratio of 7 : 1 after the described changes. Common Pitfalls:Some learners may incorrectly subtract 30 only from selected or only from unselected instead of from the total. Others may forget to recalculate the new unselected count from the new total and new selected. Keeping track of each adjusted number and carefully forming the ratio equation is essential to avoid mistakes. Final Answer:The original number of candidates who applied for the process is 910.

Question 5

Of three positive numbers, the ratio of the first to the second is 3 : 4 and the ratio of the second to the third is 5 : 6. The product of the second and third numbers is 4320. What is the sum of the three numbers?

Accepted Answer

Introduction / Context:This question combines ratio relationships with a given product of two of the numbers. It is a typical aptitude problem where we are asked to find the sum of three numbers when multiple ratio links and a product condition are provided. It tests algebraic manipulation and understanding of how ratios translate into actual values using a common multiplier. Given Data / Assumptions: First : Second = 3 : 4. Second : Third = 5 : 6. Product of Second and Third = 4320. The numbers are positive. We need the sum of all three numbers. Concept / Approach:We first express the three numbers in terms of a single parameter by combining the two given ratios. Once all three are expressed as multiples of one common variable, we will use the condition on the product of the second and third to solve for that variable. After we find the variable, we compute the individual numbers and then add them to obtain the sum. Step-by-Step Solution: Let the first and second be 3x and 4x (from ratio 3 : 4). Let the second and third be 5y and 6y (from ratio 5 : 6). The second number appears in both expressions: 4x = 5y. So, y = 4x / 5. Now express all numbers in terms of x. Second number = 4x. Third number = 6y = 6 * (4x / 5) = 24x / 5. Given that second * third = 4320: (4x) * (24x / 5) = 4320. Compute the left side: (96x^2) / 5 = 4320. Multiply both sides by 5: 96x^2 = 4320 * 5 = 21600. So x^2 = 21600 / 96 = 225. Thus x = 15 (positive root, since numbers are positive). First number = 3x = 3 * 15 = 45. Second number = 4x = 4 * 15 = 60. Third number = 24x / 5 = 24 * 15 / 5 = 72. Sum = 45 + 60 + 72 = 177. Verification / Alternative check:Check the product condition: second * third = 60 * 72 = 4320, which matches the given value. Check ratios: first : second = 45 : 60 = 3 : 4 and second : third = 60 : 72 = 5 : 6. Both conditions are satisfied, confirming correctness. Why Other Options Are Wrong: 167, 187, and 197 do not correspond to any combination of three numbers that satisfy all the given ratio and product conditions. If we tried to adjust the numbers to reach those sums, at least one ratio or the product of the second and third would be violated. Common Pitfalls:Students may attempt to directly guess the numbers or may incorrectly combine ratios without resolving them to a single common variable. Another frequent error is to mis-handle the product equation and division operations, giving a wrong value for x. Carefully solving the algebra and checking the ratios at the end is a good safeguard. Final Answer:The sum of the three numbers is 177.

Question 6

If 15% of x is equal to three times 10% of y, then what is the ratio x : y?

Accepted Answer

Introduction / Context:This is a basic percentage and ratio problem that checks the ability to translate verbal percentage relationships into algebraic equations. Once we write the relationship between 15% of x and 10% of y, the ratio x : y can be easily found by simple algebraic manipulation. Such questions are very common in quantitative aptitude tests. Given Data / Assumptions: 15% of x is equal to three times 10% of y. We must find the simplified ratio x : y. Both x and y are assumed to be positive real numbers. Concept / Approach:We will write the given statement as an equation using percentage to decimal conversion. Then we isolate the ratio x/y. Finally, we simplify the fraction obtained to get the ratio in simplest whole number form. Understanding that percent just means “per hundred” is enough for this question. Step-by-Step Solution: Step 1: Convert percentages to decimal form. 15% of x = 0.15x. 10% of y = 0.10y. Given: 15% of x = 3 times 10% of y. So, 0.15x = 3 * 0.10y. Therefore, 0.15x = 0.30y. Step 2: Solve for x/y. Divide both sides by 0.15: x = (0.30 / 0.15) * y. So, x = 2y. Hence, x / y = 2 / 1. Thus, x : y = 2 : 1. Verification / Alternative check:Choose a simple value to test the ratio. Let y = 1, so x = 2. Then 15% of x = 15% of 2 = 0.30. Ten percent of y = 0.10. Three times 10% of y = 3 * 0.10 = 0.30. Both sides are equal, confirming that the relationship is satisfied when x : y = 2 : 1. Why Other Options Are Wrong: 1 : 2 would imply x is smaller than y, which contradicts x = 2y. 3 : 2 and 2 : 3 do not satisfy the equation 0.15x = 0.30y when substituted. Common Pitfalls:Common mistakes include mixing up which value is multiplied by 3 or misreading “three times 10% of y” as “30% of y” without proper algebra. While 3 * 10% does equal 30%, it is important to clearly write it as 3 * 0.10y before simplifying. Another error is to forget to simplify the ratio fully, but here 2 : 1 is already in simplest form. Final Answer:The required ratio is 2 : 1.

Question 7

The ratio of the ages of two boys is 3 : 4 at present. After 3 years, the ratio of their ages will become 4 : 5. What will be the ratio of their ages after 21 years from now?

Accepted Answer

Introduction / Context:Age problems with ratios at different times are very common in competitive exams. Here, we know the current ratio of ages and the ratio after a certain number of years. We must find the ratio after an even later time. This tests setting up linear equations from age ratios and understanding how adding the same number of years affects each age but preserves a new ratio relationship. Given Data / Assumptions: Current ratio of ages of two boys = 3 : 4. After 3 years, the ratio will be 4 : 5. We must find the ratio of their ages after 21 years from now. Ages are positive whole numbers. Concept / Approach:We start by expressing their present ages as 3k and 4k using the given current ratio. After 3 years, their ages become 3k + 3 and 4k + 3, and this ratio is given as 4 : 5. That gives an equation which we can solve for k. Once k is known, we can calculate their actual present ages. Finally, we add 21 years to each age and simplify the resulting ratio to answer the question. Step-by-Step Solution: Let present ages of the two boys be 3k and 4k. After 3 years, their ages will be 3k + 3 and 4k + 3. We are told that (3k + 3) : (4k + 3) = 4 : 5. So, (3k + 3) / (4k + 3) = 4 / 5. Cross-multiply: 5(3k + 3) = 4(4k + 3). This gives 15k + 15 = 16k + 12. Rearrange: 16k - 15k = 15 - 12. So, k = 3. Therefore, present ages are 3k = 9 and 4k = 12 years. After 21 years, their ages will be 9 + 21 = 30 and 12 + 21 = 33 years. So the ratio after 21 years is 30 : 33. Simplify by dividing both by 3: 30 : 33 = 10 : 11. Verification / Alternative check:Check the intermediate condition. After 3 years, ages become 12 and 15. The ratio 12 : 15 simplifies to 4 : 5, which matches the given condition. This confirms that k = 3 is correct. Then, after 21 years, ages become 30 and 33, and their ratio is indeed 10 : 11. Why Other Options Are Wrong: 14 : 17, 17 : 19, and 11 : 12 do not match the ages 30 and 33 when simplified. For example, 30 : 33 cannot simplify to 11 : 12 or 14 : 17, so those options are incorrect. Common Pitfalls:Students sometimes use 3 years directly in the final step instead of 21 years, or they incorrectly form the equation by mixing present and future ages. Another error is to forget to simplify the final ratio, even when the numbers have a common factor. Carefully separating the stages (present, after 3 years, after 21 years) and checking each ratio helps avoid confusion. Final Answer:The ratio of their ages after 21 years will be 10 : 11.

Question 8

If A : B = 2 : 5, B : C = 4 : 3 and C : D = 2 : 1, then what is the combined ratio A : C : D?

Accepted Answer

Introduction / Context:This problem links several pairwise ratios A : B, B : C and C : D and asks for a combined three-term ratio A : C : D. Such questions are used in aptitude tests to check whether the student can systematically combine multiple ratios and express all quantities in terms of a single scale, then simplify the final ratio correctly. Given Data / Assumptions: A : B = 2 : 5. B : C = 4 : 3. C : D = 2 : 1. We need to find A : C : D in simplest integer form. Concept / Approach:The strategy is to express A, B, C and D in terms of a single parameter by successively using each ratio. We start from A : B, then use B : C to connect C, and finally use C : D to include D. After writing all variables in terms of a single constant, we can read off the relative values of A, C and D and simplify them to get the final ratio. Step-by-Step Solution: From A : B = 2 : 5, let A = 2k and B = 5k. From B : C = 4 : 3, let B = 4m and C = 3m. Since B is common, equate 5k = 4m. So, m = (5k) / 4. Then C = 3m = 3 * (5k / 4) = 15k / 4. From C : D = 2 : 1, let C = 2n and D = n. But C is already 15k / 4, so set 2n = 15k / 4. Thus, n = (15k) / 8 and D = n = 15k / 8. Now express A, C, D all in terms of k with a common denominator 8. A = 2k = 16k / 8. C = 15k / 4 = 30k / 8. D = 15k / 8. So A : C : D = 16k/8 : 30k/8 : 15k/8. Cancel common factor k/8 to get 16 : 30 : 15. Verification / Alternative check:We can verify by choosing k = 8 for convenience. Then A = 16, B = 5k = 40, C = 30 and D = 15. Check original ratios: A : B = 16 : 40 = 2 : 5, B : C = 40 : 30 = 4 : 3, and C : D = 30 : 15 = 2 : 1. All conditions are satisfied, confirming that A : C : D = 16 : 30 : 15 is correct. Why Other Options Are Wrong: 6 : 5 : 2, 7 : 20 : 10 and 8 : 30 : 15 do not match the combined relationships when checked against the original ratios. For example, if A : C : D were 8 : 30 : 15, then C : D = 30 : 15 = 2 : 1 is correct, but that set does not maintain the correct A : B or B : C ratios when B is reconstructed. Common Pitfalls:Some students mistakenly try to directly multiply or add the ratios instead of aligning the common terms. Others forget to maintain a consistent scale when switching from one ratio to another, leading to wrong relative values. The safe method is always to introduce variables, equate the common terms and simplify at the end. Final Answer:The combined ratio is 16 : 30 : 15.

Question 9

What single number should be added to each of the numbers 35, 115, 53 and 165 so that the resulting four numbers are in proportion, that is, 35 + x : 115 + x = 53 + x : 165 + x?

Accepted Answer

Introduction / Context:This question asks for a constant to be added to four given numbers so that they form a proportion. In proportion notation, a : b = c : d means a/b = c/d. When the same number x is added to all of them, we want (35 + x) : (115 + x) = (53 + x) : (165 + x). This type of problem tests setting up and solving a proportion equation in one variable. Given Data / Assumptions: We have four base numbers: 35, 115, 53 and 165. We add the same number x to each: 35 + x, 115 + x, 53 + x, 165 + x. These must be in proportion: (35 + x) : (115 + x) = (53 + x) : (165 + x). We must find the value of x. Concept / Approach:When four numbers A, B, C and D are in proportion, A/B = C/D. So we set (35 + x)/(115 + x) equal to (53 + x)/(165 + x) and cross-multiply to get a linear equation in x. Solving this equation gives the required number that must be added to all four original numbers to make them proportional. Step-by-Step Solution: We require (35 + x) / (115 + x) = (53 + x) / (165 + x). Cross-multiply: (35 + x)(165 + x) = (115 + x)(53 + x). Expand the left side: (35 + x)(165 + x) = 35 * 165 + 35x + 165x + x^2. Compute 35 * 165 = 5775, so left side = x^2 + 200x + 5775. Expand the right side: (115 + x)(53 + x) = 115 * 53 + 115x + 53x + x^2. Compute 115 * 53 = 6095, so right side = x^2 + 168x + 6095. Set them equal: x^2 + 200x + 5775 = x^2 + 168x + 6095. Cancel x^2 from both sides: 200x + 5775 = 168x + 6095. Rearrange: 200x - 168x = 6095 - 5775. So 32x = 320. Thus x = 320 / 32 = 10. Verification / Alternative check:Substitute x = 10 back. New numbers: 45, 125, 63, 175. Check proportion: 45/125 = 0.36 and 63/175 = 0.36. Since both ratios are equal, the four numbers 45, 125, 63 and 175 are proportional, confirming that x = 10 is correct. Why Other Options Are Wrong: Values 12, 8 and 6 do not make the two ratios equal when substituted for x. For example, if x = 8, we get 43/123 and 61/173, which are clearly not equal. Common Pitfalls:Common errors include mis-expanding the products when cross-multiplying or incorrectly simplifying terms, leading to a wrong value of x. Some students may also misinterpret the phrase “continued proportion” and try to impose extra conditions. The key is to stick to the clear proportional equation and perform algebra carefully. Final Answer:The number that must be added is 10.

Question 10

Select the related number from the given alternatives: 0.02 : 0.002 :: 1/3 : ?

Accepted Answer

Introduction / Context:This is a simple analogy question involving numbers. We are given a pair of numbers 0.02 and 0.002 and asked to find what corresponds to 1/3 in the same way. Such questions test pattern recognition, especially recognizing multiplicative relationships between the first pair and then applying that same relationship to the second number. Given Data / Assumptions: First pair: 0.02 and 0.002. Second pair: 1/3 and an unknown number. We must find the number that stands in the same relationship to 1/3 as 0.002 does to 0.02. Concept / Approach:We observe how 0.02 changes to 0.002. The operation is straightforward: 0.002 = 0.02 / 10. Therefore, the relationship appears to be “divide by 10”. To preserve the same relationship, we must divide 1/3 by 10 to find the corresponding number. This uses the concept of proportional transformation between the two pairs. Step-by-Step Solution: Step 1: Identify the relationship between 0.02 and 0.002. Compute 0.02 / 10 = 0.002. So the transformation is: multiply by 1/10 or divide by 10. Step 2: Apply the same operation to 1/3. We need (1/3) / 10 = 1 / (3 * 10) = 1 / 30. Thus, the required number is 1/30. Verification / Alternative check:We can check the ratio form: 0.02 : 0.002 = 0.02 / 0.002 = 10. Similarly, 1/3 : (1/30) = (1/3) / (1/30) = (1/3) * (30/1) = 10. Since both pairs have the same ratio 10, the analogy holds correctly. Why Other Options Are Wrong: 0.333 is approximately 1/3, so it does not represent applying the “divide by 10” operation. 0.003 is 0.03 / 10, not 1/3 / 10. 3 is the reciprocal of 1/3, corresponding to multiply by 9 rather than divide by 10. Common Pitfalls:Some students misread the pattern and attempt to multiply instead of divide, or they try to relate decimal places directly without checking the exact numeric operation. Always check the factor between the numbers (in this case, dividing by 10) and then apply the exact same factor to the new term in the analogy. Final Answer:The related number is 1/30.

Question 11

If 7A = 5B = 2C, then what is the ratio A : B : C?

Accepted Answer

Introduction / Context:This problem involves three variables A, B and C linked by equalities of the form 7A = 5B = 2C. We are asked to find the ratio A : B : C. Such questions test the ability to introduce a common constant and express each variable in terms of that constant, then extract the required ratio in simplest form. Given Data / Assumptions: 7A = 5B = 2C. We must determine A : B : C. All variables are assumed to be positive. Concept / Approach:When several expressions are equal, we can set each equal to a common constant, say k. Then 7A = k, 5B = k and 2C = k. From this, we can solve for A, B and C in terms of k and then write their ratio. The ratios naturally simplify when we clear denominators and remove any common factors. Step-by-Step Solution: Let 7A = 5B = 2C = k. Then A = k / 7, B = k / 5 and C = k / 2. So A : B : C = (k / 7) : (k / 5) : (k / 2). Cancel k from all three terms: A : B : C = 1/7 : 1/5 : 1/2. To remove fractions, multiply each term by the least common multiple of denominators 7, 5 and 2, which is 70. So we get A : B : C = 70 * (1/7) : 70 * (1/5) : 70 * (1/2). Compute each term: 70/7 = 10, 70/5 = 14, 70/2 = 35. Thus A : B : C = 10 : 14 : 35. Verification / Alternative check:Choose k = 70 for convenience. Then A = 70/7 = 10, B = 70/5 = 14, C = 70/2 = 35. Check the original relation: 7A = 7 * 10 = 70, 5B = 5 * 14 = 70 and 2C = 2 * 35 = 70. All three match 70, so our values are consistent and the ratio 10 : 14 : 35 is correct. Why Other Options Are Wrong: 35 : 14 : 10 reverses the correct order and would only hold if the variables were assigned differently. 14 : 10 : 35 does not follow from the equalities because it would give 7A different from 5B or 2C. 2 : 5 : 7 reflects the denominators rather than the actual values of A, B and C. Common Pitfalls:Some students incorrectly take 7A, 5B and 2C themselves as the ratio terms (7 : 5 : 2), which is wrong because those are equal, not proportional. The actual ratio is formed from A, B and C, which are inversely proportional to the coefficients 7, 5 and 2 when equated to a common constant. Final Answer:The correct ratio is 10 : 14 : 35.

Question 12

The ratio of two numbers is 6 : 11 and their difference is 65. What is the smaller of the two numbers?

Accepted Answer

Introduction / Context:This ratio and difference question asks us to find the actual values of two numbers when their ratio and difference are known. It is a standard type in aptitude tests and is solved using a simple linear equation in one variable derived from the ratio representation of the numbers. Given Data / Assumptions: Ratio of two numbers = 6 : 11. Difference between the two numbers = 65. We need to find the smaller of the two numbers. Concept / Approach:We represent the two numbers as 6k and 11k, where k is a positive constant. The difference between them is (11k - 6k) = 5k. We are told this difference equals 65, so we can solve for k. Once k is known, we compute each number and identify the smaller one as 6k. Step-by-Step Solution: Let the two numbers be 6k and 11k. Their difference is 11k - 6k = 5k. Given that the difference is 65, so 5k = 65. Solve for k: k = 65 / 5 = 13. Smaller number = 6k = 6 * 13 = 78. Larger number = 11k = 11 * 13 = 143. Verification / Alternative check:Check the ratio: 78 : 143. Divide both terms by 13 to get 6 : 11, which matches the given ratio. Check the difference: 143 - 78 = 65, which matches the given difference. So the calculations are consistent and confirm the smaller number as 78. Why Other Options Are Wrong: 58, 65 and 52 do not satisfy both the given ratio and difference when paired with any corresponding second number. For example, if the smaller number were 58, the larger would be 58 * 11/6 (from the ratio), which is not an integer and does not produce a difference of 65. Common Pitfalls:Some students misinterpret the ratio and directly subtract 11 - 6 = 5 and then equate it to the difference, forgetting to multiply by the common factor k. Another error is to mistakenly identify the larger number instead of the smaller one when answering. Keeping track that 6k is the smaller and 11k is the larger is important. Final Answer:The smaller number is 78.

Question 13

If two successive discounts of 25% and 30% are given on the marked price of an item, then what is the ratio of the marked price to the final selling price?

Accepted Answer

Introduction / Context:This question concerns successive discounts on a marked price and asks for the ratio of the original marked price to the final selling price. Successive discounts do not add directly as percentages; instead, each discount applies on the reduced price. This is a common profit and loss concept in aptitude exams. Given Data / Assumptions: First discount = 25% on marked price. Second discount = 30% on the reduced price. We need the ratio Marked Price : Selling Price. No taxes or other charges are involved. Concept / Approach:We assume a convenient marked price, typically 100 units, to make percentage calculations easy. We then apply the first discount to find the new price and then apply the second discount on that reduced value. The resulting price is the final selling price. Then we express Marked Price : Selling Price as a simplified ratio. Step-by-Step Solution: Step 1: Assume Marked Price (MP) = 100 units. Step 2: Apply first discount of 25%. 25% of 100 = 25, so new price after first discount = 100 - 25 = 75. Step 3: Apply second discount of 30% on 75. 30% of 75 = 0.30 * 75 = 22.5. So final Selling Price (SP) = 75 - 22.5 = 52.5. Step 4: Marked Price : Selling Price = 100 : 52.5. Multiply both terms by 2 to remove decimal: 200 : 105. Now divide by 5: 200/5 = 40, 105/5 = 21. Thus MP : SP = 40 : 21. Verification / Alternative check:We can compute the combined discount factor directly. Price factor after first discount = 0.75. Price factor after second discount = 0.70. Total factor = 0.75 * 0.70 = 0.525. So, SP = 0.525 * MP. Therefore, MP/SP = 1 / 0.525 ≈ 1.9047, which equals 40/21 ≈ 1.9047. This confirms the ratio 40 : 21. Why Other Options Are Wrong: 20 : 7 is approximately 2.857, which would imply a much larger discount than actually given. 20 : 17 and 40 : 33 correspond to different combined discount factors and do not match the successive discounts of 25% and 30%. Common Pitfalls:One common mistake is to simply add the discounts (25% + 30% = 55%) and treat it as a single discount, which is incorrect because the second discount applies on a reduced price, not the original marked price. Another mistake is to mis-handle decimal calculations when computing 30% of 75. Working carefully or using fractions avoids these problems. Final Answer:The ratio of marked price to selling price is 40 : 21.

Question 14

Two numbers have a sum of 84. Which of the following ratios can they not be in: 5 : 7, 13 : 8, 1 : 3 or 3 : 2?

Accepted Answer

Introduction / Context:This question checks whether a given ratio is compatible with a specified sum of two numbers. Such problems involve expressing the numbers in terms of a ratio and verifying whether the sum matches the required total with integer values. This is important in ratio and proportion topics where sums and ratios are linked. Given Data / Assumptions: The sum of two numbers is 84. Possible ratios to test: 5 : 7, 13 : 8, 1 : 3, 3 : 2. We must determine which ratio the two numbers cannot have if they are to sum to 84 as whole numbers. Concept / Approach:For each ratio m : n, we can represent the two numbers as m*k and n*k for some constant k. The sum is then (m + n)k. For the sum to be exactly 84, k must be 84 / (m + n). To get integer numbers, k must be an integer. Therefore, we check for each ratio whether 84 is divisible by (m + n). If it is not divisible, that ratio is not possible for two whole numbers summing to 84. Step-by-Step Solution: 1) Ratio 5 : 7. Sum of ratio parts = 5 + 7 = 12. k = 84 / 12 = 7, which is an integer. So numbers can be 35 and 49. 2) Ratio 13 : 8. Sum of ratio parts = 13 + 8 = 21. k = 84 / 21 = 4, which is an integer. So numbers can be 52 and 32. 3) Ratio 1 : 3. Sum of ratio parts = 1 + 3 = 4. k = 84 / 4 = 21, which is an integer. So numbers can be 21 and 63. 4) Ratio 3 : 2. Sum of ratio parts = 3 + 2 = 5. k = 84 / 5 = 16.8, which is not an integer. Therefore, the numbers cannot be in the ratio 3 : 2 while summing to 84 as whole numbers. Verification / Alternative check:We can explicitly compute pairs from the valid ratios and check sums. For 5 : 7, 35 + 49 = 84; for 13 : 8, 52 + 32 = 84; for 1 : 3, 21 + 63 = 84. In contrast, attempting 3 : 2 gives numbers 3k and 2k with sum 5k = 84, leading to k = 84/5 = 16.8, so one or both numbers would not be integers. Hence 3 : 2 is the only invalid ratio if we require integer values. Why Other Options Are Wrong: 5 : 7, 13 : 8 and 1 : 3 all yield integer-valued pairs whose sum is 84, so they are possible. They therefore cannot be the “not possible” ratio asked by the question. Common Pitfalls:Some students may assume that any ratio can work without checking the divisibility of 84 by the sum of ratio parts. Others may make arithmetic mistakes when dividing 84, especially if done mentally. Always compute (m + n) and then see if 84 divides evenly to ensure integer solutions. Final Answer:The numbers cannot be in the ratio 3 : 2 if their sum is 84.

Question 15

Two numbers x and y are such that their mean proportion (geometric mean) is 9 and their third proportion is 243. What are the values of x and y?

Accepted Answer

Introduction / Context:This question uses the concepts of mean proportion (geometric mean) and third proportion of two numbers. It tests understanding of proportional relationships and the ability to set up and solve simultaneous equations using these geometric definitions. Such questions are often seen in quantitative aptitude sections under the topic of ratio and proportion. Given Data / Assumptions: Mean proportion (geometric mean) of x and y is 9. Third proportion to x and y is 243. We must find x and y. x and y are positive numbers. Concept / Approach:The geometric mean of x and y is sqrt(xy), so we have sqrt(xy) = 9, which implies xy = 81. The third proportion to x and y is defined as a number t such that x : y = y : t, or equivalently y^2 = x * t. Here, t is given as 243, so y^2 = 243x. We now have two equations: xy = 81 and y^2 = 243x. Solving these simultaneously yields x and y. Step-by-Step Solution: From mean proportion condition: sqrt(xy) = 9. Square both sides: xy = 81. (Equation 1) From third proportion condition: x : y = y : 243. So y^2 = 243x. (Equation 2) From Equation 1, express x as x = 81 / y. Substitute into Equation 2: y^2 = 243 * (81 / y). Simplify: y^2 = (243 * 81) / y. Multiply both sides by y: y^3 = 243 * 81. Compute 243 * 81 = 19683. So y^3 = 19683. Take cube root: y = 27 (since 27^3 = 19683). Use Equation 1: xy = 81, so x = 81 / 27 = 3. Thus, x = 3 and y = 27. Verification / Alternative check:Check mean proportion: sqrt(xy) = sqrt(3 * 27) = sqrt(81) = 9, which matches the given value. Check third proportion: with x = 3 and y = 27, the ratio x : y = 3 : 27 = 1 : 9, and y : t should be 1 : 9 as well, so t must be 243 (since 27 : 243 = 1 : 9). This confirms the correctness of x and y. Why Other Options Are Wrong: 3 and 9 give xy = 27, not 81, and do not satisfy both conditions. 6 and 27 give xy = 162, not 81. 6 and 81 give xy = 486, again not 81, and they also fail the third proportion condition. Common Pitfalls:Some students mix up the definitions of mean and third proportion or think both are arithmetic instead of geometric. Others may incorrectly square or cube during manipulation and introduce algebraic errors. Writing down the precise definitions and carefully substituting from one equation into another is the safest route. Final Answer:The values of the numbers are 3 and 27.

Question 16

The bus fare between two cities is increased in the ratio 11 : 18. What will be the increase in the fare if the original fare was Rs. 550?

Accepted Answer

Introduction / Context:This is a simple ratio-based price increase problem. We are told that the fare between two cities increases such that the ratio of the old fare to the new fare is 11 : 18. Given the original fare, we must compute the new fare and then the absolute increase. This tests the ability to interpret fare changes expressed in ratio form instead of a direct percentage. Given Data / Assumptions: Original fare : New fare = 11 : 18. Original fare = Rs. 550. We must find the increase in fare in rupees. Concept / Approach:Since original fare corresponds to the ratio part 11 and new fare corresponds to ratio part 18, we first determine what one ratio unit corresponds to in rupees. That is done by dividing the original fare by 11. Then we multiply this unit by 18 to get the new fare. The increase is new fare minus original fare. This is a standard way of using ratios to scale from one value to another. Step-by-Step Solution: Let the common unit for the ratio be k. Original fare = 11k. New fare = 18k. Given original fare = Rs. 550, so 11k = 550. Solve for k: k = 550 / 11 = 50. New fare = 18k = 18 * 50 = Rs. 900. Increase in fare = New fare - Original fare = 900 - 550 = Rs. 350. Verification / Alternative check:Check that the ratio of old to new fare matches the given ratio. Old : New = 550 : 900. Divide by 50 to simplify: 11 : 18, which is exactly the ratio specified in the question. Therefore, our value for the new fare and the increase is consistent. Why Other Options Are Wrong: Rs. 900 is the new fare itself, not the increase. Rs. 180 and Rs. 360 do not correspond to the difference between 550 and any new fare that is in the ratio 11 : 18 with 550. Common Pitfalls:Some students mistakenly treat 11 : 18 as a percentage and try to compute a percentage increase directly, which leads to incorrect values. Another error is to incorrectly scale the fare by 18/11 but then forget to subtract the original fare to find the increase. Always separate the steps: compute the new fare from the ratio, then subtract to get the increase. Final Answer:The increase in fare is Rs. 350.

Question 17

A bag contains Rs. 43 in the form of 5-rupee coins, 50-paise coins and 10-paise coins in the ratio 1 : 5 : 11 respectively. What is the total number of 50-paise coins in the bag?

Accepted Answer

Introduction / Context:This is a coin and ratio problem involving different denominations with a known total value. The numbers of 5-rupee, 50-paise and 10-paise coins are in a certain ratio. The total amount of money in the bag is given, and we must find how many 50-paise coins are present. This tests the ability to convert coin counts to rupee values and solve a simple linear equation based on ratios. Given Data / Assumptions: Denominations: Rs. 5, Rs. 0.50 (50 paise), Rs. 0.10 (10 paise). Counts are in the ratio 1 : 5 : 11 respectively. Total money in the bag = Rs. 43. We must find the number of 50-paise coins. Concept / Approach:We represent the numbers of 5-rupee, 50-paise and 10-paise coins as k, 5k and 11k respectively, using the given ratio. We then write an equation for the total value in rupees by multiplying each coin count by its denomination and summing. This total must equal Rs. 43. Solving for k gives the exact counts of each coin type. The number of 50-paise coins is simply 5k. Step-by-Step Solution: Let the number of 5-rupee coins = k. Number of 50-paise coins = 5k. Number of 10-paise coins = 11k. Compute the total value in rupees. Value from 5-rupee coins = 5 * k = 5k rupees. Value from 50-paise coins = 0.50 * 5k = 2.5k rupees. Value from 10-paise coins = 0.10 * 11k = 1.1k rupees. Total value = 5k + 2.5k + 1.1k = 8.6k rupees. Given total = Rs. 43, so 8.6k = 43. Solve for k: k = 43 / 8.6 = 5. Therefore, number of 50-paise coins = 5k = 5 * 5 = 25. Verification / Alternative check:Check all coin counts and total. With k = 5, 5-rupee coins = 5, 50-paise coins = 25, 10-paise coins = 55. Compute total value: 5 * 5 = Rs. 25; 25 coins of 0.50 each = Rs. 12.5; 55 coins of 0.10 each = Rs. 5.5. Sum = 25 + 12.5 + 5.5 = Rs. 43, which matches the given total. Therefore, the answer is consistent. Why Other Options Are Wrong: 5, 55 and 50 would correspond to different values of k and would not lead to a total of Rs. 43 when combined with the corresponding numbers of 5-rupee and 10-paise coins. For example, if there were 50 coins of 50-paise, the value from them alone would be Rs. 25, and it would be difficult to adjust other coins to keep the ratio and still total Rs. 43. Common Pitfalls:One common mistake is to confuse the value ratio with the count ratio and directly assign rupee values like 1, 5, 11 rupees. Another is mishandling decimal calculations for 0.50 and 0.10, leading to an incorrect equation. Carefully converting each type of coin to rupees and summing is essential. Final Answer:The bag contains 25 coins of 50 paise.

Question 18

Rs 18,200 is divided among three persons X, Y and Z in the ratio 1/3 : 1/4 : 1/2. What is the share of X in rupees?

Accepted Answer

Introduction / Context: This aptitude question is based on the concept of ratio and proportion, specifically on how a fixed sum of money is divided among three people according to a given ratio. Such problems commonly appear in competitive exams to test understanding of how to convert fractional ratios into whole number ratios and then calculate individual shares from the total amount. Given Data / Assumptions: Total amount to be divided = Rs 18,200. The ratio of shares of X, Y and Z is 1/3 : 1/4 : 1/2. Concept / Approach: To handle ratios containing fractions, we first convert them into whole number ratios. This is done by taking the LCM of the denominators of the fractions and multiplying each term by this LCM. After obtaining a clean whole number ratio, we treat the sum of the ratio terms as the total number of parts into which the amount is divided. Then each person’s share is calculated as (individual parts / total parts) * total amount. Step-by-Step Solution: 1) The ratio is given as 1/3 : 1/4 : 1/2 for X : Y : Z. 2) Denominators are 3, 4 and 2. The LCM of 3, 4 and 2 is 12. 3) Multiply each term by 12 to convert into whole numbers: (1/3)*12 = 4, (1/4)*12 = 3, (1/2)*12 = 6. 4) So the equivalent ratio becomes X : Y : Z = 4 : 3 : 6. 5) Total number of parts = 4 + 3 + 6 = 13 parts. 6) Each part is equal to 18,200 / 13 = 1,400 rupees. 7) Share of X = 4 parts = 4 * 1,400 = Rs 5,600. Verification / Alternative check: We can also compute the shares of Y and Z to verify that the total adds back to 18,200. Y gets 3 * 1,400 = Rs 4,200. Z gets 6 * 1,400 = Rs 8,400. Adding them together: 5,600 + 4,200 + 8,400 = 18,200, which matches the original total. This confirms that our conversion of the fractional ratio into whole number parts and the subsequent calculations are consistent and correct. Why Other Options Are Wrong: Option A (Rs 4,400) and option B (Rs 4,200) are actually closer to the amount that Y receives, not X. Option C (Rs 7,000) is larger than the correct value and does not correspond to any correct part-based calculation. Option E (Rs 6,000) also does not match any exact multiple of 1,400, so it cannot represent a valid share based on the derived ratio 4 : 3 : 6. Common Pitfalls: Students often forget to convert fractional ratios into whole numbers first and may try to work directly with fractions, which increases the chance of arithmetic errors. Another frequent mistake is to miscalculate the LCM of denominators, which leads to an incorrect whole number ratio and wrong final answers. Some learners also divide the total amount by an incorrect number of ratio parts or mix up whose share corresponds to which part in the final ratio. Final Answer: Therefore, the share of X from the amount of Rs 18,200 is Rs 5,600.

Question 19

A bag contains Rs 51.25 in 2-rupee, 50-paise and 25-paise coins in the ratio 3 : 5 : 7. How many 50-paise coins are there in the bag?

Accepted Answer

Introduction / Context: This question checks understanding of ratio, proportion and handling of mixed denominations in rupees and paise. You are given the total value of coins of three different types and the ratio of their counts. The task is to find the exact number of 50-paise coins, which is a standard type of aptitude problem involving both ratios and basic money calculations. Given Data / Assumptions: Total money in the bag = Rs 51.25. Coin types: 2-rupee coins, 50-paise coins and 25-paise coins. Ratio of number of coins (2-rupee : 50-paise : 25-paise) = 3 : 5 : 7. We assume all coins are of these three denominations only. Concept / Approach: Let the common multiplying factor be x. Then the number of 2-rupee coins is 3x, of 50-paise coins is 5x and of 25-paise coins is 7x. We then express the total monetary value in terms of x and equate it to the given total amount. Since 1 rupee = 100 paise, we convert paise to rupees where needed. Solving the resulting linear equation gives x, and then we can find the required number of 50-paise coins which is 5x. Step-by-Step Solution: 1) Let the number of 2-rupee coins be 3x, 50-paise coins be 5x and 25-paise coins be 7x. 2) Value of 2-rupee coins = 3x * 2 = 6x rupees. 3) Value of 50-paise coins = 5x * 0.5 = 2.5x rupees. 4) Value of 25-paise coins = 7x * 0.25 = 1.75x rupees. 5) Total value = 6x + 2.5x + 1.75x = 10.25x rupees. 6) Given total value is Rs 51.25, so 10.25x = 51.25. 7) Therefore x = 51.25 / 10.25 = 5. 8) Number of 50-paise coins = 5x = 5 * 5 = 25 coins. Verification / Alternative check: We can quickly verify by recomputing the total value using x = 5. Number of 2-rupee coins = 15, value = 30 rupees. Number of 50-paise coins = 25, value = 12.5 rupees. Number of 25-paise coins = 35, value = 8.75 rupees. Sum = 30 + 12.5 + 8.75 = 51.25 rupees, which matches the given total amount. Therefore the value of x and thus the number of 50-paise coins is correct. Why Other Options Are Wrong: Options A (15) and D (5) are simply smaller multiples of 5 and do not satisfy the total value condition when you recompute. Option B (35) would make the value of 50-paise coins alone too large. Option E (30) also results in a total value different from Rs 51.25. Only 25 coins of 50 paise each fits consistently with both the ratio and the total amount. Common Pitfalls: Common mistakes include confusing the ratio of number of coins with the ratio of values, or forgetting to convert paise to rupees correctly. Some learners also mistakenly distribute the total amount directly in the ratio 3 : 5 : 7 as rupees without considering that the denominations are different, which leads to incorrect answers. Always work with the number of coins first, and then convert to value using the denomination of each coin type. Final Answer: Thus, the bag contains 25 coins of 50 paise each.

Question 20

A certain sum of money is divided among A, B and C in the ratio 5 : 6 : 9. If A receives Rs 450, what is the total sum that was divided?

Accepted Answer

Introduction / Context: This question is a direct application of the ratio concept used to share a total sum among several people. You are given the ratio of shares of A, B and C and the actual amount received by one person. From this information, you need to reconstruct the total amount. This type of question frequently appears in banking, SSC and other competitive examinations. Given Data / Assumptions: Ratio of A : B : C = 5 : 6 : 9. Amount received by A = Rs 450. The entire sum is fully distributed among A, B and C according to this ratio. Concept / Approach: When amounts are divided in a ratio, each term in the ratio corresponds to a proportional part of the total. If A gets 5 parts, B gets 6 parts and C gets 9 parts, then the total number of parts is 5 + 6 + 9. If we know the amount corresponding to one person’s parts, we can find the value of a single part and then the total sum. The key idea is to treat the ratio numbers as multiples of some common factor k and solve for k using the known share. Step-by-Step Solution: 1) Let the common multiplying factor be k. 2) Then the share of A = 5k, share of B = 6k and share of C = 9k. 3) According to the question, A receives Rs 450, so 5k = 450. 4) Solve for k: k = 450 / 5 = 90. 5) Total number of parts = 5 + 6 + 9 = 20 parts. 6) Total sum = 20k = 20 * 90 = Rs 1,800. Verification / Alternative check: We can calculate the amount for B and C using k = 90 as a check. B gets 6k = 6 * 90 = Rs 540. C gets 9k = 9 * 90 = Rs 810. Now sum these: 450 + 540 + 810 = 1,800. This equals the total sum we obtained, validating that the ratio and the computations are all correct. This balance check is useful in exam conditions to avoid silly mistakes. Why Other Options Are Wrong: Option A (Rs 2,000), option C (Rs 2,250), option D (Rs 1,000) and option E (Rs 1,500) do not satisfy the condition that A’s share is exactly 5 parts equal to Rs 450. For example, if the total were Rs 2,000, each part would be Rs 100, so A would get 5 * 100 = Rs 500, which is incorrect. Similar mismatches happen with the other totals. Only Rs 1,800 leads to A receiving Rs 450 when using the ratio 5 : 6 : 9. Common Pitfalls: A common mistake is to directly divide A’s share by the total number of ratio terms instead of using only A’s ratio term. Another error is adding the ratio numbers incorrectly or miscomputing the value of k. Carefully setting up the relationship between the known share and the ratio term (5k = 450 here) is essential to get the correct result. Final Answer: The total sum of money that was divided among A, B and C is Rs 1,800.

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