Question 1

The diagonals of two squares are in the ratio 5 : 2. What is the ratio of the areas of these two squares?

Accepted Answer

Introduction / Context: This problem tests understanding of the relationship between the diagonal and the area of a square. Ratio questions involving geometric figures often focus on how linear dimensions such as side or diagonal affect corresponding measures like area or volume. Here, you are given the ratio of the diagonals of two squares and asked to find the ratio of their areas, which is a classic geometry and ratio concept useful for many competitive exams. Given Data / Assumptions: Two squares are considered. The ratio of their diagonals is 5 : 2. We assume both are standard squares with all interior angles equal and sides equal. Concept / Approach: For a square, if the side length is s, the diagonal d is given by d = s * sqrt(2). Therefore, the side length is directly proportional to the diagonal. Since the area of a square is s^2, and s is proportional to the diagonal, the area becomes proportional to the square of the diagonal. So if the ratio of diagonals is d1 : d2, then the ratio of areas will be (d1^2) : (d2^2). Step-by-Step Solution: 1) Let the diagonals of the two squares be d1 and d2 with d1 : d2 = 5 : 2. 2) Since area of a square is proportional to the square of its diagonal, area ratio = d1^2 : d2^2. 3) Compute the squares of the ratio terms: 5^2 = 25 and 2^2 = 4. 4) Therefore, the ratio of areas of the two squares is 25 : 4. Verification / Alternative check: We can assume convenient values to verify. Let d2 = 2 units, then d1 = 5 units. For a square, side s = d / sqrt(2). So s1 = 5 / sqrt(2) and s2 = 2 / sqrt(2). Then area1 = (5 / sqrt(2))^2 = 25 / 2, and area2 = (2 / sqrt(2))^2 = 4 / 2 = 2. The ratio area1 : area2 = (25 / 2) : 2 = 25 : 4. This confirms our earlier reasoning and shows that choosing actual numerical values is a valid alternative method. Why Other Options Are Wrong: Option A (5 : 6) and option C (5 : 4) treat the ratio as if area were directly proportional to the diagonal instead of the square of the diagonal. Option D (125 : 8) would correspond to cubing the ratio rather than squaring it. Option E (4 : 25) is simply the inverse of the correct ratio and would only be valid if the question asked for the ratio of the smaller area to the larger area in reversed order. Thus, only 25 : 4 correctly represents the ratio of the larger square’s area to the smaller square’s area. Common Pitfalls: Many learners mistakenly assume that the ratio of areas is the same as the ratio of diagonals, forgetting that area depends on the square of a linear dimension. Another common pitfall is to confuse diagonal-based formulas with side-based formulas and perform unnecessary calculations. Always remember that when a linear dimension changes in ratio a : b, the area changes in the ratio a^2 : b^2 for similar figures like squares. Final Answer: The ratio of the areas of the two squares is 25 : 4.

Question 2

The ratio of two positive numbers is 3 : 4 and the sum of their squares is 400. What is the sum of the two numbers?

Accepted Answer

Introduction / Context: This question combines the concept of ratio with algebraic manipulation involving squares of numbers. You are given the ratio between two positive numbers and the sum of their squares, and you are asked to determine the sum of the original numbers. This kind of problem is frequently tested to see if candidates can translate ratio information into algebraic expressions and then solve a simple equation involving squares. Given Data / Assumptions: Let the two positive numbers be x and y. Their ratio is x : y = 3 : 4. The sum of their squares is x^2 + y^2 = 400. Both numbers are positive. Concept / Approach: When two numbers are in the ratio 3 : 4, we can write x = 3k and y = 4k for some positive constant k. Substituting these expressions into the condition about the sum of squares gives an equation in k. Solving this equation yields the value of k, from which we can compute both numbers and then their sum. This approach is systematic and avoids guesswork. Step-by-Step Solution: 1) Let the two numbers be x and y with x : y = 3 : 4. 2) Express x and y in terms of k: x = 3k and y = 4k. 3) Given that x^2 + y^2 = 400. 4) Substitute: (3k)^2 + (4k)^2 = 400. 5) This gives 9k^2 + 16k^2 = 400. 6) Combine like terms: 25k^2 = 400. 7) Solve for k^2: k^2 = 400 / 25 = 16, so k = 4 (positive root because numbers are positive). 8) Then x = 3k = 3 * 4 = 12 and y = 4k = 4 * 4 = 16. 9) Sum of the numbers = x + y = 12 + 16 = 28. Verification / Alternative check: We can check the solution by verifying the sum of squares condition with x = 12 and y = 16. Compute x^2 + y^2 = 12^2 + 16^2 = 144 + 256 = 400, which matches the given condition exactly. This confirms that both numbers and their sum have been calculated correctly, and that no sign or arithmetic mistakes have been made. Why Other Options Are Wrong: Any other sum choice such as 22, 24, 26 or 30 would correspond to different values of x and y, which would not satisfy both the given ratio and the required sum of squares. For example, if the sum were 24 and the ratio is 3 : 4, the numbers would be 9 and 15, whose squares sum to 81 + 225 = 306, not 400. Similar checks show that none of the other options satisfy both conditions simultaneously. Common Pitfalls: Students sometimes confuse the sum of numbers with the sum of squares and try to directly assign values based on the ratio without forming the correct equation. Another common error is taking the negative root of k even though the question specifies positive numbers. Always pay attention to the word positive in such questions, and make sure you substitute back to check that your final numbers satisfy all original conditions. Final Answer: The sum of the two positive numbers is 28.

Question 3

If A = 2B = 4C, what is the simplest ratio A : B : C?

Accepted Answer

Introduction / Context: This question is a standard example of working with compound ratio expressions where a single quantity is given equal to multiple expressions involving three variables. Such questions test your ability to rewrite relationships like A = 2B = 4C into a clean ratio A : B : C. This is very useful in algebra and in many quantitative aptitude tests. Given Data / Assumptions: We are given A = 2B = 4C. We assume A, B and C are positive real numbers. The task is to express A : B : C in simplest integer form. Concept / Approach: When we have an expression like A = 2B = 4C, we introduce a common value, say k, to represent the equal quantity. That is, we set A = k, 2B = k and 4C = k. This allows us to express B and C in terms of k and then find the ratio A : B : C. To convert the resulting fractional expressions into whole numbers, we multiply all terms by a suitable common multiple of the denominators. Step-by-Step Solution: 1) Let A = 2B = 4C = k for some positive constant k. 2) From A = k, we have A = k. 3) From 2B = k, we get B = k / 2. 4) From 4C = k, we get C = k / 4. 5) So A : B : C = k : (k / 2) : (k / 4). 6) To clear denominators, multiply all three terms by 4. 7) This gives A : B : C = 4k : 2k : 1k. 8) Therefore, the required simplest ratio is 4 : 2 : 1. Verification / Alternative check: We can pick a specific value for k to verify. Let k = 4. Then A = 4, B = 4 / 2 = 2, C = 4 / 4 = 1. Check the original relationship: 2B = 2 * 2 = 4 and 4C = 4 * 1 = 4, both equal A = 4. So 4 : 2 : 1 is consistent with A = 2B = 4C. This numerical example confirms that our derived ratio is correct. Why Other Options Are Wrong: Option B (1 : 2 : 4) reverses the pattern and does not satisfy A = 2B = 4C when tested with actual values. Option C (8 : 4 : 1) correctly preserves proportions but is not in simplest form, since it can be reduced by dividing all terms by 2 to obtain 4 : 2 : 1. Option D (16 : 4 : 1) and option E (2 : 1 : 4) also fail to meet the original condition when substituted back into the relationship A = 2B = 4C. Common Pitfalls: A frequent error is inverting the relationships, such as assuming A : B : C = 1 : 2 : 4 without carefully solving for each variable. Another mistake is failing to clear denominators correctly, which leads to fractional ratios that are not simplified. Always introduce a common constant k, express each variable in terms of k, and then carefully reduce the resulting ratio to its simplest integer form. Final Answer: The simplest ratio of A : B : C that satisfies A = 2B = 4C is 4 : 2 : 1.

Question 4

The flight fare between two cities is increased in the ratio 11 : 13. If the original fare was Rs 12,100, what is the increase in the fare in rupees?

Accepted Answer

Introduction / Context: This question tests a basic but important application of ratios to percentage or absolute changes. Instead of directly specifying a percentage increase, the problem states that the flight fare increases in the ratio 11 : 13. You need to translate this ratio information into an actual increase in rupees, given the original fare. Such questions help develop comfort with using ratios to compare old and new values. Given Data / Assumptions: Original fare between two cities = Rs 12,100. The ratio of old fare to new fare is 11 : 13. The fare is increased, so the new fare is higher than the old fare. We want the increase in rupees, not the new fare itself. Concept / Approach: When old : new = 11 : 13, it means that if the old fare is treated as 11 equal parts, the new fare is 13 of the same parts. Therefore the increase corresponds to the difference between 13 parts and 11 parts, that is 2 parts. Using the actual old fare, we compute the value of one part and then calculate the increase as 2 times that part size. Step-by-Step Solution: 1) Let the common unit part of the fare be k. 2) Old fare = 11k and new fare = 13k. 3) We are given that the old fare is Rs 12,100, so 11k = 12,100. 4) Solve for k: k = 12,100 / 11 = 1,100. 5) Increase in fare = new fare minus old fare = 13k - 11k = 2k. 6) Therefore, increase = 2 * 1,100 = Rs 2,200. Verification / Alternative check: We can compute the new fare explicitly as 13k = 13 * 1,100 = Rs 14,300. Then the increase is 14,300 - 12,100 = 2,200. This matches the earlier calculation using the difference in ratio parts. Both methods agree, confirming that our understanding of the ratio 11 : 13 and the resulting increase is correct. Why Other Options Are Wrong: Option A (Rs 14,300) is the new fare, not the increase, and therefore answers the wrong quantity. Option C (Rs 22,000) and option E (Rs 1,100) are multiples of the base unit but do not represent the difference between the old and new fares. Option D (Rs 1,430) appears to be a miscalculation, possibly confusing the 13 in the ratio with a percentage. Only Rs 2,200 correctly represents the actual increase based on the ratio 11 : 13 with an original fare of Rs 12,100. Common Pitfalls: A common error is to treat the ratio 11 : 13 as direct fares and forget that the original fare must correspond to the 11-part value. Another pitfall is confusing the new fare with the increase, or incorrectly assuming that an 11 : 13 ratio describes a 13 over 11 percent increase. Always distinguish carefully between the absolute amounts (old and new) and the increment between them, and use the difference between the ratio parts (13 - 11) to represent the increase proportion. Final Answer: The increase in the flight fare is Rs 2,200.

Question 5

In a school of 432 pupils, the ratio of the number of boys to the number of girls is 5 : 4. After some new admissions, the number of boys increases by 12 and the ratio of boys to girls becomes 7 : 6. How many new girls were admitted?

Accepted Answer

Introduction / Context: This problem involves ratios and changes in a population after new admissions. You are given the initial ratio and total strength of students, then told how the ratio changes after some additional boys and girls are admitted. The task is to determine how many new girls joined. This blends ratio, basic algebra and understanding of how changes affect proportions. Given Data / Assumptions: Initial total number of pupils = 432. Initial ratio of boys : girls = 5 : 4. After some admissions, number of boys increases by 12. New ratio of boys : girls becomes 7 : 6. The number of new girls admitted is to be found. Concept / Approach: First, use the initial ratio and total strength to find the original numbers of boys and girls. Then incorporate the changes: boys increase by a fixed number, and girls increase by some unknown x. Apply the new ratio to form an equation in x. Solving this equation gives the number of new girls admitted. This is a typical approach whenever a ratio changes after some additions or subtractions. Step-by-Step Solution: 1) Let initial numbers be B boys and G girls with B : G = 5 : 4 and B + G = 432. 2) From the ratio, let B = 5k and G = 4k. 3) Then 5k + 4k = 432, so 9k = 432. 4) Solve for k: k = 432 / 9 = 48. 5) Therefore, B = 5 * 48 = 240 and G = 4 * 48 = 192 initially. 6) After admissions, the number of boys becomes B' = 240 + 12 = 252. 7) Let the number of new girls admitted be x. Then G' = 192 + x. 8) New ratio is B' : G' = 7 : 6, so 252 : (192 + x) = 7 : 6. 9) Cross-multiply: 252 * 6 = 7 * (192 + x). 10) This gives 1,512 = 1,344 + 7x. 11) So 7x = 1,512 - 1,344 = 168, hence x = 168 / 7 = 24. Verification / Alternative check: After admissions, the number of girls becomes 192 + 24 = 216. Now check the final ratio: B' : G' = 252 : 216. Divide both terms by 36 to simplify, we get 7 : 6, which matches the given new ratio. This confirms that taking 24 new girls is consistent with both the initial conditions and the final ratio after admissions. Why Other Options Are Wrong: If we try option A (12), the new number of girls would be 204 and the ratio 252 : 204 would simplify to 21 : 17, not 7 : 6. Similarly, option B (14), option D (20) and option E (18) lead to simplified ratios that do not match 7 : 6. Only when x = 24 does the ratio of boys to girls precisely become 7 : 6, so 24 is the only correct choice. Common Pitfalls: One common mistake is to apply the ratio to the increase instead of to the final numbers. Another common error is forgetting that the total number of pupils also changes, so using the old total of 432 after admissions is incorrect. Always recompute the new numbers and then apply the new ratio. Careful algebraic setup is the key to handling such multi-step ratio problems correctly. Final Answer: The number of new girls admitted to the school is 24.

Question 6

Sanjay’s marks in English and Hindi are in the ratio 7 : 11. If he scored 20 marks more in Hindi than in English, what are his marks in English?

Accepted Answer

Introduction / Context: This question illustrates how ratios can represent comparisons between two exam scores. By combining the ratio with an absolute difference, we form a simple linear equation. Such questions test the ability to convert word statements into algebraic expressions and solve them systematically, which is a core skill in quantitative aptitude. Given Data / Assumptions: Marks in English : marks in Hindi = 7 : 11. Marks in Hindi are 20 more than marks in English. We need to find Sanjay’s marks in English. Concept / Approach: Let us denote the common multiplying factor of the ratio by k, so that English marks and Hindi marks become 7k and 11k respectively. The problem states that Hindi marks exceed English marks by 20. This gives an equation involving k. Solving that equation provides the value of k, from which we can find the individual marks in each subject. Step-by-Step Solution: 1) Let Sanjay’s marks in English be 7k and in Hindi be 11k. 2) According to the question, Hindi marks are 20 higher than English marks. 3) Therefore, 11k - 7k = 20. 4) Simplify the left side: 4k = 20. 5) Solve for k: k = 20 / 4 = 5. 6) English marks = 7k = 7 * 5 = 35. 7) Hindi marks = 11k = 11 * 5 = 55. Verification / Alternative check: We verify both the ratio and the difference condition. First, check the ratio: 35 : 55 simplifies by dividing both numbers by 5 to 7 : 11, which matches the given ratio. Second, check the difference: Hindi minus English = 55 - 35 = 20, which also matches the given extra marks in Hindi. Since both conditions are satisfied, our solution is consistent and correct. Why Other Options Are Wrong: Option B (55) is the Hindi score, not the English score. Options C (45), D (65) and E (40) do not yield the ratio 7 : 11 with a difference of 20 when paired with some corresponding Hindi marks. For example, if English were 45, then Hindi would be 65 to maintain a difference of 20, but 45 : 65 does not reduce to 7 : 11. Therefore, only 35 fits both the ratio and the difference conditions. Common Pitfalls: A common mistake is to confuse the direction of the difference and write English minus Hindi equals 20, or to add 20 to the wrong subject. Another typical error is trying to guess numbers without using the ratio method, which may lead to inconsistent pairs that do not satisfy both conditions. Always express the ratio using a variable multiplier, set up the difference equation carefully, and solve step by step. Final Answer: Sanjay’s marks in English are 35.

Question 7

What is the third proportional to the numbers 10 and 20?

Accepted Answer

Introduction / Context: This question is about the concept of third proportional between two numbers, which is a common topic in ratio and proportion. Understanding proportionality is important for solving many geometry, algebra and word problems. The question asks for a single number that completes a specific proportional relationship with 10 and 20. Given Data / Assumptions: First number = 10. Second number = 20. We need to find the third proportional to 10 and 20. Concept / Approach: 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. From a : b = b : c, we can write a / b = b / c and then cross-multiply to get a * c = b^2. Therefore, c = b^2 / a. This formula can be applied directly to find the third proportional when the first two numbers are known. Step-by-Step Solution: 1) Let a = 10 and b = 20. 2) We want c such that 10 : 20 = 20 : c. 3) Write the proportion in fraction form: 10 / 20 = 20 / c. 4) Simplify 10 / 20 to 1 / 2. 5) So 1 / 2 = 20 / c. 6) Cross-multiply to get c = 20 * 2 = 40. 7) Therefore, the third proportional to 10 and 20 is 40. Verification / Alternative check: We can verify by checking the original proportional relation. If a = 10, b = 20 and c = 40, then 10 : 20 = 1 : 2 and 20 : 40 = 1 : 2 as well. Since both ratios are equal, 10 : 20 = 20 : 40, confirming that 40 is indeed the correct third proportional. Another way is to use the formula c = b^2 / a = 20^2 / 10 = 400 / 10 = 40, which gives the same result. Why Other Options Are Wrong: Option A (30), option B (25), option C (50) and option E (20) do not satisfy the condition 10 : 20 = 20 : c. For example, with c = 50, 20 : 50 simplifies to 2 : 5, which is not the same as 10 : 20 = 1 : 2. Similarly, none of the other proposed values make the two ratios identical. Only c = 40 ensures equality of the two ratios, as required for a third proportional. Common Pitfalls: Learners often confuse the definition of third proportional with mean proportional (geometric mean) or with the idea of a simple arithmetic progression. Some also mistakenly use c = a * b instead of c = b^2 / a. Carefully remembering that in a : b = b : c, the middle term b appears in both ratios is crucial to setting up and solving the correct equation for the third proportional. Final Answer: The third proportional to 10 and 20 is 40.

Question 8

If 2A = 3B = 8C, what is the simplest ratio A : B : C?

Accepted Answer

Introduction / Context: This question is another example of solving compound proportional relationships, similar to earlier problems but with a different numeric structure. You are given that 2A, 3B and 8C are all equal to one common value. The task is to express A : B : C as a simple integer ratio. Such problems strengthen your ability to translate equalities into ratios and to handle denominators correctly. Given Data / Assumptions: We are told 2A = 3B = 8C. A, B and C are all assumed to be positive real numbers. We want to find A : B : C in simplest integer form. Concept / Approach: Introduce a common constant k such that 2A = 3B = 8C = k. Then express each variable in terms of k. From there, A, B and C are written as fractions involving k. To convert these fractional expressions into an integer ratio, multiply through by the least common multiple of the denominators. Finally, simplify the resulting numbers to get the simplest form of A : B : C. Step-by-Step Solution: 1) Let 2A = 3B = 8C = k for some positive constant k. 2) From 2A = k, A = k / 2. 3) From 3B = k, B = k / 3. 4) From 8C = k, C = k / 8. 5) So A : B : C = (k / 2) : (k / 3) : (k / 8). 6) Cancel k from all terms, giving 1 / 2 : 1 / 3 : 1 / 8. 7) To remove denominators, multiply each term by the LCM of 2, 3 and 8, which is 24. 8) This gives (1 / 2) * 24 : (1 / 3) * 24 : (1 / 8) * 24 = 12 : 8 : 3. 9) Therefore, the simplest ratio A : B : C is 12 : 8 : 3. Verification / Alternative check: Select k = 24 as a convenient value. Then A = 24 / 2 = 12, B = 24 / 3 = 8 and C = 24 / 8 = 3. Check the original condition: 2A = 2 * 12 = 24, 3B = 3 * 8 = 24 and 8C = 8 * 3 = 24. All three expressions equal 24, which confirms that 2A = 3B = 8C. Therefore 12 : 8 : 3 is consistent with the given equation and is the correct ratio. Why Other Options Are Wrong: Option A (8 : 3 : 2) and option B (8 : 4 : 3) do not satisfy the relationship 2A = 3B = 8C when you test them with any common multiple. Option C (2 : 3 : 8) reverses the structure and clearly does not meet the required equality. Option E (3 : 8 : 12) is simply the reverse of the correct order and again fails to meet the original condition when substituted. Only 12 : 8 : 3 produces equal values for 2A, 3B and 8C. Common Pitfalls: Typical mistakes include directly writing A : B : C as 2 : 3 : 8 or 1 : 2 : 4 without carefully forming the equations. Another common error is forgetting to use the least common multiple when clearing denominators, which can lead to incorrect or non-simplified ratios. Always express each variable in terms of a common constant, cancel that constant and then scale up to remove fractions. Final Answer: The simplest ratio A : B : C that satisfies 2A = 3B = 8C is 12 : 8 : 3.

Question 9

The ratio of selected candidates to unselected candidates in a recruitment is 14 : 25. If 35 fewer candidates had applied and 10 fewer were selected, the ratio of selected to unselected would have been 3 : 5. What was the total number of candidates …

Accepted Answer

Introduction / Context: This is a classic ratio and algebra problem in the context of recruitment. You are given an initial ratio of selected to unselected candidates, and a hypothetical scenario that changes both the total applicants and the number selected. From this changed scenario you must work backwards to find the original total number of applicants. This type of problem appears frequently in aptitude tests, especially in government and banking exams. Given Data / Assumptions: Initial ratio of selected candidates (S) to unselected candidates (U) = 14 : 25. If 35 fewer candidates had applied, and 10 fewer were selected, the new ratio S : U would be 3 : 5. The total number of candidates is S + U. We assume all numbers are positive integers. Concept / Approach: First, express the initial numbers of selected and unselected candidates in terms of a single variable using the ratio 14 : 25. Then represent the new scenario: the total applicants decrease by 35, selected candidates decrease by 10, and therefore unselected candidates change accordingly. Use the new ratio 3 : 5 to form a second equation. Solving this equation yields the scaling factor and the original numbers of selected and unselected candidates, from which we compute the total applicants. Step-by-Step Solution: 1) Let initial selected candidates be S and unselected candidates be U with S : U = 14 : 25. 2) So S = 14k and U = 25k for some positive integer k. 3) Total initial applicants = S + U = 14k + 25k = 39k. 4) In the hypothetical scenario, 35 fewer candidates apply, so new total = 39k - 35. 5) Also, 10 fewer candidates are selected, so new selected = S - 10 = 14k - 10. 6) New unselected = (new total) - (new selected) = (39k - 35) - (14k - 10) = 25k - 25. 7) The new ratio of selected to unselected is given as 3 : 5, so (14k - 10) : (25k - 25) = 3 : 5. 8) Write as a fraction: (14k - 10) / (25k - 25) = 3 / 5. 9) Cross-multiply: 5(14k - 10) = 3(25k - 25). 10) Expand: 70k - 50 = 75k - 75. 11) Rearrange: 75k - 70k = -50 + 75, so 5k = 25, hence k = 5. 12) Therefore S = 14k = 70 and U = 25k = 125. 13) Total number of candidates who applied = S + U = 70 + 125 = 195. Verification / Alternative check: Check the hypothetical scenario with these values. New total applicants = 195 - 35 = 160. New selected = 70 - 10 = 60. New unselected = 160 - 60 = 100. The new ratio is 60 : 100, which simplifies to 3 : 5 when divided by 20. This perfectly matches the condition given in the hypothetical scenario, confirming that the original total of 195 applicants is correct. Why Other Options Are Wrong: Other totals like 205, 185, 175 or 165 do not satisfy both the original and the hypothetical ratio conditions when you try to split them into selected and unselected candidates with the given changes. Only 195 allows the numbers S and U to fit the ratio 14 : 25, while still satisfying the new 3 : 5 ratio after reducing the number of applicants and selected candidates as specified. Common Pitfalls: A usual mistake is to incorrectly express the new unselected candidates or to assume that the decrease of 35 applicants affects only unselected candidates. Another pitfall is to form the wrong ratio equation, such as mixing up the order of selected and unselected. Always define variables clearly, write down expressions for both scenarios and then apply the ratios carefully to avoid such errors. Final Answer: The total number of candidates who had applied for the recruitment is 195.

Question 10

What is the fourth proportional to the numbers 6, 24 and 83?

Accepted Answer

Introduction / Context: This question tests the concept of fourth proportional in a proportion relation. Proportional numbers are a common theme in quantitative aptitude, and knowing how to find the third or fourth proportional quickly is a useful exam skill. Here, you are given three numbers and asked to find the fourth proportional that completes the proportion based on the standard definition. Given Data / Assumptions: First number a = 6. Second number b = 24. Third number c = 83. We are required to find the fourth proportional d. Concept / Approach: If a, b, c and d are in proportion, we write a : b = c : d. In such a case, d is called the fourth proportional to a, b and c. From a : b = c : d, we get a / b = c / d. Cross-multiplying gives a * d = b * c, so d = (b * c) / a. We can apply this formula directly to the given numbers once we identify a, b and c correctly. Step-by-Step Solution: 1) We are given a = 6, b = 24 and c = 83. 2) For the fourth proportional d, we use a : b = c : d. 3) Therefore, 6 : 24 = 83 : d. 4) Write this in fraction form: 6 / 24 = 83 / d. 5) Simplify 6 / 24 to 1 / 4. 6) So 1 / 4 = 83 / d. 7) Cross-multiply to obtain d = 83 * 4. 8) Compute d = 332. 9) Therefore the fourth proportional to 6, 24 and 83 is 332. Verification / Alternative check: We verify by checking whether 6 : 24 equals 83 : 332. The ratio 6 : 24 simplifies to 1 : 4. The ratio 83 : 332 simplifies by dividing both terms by 83, giving 1 : 4. Since both ratios are identical, they confirm that 6 : 24 = 83 : 332, and therefore 332 is indeed the correct fourth proportional. Why Other Options Are Wrong: Option A (249) would give 83 : 249 which does not simplify to 1 : 4. Option C (166) gives 83 : 166 which is 1 : 2, not 1 : 4. Option D (498) would make 83 : 498 approximately 1 : 6, again incorrect. Option E (124.5) leads to a non-integer ratio. Only 332 maintains the correct proportional relationship with the given numbers. Common Pitfalls: Students sometimes mistakenly set up the proportion as 6 : 24 = d : 83, which would lead to the wrong formula. Another common issue is forgetting to simplify the initial ratio 6 : 24 and thus making the arithmetic unnecessarily complicated. Always carefully set the proportion as a : b = c : d when finding the fourth proportional and then cross-multiply correctly. Final Answer: The fourth proportional to 6, 24 and 83 is 332.

Question 11

According to a will, wealth of Rs 11,50,000 is to be divided between a son and a daughter in the ratio 2/3 : 5/4. What is the son’s share in rupees (in lakhs)?

Accepted Answer

Introduction / Context: This question involves division of an amount using a ratio that itself is given in fractional form. Ratios with fractions can appear confusing at first, but they can be converted to simple whole number ratios with an appropriate method. The setting of a will dividing wealth between a son and a daughter is common in aptitude questions and tests your ability to handle fractional ratios and large sums together. Given Data / Assumptions: Total wealth to be divided = Rs 11,50,000. Ratio of son : daughter = 2/3 : 5/4. We need the son’s share, reported in lakhs of rupees. Concept / Approach: First, convert the fractional ratio 2/3 : 5/4 into a whole number ratio by eliminating denominators. This is done by multiplying both fractions by the least common multiple of their denominators. Once we obtain an integer ratio, we can treat it as the usual a : b sharing ratio, where the total is divided in proportion to the sum of the parts. Then the son’s share is calculated as son’s parts over total parts, multiplied by the total wealth. Step-by-Step Solution: 1) Given ratio of son : daughter = 2/3 : 5/4. 2) Denominators are 3 and 4; their LCM is 12. 3) Multiply each term by 12 to clear denominators. 4) Son’s part = (2/3) * 12 = 8. 5) Daughter’s part = (5/4) * 12 = 15. 6) So the equivalent whole number ratio is son : daughter = 8 : 15. 7) Total number of parts = 8 + 15 = 23. 8) Son’s share = (8 / 23) * 11,50,000 rupees. 9) Compute 11,50,000 * 8 = 92,00,000. 10) Divide by 23: 92,00,000 / 23 = 4,00,000 rupees. 11) Convert to lakhs: 4,00,000 rupees = 4 lakhs. Verification / Alternative check: We can also calculate the daughter’s share to confirm the total. Daughter’s share = (15 / 23) * 11,50,000 = (15 * 11,50,000) / 23. This equals 11,50,000 - 4,00,000 = 7,50,000 rupees by complementary calculation. Sum of shares = 4,00,000 + 7,50,000 = 11,50,000 rupees, exactly the given total wealth. The ratio of 4,00,000 : 7,50,000 simplifies by dividing by 50,000 to 8 : 15, which matches the converted ratio, confirming that our calculations are correct. Why Other Options Are Wrong: Options A (5), B (6) and C (7) lakhs represent larger shares than what correctly arises from the ratio 8 : 15 when applied to Rs 11,50,000. Option E (3.5 lakhs) is smaller than the correct value and would not yield the given ratio when paired with the daughter’s corresponding share. Only 4 lakhs for the son produces the correct fraction of the total wealth and maintains the converted ratio. Common Pitfalls: A very common mistake is to mis-handle the fractional ratio and treat 2/3 and 5/4 directly as integers, or to simply add the fractions without converting them into an integer ratio. Another error is miscalculating the LCM of the denominators or incorrectly multiplying the fractions. To avoid these issues, always clear the denominators first and convert all fractional ratios into simple integer parts before distributing the total amount. Final Answer: The son’s share of the wealth is 4 lakhs (Rs 4,00,000).

Question 12

If 5A = 13B = 7C, what is the simplest ratio A : B : C?

Accepted Answer

Introduction / Context: This question is yet another variant of equal expressions involving three variables, where multiple scaled versions of the variables are equal. This time the coefficients are 5, 13 and 7, which are all different primes. The aim is to determine the ratio A : B : C. These problems help strengthen your algebraic manipulation skills and understanding of proportional relationships. Given Data / Assumptions: We have the equality 5A = 13B = 7C. Variables A, B and C are positive real numbers. We need to find A : B : C in simplest integer form. Concept / Approach: Introduce a common equal value k so that 5A = 13B = 7C = k. Then express each variable in terms of this common value. The resulting expressions will involve fractions with denominators 5, 13 and 7. To convert these into whole numbers in a ratio, use the least common multiple of those denominators. This standard method ensures that A, B and C are expressed with a common scaling, giving an integer ratio. Step-by-Step Solution: 1) Let 5A = 13B = 7C = k. 2) From 5A = k, A = k / 5. 3) From 13B = k, B = k / 13. 4) From 7C = k, C = k / 7. 5) So A : B : C = (k / 5) : (k / 13) : (k / 7). 6) Cancel k from each term to obtain 1 / 5 : 1 / 13 : 1 / 7. 7) The denominators are 5, 13 and 7, whose LCM is 5 * 13 * 7 = 455. 8) Multiply each term by 455 to clear denominators. 9) A term: (1 / 5) * 455 = 91. 10) B term: (1 / 13) * 455 = 35. 11) C term: (1 / 7) * 455 = 65. 12) Thus, A : B : C = 91 : 35 : 65. Verification / Alternative check: Choose k = 455 as a convenient common value. Then A = 455 / 5 = 91, B = 455 / 13 = 35 and C = 455 / 7 = 65. Check the original equation: 5A = 5 * 91 = 455, 13B = 13 * 35 = 455 and 7C = 7 * 65 = 455. All three are equal, confirming that 5A = 13B = 7C. This verifies our ratio 91 : 35 : 65 is correct and consistent with the given condition. Why Other Options Are Wrong: Option B (65 : 35 : 91) and option C (35 : 91 : 65) are permutations of the numbers that do not match the positions of A, B and C as per the given equality. Option D (7 : 13 : 5) and option E (13 : 5 : 7) incorrectly treat the coefficients in the equality as direct ratio terms, which is not valid. Only 91 : 35 : 65 gives values of A, B and C that satisfy 5A = 13B = 7C. Common Pitfalls: A frequent mistake is to wrongly assume that A : B : C is simply 5 : 13 : 7 or its inverse, without doing the full conversion process. Another error is miscomputing the least common multiple of 5, 13 and 7, which are all prime and therefore multiply directly. Always carefully express each variable in terms of the common constant and then clear denominators properly to get the correct ratio. Final Answer: The simplest ratio A : B : C that satisfies 5A = 13B = 7C is 91 : 35 : 65.

Question 13

In what ratio does the y-axis divide the line segment joining the points (2, 5) and (6, 10)? (Consider the ratio in which it divides the segment externally.)

Accepted Answer

Introduction / Context: This question tests understanding of coordinate geometry and section formula concepts. You are asked to find the ratio in which the y-axis divides the line joining two points in the plane. Since both given points lie on the right side of the y-axis, the division is external, and the question explicitly asks you to consider this external division. This type of problem commonly appears in geometry sections of aptitude exams. Given Data / Assumptions: Point A has coordinates (2, 5). Point B has coordinates (6, 10). The y-axis is the line x = 0. We are to find the ratio in which x = 0 divides line AB, considering external division. Concept / Approach: For external division, if a point P divides the line segment joining A(x1, y1) and B(x2, y2) externally in the ratio m : n, the x-coordinate of P is given by x = (m x2 - n x1) / (m - n). Since the dividing line is the y-axis (x = 0), we set this x-coordinate to zero and use the relation to determine m : n. For a vertical line x = 0, this leads to a simple relationship between m, n and the x-coordinates of A and B. Step-by-Step Solution: 1) Let A(2, 5) and B(6, 10). 2) Suppose the y-axis (x = 0) divides the line joining A and B externally at point P in the ratio m : n, where AP : PB = m : n. 3) For external division, the x-coordinate of P is given by x = (m x2 - n x1) / (m - n). 4) Here x1 = 2 and x2 = 6. 5) Since P lies on the y-axis, its x-coordinate is 0, so (m * 6 - n * 2) / (m - n) = 0. 6) The numerator must be zero, so 6m - 2n = 0. 7) Rearranging gives 6m = 2n, or n = 3m. 8) Therefore the ratio m : n = m : 3m = 1 : 3. Verification / Alternative check: We can interpret this as the y-axis dividing the line externally so that A is one part away from P and B is three parts away on the line extended. The ratio of distances |AP| : |PB| is therefore 1 : 3, which is what we obtained algebraically. Although we do not need to compute the exact coordinates of P here, the ratio 1 : 3 is consistent with the requirement that P lies on x = 0 when using the external section formula. Why Other Options Are Wrong: Option A (3 : 1) is simply the reverse of the correct ratio and would correspond to n : m rather than m : n. Options C (2 : 5), D (5 : 2) and E (4 : 1) do not satisfy the equation 6m = 2n, so they cannot yield an x-coordinate of zero when substituted into the external section formula. Only 1 : 3 makes the numerator 6m - 2n equal to zero for some positive m, n and thus correctly places the division point on the y-axis. Common Pitfalls: A common mistake is to apply the internal division formula even when the division is external, which leads to incorrect equations. Another pitfall is to forget that for the line x = 0, the relevant condition is on the x-coordinate of the dividing point. Some students also confuse the order of the ratio (m : n versus n : m), so it is important to stay consistent with the formula used. Carefully identify whether the division is internal or external and substitute coordinates correctly. Final Answer: The y-axis divides the line joining (2, 5) and (6, 10) externally in the ratio 1 : 3.

Question 14

The bus fare between two cities is increased in the ratio 1 : 2. If the original fare is Rs 175, what is the increase in the fare in rupees?

Accepted Answer

Introduction / Context: This is a straightforward ratio-based fare increase question. Instead of specifying a percentage, the problem tells you that the bus fare increases in the ratio 1 : 2. With the original fare known, you must determine how much the fare has increased in rupee terms. Such problems reinforce the interpretation of ratios as comparisons between old and new values. Given Data / Assumptions: Old fare between two cities = Rs 175. Ratio of old fare to new fare = 1 : 2. The fare is increased, so the new fare is higher than the old fare. We need the increase in rupees, not simply the new fare. Concept / Approach: If old : new = 1 : 2, it means that the new fare is twice the old fare. The increase in fare is the difference between the new and old values. Therefore, after computing the new fare from the ratio, we subtract the old fare to find the increase. This is a direct application of interpreting ratios as multiplicative comparisons. Step-by-Step Solution: 1) Let the common unit part of the fare be k. 2) Old fare = 1k and new fare = 2k. 3) Given that old fare = Rs 175, so 1k = 175, hence k = 175. 4) New fare = 2k = 2 * 175 = Rs 350. 5) Increase in fare = new fare - old fare = 350 - 175. 6) Therefore, increase in fare = Rs 175. Verification / Alternative check: We can interpret the ratio 1 : 2 as indicating the new fare is 100 percent more than the old fare. Since 100 percent of 175 is again 175, the increase should be Rs 175. Adding this to the original fare 175 gives 350, which matches our earlier calculation of the new fare. Both the ratio interpretation and the direct multiplication agree on the same increase value. Why Other Options Are Wrong: Option A (Rs 350) is the new fare itself, not the increase. Option B (Rs 70) would correspond to an increase of only 40 percent, which does not match the ratio 1 : 2. Option C (Rs 140) is also inconsistent, as that would give a new fare of 315, not in the ratio 1 : 2 with 175. Option E (Rs 90) is a random value that does not support the given ratio. Only Rs 175 correctly represents the increase implied by the ratio 1 : 2 applied to an original fare of 175. Common Pitfalls: One common error is confusing the ratio 1 : 2 with a 2 percent or small percentage increase, or misinterpreting the ratio as 1 part extra instead of the entire new value being 2 parts. Learners may also forget to subtract the old fare to find the increase and instead report the new fare directly. Always separate the steps of finding the new value from finding the increase or decrease, and interpret ratios clearly. Final Answer: The increase in bus fare is Rs 175.

Question 15

By increasing the price of an entry ticket to a fair in the ratio 17 : 19, the number of visitors decreases in the ratio 11 : 10. In what ratio has the total collection changed (old collection to new collection)?

Accepted Answer

Introduction / Context: This question combines the ideas of ratio, price change and demand change to find the effect on total revenue or collection. Ticket price and visitor count are both changed in given ratios, and we must determine how the total collection changes. Such problems are important for understanding the combined effect of price and quantity changes in simple economic scenarios. Given Data / Assumptions: Price of entry ticket increases in the ratio 17 : 19. Number of visitors decreases in the ratio 11 : 10. Total collection is defined as ticket price multiplied by number of visitors. We are to find the ratio of old collection to new collection. Concept / Approach: Let the original price be P and the original number of visitors be N. The new price then becomes (19/17) * P, and the new number of visitors becomes (10/11) * N. Total collection originally is P * N. New collection is the product of new price and new visitors. The ratio of new collection to old collection is obtained by multiplying the price ratio and visitor ratio. Then we interpret this ratio and compare to the options which describe whether the collection has increased or decreased and by what ratio. Step-by-Step Solution: 1) Let original ticket price be P and original number of visitors be N. 2) New price = (19/17) * P because price increases in the ratio 17 : 19. 3) New visitors = (10/11) * N because visitors decrease in the ratio 11 : 10. 4) Original collection = P * N. 5) New collection = (19/17 * P) * (10/11 * N) = (19 * 10) / (17 * 11) * P * N. 6) Simplify the factor: (19 * 10) / (17 * 11) = 190 / 187. 7) So new collection : old collection = 190 : 187. 8) This means new collection is slightly larger than old collection, since 190 > 187. 9) Therefore old collection : new collection = 187 : 190. Verification / Alternative check: We can assume convenient numbers to verify. Let original price P = Rs 17 and original visitors N = 11, so original collection = 17 * 11 = 187. New price becomes Rs 19 and new visitors become 10, so new collection = 19 * 10 = 190. The ratio of old to new collection is 187 : 190, consistent with our calculation. This confirms that the total collection has increased and not decreased. Why Other Options Are Wrong: Option A states that the collection decreased in the ratio 190 : 187, which is incorrect both in direction (it is actually an increase) and ratio order. Options C and D use ratios 170 : 209 or 209 : 170, which do not match the factor 190 : 187 obtained from multiplying the price and visitor ratios. Option E assumes no change in collection, which is also wrong because the product factor is not equal to 1. Only option B correctly describes the change as an increase with old collection : new collection = 187 : 190. Common Pitfalls: Learners often invert ratios or confuse which ratio is old : new versus new : old. Another common mistake is to subtract the ratios instead of multiplying them, or to treat ratios as simple percentage changes without performing proper multiplication. Always express both price and quantity changes as multiplicative factors, multiply them to get the revenue factor, and then convert that into a ratio in the correct order required by the question. Final Answer: The total collection has increased in the ratio 187 : 190 (old collection : new collection).

Question 16

Lokesh starts a business with Rs 2,40,000. After 3 months, Vishal joins with Rs 2,10,000. At the end of the year, in what ratio should their profits be shared?

Accepted Answer

Introduction / Context: This is a standard partnership problem from profit and loss and ratio topics. Profits in a partnership business are shared in the ratio of the products of capital and time for which the capital is invested. Here, Lokesh and Vishal invest different amounts and also invest for different durations. You must combine both factors to determine the correct profit-sharing ratio at the end of one year. Given Data / Assumptions: Lokesh invests Rs 2,40,000 and stays invested for 12 months. Vishal invests Rs 2,10,000 and joins after 3 months, so he stays invested for 9 months. Total business duration considered = 12 months. Profits are shared in the ratio of capital multiplied by time. Concept / Approach: In a partnership where profit is shared according to investment and duration, each partner’s share of profit is proportional to capital * time. For example, if one partner invests more money or invests for a longer period, their profit share increases accordingly. To find the ratio of their profits, calculate the product of capital and time for each partner and express these products as a simplified ratio. Step-by-Step Solution: 1) Lokesh’s capital = Rs 2,40,000; time invested = 12 months. 2) Vishal’s capital = Rs 2,10,000; he joins after 3 months and thus invests for 9 months. 3) Lokesh’s capital-time product = 2,40,000 * 12. 4) Vishal’s capital-time product = 2,10,000 * 9. 5) Cancel common zeros to simplify: consider 240 * 12 and 210 * 9. 6) Compute 240 * 12 = 2,880. 7) Compute 210 * 9 = 1,890. 8) So initial profit ratio = 2,880 : 1,890. 9) Divide both by 30 to simplify: (2,880 / 30) : (1,890 / 30) = 96 : 63. 10) Further divide both by 3: 96 : 63 becomes 32 : 21. 11) Therefore, Lokesh and Vishal should share profits in the ratio 32 : 21. Verification / Alternative check: We can express the ratio directly in terms of capital and relative time without full multiplication. Lokesh’s effective investment factor is 2,40,000 * 12. Vishal’s effective factor is 2,10,000 * 9. Divide both capital-time products by 30,000 to reduce mental arithmetic: Lokesh gets (2,40,000 * 12) / 30,000 = 8 * 12 = 96; Vishal gets (2,10,000 * 9) / 30,000 = 7 * 9 = 63. Ratio 96 : 63 simplifies to 32 : 21, confirming the same result with less computation. Why Other Options Are Wrong: Option B (16 : 13) and option C (8 : 5) underestimate Lokesh’s advantage in either capital or time. Option D (45 : 39) and option E (4 : 3) do not match the correct capital-time products when simplified. It is only with the ratio 32 : 21 that the relative contributions of both capital and time for Lokesh and Vishal are reflected correctly in the profit share. Common Pitfalls: A common mistake is to ignore the time factor and share profits simply in the ratio of the invested amounts 2,40,000 : 2,10,000. Another error is to apply the time adjustment incorrectly, such as using 3 months for Vishal instead of the 9 months he is actually invested. Always carefully read when each partner joins and multiply the capital by the exact period for which it remains in the business. Final Answer: Lokesh and Vishal should share the business profit in the ratio 32 : 21.

Question 17

In the analogy 8 : 27 :: 64 : ?, which number should replace the question mark?

Accepted Answer

Introduction / Context: This question is from the number analogy type, where you must identify the pattern linking the first pair of numbers and then apply the same pattern to find the missing number in the second pair. Here, 8 is related to 27 and 64 is related to an unknown number. Many such patterns involve squares, cubes or simple arithmetic operations, and recognizing powers is a key skill. Given Data / Assumptions: First pair: 8 and 27. Second pair: 64 and ? The relationship between the first two numbers should be applied consistently to the second pair. Concept / Approach: Observe that 8 = 2^3 and 27 = 3^3, so both numbers are consecutive cubes. This suggests that the pattern may involve cubes of consecutive integers. Check 64 to see whether it also fits a cube pattern. Indeed, 64 = 4^3. Therefore we expect the number paired with 64 to be the next cube, namely 5^3. This reasoning follows a simple power sequence of natural numbers. Step-by-Step Solution: 1) Identify 8 in the first pair and express it as a power: 8 = 2^3. 2) Identify 27 in the first pair and express it as a power: 27 = 3^3. 3) Notice that the bases 2 and 3 are consecutive integers, and both numbers are cubes. 4) Now look at the second pair: 64 and ?. 5) Express 64 as a cube: 64 = 4^3. 6) Following the same pattern, the number paired with 64 should be the cube of the next integer after 4, that is 5^3. 7) Compute 5^3 = 5 * 5 * 5 = 125. 8) Therefore, the missing number in the analogy is 125. Verification / Alternative check: Another way to verify is to think in terms of pairs of cubes of consecutive numbers. The first pair (8, 27) is (2^3, 3^3). The next pair in the same sequence would be (3^3, 4^3) = (27, 64). Shifting the view slightly, the problem has (8, 27) and (64, ?), which can be interpreted as the pattern progressing in the base numbers. To keep the progression consistent from base 2 to base 5, after 4^3 the next term must be 5^3 = 125. This supports our earlier conclusion. Why Other Options Are Wrong: Options A (190), B (144), D (72) and E (216) do not fit the simple cube pattern of consecutive integers. For example, 144 is 12^2 and 216 is 6^3, neither of which align with a cube of a number that makes the sequence 2^3, 3^3, 4^3, 5^3. Only 125 matches 5^3 and preserves the clear relationship between the numbers in both pairs. Common Pitfalls: Some learners may search for complicated arithmetic operations such as adding or multiplying constants instead of checking simple powers. Others might focus on differences like 27 - 8 = 19 and try to apply a similar difference to 64, which does not produce an integer from the options. Always first test common patterns like squares and cubes in number analogy questions, because they are frequently used and easy to verify. Final Answer: The number that should replace the question mark is 125.

Question 18

A company reduces the number of its employees in the ratio 14 : 12 and increases their individual wages in the ratio 16 : 18. Determine whether the total wage bill increases or decreases, and in what ratio the new wage bill compares to the original …

Accepted Answer

Introduction / Context: In this aptitude question on ratio and proportion, we analyse how simultaneous changes in the number of employees and their individual wages affect the total wage bill of a company. The core idea is that the overall wage bill equals the number of employees multiplied by the wage per employee, so any percentage or ratio change in these components multiplies together. Given Data / Assumptions: The number of employees is reduced in the ratio 14 : 12 (original : new). The individual wages are increased in the ratio 16 : 18 (original : new). We assume the original total wage bill is some convenient value, for example 1 unit, to compare old and new situations. We need to decide whether the wage bill increases or decreases and find the ratio new : old. Concept / Approach: The total wage bill is proportional to (number of employees) * (wage per employee). When data is given in ratios, we treat them as multiplicative factors. If the number of employees changes from 14 to 12, the factor is 12/14. If wages change from 16 to 18, the factor is 18/16. The combined effect is the product of these two factors. If the product is less than 1, the wage bill decreases; if greater than 1, it increases. Step-by-Step Solution: Let original number of employees be 14 units and original wage per employee be 16 units. Then original total wage bill = 14 * 16. New number of employees = 12 units (from the ratio 14 : 12). New wage per employee = 18 units (from the ratio 16 : 18). New total wage bill = 12 * 18. Ratio of new wage bill to old wage bill = (12 * 18) / (14 * 16). Simplify: (12 * 18) = 216 and (14 * 16) = 224. So ratio = 216 / 224 = 27 / 28. Since 27 / 28 is less than 1, the wage bill has decreased. Verification / Alternative check: We can also think in percentage terms. The number of employees moves from 14 to 12, which is a factor of 12 / 14 = 6 / 7, approximately 0.857. The wages move from 16 to 18, which is 18 / 16 = 9 / 8, equal to 1.125. The combined factor is (6 / 7) * (9 / 8) = 54 / 56 = 27 / 28, which confirms the earlier calculation and again shows that the wage bill reduces slightly, because the reduction due to fewer employees dominates the increase due to higher wages. Why Other Options Are Wrong: “Increases, 28 : 27” and “Increases, 27 : 28” are incorrect because the product of the two changes is less than 1, not greater. “Decreases, 28 : 27” is wrong because it reverses the correct ratio; 28 : 27 would mean the new bill is larger than the old one. “No change in total wage bill” is incorrect because the combined factor is not equal to 1. Common Pitfalls: Students often mistakenly add or subtract percentage or ratio changes instead of multiplying them. Here, one might wrongly think that a reduction and an increase of similar size cancel out. Another error is to treat 14 : 12 and 16 : 18 as percentage changes directly, rather than as multiplicative ratios. Careful use of ratio multiplication avoids these mistakes. Final Answer: The total wage bill decreases, and the ratio of the new wage bill to the old wage bill is 27 : 28.

Question 19

In a garden there are a total of 361 saplings consisting only of mango tree saplings and neem tree saplings. The ratio of the number of mango saplings to the number of neem saplings is 11 : 8. How many mango tree saplings are there in the garden?

Accepted Answer

Introduction / Context: This problem is a straightforward application of ratios in a real life setting. We are given the total number of saplings in a garden and the ratio between two types of trees, mango and neem. The goal is to convert the ratio into actual numbers that add up to the total. Such questions are very common in aptitude tests because they check a candidate understanding of basic ratio and proportion ideas. Given Data / Assumptions: Total number of saplings (mango plus neem) = 361. Ratio of mango saplings to neem saplings = 11 : 8. Let mango saplings be represented by 11 parts and neem saplings by 8 parts. We assume that no other type of tree is present in the garden. Concept / Approach: If two quantities are in the ratio a : b, then they can be written as a * k and b * k, where k is a common multiplier. The sum of the parts a + b multiplied by k gives the total. Therefore, we find k by dividing the total quantity by the sum of the ratio parts. Once k is known, individual quantities are easily obtained by multiplication. This is a core property of ratios and proportions that appears in many competitive exams. Step-by-Step Solution: Let the number of mango saplings be 11k and the number of neem saplings be 8k. Total number of saplings = 11k + 8k = 19k. We are told that 19k = 361. So k = 361 / 19. Compute 361 / 19 = 19. Therefore, number of mango saplings = 11k = 11 * 19 = 209. Number of neem saplings = 8k = 8 * 19 = 152 (this also checks that 209 + 152 = 361). Verification / Alternative check: We can check the ratio directly. If there are 209 mango saplings and 152 neem saplings, their ratio is 209 : 152. Both numbers are divisible by 19, which gives 209 / 19 = 11 and 152 / 19 = 8. So the simplified ratio is 11 : 8, which matches the given condition. Also, the sum 209 + 152 equals 361, confirming that the total saplings count is maintained. Why Other Options Are Wrong: 171 would require the remaining 190 saplings to be neem, giving a ratio 171 : 190, which does not simplify to 11 : 8. 57 would imply 304 neem saplings, and 57 : 304 does not match the required ratio. 152 is the number of neem saplings, not mango, so it answers the wrong quantity. 190 also leads to a mismatch when compared with the ratio 11 : 8. Common Pitfalls: A frequent mistake is to try to guess numbers that add to 361 without respecting the ratio. Another error is to treat 11 and 8 as percentages and directly use them on the total, which is not correct. Always remember that ratio parts are relative units, not percentages. The correct method is to sum the ratio parts, divide the total by this sum to find the value of one part, and then multiply by individual ratio numbers. Final Answer: The garden contains 209 mango tree saplings.

Question 20

The incomes of persons C and D are in the ratio 3 : 2, and the incomes of D and E are in the ratio 5 : 4. One third of the income of C is Rs 4000 more than one half of the income of E. What is the monthly income of D in rupees?

Accepted Answer

Introduction / Context: This question tests combined use of ratios and linear equations in the context of monthly incomes. We are given two different ratios involving three people and an extra condition relating parts of two incomes. The task is to convert ratio information into algebraic expressions and then use the given difference condition to find the actual numerical values of the incomes, especially that of person D. Given Data / Assumptions: Income of C : income of D = 3 : 2. Income of D : income of E = 5 : 4. One third of income of C is Rs 4000 more than one half of income of E. All incomes are assumed to be positive real numbers, expressed in rupees per month. Concept / Approach: When ratios involve three related quantities, we often express them in terms of a common base variable. First we combine the two ratios into a single three way ratio C : D : E. Then we convert the statement about one third of C and one half of E into an equation. Solving this equation gives us the common multiplier for the ratio and thus the exact incomes of C, D and E. This approach uses proportional reasoning along with basic algebra. Step-by-Step Solution: From C : D = 3 : 2, let C = 3k and D = 2k. From D : E = 5 : 4, let D = 5m and E = 4m. Since D is the same person, 2k = 5m, so k = (5m) / 2. Then C = 3k = 3 * (5m / 2) = (15m) / 2 and E = 4m. Multiply all by 2 to clear the denominator: C : D : E = 15m : 10m : 8m, so the combined ratio is 15 : 10 : 8. Let C = 15u, D = 10u and E = 8u for some u. Given: one third of C is 4000 more than one half of E. So (1 / 3) * 15u = (1 / 2) * 8u + 4000. This gives 5u = 4u + 4000, so u = 4000. Therefore, D = 10u = 10 * 4000 = Rs 40000. Verification / Alternative check: Compute C and E to confirm the condition. C = 15u = 60000 and E = 8u = 32000. One third of C is 60000 / 3 = 20000. One half of E is 32000 / 2 = 16000. The difference is 20000 − 16000 = 4000, which matches the problem statement. This confirms that u = 4000 and therefore D = 40000 is consistent. Why Other Options Are Wrong: 43000, 50000, 60000 and 36000 do not satisfy the combined ratios 3 : 2 and 5 : 4 when checked against the given difference condition between one third of C and one half of E. Any other value for D would require a different u and then the numerical difference would not be exactly Rs 4000. Common Pitfalls: One common mistake is to treat the two given ratios independently without merging them into a single ratio for C, D and E. Another typical error is to misinterpret “one third of C is Rs 4000 more than one half of E” and reverse the direction of the inequality. Also, candidates sometimes directly assign numbers to C and E without checking consistency with both ratios, which leads to contradictions. Systematically forming and solving the equation prevents these errors. Final Answer: The monthly income of D is Rs 40000.

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