Question 1

Three partners A, B and C start a business together and invest their capitals in the ratio 8 : 5 : 3. At the end of the business period the ratio of their profits is 4 : 15 : 6. What is the ratio of the time periods for which A, B and C invested the…

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Introduction / Context: This problem involves partnership, where profit sharing depends on both the amount of capital invested and the duration of investment. The question gives the ratio of capitals and the ratio of final profits and asks us to determine the ratio of the time periods for which each partner kept their money in the business. This is a classic concept in partnership questions seen in aptitude exams. Given Data / Assumptions: Capital ratio of A : B : C = 8 : 5 : 3. Profit ratio of A : B : C = 4 : 15 : 6. Let the time periods of investment for A, B and C be t1, t2 and t3 respectively. We assume that profit is directly proportional to (capital * time). Concept / Approach: In partnership problems, the share of profit for each partner is proportional to the product of their capital and investment time. In symbolic form, Profit ∝ Capital * Time. Therefore, for each partner, Capital * Time is proportional to the given profit ratio values. We can set equations based on these proportionalities and solve for the relative times. After that we convert them to the simplest integer ratio form. Step-by-Step Solution: Let the time periods be t1, t2, t3 for A, B and C. Then 8 * t1 : 5 * t2 : 3 * t3 = 4 : 15 : 6. This means 8 * t1 / 4 = 5 * t2 / 15 = 3 * t3 / 6. Simplify each fraction: 8 * t1 / 4 = 2 * t1; 5 * t2 / 15 = t2 / 3; 3 * t3 / 6 = t3 / 2. So we have 2 * t1 = t2 / 3 = t3 / 2 = k, for some constant k. From 2 * t1 = k, we get t1 = k / 2. From t2 / 3 = k, we get t2 = 3k. From t3 / 2 = k, we get t3 = 2k. Therefore, t1 : t2 : t3 = (k / 2) : 3k : 2k. Multiply all terms by 2 to remove the denominator: t1 : t2 : t3 = 1 : 6 : 4. Verification / Alternative check: We can verify by forming the effective capital time products. If we assume k = 2 for convenience, then t1 = 1, t2 = 6 and t3 = 4. Capital products are: A → 8 * 1 = 8, B → 5 * 6 = 30, C → 3 * 4 = 12. The ratio 8 : 30 : 12 simplifies by dividing by 2 to 4 : 15 : 6, which exactly matches the given profit ratio. This confirms that the time ratio 1 : 6 : 4 is consistent. Why Other Options Are Wrong: Ratios like 1 : 6 : 5 or 2 : 3 : 4 do not give profit ratios equal to 4 : 15 : 6 when multiplied with the given capital ratio 8 : 5 : 3. The options 2 : 6 : 3 and 3 : 2 : 1 similarly produce different profit distributions and therefore do not satisfy the condition. Common Pitfalls: Students commonly confuse the relationship and try to directly equate the profit ratio with the capital ratio, ignoring time. Another frequent mistake is incorrect handling of proportion when solving 8 * t1 : 5 * t2 : 3 * t3 = 4 : 15 : 6. Writing and equating each part systematically and simplifying fractions carefully avoids errors. It is also important to simplify the final ratio completely to its lowest terms. Final Answer: The ratio of the time periods for which A, B and C invested their capital is 1 : 6 : 4.

Question 2

The cash difference between the selling price of an article when it is sold at a profit of 4% and when it is sold at a profit of 6% is Rs 3. What is the ratio of the two selling prices corresponding to these profit percentages?

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Introduction / Context: This question is about profits and selling prices. The cost price is not known directly, but we are given the profit percentages and the absolute difference between the two selling prices. Using the definition of profit percentage in terms of cost price and selling price, we can set up an equation for the unknown cost price, find the two selling prices, and then form their ratio. Given Data / Assumptions: Profit in first case = 4% of cost price. Profit in second case = 6% of cost price. Difference between the two selling prices = Rs 3. We assume the cost price is the same in both situations. Concept / Approach: Let the cost price be C. A profit of 4% means the selling price is C * (1 + 4 / 100) = 1.04 * C. A profit of 6% means the selling price is 1.06 * C. The difference between these two selling prices is 1.06 * C − 1.04 * C = 0.02 * C. This difference is given as Rs 3, so we can solve for C. Once we know C, we compute both selling prices and then simplify their ratio. Step-by-Step Solution: Let cost price = C. Selling price at 4% profit, S1 = 1.04 * C. Selling price at 6% profit, S2 = 1.06 * C. Given S2 − S1 = 3. So 1.06 * C − 1.04 * C = 3. This gives 0.02 * C = 3. Therefore C = 3 / 0.02 = 150. Then S1 = 1.04 * 150 = 156. S2 = 1.06 * 150 = 159. Required ratio S1 : S2 = 156 : 159. Divide both by 3 to simplify: 156 / 3 = 52 and 159 / 3 = 53. So the ratio is 52 : 53. Verification / Alternative check: To verify, compute the profits directly. At selling price 156 with cost price 150, profit is 6, which is 6 / 150 * 100 = 4%. At selling price 159, profit is 9, which is 9 / 150 * 100 = 6%. The difference 159 − 156 = 3 matches the given condition. This confirms that our value of C and the final ratio are correct. Why Other Options Are Wrong: 51 : 53, 55 : 53 and 54 : 53 would correspond to different numerical selling prices which do not yield a difference of exactly Rs 3 when combined with a common cost price. 25 : 26 is unrelated to the specific 4% and 6% profit structure in this problem and does not match the computed ratio from the actual selling prices. Common Pitfalls: A common mistake is to treat the 2% difference in profit as directly equal to Rs 3 and forget that this 2% is of the cost price, not of any arbitrary number. Some learners also incorrectly assume that the ratio of selling prices equals the ratio of percentage profits, which is not true because both profits are based on the same cost price. Correct handling of percentages and careful algebraic steps prevent these errors. Final Answer: The ratio of the two selling prices corresponding to 4% and 6% profit is 52 : 53.

Question 3

In triangle PQR, a straight line is drawn parallel to the base QR and it cuts side PQ at point X and side PR at point Y. If the ratio PX : XQ is 5 : 6, then what is the ratio of the length XY to the length QR?

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Introduction / Context: This geometry question is based on similar triangles created by a line drawn parallel to one side of a triangle. When a line is parallel to the base of a triangle and cuts the other two sides, the smaller triangle formed near the vertex is similar to the original triangle. This similarity allows us to relate corresponding side lengths using simple ratios. Given Data / Assumptions: Triangle PQR has base QR. A line parallel to QR meets side PQ at X and side PR at Y. The ratio PX : XQ is given as 5 : 6. We need to find the ratio XY : QR. All segments are straight and the triangle is non degenerate. Concept / Approach: If a line is drawn parallel to one side of a triangle and intersects the other two sides, then the smaller triangle formed is similar to the entire triangle by the basic proportionality theorem and similarity rules. The ratio of similarity is given by corresponding sides from the vertex to the cut point on each side. In this case, PX corresponds to PQ, and XY corresponds to QR. Therefore, XY / QR equals PX / PQ. We express PX / PQ in terms of the given ratio PX : XQ. Step-by-Step Solution: Given PX : XQ = 5 : 6. Let PX = 5k and XQ = 6k for some positive k. Then PQ = PX + XQ = 5k + 6k = 11k. The small triangle PXY is similar to triangle PQR, so the linear scale factor is PX / PQ. Thus, scale factor = PX / PQ = 5k / 11k = 5 / 11. Corresponding sides in similar triangles are proportional, so XY / QR = 5 / 11. Therefore, the ratio XY : QR = 5 : 11. Verification / Alternative check: We can check with assumed lengths. Suppose PQ = 11 units and PR is some length consistent with a general triangle. Take PX = 5 units and XQ = 6 units as per the ratio. Then triangle PXY is 5 / 11 scaled down compared to PQR. If QR is, say, 22 units, then XY should be 5 / 11 of 22, which is 10 units. The ratio XY : QR = 10 : 22 simplifies to 5 : 11, confirming the result. The specific length chosen for QR does not matter, only the proportionality does. Why Other Options Are Wrong: 6 : 5 and 11 : 5 invert or otherwise distort the correct similarity ratio and would imply that the inner segment XY is longer than the entire base QR, which is impossible. 11 : 6 and 4 : 11 also do not follow from the similarity factor PX / PQ = 5 / 11 and are therefore inconsistent with the geometry of the figure. Common Pitfalls: A major source of error is to misinterpret PX : XQ as PX : PQ or to treat PX and XQ as directly proportional to XY and QR. Another pitfall is forgetting that PQ is the sum PX + XQ. Some students also mistakenly apply area ratios instead of length ratios. Carefully identifying which sides correspond in similar triangles and working step by step with the given ratio avoids these mistakes. Final Answer: The required ratio of XY to QR is 5 : 11.

Question 4

The monthly salaries of Pia and Som are in the ratio 5 : 4. From her monthly salary, Pia gives 25% to her mother, 15% towards her sister tuition fees, and 18% towards repayment of a loan. She spends the remaining amount, which equals Rs 2100. What i…

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Introduction / Context: This question combines the ideas of ratios and percentage allocation of income. We know how Pia and Som compare in terms of salary, and we are told how Pia spends her salary in several percentage categories, leaving a fixed amount. We need to work backwards from the remaining amount to find Pia salary, and then use the salary ratio to obtain Som salary. This is a common style of problem in bank exams and other aptitude tests. Given Data / Assumptions: Ratio of salaries of Pia : Som = 5 : 4. Pia gives 25% of her salary to her mother. Pia spends 15% on her sister tuition fees. Pia spends 18% on repayment of a loan. The remaining part of her salary equals Rs 2100. We assume all percentages are of Pia gross monthly salary. Concept / Approach: Let Pia monthly salary be P. Then we convert each percentage into a fraction of P. The total of all percentage allocations plus the remaining amount must equal P. The remaining fraction of her salary, after the given percentages are deducted, multiplied by P should equal Rs 2100. From this, we can solve for P. Once we know P, we use the given ratio 5 : 4 to compute Som salary as (4 / 5) * P. Step-by-Step Solution: Let Pia monthly salary be P. She gives 25% to her mother, which is 0.25 * P. She spends 15% on sister fees, which is 0.15 * P. She spends 18% on a loan, which is 0.18 * P. Total percentage spent on these three heads = 25% + 15% + 18% = 58%. Therefore, remaining percentage of salary = 100% − 58% = 42%. So remaining amount = 0.42 * P. Given that this remaining amount is Rs 2100, we have 0.42 * P = 2100. Thus P = 2100 / 0.42 = 5000. So Pia salary is Rs 5000 per month. From the ratio 5 : 4, Som salary = (4 / 5) * 5000 = Rs 4000. Verification / Alternative check: We can verify by checking the actual distribution of Pia salary. Twenty five percent of 5000 is 1250, fifteen percent is 750 and eighteen percent is 900. The sum of these three is 1250 + 750 + 900 = 2900. The remainder is 5000 − 2900 = 2100, which matches the given condition. This confirms that P = 5000 is correct and therefore Som salary equal to Rs 4000 is also correct based on the 5 : 4 ratio. Why Other Options Are Wrong: If Som salary were 3500, then Pia salary would be 4375, and 42% of that is not 2100. Values 4500, 5000 or 3200 for Som do not match the requirement that Pia remainder be exactly Rs 2100 when the ratio 5 : 4 is applied to connect their salaries. Common Pitfalls: A frequent mistake is misreading the percentages and adding or subtracting incorrectly. Another common error is to apply the 25%, 15% and 18% directly to the 2100 instead of to Pia full salary. Some students also forget to use the salary ratio at the end to find Som salary and instead report Pia salary as the final answer. Carefully tracking which quantity each percentage is based on and using the ratio only at the final step avoids these issues. Final Answer: The monthly salary of Som is Rs 4000.

Question 5

Each question below contains two quantities, Quantity I and Quantity II. Read the information and decide the relationship between the two quantities. Quantity I: Three numbers are in the ratio 5 : 6 : 10. The sum of the largest and the smallest numb…

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Introduction / Context: This is a comparison type question where two different mathematical situations are described as Quantity I and Quantity II. We need to compute the numerical value in each case and then compare them. Questions like this appear in competitive exams to check a candidate ability to handle multiple small problems and then make a logical comparison rather than only finding individual answers. Given Data / Assumptions: Quantity I: Three numbers are in the ratio 5 : 6 : 10. The sum of the largest and smallest numbers is 126 more than the middle number. Quantity II: Twelve percent of the first number equals twenty five percent of the second number. The difference between these two numbers is 78. All numbers are real and positive. Concept / Approach: For Quantity I we use the idea that numbers in a ratio can be written as multiples of a common variable. We then translate the given condition into an equation in that variable and solve. For Quantity II we interpret the percentage statements as algebraic equations linking two unknown numbers. After solving each situation, we compare the final numerical values of the required largest numbers in both quantities. Step-by-Step Solution: For Quantity I, let the three numbers be 5k, 6k and 10k. Smallest = 5k, middle = 6k, largest = 10k. Given: largest + smallest is 126 more than the middle number. So 10k + 5k = 6k + 126. This gives 15k = 6k + 126, so 9k = 126 and k = 14. Largest number in Quantity I = 10k = 10 * 14 = 140. For Quantity II, let the two numbers be A and B, with A larger than B. Given 12% of A = 25% of B. So 0.12 * A = 0.25 * B. Hence A / B = 0.25 / 0.12 = 25 / 12. Let B = 12x and A = 25x. Difference A − B = 25x − 12x = 13x = 78. So x = 78 / 13 = 6. Therefore A = 25x = 25 * 6 = 150. Largest number in Quantity II = 150. Verification / Alternative check: Check the conditions for Quantity II: 12% of A = 12% of 150 = 18 and 25% of B = 25% of 72 = 18, so the percentage relation holds. The difference A − B = 150 − 72 = 78, matching the statement. For Quantity I, with k = 14 we have numbers 70, 84 and 140. Largest plus smallest = 210 and middle + 126 = 84 + 126 = 210, so that condition also holds. Thus both values 140 and 150 are correct, and the comparison is valid. Why Other Options Are Wrong: Quantity I > Quantity II is wrong because 140 is less than 150. Quantity I = Quantity II is incorrect since the values are not equal. Relationship cannot be determined is incorrect because both values can be uniquely computed. Both quantities are zero is clearly not true given the positive values obtained. Common Pitfalls: Typical errors include misidentifying which number is largest in Quantity I or mishandling percentages in Quantity II. Students sometimes invert the ratio when moving from 12% of A equals 25% of B to A : B, which leads to wrong values. Some also forget that the numbers in a ratio must be multiplied by the same factor and instead treat them as final values. A careful step-by-step algebraic approach avoids these mistakes. Final Answer: Quantity I is less than Quantity II, so Quantity I < Quantity II.

Question 6

Three partners S, T and U start a business together. Their capitals are in the ratio 3 : 4 : 6. At the end, the ratio of their shares in profit is 1 : 2 : 3. What is the respective ratio of the time periods for which S, T and U invested their capita…

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Introduction / Context: This partnership problem again focuses on how profit sharing depends on both capital and time. We are given the ratio of capital contributions by S, T and U, and the final ratio of their profits. Using the relationship Profit ∝ Capital * Time, we can determine how long each partner kept their money invested relative to the others. This type of question is typical in quantitative aptitude sections of exams. Given Data / Assumptions: Capital ratio S : T : U = 3 : 4 : 6. Profit ratio S : T : U = 1 : 2 : 3. Let the time periods of investment be tS, tT and tU. Profit for each partner is directly proportional to (capital * time). Concept / Approach: If capital and time are known in ratio form for partners in a business, the profit share is proportional to the product of capital and time. Here we know the capital ratio and the resulting profit ratio, but not the time ratio. By setting up proportional relationships between capital time products and the observed profit ratios, we can solve for the relative times and simplify them to their lowest integer ratio form. Step-by-Step Solution: Let tS, tT, tU be the time periods for S, T and U. Capital time products are: S → 3 * tS, T → 4 * tT, U → 6 * tU. These are proportional to the profit shares 1 : 2 : 3. So 3 * tS : 4 * tT : 6 * tU = 1 : 2 : 3. This implies 3 * tS / 1 = 4 * tT / 2 = 6 * tU / 3. Simplify: 3 * tS / 1 = 3 * tS, 4 * tT / 2 = 2 * tT, 6 * tU / 3 = 2 * tU. Therefore 3 * tS = 2 * tT = 2 * tU = k, for some constant k. From 3 * tS = k, we get tS = k / 3. From 2 * tT = k, we get tT = k / 2. From 2 * tU = k, we get tU = k / 2. So tS : tT : tU = (k / 3) : (k / 2) : (k / 2). Multiply through by 6 to clear denominators: tS : tT : tU = 2 : 3 : 3. Verification / Alternative check: Take k = 6 for convenience. Then tS = 2, tT = 3 and tU = 3. Capital time products: S → 3 * 2 = 6, T → 4 * 3 = 12, U → 6 * 3 = 18. The ratio 6 : 12 : 18 simplifies by dividing by 6 to 1 : 2 : 3, which matches the given profit ratio. This confirms that the time ratio 2 : 3 : 3 is correct. Why Other Options Are Wrong: Ratios like 3 : 2 : 2 or 4 : 5 : 3 give capital time products that do not simplify to 1 : 2 : 3. Similarly, 2 : 2 : 3 and 1 : 2 : 3 do not reproduce the correct profit distribution when combined with the original capital ratio 3 : 4 : 6. Common Pitfalls: A frequent error is to assume that the ratio of profit is the same as the ratio of capital, ignoring time altogether. Another mistake is algebraic: not setting up the fractions correctly when equating 3 * tS : 4 * tT : 6 * tU to 1 : 2 : 3. Some candidates also forget to fully simplify the final ratio. Working methodically with the capital time products and checking with a quick verification prevents these issues. Final Answer: The respective ratio of the time periods for which S, T and U invested their capital is 2 : 3 : 3.

Question 7

The sides of a triangular park are in the ratio 3 : 4 : 5 and the perimeter of the park is 170 metres. What is the length, in metres, of the perpendicular drawn from the vertex opposite the largest side to that largest side?

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Introduction / Context: This geometry question involves a triangle whose sides are in the ratio 3 : 4 : 5 and whose perimeter is known. The ratio 3 : 4 : 5 is characteristic of a right angled triangle, where the side with length 5 parts is the hypotenuse and the other two sides are the perpendicular legs. We are asked to find the perpendicular distance from the vertex opposite the largest side (the hypotenuse) to that side. In a right triangle, this distance is simply the altitude to the hypotenuse, which can be found using area relationships. Given Data / Assumptions: Side ratios of the triangle are 3 : 4 : 5. Perimeter of the triangle = 170 metres. The largest side is the one corresponding to 5 parts. We need the perpendicular distance from the opposite vertex to this largest side. The triangle is assumed to be a 3 : 4 : 5 right triangle. Concept / Approach: If the sides of a triangle are in the ratio 3 : 4 : 5, it represents a right triangle with the largest side as the hypotenuse. Let the common multiplying factor be k. Then the sides are 3k, 4k and 5k. The perimeter then is 12k. From the given perimeter we can find k and therefore the actual side lengths. The area of a right triangle is (1 / 2) * (product of the two legs). The altitude from the right angle to the hypotenuse can then be found using the formula: altitude = (2 * area) / hypotenuse. Step-by-Step Solution: Let the sides be 3k, 4k and 5k metres. Perimeter = 3k + 4k + 5k = 12k. Given perimeter = 170 metres, so 12k = 170. Thus k = 170 / 12 = 85 / 6. Then the three sides are: 3k = 3 * 85 / 6 = 255 / 6 metres, 4k = 340 / 6 metres and 5k = 425 / 6 metres. The triangle is right angled with legs 3k and 4k and hypotenuse 5k. Area of triangle = (1 / 2) * (3k) * (4k) = 6k^2. Altitude from the right angle to the hypotenuse (largest side) = (2 * area) / hypotenuse. So altitude = (2 * 6k^2) / (5k) = (12k^2) / (5k) = 12k / 5. Substitute k = 85 / 6: altitude = 12 * (85 / 6) / 5. Simplify: 12 * (85 / 6) = 2 * 85 = 170, so altitude = 170 / 5 = 34 metres. Verification / Alternative check: We can approximate the side lengths numerically: 3k ≈ 42.5, 4k ≈ 56.67 and 5k ≈ 70.83 metres. Using the formula for area from base and height, area = (1 / 2) * hypotenuse * altitude. With altitude 34 and hypotenuse about 70.83, area ≈ (1 / 2) * 70.83 * 34 ≈ 1202. The area from legs is (1 / 2) * 42.5 * 56.67 ≈ 1202 as well, within rounding. This confirms that altitude 34 metres is consistent. Why Other Options Are Wrong: Values like 24, 30, 36 or 40 metres do not satisfy the relationship area = (1 / 2) * base * height when used with the actual hypotenuse and the legs derived from the given perimeter. Only altitude 34 metres makes the two different area expressions match. Common Pitfalls: Some candidates mistake the largest side as one of the legs rather than the hypotenuse, leading to wrong use of the Pythagoras theorem. Others incorrectly use half the perimeter as the hypotenuse or mix up the altitude with a leg length. Another error is to ignore that the ratio 3 : 4 : 5 implies a right triangle. Proper recognition of the 3 : 4 : 5 pattern and careful algebra with k removes confusion. Final Answer: The perpendicular distance from the vertex to the largest side is 34 metres.

Question 8

In an army recruitment process, the ratio of selected candidates to unselected candidates was 9 : 2. If 80 fewer candidates had applied in total and 20 fewer candidates had been selected, the ratio of selected to unselected would have become 5 : 1. …

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Introduction / Context: This problem uses ratios to relate selected and unselected candidates in an army selection process. We have two scenarios: the original one and a hypothetical one with different totals and selections. By converting the verbal description of the ratios into algebraic equations, we can determine the original total number of applicants. Such ratio based word problems are common in aptitude exams. Given Data / Assumptions: Initially, ratio of selected to unselected candidates = 9 : 2. If 80 fewer candidates had applied and 20 fewer candidates had been selected, the new ratio of selected to unselected would be 5 : 1. We assume only two categories: selected and unselected candidates. We are asked to find the original total number of applicants. Concept / Approach: Let the numbers of selected and unselected candidates initially be 9k and 2k respectively, for some positive integer k. Then the total number of applicants is 11k. In the hypothetical situation, total applicants become 11k − 80 and selected candidates become 9k − 20. The new number of unselected candidates is total minus selected. Using the new ratio 5 : 1, we set up an equation and solve for k. Finally we compute 11k to get the original total applicants. Step-by-Step Solution: Let initially selected = 9k and unselected = 2k. Total initial applicants = 9k + 2k = 11k. In the second scenario, 80 fewer apply, so total becomes 11k − 80. Also, 20 fewer are selected, so selected become 9k − 20. Hence unselected in the second scenario = (11k − 80) − (9k − 20) = 11k − 80 − 9k + 20 = 2k − 60. Given that the new ratio selected : unselected is 5 : 1, we have: (9k − 20) / (2k − 60) = 5 / 1. So 9k − 20 = 5 * (2k − 60). 9k − 20 = 10k − 300. Bring like terms together: −k = −280, so k = 280. Original total applicants = 11k = 11 * 280 = 3080. Verification / Alternative check: Using k = 280, initial selected = 9 * 280 = 2520 and unselected = 2 * 280 = 560. The ratio 2520 : 560 simplifies to 9 : 2. New total after removing 80 is 3080 − 80 = 3000. New selected = 2520 − 20 = 2500, so new unselected = 3000 − 2500 = 500. The new ratio is 2500 : 500 = 5 : 1, which matches the condition. Hence the value 3080 is consistent with both scenarios. Why Other Options Are Wrong: Values like 6160, 1540, 9240 or 2200 do not satisfy both ratio conditions when we recompute selected and unselected numbers using the described changes. Only 3080 leads to integer values of selected and unselected candidates that fit both given ratios. Common Pitfalls: Students sometimes incorrectly set up the second scenario, especially when determining the new number of unselected candidates. Another common mistake is to forget that both selected and unselected counts must remain positive when solving for k. Careful bookkeeping of each change (fewer applicants, fewer selected) and using total = selected + unselected at every stage avoids these errors. Final Answer: The original number of candidates who applied for the process was 3080.

Question 9

Two positive numbers x and y are such that their mean proportional (geometric mean) is 16 and the third proportional to x and y is 128. What are the values of x and y?

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Introduction / Context: This problem involves the classical concepts of mean proportional (geometric mean) and third proportional between two numbers. These terms often appear in ratio and proportion topics in aptitude and school mathematics. The question gives the value of the geometric mean and the third proportional, and asks for the original pair of numbers x and y that satisfy both conditions. Given Data / Assumptions: Mean proportional (geometric mean) of x and y is 16. Third proportional to x and y is 128. x and y are positive real numbers. By definition, if a : b = b : c, then b is the mean proportional and c is the third proportional. Concept / Approach: For two positive numbers x and y, their geometric mean is sqrt(x * y). So the first condition gives x * y = 16^2 = 256. The third proportional condition means x : y = y : 128. This gives a second relationship between x and y. Using these two equations together, we can solve for x and y. Knowledge of these special proportional definitions is the key to solving the question quickly. Step-by-Step Solution: From the geometric mean condition: sqrt(x * y) = 16. Squaring both sides gives x * y = 256. (Equation 1) From the third proportional condition: x : y = y : 128. This means x / y = y / 128. Cross multiply to get 128 * x = y^2. (Equation 2) From Equation 1, x = 256 / y. Substitute into Equation 2: 128 * (256 / y) = y^2. This simplifies to (128 * 256) / y = y^2. So y^3 = 128 * 256. Compute 128 * 256 = 2^7 * 2^8 = 2^15 = 32768. Hence y^3 = 32768, so y = cube root of 32768 = 32. From x * y = 256, x = 256 / 32 = 8. Therefore, x = 8 and y = 32. Verification / Alternative check: Check the geometric mean: sqrt(8 * 32) = sqrt(256) = 16, which matches the given mean proportional. Check the third proportional condition: x : y = 8 : 32 simplifies to 1 : 4. Also, y : 128 = 32 : 128, which simplifies to 1 : 4. Therefore 8 : 32 = 32 : 128, confirming that 128 is indeed the third proportional to 8 and 32. Both conditions are satisfied. Why Other Options Are Wrong: Pairs such as 8 and 16 or 16 and 32 do not produce both a geometric mean of 16 and a third proportional equal to 128 when tested. For example, 8 and 16 give geometric mean sqrt(128), not 16; 16 and 16 give third proportional 16, not 128. Common Pitfalls: Learners often confuse arithmetic mean with geometric mean and incorrectly use (x + y) / 2 instead of sqrt(x * y). Some also misremember the definition of third proportional, mistakenly using x : y = 128 : y rather than x : y = y : 128. Careful recall of definitions and systematic use of algebraic equations ensures correct results. Final Answer: The two numbers are x = 8 and y = 32.

Question 10

Select the related number from the given alternatives to complete the analogy: 1 : 8 :: ? : 64

Accepted Answer

Introduction / Context: This is a number analogy question where we need to identify the pattern linking the first pair of numbers and then apply the same pattern to find the missing number in the second pair. Such problems test quick observation of numerical relationships, such as squares, cubes or other simple operations. Given Data / Assumptions: The given analogy is 1 : 8 :: ? : 64. We must choose the correct number from the options that maintains the same relationship between the pair as in the first pair. All numbers are positive integers. Concept / Approach: We first analyse the relationship between 1 and 8. Notice that 1 = 1^3 and 8 = 2^3. The sequence 1, 8, 27, 64, 125, and so on are cubes of natural numbers 1, 2, 3, 4, 5, etc. It is reasonable to suspect that the pattern involves consecutive cubes. If 1 corresponds to 8 (1^3 and 2^3), then the missing number should correspond to 64, which is 4^3. The natural candidate is 27, which is 3^3, forming a smooth progression 1^3, 2^3, 3^3, 4^3. Step-by-Step Solution: Observe that 1 = 1^3 and 8 = 2^3. So the first pair can be seen as (1^3, 2^3). We are given 1 : 8 :: ? : 64. Here 64 is 4^3. To maintain a pattern of consecutive cubes, the missing number should be 3^3, which is 27. Then we have a chain: 1^3, 2^3, 3^3, 4^3 → 1, 8, 27, 64. Thus the correct completion is 1 : 8 :: 27 : 64. Verification / Alternative check: We can check if any simple multiplication or ratio pattern also works. The ratio 8 / 1 = 8 and 64 / 27 is not 8, so the pair is not directly in the same ratio. Instead, the pattern of cubes is consistent and natural for both pairs. No other simple operation, such as squaring or adding a fixed value, arranges the numbers as neatly as consecutive cubes. Therefore the cube based pattern is the most reasonable and mathematically sound explanation. Why Other Options Are Wrong: 25 is 5^2, 36 is 6^2, 30 is not a perfect square or cube and 49 is 7^2; none of these fit a consecutive cube pattern with 64 as 4^3. Only 27 is a perfect cube (3^3) and fits between 8 (2^3) and 64 (4^3) in the natural cube sequence. Common Pitfalls: A common mistake is to look for constant differences (like 8 − 1 = 7) and then try to apply the same difference to 64, which would suggest 57, a number not present in the options. Another error is to assume a direct ratio, such as 1 : 8 = x : 64, which would give x = 8. Missing the fact that 8 and 64 are cubes prevents recognition of the intended pattern. Always check for powers (squares, cubes) in analogy questions involving small integers. Final Answer: The missing number in the analogy 1 : 8 :: ? : 64 is 27.

Question 11

A box of sweets is distributed between persons A and B in the ratio 3 : 4. If A receives 36 sweets, how many sweets were there in the box in total?

Accepted Answer

Introduction / Context: This is a basic ratio and proportion problem where the total quantity is divided between two people in a given ratio. One person share is known, and we are asked to find the total number of items. Such questions are fundamental in ratio topics and appear frequently in competitive exams and school tests. Given Data / Assumptions: Sweets are divided between A and B in the ratio 3 : 4. A receives 36 sweets. We assume all sweets are counted correctly and there is no wastage. We are asked to find the total number of sweets in the box. Concept / Approach: If two quantities are in ratio a : b, they can be represented as a * k and b * k for some common multiplier k. Here A share corresponds to 3 parts and B share to 4 parts. Since A share is known numerically, we can find k by dividing that share by the number of ratio parts associated with A. Once k is known, the total is (3k + 4k) = 7k. Step-by-Step Solution: Let A share = 3k and B share = 4k for some positive k. Given that A receives 36 sweets, we have 3k = 36. So k = 36 / 3 = 12. Total sweets = 3k + 4k = 7k. Substitute k = 12: total sweets = 7 * 12 = 84. Therefore, the box contained 84 sweets. Verification / Alternative check: Using the total 84 sweets, A share should be (3 / 7) of 84 and B share should be (4 / 7) of 84. Compute (3 / 7) * 84 = 36, which matches the given amount for A. B share is (4 / 7) * 84 = 48. Therefore the distribution 36 and 48 is in ratio 3 : 4 and sums to 84, confirming the correctness of the total. Why Other Options Are Wrong: 12 is only part of A share and is far less than the required total. 144 and 27 do not produce a share of exactly 36 for A when the ratio 3 : 4 is applied. 63 would give A share (3 / 7) * 63 = 27, not 36, so it does not satisfy the condition. Common Pitfalls: A common error is to divide 36 by 4 instead of 3, or to use 3 + 4 = 7 incorrectly. Another mistake is to assume that 36 is the total and then try to split it in ratio 3 : 4, which is not what the problem states. Carefully identifying which person share is given and which ratio part belongs to that person is essential for solving these problems correctly. Final Answer: The total number of sweets in the box is 84.

Question 12

In a rhombus, one of the diagonals is 70% of the other diagonal. What is the ratio of the area of the rhombus to the square of the length of the larger diagonal?

Accepted Answer

Introduction / Context: This question applies the formula for the area of a rhombus in terms of its diagonals and asks for a ratio involving the square of the longer diagonal. It combines knowledge of geometry and percent relationships. The rhombus is a quadrilateral with all sides equal, and its diagonals intersect at right angles. The area can be conveniently expressed using the product of the diagonals. Given Data / Assumptions: In the rhombus, let the longer diagonal be denoted by D. The shorter diagonal is 70% of the longer one, that is 0.7 * D. Area of a rhombus is (1 / 2) * (product of the diagonals). We need the ratio: (area of rhombus) : (D^2). Concept / Approach: We use the formula for the area of a rhombus: Area = (1 / 2) * d1 * d2, where d1 and d2 are the diagonals. Here, d1 = D (larger diagonal) and d2 = 0.7 * D. Substituting these into the area formula, we obtain area in terms of D^2. Then we form the ratio of this area to D^2. Since D is positive, it cancels out properly in the ratio, leaving a numerical fraction which we convert into a ratio form. Step-by-Step Solution: Let the longer diagonal be D. Then the shorter diagonal is 70% of D, that is 0.7 * D. Area of the rhombus = (1 / 2) * d1 * d2 = (1 / 2) * D * (0.7 * D). So area = (1 / 2) * 0.7 * D^2 = 0.35 * D^2. We need the ratio (area) : (D^2). So ratio = 0.35 * D^2 : D^2. Cancel D^2 to get 0.35 : 1. Write 0.35 as 35 / 100, which simplifies to 7 / 20. Hence the required ratio = 7 : 20. Verification / Alternative check: Choose a convenient value for D, for example D = 20 units. Then the shorter diagonal is 70% of 20, which is 14 units. Area = (1 / 2) * 20 * 14 = 140 square units. The square of the longer diagonal is D^2 = 400. The ratio area : D^2 is then 140 : 400. Dividing both by 20 gives 7 : 20, confirming our earlier algebraic result. Why Other Options Are Wrong: Ratios like 3 : 10, 3 : 20, 7 : 10 or 1 : 2 do not match the fraction 0.35 when expressed as a ratio, and therefore do not accurately represent area : D^2 in this configuration. Only 7 : 20 corresponds to 0.35 in simplest ratio form. Common Pitfalls: Some students mistakenly interpret 70% as 0.7 added to 1, using 1.7 * D instead of 0.7 * D for the shorter diagonal. Others plug the percentage incorrectly into the area formula or attempt to square 0.7 incorrectly. Another frequent mistake is to forget to simplify the decimal 0.35 into a clean ratio. Always convert percentages into decimals or fractions carefully and follow the formula for area step by step. Final Answer: The ratio of the area of the rhombus to the square of the larger diagonal is 7 : 20.

Question 13

A car travels 80 km in 2 hours and a train travels 180 km in 3 hours. What is the ratio of the speed of the car to the speed of the train?

Accepted Answer

Introduction / Context: This is a basic speed, distance and time question that asks for the ratio of two speeds. Each speed can be computed using the formula speed = distance / time. Then we form the ratio of the car speed to the train speed and simplify it. Such questions test fundamental understanding of motion and proportional reasoning. Given Data / Assumptions: Car travels 80 km in 2 hours. Train travels 180 km in 3 hours. We assume both move at constant speed over the given distances. We are required to find (speed of car) : (speed of train). Concept / Approach: Speed is defined as distance divided by time. Once we compute each speed numerically, we express the two speeds as a ratio and then reduce that ratio to its simplest form by dividing both parts by their greatest common divisor. This uses straightforward arithmetic and is a direct application of the definition of speed. Step-by-Step Solution: Speed of car = distance / time = 80 km / 2 hours. So speed of car = 40 km per hour. Speed of train = 180 km / 3 hours. So speed of train = 60 km per hour. Required ratio = speed of car : speed of train = 40 : 60. Divide both sides by 20 to simplify: 40 / 20 = 2 and 60 / 20 = 3. Hence the ratio is 2 : 3. Verification / Alternative check: We can check using proportional thinking. Car covers 80 km in 2 hours, so in 3 hours at the same speed it would cover 120 km. Train covers 180 km in 3 hours, which is clearly faster. The ratio 120 : 180 also simplifies to 2 : 3, consistent with the earlier calculation. This confirms that our ratio is correct. Why Other Options Are Wrong: 3 : 2 or 4 : 3 would suggest the car is faster than the train, which is not the case because 40 km per hour is less than 60 km per hour. Ratios like 3 : 4 or 5 : 4 do not match the true speed ratio when the distances and times are checked. Common Pitfalls: A common mistake is to compare distances directly (80 and 180) without adjusting for the different times, or to compare times without considering distances. Another error is to forget to simplify the ratio after calculating the speeds. Always compute speed first using distance over time, then form and simplify the ratio to get the final answer. Final Answer: The ratio of the speed of the car to the speed of the train is 2 : 3.

Question 14

If x and y are two numbers such that x / y = 2 / 3, then what is the ratio of (3x + 2y) to (3x − y)?

Accepted Answer

Introduction / Context: This algebra question uses a given ratio between two variables x and y and asks for the ratio of two linear expressions in x and y. Instead of trying arbitrary numbers, we can assign convenient values to x and y that preserve the given ratio and then evaluate the expressions. This is a standard technique in ratio problems involving algebraic expressions. Given Data / Assumptions: x / y = 2 / 3. x and y are non zero real numbers. We need the ratio (3x + 2y) : (3x − y). Concept / Approach: From x / y = 2 / 3, we can let x = 2k and y = 3k for some non zero constant k. This representation keeps the ratio intact while making calculations simple. Substituting these into the expressions 3x + 2y and 3x − y converts them into expressions in terms of k alone, which then cancel out in the ratio. Finally we simplify the resulting numeric ratio. Step-by-Step Solution: Given x / y = 2 / 3, let x = 2k and y = 3k. Compute 3x + 2y: 3x + 2y = 3 * (2k) + 2 * (3k) = 6k + 6k = 12k. Compute 3x − y: 3x − y = 3 * (2k) − 3k = 6k − 3k = 3k. Required ratio = (3x + 2y) : (3x − y) = 12k : 3k. Divide both parts by 3k: 12k / 3k = 4 and 3k / 3k = 1. Therefore the ratio is 4 : 1. Verification / Alternative check: We can choose explicit numbers for x and y that satisfy x / y = 2 / 3. Take x = 2 and y = 3. Then 3x + 2y = 3 * 2 + 2 * 3 = 6 + 6 = 12. Also, 3x − y = 3 * 2 − 3 = 6 − 3 = 3. The ratio 12 : 3 simplifies exactly to 4 : 1, confirming the result. Any other pair (x, y) that satisfies the original ratio, such as (4, 6), will produce the same final ratio. Why Other Options Are Wrong: 3 : 1 or 2 : 1 would require the numerator or denominator to differ by some factor not supported by the substitution from x / y = 2 / 3. 4 : 3 and 5 : 2 do not match calculations for any consistent value of k and therefore are incorrect. Common Pitfalls: A common mistake is to attempt to manipulate the expression (3x + 2y) / (3x − y) symbolically but misapply the given ratio. Another error is to incorrectly set x = 2 and y = 3 without recognizing that they could be multiplied by a common factor, though that does not actually change the outcome if used correctly. The safest method is to express x and y as 2k and 3k and then simplify systematically. Final Answer: The ratio of (3x + 2y) to (3x − y) is 4 : 1.

Question 15

A profit of Rs 12,400 has to be divided among three partners A, B and C in the ratio 5 : 7 : 8. How much money, in rupees, does partner B receive?

Accepted Answer

Introduction / Context: This is a direct application of ratios in distributing a given total amount among three partners. The total profit and the ratio of shares are known, and we are asked to find the numerical share of one particular partner, B. Such problems are common in partnership and ratio topics in aptitude exams. Given Data / Assumptions: Total profit to be distributed = Rs 12,400. Ratio of shares of A : B : C = 5 : 7 : 8. We assume the entire profit is distributed with no remainder. We need the exact amount received by B. Concept / Approach: If three quantities are in ratio a : b : c and their sum is S, then each quantity is given by (a / (a + b + c)) * S, (b / (a + b + c)) * S and (c / (a + b + c)) * S respectively. Here, a = 5, b = 7 and c = 8. We first find the sum of the ratio numbers, then compute B share as (7 / total parts) multiplied by 12400. Step-by-Step Solution: Ratio of shares A : B : C = 5 : 7 : 8. Total number of ratio parts = 5 + 7 + 8 = 20. Total profit = Rs 12,400. Share of B = (7 / 20) * 12,400. Compute (1 / 20) of 12,400: this is 12,400 / 20 = 620. So B share = 7 * 620 = 4340. Therefore, B receives Rs 4340. Verification / Alternative check: We can also compute the shares of A and C to verify the entire distribution. A share = (5 / 20) * 12,400 = 5 * 620 = 3100. C share = (8 / 20) * 12,400 = 8 * 620 = 4960. The sum of all three shares is 3100 + 4340 + 4960 = 12,400, which matches the total profit given, thus confirming that B share is correctly calculated as Rs 4340. Why Other Options Are Wrong: 3440, 3340, 4430 and 3740 do not match (7 / 20) of 12,400. If any of these values were used instead of 4340, the total of the three shares would not equal Rs 12,400 when keeping the same ratio 5 : 7 : 8. Common Pitfalls: A frequent mistake is to divide 12,400 directly by 7 or to forget to sum the ratio numbers first. Another error is to miscalculate (7 / 20) of 12,400 due to arithmetic slips. Carefully adding the ratio parts and working step by step through the fraction helps avoid such errors. Final Answer: Partner B receives Rs 4340.

Question 16

Find two positive numbers such that their mean proportional (geometric mean) is 8 and the third proportional to these two numbers is 64.

Accepted Answer

Introduction / Context: This question is similar to an earlier one on mean and third proportional, but now a different geometric mean and third proportional are given. We must once again apply the definitions of these special proportions to determine the original pair of numbers. This reinforces conceptual understanding of how proportional relationships translate into algebraic equations. Given Data / Assumptions: The mean proportional (geometric mean) of the two numbers is 8. The third proportional to these two numbers is 64. Both numbers are positive. By definition, if a : b = b : c, then b is the mean proportional between a and c, and c is the third proportional to a and b. Concept / Approach: Let the two numbers be a and b. The geometric mean condition gives sqrt(a * b) = 8, so a * b = 8^2 = 64. The third proportional condition tells us that a : b = b : 64, or a / b = b / 64. This yields a second equation. Solving these two equations simultaneously allows us to find unique positive values of a and b that satisfy both conditions. Step-by-Step Solution: Let the two numbers be a and b. Given that their geometric mean is 8, so sqrt(a * b) = 8. Square both sides: a * b = 64. (Equation 1) Given that 64 is the third proportional to a and b, so a : b = b : 64. Thus a / b = b / 64, giving b^2 = 64 * a. (Equation 2) From Equation 1, a = 64 / b. Substitute into Equation 2: b^2 = 64 * (64 / b) = (64 * 64) / b. So b^3 = 64 * 64 = 4096. Hence b = cube root of 4096 = 16. Now from a * b = 64, a = 64 / 16 = 4. Therefore the two numbers are 4 and 16. Verification / Alternative check: Check the geometric mean: sqrt(4 * 16) = sqrt(64) = 8, which satisfies the first condition. Check the third proportional condition: a : b = 4 : 16 simplifies to 1 : 4. Also, b : 64 = 16 : 64 simplifies to 1 : 4. Hence 4 : 16 = 16 : 64, confirming that 64 is indeed the third proportional to 4 and 16. Both conditions are met, so the solution is consistent. Why Other Options Are Wrong: Pairs such as 4 and 8, 8 and 16, 8 and 8 or 2 and 32 either fail to give a geometric mean of 8 or do not yield 64 as the third proportional when tested. Only the pair 4 and 16 satisfies both conditions simultaneously. Common Pitfalls: As before, confusion between arithmetic mean and geometric mean can lead to incorrect equations. Another common mistake is misapplying the third proportional definition as a : b = 64 : b instead of a : b = b : 64. Ensuring that definitions are clearly remembered and carefully translating them into algebraic form prevents such errors. Final Answer: The required two numbers are 4 and 16.

Question 17

Two whole numbers are such that the cube of the first number exceeds the cube of the second number by 61 and the ratio of the first number to the second number is 5 : 4. What is the value of the larger number?

Accepted Answer

Introduction / Context: This problem combines ratios with cubic expressions. The relationship between the cubes of two numbers is given, along with the ratio of the numbers themselves. The task is to find the larger number. Since cube values grow rapidly, the given difference of 61 strongly suggests small integer values for the numbers, and the ratio condition helps fix them uniquely. Given Data / Assumptions: The cube of the first number minus the cube of the second number equals 61. The ratio of the first number to the second number is 5 : 4. Both numbers are whole numbers (non negative integers). We are asked for the larger of the two numbers. Concept / Approach: From the ratio 5 : 4, we express the numbers as 5k and 4k. We substitute these expressions into the given condition involving cubes: (5k)^3 − (4k)^3 = 61. This becomes an equation in k. Because 61 is a prime number and relatively small, we expect k to be a small integer. Solving the resulting cubic equation allows us to find k and then the actual values of the numbers. Step-by-Step Solution: Let the larger number be 5k and the smaller number be 4k, since their ratio is 5 : 4. Given that (5k)^3 − (4k)^3 = 61. Compute the cubes: (5k)^3 = 125k^3 and (4k)^3 = 64k^3. So 125k^3 − 64k^3 = 61. This simplifies to 61k^3 = 61. Divide both sides by 61 to obtain k^3 = 1. Thus k = 1 (since we are working with real whole numbers). Therefore the two numbers are 5k = 5 and 4k = 4. The larger number is 5. Verification / Alternative check: Substitute the numbers directly into the condition. The cubes are 5^3 = 125 and 4^3 = 64. The difference is 125 − 64 = 61, which matches the given difference. The ratio 5 : 4 also matches the specified ratio of the first number to the second number. Thus both conditions are satisfied, confirming that the larger number is indeed 5. Why Other Options Are Wrong: If the larger number were 3, 4, 6 or 7, it would not be possible to find a second whole number such that the cube difference is exactly 61 while also maintaining a ratio of 5 : 4. Testing 6 and 7, for example, leads to cube differences very different from 61 and violates the ratio constraint. Common Pitfalls: Some students try to guess both numbers randomly without using the given ratio, which is inefficient. Others expand the cubes but forget to factor out k^3, leading to algebraic mistakes. Recognising that 61k^3 = 61 immediately implies k = 1 makes the problem very simple. Also, checking that the numbers are whole and that both conditions hold is essential. Final Answer: The value of the larger number is 5.

Question 18

Three numbers are in the ratio 5 : 6 : 8 and their total sum is 380. What is the value of the smallest of these three numbers?

Accepted Answer

Introduction / Context: This aptitude question on ratio and proportion tests how to convert a given ratio of three numbers into actual values when their total sum is known. It also checks basic understanding of how to identify the smallest number once all the individual values are calculated. Given Data / Assumptions: • The ratio of the three numbers is 5 : 6 : 8.• The sum of the three numbers is 380.• The numbers are positive real numbers and follow the given ratio exactly.• We are asked to find the smallest of these three numbers. Concept / Approach: The core idea is that if three numbers are in the ratio a : b : c, then they can be written as a*k, b*k, c*k for some common multiplier k. Their sum is then (a + b + c) * k. By equating this to the given total, we can solve for k. Once k is known, each individual number can be obtained by multiplying the ratio parts by this k. Finally, we compare the three values to identify the smallest number. Step-by-Step Solution: Step 1: Let the three numbers be 5k, 6k and 8k.Step 2: Their sum is 5k + 6k + 8k = 19k.Step 3: We are told that the sum of the three numbers is 380, so 19k = 380.Step 4: Solve for k by dividing both sides of the equation by 19: k = 380 / 19 = 20.Step 5: Now compute each number:• First number = 5k = 5 * 20 = 100.• Second number = 6k = 6 * 20 = 120.• Third number = 8k = 8 * 20 = 160.Step 6: Compare the three values. The smallest among 100, 120 and 160 is 100. Verification / Alternative check: We can quickly verify the result by checking the ratio and the sum. The three numbers we found are 100, 120 and 160. Their ratio is 100 : 120 : 160. Dividing each by 20 gives 5 : 6 : 8, which matches the given ratio. Also, the sum is 100 + 120 + 160 = 380, which matches the given total. Both the ratio and the sum are correct, so the calculation is consistent and the smallest number must indeed be 100. Why Other Options Are Wrong: • 80: If 80 were the smallest number, the other numbers scaled from the given ratio would not sum to 380 and the ratio would not remain 5 : 6 : 8.• 120: This is actually the middle number, not the smallest. The ratio and total confirm that 120 is the second value.• 140: This value does not appear among the three numbers obtained from the correct multiplier and therefore cannot be the smallest number. Common Pitfalls: Students sometimes try to guess numbers instead of using the multiplier method. Another common mistake is to divide the total by the wrong sum of ratio parts or to miscalculate the sum as something other than 19. Some learners also forget that the ratio parts represent relative sizes and must all be multiplied by the same k. Finally, a few learners might accidentally identify the largest or middle number instead of carefully choosing the smallest one after calculation. Final Answer: The smallest of the three numbers is 100.

Question 19

The flight fare between two cities is increased in the ratio 9 : 11. If the original one way air fare was Rs. 18,000, what is the increase in the fare in rupees?

Accepted Answer

Introduction / Context: This question is about ratio and proportion applied to a real life situation involving an increase in flight fare. The idea is to understand how a price changes when it is scaled by a given ratio and then compute the absolute increase in rupee terms, which is a very common type of aptitude test question. Given Data / Assumptions: • Original flight fare between the two cities is Rs. 18,000.• The fare is increased in the ratio 9 : 11.• The ratio 9 : 11 means the new fare corresponds to 11 parts while the old fare corresponds to 9 parts.• We are required to find the increase in fare, that is, new fare minus old fare. Concept / Approach: If a quantity changes from a ratio of a to b, it means the new value is (b / a) times the old value. Here, the fare changes from 9 parts to 11 parts. So, new fare = old fare * (11 / 9). Once we find the new fare using this scaling factor, the increase is simply new fare minus old fare. This method avoids any confusion between percentage increase and ratio increase by directly using the proportional relation. Step-by-Step Solution: Step 1: Let the old fare correspond to 9 parts, so new fare corresponds to 11 parts.Step 2: The scaling factor for the new fare is 11 / 9.Step 3: Compute the new fare using the old fare of Rs. 18,000.Step 4: New fare = 18,000 * 11 / 9.Step 5: First divide 18,000 by 9 to simplify: 18,000 / 9 = 2,000.Step 6: Now multiply 2,000 by 11: 2,000 * 11 = 22,000.Step 7: Therefore, the new fare is Rs. 22,000.Step 8: The increase in fare = new fare − old fare = 22,000 − 18,000 = Rs. 4,000. Verification / Alternative check: We can check whether the new fare and old fare are indeed in the ratio 9 : 11. Divide both fares by 2,000. Old fare 18,000 / 2,000 = 9 and new fare 22,000 / 2,000 = 11. So the ratio is 9 : 11 as required. This confirms that the calculation is correct and the increase of Rs. 4,000 is consistent with the described change in ratio. Why Other Options Are Wrong: • Rs. 3,600: This value might be obtained if someone incorrectly applies a percentage calculation instead of the exact ratio scaling.• Rs. 2,000: This would correspond to a smaller increase and does not satisfy the required 9 : 11 ratio for old and new fares.• Rs. 2,200: This is also not consistent with the proportional relation between 18,000 and 22,000 implied by 9 : 11. Common Pitfalls: Many learners confuse ratio increase with a simple percentage increase and may try to compute the percentage change directly from 9 to 11 as if it were a percentage. Another common mistake is to add or subtract numbers from the ratio instead of using them as multiplicative factors. Some candidates also forget to compute the difference between new and old fare and instead mark the new fare itself as the answer. Being clear about whether the question asks for the new value or just the increase is very important. Final Answer: The increase in flight fare is Rs. 4,000.

Question 20

A bag contains 1 rupee coins, 50 paise coins and 10 paise coins in the ratio 3 : 2 : 3, and the total amount of money is Rs. 12.90. How many 50 paise coins are there in the bag?

Accepted Answer

Introduction / Context: This question on ratio and proportion involves different coin denominations contributing to a total amount. It checks whether the student can translate a ratio of counts into an algebraic expression for total money and then solve for the number of coins of a particular denomination, which is a very practical skill for handling mixtures of values. Given Data / Assumptions: • The bag contains three types of coins: 1 rupee coins, 50 paise coins, and 10 paise coins.• The ratio of the number of these coins is 3 : 2 : 3.• The total amount of money in the bag is Rs. 12.90.• We must find the number of 50 paise coins only. Concept / Approach: The ratio 3 : 2 : 3 represents the relative counts of each type of coin, not their values. Let the common multiplier be x, so the counts become 3x, 2x and 3x. Then we convert these counts into rupees by multiplying each count by the respective coin value: 1 rupee for the first type, 0.50 rupee for the second type, and 0.10 rupee for the third type. Summing these values gives a total expression in terms of x, which we equate to Rs. 12.90. Solving for x allows us to find the number of each coin, especially the 50 paise coins. Step-by-Step Solution: Step 1: Let the number of 1 rupee coins be 3x, the number of 50 paise coins be 2x, and the number of 10 paise coins be 3x.Step 2: Convert each type into rupees.• Value from 1 rupee coins = 3x * 1 = 3x rupees.• Value from 50 paise coins = 2x * 0.50 = 1x rupee.• Value from 10 paise coins = 3x * 0.10 = 0.30x rupee.Step 3: Total money in the bag = 3x + x + 0.30x = 4.30x rupees.Step 4: We are told that this total is Rs. 12.90, so 4.30x = 12.90.Step 5: Solve for x by dividing both sides by 4.30: x = 12.90 / 4.30 = 3.Step 6: Number of 50 paise coins = 2x = 2 * 3 = 6 coins. Verification / Alternative check: We can verify by computing the actual total value using x = 3. The counts are: 1 rupee coins = 3 * 3 = 9; 50 paise coins = 2 * 3 = 6; 10 paise coins = 3 * 3 = 9. Total value = 9 * 1 + 6 * 0.50 + 9 * 0.10 = 9 + 3 + 0.90 = 12.90 rupees. This matches the given total amount, confirming that x has been calculated correctly and that the number of 50 paise coins is 6. Why Other Options Are Wrong: • 3: This would correspond to x = 1.5, which does not produce the correct total amount of Rs. 12.90.• 9: This would require a different ratio multiplier and would give a total value larger than Rs. 12.90.• 12: This would drastically increase the total number of coins and hence the total amount, making it inconsistent with Rs. 12.90. Common Pitfalls: Learners often confuse the ratio of numbers of coins with the ratio of values of coins. Another mistake is to forget to convert paise into rupees properly and instead mix units such as treating 50 paise as 50 rupees by mistake. Some also forget that the ratio represents a common multiplier and try to solve directly without introducing the variable x. Finally, errors may occur if students round intermediate values incorrectly, though in this case the division is exact. Final Answer: The number of 50 paise coins in the bag is 6.

Ratio and Proportion Questions