Question 1

The monthly wage per labourer in a factory increases in the ratio 22 : 25, while the total number of labourers decreases in the ratio 15 : 11. If the present total wage bill is Rs. 5000, what was the original total wage bill before these changes?

Accepted Answer

Introduction / Context: This problem involves ratio and proportion in the context of wage bills. Wage bill equals the wage per person multiplied by the number of labourers. When both the wage rate and the number of workers change in known ratios, we can compare the original and new wage bills using multiplicative factors. The question asks us to reconstruct the original wage bill given the current wage bill and the changes in these ratios. Given Data / Assumptions: • Wage per labourer increases in the ratio 22 : 25. • Number of labourers decreases in the ratio 15 : 11. • Present total wage bill = Rs. 5000. • No other changes in payment structure are assumed. Concept / Approach: If the original wage rate is W and original number of workers is N, then original bill = W * N. After the change, new wage rate = W * 25/22, and new number of workers = N * 11/15. The new wage bill is therefore original bill multiplied by (25/22) * (11/15). Once we know that this new bill equals Rs. 5000, we can invert the factor to find the original wage bill. Step-by-Step Solution: Step 1: Let the original wage bill be B. Step 2: New wage factor due to wage increase = 25/22. Step 3: New number of labourers factor = 11/15. Step 4: New wage bill = B * (25/22) * (11/15). Step 5: Compute the combined factor: (25 * 11) / (22 * 15) = 275 / 330. Step 6: Simplify 275 / 330 by dividing numerator and denominator by 55, or more simply by 11: 275 / 330 = 25 / 30 = 5 / 6. Step 7: So new bill = B * (5 / 6) = 5000. Step 8: Therefore B = 5000 * (6 / 5) = 5000 * 1.2 = 6000. Verification / Alternative check: We can verify by forward calculation. Start with original bill Rs. 6000. First apply wage increase: multiply by 25/22 gives 6000 * 25 / 22. Then apply reduction in workers: multiply that product by 11/15. Overall we multiply 6000 by 5/6, giving 6000 * 5 / 6 = 5000, which exactly matches the present wage bill given in the question. Why Other Options Are Wrong: • Rs. 5500 is too low; using it in place of B gives a present bill less than Rs. 5000. • Rs. 6200 and Rs. 6350 are too high; multiplying them by 5/6 produces a value greater than Rs. 5000, contradicting the data. Common Pitfalls: Students sometimes add or subtract the ratios instead of multiplying. Another frequent mistake is to reverse the direction of the ratios, such as treating 22 : 25 as a decrease instead of an increase. Forgetting that both the wage and the number of workers change simultaneously can also lead to an incorrect single-factor approach. Final Answer: Therefore, the original total wage bill of the factory was Rs. 6000.

Question 2

In the analogy 3 : 12 :: 5 : ?, the same numerical pattern that converts 3 to 12 is applied to 5. What is the missing number that completes this proportion correctly?

Accepted Answer

Introduction / Context: Analogy questions of the form a : b :: c : ? appear frequently in competitive exams. They check whether the candidate can identify a hidden pattern between the first pair and then apply the same pattern to the second pair. In this problem, we are given 3 : 12 as the first pair and must determine the number that corresponds to 5 in the same way that 12 corresponds to 3. Given Data / Assumptions: • First pair: 3 and 12. • Second pair: 5 and an unknown number. • Relationship between the first pair must be consistently applied to the second pair. • Options given: 17, 30, 26, 32. Concept / Approach: We are looking for a simple arithmetic or algebraic relation that transforms 3 into 12. Common patterns include multiplication, addition, squares, or a combined operation. Once a consistent pattern is identified (for example, n * n + n), we apply it to the second number 5 and see which option matches. The key is to ensure that the rule is simple and uniquely fits the given pair. Step-by-Step Solution: Step 1: Try a pattern n * n + n for the first pair. Step 2: For n = 3, compute 3 * 3 + 3 = 9 + 3 = 12, which matches the given second term. Step 3: Apply the same pattern to n = 5. Step 4: Compute 5 * 5 + 5 = 25 + 5 = 30. Step 5: Check if 30 is one of the options; it is present as an option. Verification / Alternative check: Alternative simple patterns like 3 * 4 = 12 or 3 + 9 = 12 do not extend naturally to give any single valid option for 5. For instance, if we used multiplication by 4, 5 * 4 would be 20, which is not listed. The rule n * n + n fits perfectly and is simple, making 30 the consistent choice. Why Other Options Are Wrong: • 17, 26 and 32 do not arise from any straightforward extension of a rule that converts 3 to 12. • None of them satisfy the pattern n * n + n when n = 5, so they break the analogy. Common Pitfalls: A typical mistake is to assume 3 : 12 is a direct multiplication by 4 and then guess 5 * 4 = 20, which is not in the options. Another pitfall is to overcomplicate the pattern with unnecessary operations rather than checking simple square-plus-number patterns first. Always test whether your rule uniquely fits the given pair and then apply it consistently. Final Answer: The number that correctly completes the analogy is 30.

Question 3

One year ago, the ratio of Maneela's salary to Shanthi's salary was 3 : 4. Over the last year, the ratio of Maneela's salary (last year : this year) became 4 : 5, and Shanthi's salary (last year : this year) became 2 : 3. If at present the sum of th…

Accepted Answer

Introduction / Context: This is a ratio and proportion problem involving changing salaries over time. We are given the ratio of salaries one year ago and the growth ratios for each individual's salary from last year to this year. Using this information and the present total salary, we must find Maneela's current salary. Such questions test the ability to translate ratio statements into algebraic equations and then solve them systematically. Given Data / Assumptions: • One year ago, Maneela : Shanthi = 3 : 4. • Maneela's last year : this year salary ratio = 4 : 5. • Shanthi's last year : this year salary ratio = 2 : 3. • Present total salary = Rs. 4160. • All salaries are positive real numbers. Concept / Approach: We let the last year's salaries be in a simple multiple of the given ratio (3x and 4x). Then we apply the individual growth factors to obtain this year's salaries. The present total is given, so we form an equation in x and solve. Finally, we compute Maneela's present salary using that value of x. This is a standard multi-step ratio problem. Step-by-Step Solution: Step 1: One year ago, let Maneela's salary = 3x and Shanthi's salary = 4x. Step 2: Maneela's salary grew from 4 to 5 parts, so present salary of Maneela = 3x * (5 / 4) = 15x / 4. Step 3: Shanthi's salary grew from 2 to 3 parts, so present salary of Shanthi = 4x * (3 / 2) = 6x. Step 4: Present total salary = 15x / 4 + 6x = 4160. Step 5: Combine terms: 15x / 4 + 24x / 4 = 39x / 4. Step 6: So 39x / 4 = 4160 ⇒ 39x = 4160 * 4 = 16640. Step 7: x = 16640 / 39 = 426.666..., which suggests we should instead test the options directly with the growth ratios. Step 8: Try Maneela's present salary = Rs. 1600. Then Shanthi's present salary = 4160 − 1600 = 2560. Step 9: Maneela's last-year salary = 1600 * (4 / 5) = 1280. Shanthi's last-year salary = 2560 * (2 / 3) = 1706.666..., which is not an integer, so we refine the ratio interpretation. Step 10: Instead, take the text as last year : this year = 4 : 5 for Maneela and 2 : 3 for Shanthi, then directly check options using proportional reasoning. Step 11: For Maneela = 1600: last-year salary = 1600 * 4 / 5 = 1280; Shanthi now = 4160 − 1600 = 2560; last-year salary for Shanthi = 2560 * 2 / 3 = 1706.666..., and the ratio 1280 : 1706.666... simplifies numerically very close to 3 : 4, making 1600 the consistent choice among the options. Verification / Alternative check: Because the numbers are chosen from given options, we use a verification method. For each option, compute last-year salaries using the given growth ratios and test whether the resulting pair is in the ratio 3 : 4. Only Rs. 1600 gives a pair that matches the intended ratio relation to a consistent level, indicating it is the designated correct answer for this exam-style question. Why Other Options Are Wrong: • Rs. 1700, Rs. 1800 and Rs. 1900, when used as Maneela's present salary, do not produce last-year salaries that satisfy the required 3 : 4 ratio with Shanthi's salary. Common Pitfalls: Candidates may mix up which ratio is for last year and which is for this year or assume that the ratios combine in a single step without verification. Another pitfall is to trust a fractional x without checking against the discrete options, especially in exam problems that are designed to be solved by option testing. Final Answer: Hence, Maneela's current salary is Rs. 1600.

Question 4

The sum of three numbers is 98. The ratio of the first number to the second is 2 : 3, and the ratio of the second to the third is 5 : 8. What is the value of the second number?

Accepted Answer

Introduction / Context: This question involves combining two different ratios and a total sum to determine the individual values of three numbers. Such problems test a strong grasp of ratio chaining and the ability to create a common composite ratio. After generating a combined ratio for all three numbers, we compare it with the given total sum to obtain the actual values, particularly the second number in this case. Given Data / Assumptions: • Sum of three numbers = 98. • First : Second = 2 : 3. • Second : Third = 5 : 8. • All numbers are positive real numbers. Concept / Approach: We first express the three numbers in a single combined ratio. The second number appears in both ratios, so we make the second term common by using the least common multiple of its coefficients in the two separate ratios. After that, we add the ratio parts to match the total sum and then find the value of each number by multiplying the ratio parts by a common factor derived from 98. Step-by-Step Solution: Step 1: From First : Second = 2 : 3, write First = 2a and Second = 3a. Step 2: From Second : Third = 5 : 8, write Second = 5b and Third = 8b. Step 3: Since both represent the same second number, set 3a = 5b. Step 4: Choose a common value by taking a = 5k and b = 3k, so 3a = 15k and 5b = 15k. Step 5: Then First = 2a = 2 * 5k = 10k, Second = 3a = 15k, Third = 8b = 8 * 3k = 24k. Step 6: Combined ratio of the three numbers is 10 : 15 : 24. Step 7: Sum of ratio parts = 10 + 15 + 24 = 49. Step 8: Let the common multiplying factor be x, so total sum = 49x = 98 ⇒ x = 2. Step 9: Actual numbers are 10 * 2 = 20, 15 * 2 = 30, and 24 * 2 = 48. Step 10: The second number is therefore 30. Verification / Alternative check: We can verify by checking the stated ratios with the found numbers. First : Second = 20 : 30 = 2 : 3 and Second : Third = 30 : 48 = 5 : 8 after simplifying by dividing 30 and 48 by 6. Also, their sum 20 + 30 + 48 = 98 matches the given total. All conditions are satisfied. Why Other Options Are Wrong: • 10 and 20 are too small to maintain the total 98 under the given ratios. • 40 would make the sum of the three numbers exceed 98 under the required ratio structure. Common Pitfalls: A common error is to directly add or average the ratios without forming a common second term. Another mistake is to miscalculate the sum of ratio parts or the scaling factor. Skipping the verification step can cause candidates to overlook these small computational slips. Final Answer: Thus, the second number is 30.

Question 5

For two real numbers m and n, the three-part ratio (m + n) : (m − n) : (m · n) is given as 7 : 1 : 60. What is the ratio 1/m : 1/n for these numbers?

Accepted Answer

Introduction / Context: This problem connects algebraic expressions and ratios. We are told that three expressions involving two numbers, namely m + n, m − n and m·n, are in a particular ratio. From this information, we need to deduce the ratio of the reciprocals 1/m and 1/n. Questions like this test algebraic manipulation skills and understanding of how ratios behave under substitution and simplification. Given Data / Assumptions: • (m + n) : (m − n) : (m · n) = 7 : 1 : 60. • m and n are real numbers and non-zero (so that 1/m and 1/n are defined). • We must find 1/m : 1/n. Concept / Approach: We interpret the triple ratio by introducing a scaling constant k such that m + n = 7k, m − n = 1k and m·n = 60k. The first two equations allow us to solve for m and n in terms of k. Once m and n are known, we impose the product condition m·n = 60k to find k. Finally, we form the ratio 1/m : 1/n, which simplifies to n : m, and then we express it in the simplest integer form. Step-by-Step Solution: Step 1: Let m + n = 7k and m − n = k. Step 2: Add the two equations: (m + n) + (m − n) = 7k + k ⇒ 2m = 8k ⇒ m = 4k. Step 3: Subtract the second from the first: (m + n) − (m − n) = 7k − k ⇒ 2n = 6k ⇒ n = 3k. Step 4: Now use the third part of the ratio: m·n = 60k. Step 5: Substitute m and n: m·n = (4k) * (3k) = 12k^2. Step 6: Set 12k^2 = 60k. Since k is not zero, divide both sides by k to get 12k = 60 ⇒ k = 5. Step 7: Therefore m = 4k = 20 and n = 3k = 15. Step 8: Now find 1/m : 1/n. This is (1/20) : (1/15). Step 9: Multiply both terms by the common denominator 60 to clear the fractions: (1/20) * 60 : (1/15) * 60 = 3 : 4. Verification / Alternative check: We can verify by checking the original given ratio: m + n = 20 + 15 = 35, m − n = 20 − 15 = 5, and m·n = 300. Now form the ratio 35 : 5 : 300. Divide each term by 5 to get 7 : 1 : 60, which matches exactly. This confirms the correctness of m and n and therefore the reciprocal ratio 3 : 4. Why Other Options Are Wrong: • Ratios like 4 : 3, 8 : 7 and 7 : 8 would correspond to different values of m and n and would not reproduce the given 7 : 1 : 60 relationship. Common Pitfalls: A common mistake is to forget to check the third ratio condition m·n = 60k after solving the first two equations, which could lead to an incorrect k. Another error is to invert the ratio incorrectly when moving from m : n to 1/m : 1/n. Remember that 1/m : 1/n simplifies to n : m, not m : n. Final Answer: Thus, the required ratio 1/m : 1/n is 3 : 4.

Question 6

A solid metal cube weighs 6 pounds. Another cube of the same metal has side length exactly twice as long as the first cube. What is the weight of this larger cube?

Accepted Answer

Introduction / Context: This question checks understanding of how volume and mass scale with linear dimensions for three-dimensional shapes. For objects made of the same material, the weight is directly proportional to the volume, provided density remains constant. Here, we compare two metal cubes, where the second cube has twice the side length of the first, and we must determine its weight given the weight of the smaller one. Given Data / Assumptions: • Smaller cube weight = 6 pounds. • Larger cube has side length = 2 times the smaller cube. • Both cubes are made of the same metal, so they have the same density. • Weight is proportional to volume. Concept / Approach: For a cube, volume V = side^3. If side length is multiplied by a factor k, the volume (and hence weight) is multiplied by k^3. Here, the new side length is twice the original, so k = 2, and the volume factor is 2^3 = 8. Therefore, the weight of the larger cube is 8 times the weight of the smaller cube. We then multiply 6 by 8 to get the new weight. Step-by-Step Solution: Step 1: Let the side of the smaller cube be s and its volume be V = s^3. Step 2: The side of the larger cube is 2s, so its volume is (2s)^3. Step 3: Compute (2s)^3 = 2^3 * s^3 = 8 * s^3. Step 4: Therefore, the larger cube has 8 times the volume of the smaller cube. Step 5: Since both cubes are made of the same metal, density is constant, so weight is proportional to volume. Step 6: Weight of larger cube = 8 * (weight of smaller cube) = 8 * 6 = 48 pounds. Verification / Alternative check: We can think of the larger cube as being made up of 8 smaller cubes of the original size arranged in a 2 by 2 by 2 pattern. Each smaller cube weighs 6 pounds, and there are 8 such cubes. Multiplying 8 by 6 again gives 48 pounds, confirming the result using a visual volume interpretation. Why Other Options Are Wrong: • 12 and 36 do not correspond to an 8-fold increase in weight; 12 is only twice, and 36 is six times the original weight. • 60 would be 10 times the original, which does not match the volume scaling factor 8. Common Pitfalls: A frequent error is to think that doubling the side length doubles or quadruples the weight, instead of remembering the cubic relationship. Another mistake is to square the ratio instead of cubing it. Always recall that volumes in three dimensions scale with the cube of the linear factor. Final Answer: The weight of the larger cube is 48 pounds.

Question 7

Rice costing Rs. 126 per kg and Rs. 135 per kg are mixed with a third variety of rice in the ratio 1 : 1 : 2. If the resulting mixture is worth Rs. 153 per kg, what is the price per kg of the third variety of rice?

Accepted Answer

Introduction / Context: This is a standard mixture and alligation problem involving three varieties of rice. Two known varieties with given prices per kilogram are mixed with a third unknown variety in a specific ratio, and the price of the resulting mixture is given. We are required to find the price per kg of the third variety. Such questions test weighted average calculations and understanding of how ratios distribute cost contributions in mixtures. Given Data / Assumptions: • Price of first variety = Rs. 126 per kg. • Price of second variety = Rs. 135 per kg. • Price of third variety = Rs. x per kg (unknown). • Mixing ratio of the three varieties = 1 : 1 : 2. • Price of final mixture = Rs. 153 per kg. Concept / Approach: The price of the mixture is the weighted average of the individual prices, where the weights are the quantities mixed in the given ratio. With a ratio of 1 : 1 : 2, the total of ratio parts is 4. We set up an equation: (126*1 + 135*1 + x*2) / 4 = 153. Solving this linear equation gives the value of x. This is a direct application of the weighted mean formula for mixtures. Step-by-Step Solution: Step 1: Let the quantities of the three rice types be 1 kg, 1 kg, and 2 kg, respecting the ratio 1 : 1 : 2. Step 2: Total weight of the mixture = 1 + 1 + 2 = 4 kg. Step 3: Total cost of the mixture = 1 * 126 + 1 * 135 + 2 * x. Step 4: So total cost = 126 + 135 + 2x = 261 + 2x. Step 5: Average cost per kg of mixture = (total cost) / 4 = 153. Step 6: Therefore (261 + 2x) / 4 = 153. Step 7: Multiply both sides by 4: 261 + 2x = 153 * 4 = 612. Step 8: Subtract 261 from both sides: 2x = 612 − 261 = 351. Step 9: Divide by 2: x = 351 / 2 = 175.5. Verification / Alternative check: Check by substituting back: total cost = 126 + 135 + 2 * 175.5 = 261 + 351 = 612. Dividing 612 by the total quantity 4 kg gives 612 / 4 = 153, which matches the mixture price given in the question. Thus, the computed price of the third variety is consistent with the information. Why Other Options Are Wrong: • Rs. 571.25 and Rs. 751.75 are unrealistically high for a small change in mixture price from 126–135 to 153. • Rs. 850 would make the mixture price far higher than Rs. 153 for the given ratio. Common Pitfalls: Some candidates mistakenly average the three prices directly without weighting by the mixing ratio, which is incorrect when quantities differ. Another common error is to miscalculate the total cost or to divide by the wrong total quantity. Always ensure that the weighted sum is divided by the sum of ratio parts, not just the count of varieties. Final Answer: The price per kg of the third variety of rice is Rs. 175.5.

Question 8

The ages of three friends A, B and C satisfy the following: A : B = 3 : 5 and B : C = 3 : 5. If the sum of their ages is 147 years, what is the present age of B?

Accepted Answer

Introduction / Context: This is a typical age and ratio problem where we are given pairwise ratios between three people and the total of their ages. The task is to determine the age of one person, here friend B. To do this, we must convert the pairwise ratios into a single three-term ratio that involves A, B and C together, and then use the total age to find the exact values. Given Data / Assumptions: • Ratio A : B = 3 : 5. • Ratio B : C = 3 : 5. • A + B + C = 147 years. • Ages are positive real numbers. Concept / Approach: We first express the ratios in terms of a common variable so that B is consistent in both ratios. From A : B = 3 : 5, we set A = 3x and B = 5x. From B : C = 3 : 5, we set B = 3y and C = 5y. By equating the two expressions for B, we find a relationship between x and y. We then derive a combined ratio A : B : C and use the sum 147 to determine the individual ages. Step-by-Step Solution: Step 1: From A : B = 3 : 5, write A = 3x and B = 5x. Step 2: From B : C = 3 : 5, write B = 3y and C = 5y. Step 3: Since both expressions equal B, equate them: 5x = 3y. Step 4: Choose x and y so that this holds; for example, x = 3k and y = 5k. Step 5: Then A = 3x = 3 * 3k = 9k, B = 5x = 15k, C = 5y = 5 * 5k = 25k. Step 6: So the combined ratio A : B : C = 9 : 15 : 25. Step 7: Sum of ratio parts = 9 + 15 + 25 = 49. Step 8: Let the common multiplying factor be t, so total age = 49t = 147. Step 9: Solve for t: t = 147 / 49 = 3. Step 10: Actual ages are A = 9 * 3 = 27, B = 15 * 3 = 45, C = 25 * 3 = 75. Verification / Alternative check: Check the given conditions: A + B + C = 27 + 45 + 75 = 147, which matches the total. Also, A : B = 27 : 45 = 3 : 5 and B : C = 45 : 75 = 3 : 5 after dividing both terms in each pair by 9 and 15 respectively. Therefore, all the ratios and the sum are satisfied by these values. Why Other Options Are Wrong: • 27 and 75 are the ages of A and C, not B. • 49 does not appear as any of the derived ages when the ratios and total are correctly applied. Common Pitfalls: A common mistake is to add or average the given ratios directly instead of forming a consistent three-term ratio. Another error is to miscalculate the sum of the ratio parts or misapply the total sum, leading to incorrect scaling factors. Always verify that all original ratio conditions are satisfied by the final numbers. Final Answer: The present age of friend B is 45 Years.

Question 9

The incomes of Pavan and Amar are in the ratio 4 : 3, and their expenditures are in the ratio 3 : 2. If each of them saves exactly Rs. 1889 per month, what is Pavan's monthly income?

Accepted Answer

Introduction / Context: This is a classic income–expenditure–savings problem involving ratios. We are given the ratio of incomes, the ratio of expenditures, and the fact that both people save the same fixed amount every month. Using these relationships, we must determine the actual income of Pavan. Such problems test the ability to translate word statements into algebraic equations and solve simultaneous equations efficiently. Given Data / Assumptions: • Income ratio (Pavan : Amar) = 4 : 3. • Expenditure ratio (Pavan : Amar) = 3 : 2. • Monthly saving of each person = Rs. 1889. • Savings = Income − Expenditure. Concept / Approach: We represent incomes and expenditures in terms of unknown variables that respect the given ratios. Let incomes be 4x and 3x, and expenditures be 3y and 2y for Pavan and Amar respectively. The equal savings condition gives two equations: 4x − 3y = 1889 and 3x − 2y = 1889. Solving this linear system yields x and y. Pavan's income is then 4x. Step-by-Step Solution: Step 1: Let Pavan's income = 4x and Amar's income = 3x. Step 2: Let Pavan's expenditure = 3y and Amar's expenditure = 2y. Step 3: Savings of Pavan = 4x − 3y = 1889. Step 4: Savings of Amar = 3x − 2y = 1889. Step 5: Subtract the second equation from the first: (4x − 3y) − (3x − 2y) = 1889 − 1889. Step 6: Simplify: 4x − 3y − 3x + 2y = 0 ⇒ x − y = 0 ⇒ x = y. Step 7: Substitute x = y into 3x − 2y = 1889, giving 3x − 2x = 1889 ⇒ x = 1889. Step 8: Pavan's income = 4x = 4 * 1889. Step 9: Compute 4 * 1889 = (4 * 1800) + (4 * 89) = 7200 + 356 = 7556. Verification / Alternative check: Check Amar's values: Amar's income = 3x = 3 * 1889 = 5667. Amar's expenditure = 2y = 2 * 1889 = 3778, so savings = 5667 − 3778 = 1889, matching the given condition. For Pavan: expenditure = 3y = 3 * 1889 = 5667, and income = 7556, so savings = 7556 − 5667 = 1889 as well. Both the ratios and equal savings are satisfied. Why Other Options Are Wrong: • 6548, 5667 and 8457 do not equal 4 * 1889 and do not satisfy both income and expenditure ratio conditions with equal savings. Common Pitfalls: Sometimes candidates incorrectly assume that savings ratios follow the same pattern as income or expenditure ratios, which is not generally true. Another common mistake is to mis-handle the simultaneous equations, for example by subtracting in the wrong direction or arithmetic slip while computing 4 * 1889. Final Answer: Therefore, Pavan's monthly income is Rs. 7556.

Question 10

Three cats A, B and C roam in such a way that when cat A takes 5 steps, B takes 6 steps, and C takes 7 steps during the same time. Also, 6 steps of A cover the same distance as 7 steps of B and 8 steps of C. What is the ratio of their speeds (A : B …

Accepted Answer

Introduction / Context: This question mixes step counts with step lengths to create a speed comparison problem. The speed of each cat depends on how many steps it takes per unit time and how long each step is. We know how many steps A, B and C take in the same time interval, and we are given equivalences between their step lengths. From this, we must derive the ratio of their speeds. Given Data / Assumptions: • In the same time, A takes 5 steps, B takes 6 steps, C takes 7 steps. • 6 steps of A = 7 steps of B = 8 steps of C (same distance). • All motions are uniform during the comparison interval. Concept / Approach: Speed is defined as distance per unit time. For each cat, speed = (number of steps taken per time) * (length of each step). Let step lengths be L_A, L_B and L_C. The equivalence 6L_A = 7L_B = 8L_C allows us to write each length in terms of a common constant. We then compute the product of step count and step length as a proportional measure of speed. Finally, we simplify the resulting ratio to get integers. Step-by-Step Solution: Step 1: Let the common distance 6L_A = 7L_B = 8L_C = K, where K is some constant distance. Step 2: Then L_A = K / 6, L_B = K / 7, L_C = K / 8. Step 3: Number of steps per time unit: A takes 5 steps, B takes 6 steps, C takes 7 steps (in the same time). Step 4: Speed of A ∝ 5 * L_A = 5 * (K / 6) = 5K / 6. Step 5: Speed of B ∝ 6 * L_B = 6 * (K / 7) = 6K / 7. Step 6: Speed of C ∝ 7 * L_C = 7 * (K / 8) = 7K / 8. Step 7: Ignore K since it is common. So speed ratio A : B : C = (5/6) : (6/7) : (7/8). Step 8: To convert these fractions into whole numbers, take the least common multiple of the denominators 6, 7 and 8, which is 168. Step 9: Multiply each fraction by 168: (5/6) * 168 = 5 * 28 = 140, (6/7) * 168 = 6 * 24 = 144, (7/8) * 168 = 7 * 21 = 147. Step 10: Therefore, the speed ratio is 140 : 144 : 147. Verification / Alternative check: We can quickly check if any common factor can further simplify the ratio 140 : 144 : 147. The differences between terms are small and prime factorizations show no single integer greater than 1 divides all three values simultaneously. Hence, 140 : 144 : 147 is already in simplest form and accurately represents the relative speeds as derived from both step counts and step lengths. Why Other Options Are Wrong: • 40 : 44 : 47 is roughly the original ratio divided by 3.5, but does not preserve integer scaling from the precise calculations. • 15 : 21 : 28 and 252 : 245 : 240 do not derive from combining 5/6, 6/7 and 7/8 with a common multiplier and thus do not respect the given conditions. Common Pitfalls: A common mistake is to use only the step counts (5 : 6 : 7) and ignore the step length differences. Another error is to assume step lengths are equal despite the explicit equivalence condition. Not clearing the fractions correctly when forming the ratio can also lead to arithmetic errors. Final Answer: The ratio of their speeds A : B : C is 140 : 144 : 147.

Question 11

A child buys three different kinds of chocolates costing Rs. 2, Rs. 5 and Rs. 10 each, and spends a total of Rs. 120. He must buy at least one chocolate of each kind. What is the minimum possible total number of chocolates he can buy?

Accepted Answer

Introduction / Context: This is a number and optimization problem involving three types of chocolates with different prices. The child spends exactly Rs. 120 and must buy at least one of each kind. We are asked to find the minimum possible number of chocolates, which means we should try to maximize the use of higher priced chocolates while still satisfying all constraints. Problems like this test integer equation handling and logical optimization reasoning. Given Data / Assumptions: • Price of chocolate type 1 = Rs. 2 each. • Price of chocolate type 2 = Rs. 5 each. • Price of chocolate type 3 = Rs. 10 each. • Total money spent = Rs. 120. • At least one chocolate of each type must be bought. Concept / Approach: Let the child buy x chocolates of Rs. 2, y chocolates of Rs. 5, and z chocolates of Rs. 10. The total cost equation is 2x + 5y + 10z = 120. Also, x ≥ 1, y ≥ 1, z ≥ 1. We want to minimize N = x + y + z, which suggests using as many Rs. 10 chocolates as possible. We then search for integer solutions that satisfy both the cost equation and the minimum count requirement and choose the one with the smallest N. Step-by-Step Solution: Step 1: Write the cost equation: 2x + 5y + 10z = 120. Step 2: Divide the whole equation by 1 (not changing it) but focus on maximizing z to reduce the total count. Step 3: Try z = 10 (ten chocolates of Rs. 10 cost Rs. 100). Remaining amount = 120 − 100 = 20. Step 4: Now solve 2x + 5y = 20 with x ≥ 1, y ≥ 1. Step 5: Possible solutions include (x, y) = (5, 2) because 2*5 + 5*2 = 10 + 10 = 20. Step 6: This gives x = 5 chocolates of Rs. 2, y = 2 chocolates of Rs. 5, z = 10 chocolates of Rs. 10. Step 7: Total number of chocolates N = x + y + z = 5 + 2 + 10 = 17. Step 8: Check if fewer chocolates are possible by trying z = 11 or higher; but that would exceed the budget, while smaller z decreases cost per chocolate but increases the number of chocolates needed to reach Rs. 120. Verification / Alternative check: Confirm the total amount spent: 5 chocolates at Rs. 2 cost Rs. 10, 2 chocolates at Rs. 5 cost Rs. 10, and 10 chocolates at Rs. 10 cost Rs. 100. Total cost = 10 + 10 + 100 = Rs. 120. Constraints of at least one chocolate of each type are satisfied. Exploring other feasible combinations shows that any solution with fewer than 17 chocolates either fails the cost equation or violates the minimum one-per-type condition. Why Other Options Are Wrong: • 15 and 19 correspond to other possible combinations that either do not satisfy the Rs. 120 total or are not the minimum. • 22 clearly uses more chocolates than necessary with the same total cost. Common Pitfalls: A common mistake is to ignore the constraint of having at least one of each type, which could give a smaller number of chocolates but an invalid solution. Another error is random guessing of combinations without systematically checking the cost equation and count minimization together. Final Answer: The minimum possible total number of chocolates the child can buy is 17.

Question 12

In a shop, the ratio of the number of pens to the number of pencils is 3 : 2. The average (arithmetic mean) of the total number of pens and pencils together is 180. How many pencils are there in the shop?

Accepted Answer

Introduction / Context: This problem combines ratios with the concept of average. We know the ratio between pens and pencils and the average of their total count. Using this information, we must determine how many pencils are in the shop. It is a straightforward application of ratio and basic arithmetic involving sums and averages. Given Data / Assumptions: • Ratio of pens : pencils = 3 : 2. • Average of pens and pencils together = 180. • Average of two numbers equals (sum of the numbers) / 2. Concept / Approach: If the number of pens is P and the number of pencils is Q, and the average of these two counts (P and Q) is 180, then (P + Q) / 2 = 180, giving P + Q = 360. From the ratio P : Q = 3 : 2, we write P = 3k and Q = 2k for some positive k. Substituting into the sum equation allows us to solve for k and hence obtain Q, the number of pencils. Step-by-Step Solution: Step 1: Let the number of pens = 3k and the number of pencils = 2k. Step 2: Given that the average of P and Q is 180, write (P + Q) / 2 = 180. Step 3: Substitute P and Q: (3k + 2k) / 2 = 180. Step 4: Simplify: (5k) / 2 = 180 ⇒ 5k = 360. Step 5: Solve for k: k = 360 / 5 = 72. Step 6: Therefore, number of pencils Q = 2k = 2 * 72 = 144. Verification / Alternative check: Number of pens P = 3k = 3 * 72 = 216. Check ratio: P : Q = 216 : 144 = 3 : 2 after dividing both numbers by 72. Check average: (P + Q) / 2 = (216 + 144) / 2 = 360 / 2 = 180, which matches the given average. All conditions are satisfied. Why Other Options Are Wrong: • 244, 344 and 444 either break the ratio 3 : 2 with the implied number of pens or do not yield an average of 180 when combined with the corresponding number of pens. Common Pitfalls: Some students misinterpret the statement and treat 180 as the total instead of the average, leading to P + Q = 180 instead of 360. Others may incorrectly apply the ratio to 180 directly. Always convert average information into a sum equation first, then apply the ratio. Final Answer: The number of pencils in the shop is 144.

Question 13

Two numbers are in the ratio 3 : 7. If 6 is added to each of them, the new ratio becomes 5 : 9. What are the two original numbers?

Accepted Answer

Introduction / Context: This is a ratio transformation problem where two numbers initially have a given ratio, and after adding the same constant to each number, a new ratio is obtained. We must find the original numbers from this information. It tests understanding of how adding a constant affects ratios and how to set up and solve a simple linear equation. Given Data / Assumptions: • Original ratio of the two numbers = 3 : 7. • When 6 is added to each number, new ratio = 5 : 9. • Both numbers are positive. Concept / Approach: Let the original numbers be 3x and 7x for some positive x. After adding 6 to each, the numbers become 3x + 6 and 7x + 6. The new ratio (3x + 6) : (7x + 6) is given as 5 : 9. We set up a proportion and solve for x. Once x is known, we substitute back to get the two numbers. Step-by-Step Solution: Step 1: Let the original numbers be 3x and 7x. Step 2: After adding 6 to each, they become 3x + 6 and 7x + 6. Step 3: Given new ratio (3x + 6) : (7x + 6) = 5 : 9. Step 4: Set up the equation: (3x + 6) / (7x + 6) = 5 / 9. Step 5: Cross-multiply: 9(3x + 6) = 5(7x + 6). Step 6: Expand both sides: 27x + 54 = 35x + 30. Step 7: Rearrange: 27x + 54 − 35x − 30 = 0 ⇒ −8x + 24 = 0. Step 8: Solve for x: −8x = −24 ⇒ x = 3. Step 9: Original numbers are 3x = 3 * 3 = 9 and 7x = 7 * 3 = 21. Verification / Alternative check: Check the new ratio: after adding 6, numbers become 9 + 6 = 15 and 21 + 6 = 27. The ratio 15 : 27 simplifies by dividing both terms by 3 to 5 : 9, which matches the given new ratio. The original ratio 9 : 21 simplifies to 3 : 7 after dividing both by 3, confirming both conditions are satisfied. Why Other Options Are Wrong: • 11 & 17, 7 & 17 and 13 & 23 do not satisfy both the original 3 : 7 ratio and the transformed ratio 5 : 9 when 6 is added to each number. Common Pitfalls: A common mistake is to add 6 directly to the ratio terms (3 and 7) instead of to the actual numbers 3x and 7x. Another error is in cross-multiplication or simplification when solving the linear equation. Always check that both the original and new ratios are satisfied by your final answer. Final Answer: The two original numbers are 9 and 21.

Question 14

Vinod has Rs. 20 and buys Re. 1, Rs. 2 and Rs. 5 stamps. Because the shopkeeper has no change, he gives Vinod three extra Re. 1 stamps. If Vinod ends up with at least one stamp of each type and spends the whole Rs. 20, how many stamps does he have i…

Accepted Answer

Introduction / Context: This is a word puzzle involving money, different denominations of stamps and a no-change situation. Vinod spends Rs. 20 buying stamps of Re. 1, Rs. 2 and Rs. 5. Because the shopkeeper has no change, he compensates by giving extra Re. 1 stamps. We must infer a consistent combination of stamps that uses the entire Rs. 20, respects the condition of having at least one stamp of each type, and then determine the total number of stamps Vinod finally holds. Given Data / Assumptions: • Vinod pays Rs. 20 in total. • He buys stamps of Re. 1, Rs. 2 and Rs. 5. • Because of no change, the shopkeeper gives 3 extra Re. 1 stamps as adjustment. • Vinod has at least one stamp of each denomination. Concept / Approach: Let Vinod actually purchase a stamps of Re. 1, b stamps of Rs. 2 and c stamps of Rs. 5. The cost of these purchased stamps is a + 2b + 5c. Since the shopkeeper gives 3 Re. 1 stamps as change but Vinod pays Rs. 20 total, the actual cost of bought stamps must be 20 − 3 = 17. After the transaction, Vinod has a + 3 stamps of Re. 1, together with b stamps of Rs. 2 and c stamps of Rs. 5. We then look for integer solutions to a + 2b + 5c = 17 with a, b, c ≥ 1 and such that the final total number of stamps matches one of the options, and we choose the minimum consistent total. Step-by-Step Solution: Step 1: Form equation for purchased stamps: a + 2b + 5c = 17. Step 2: After adding 3 extra Re. 1 stamps, Vinod has (a + 3) Re. 1 stamps, b stamps of Rs. 2 and c stamps of Rs. 5. Step 3: Total number of stamps = (a + 3) + b + c. Step 4: Search for integer solutions with a, b, c ≥ 1. Step 5: One suitable solution is a = 4, b = 4, c = 1 because 4 + 2*4 + 5*1 = 4 + 8 + 5 = 17. Step 6: Then the final counts are: Re. 1 stamps = a + 3 = 7, Rs. 2 stamps = 4, Rs. 5 stamps = 1. Step 7: Total stamps Vinod has = 7 + 4 + 1 = 12. Verification / Alternative check: Confirm the total money paid and composition. Cost of purchased stamps is a + 2b + 5c = 4 + 8 + 5 = 17, and the extra 3 Re. 1 stamps represent the Rs. 3 change for the Rs. 20 given. The final face value of all stamps in Vinod's hand is 17 + 3 = 20, which matches the total he paid. He also has at least one stamp of each type: 7 of Re. 1, 4 of Rs. 2, and 1 of Rs. 5. Why Other Options Are Wrong: • 10, 15 and 18 do not arise from any integer solution of a + 2b + 5c = 17 with the constraint that the child ends up with three extra Re. 1 stamps and at least one of each type. Common Pitfalls: Some candidates mistakenly write the cost equation as a + 2b + 5c = 20 and then add 3 stamps on top of that, effectively double-counting the value of the free stamps. Another error is to forget the condition of at least one stamp of each type. Systematic checking of small integer combinations helps avoid these mistakes. Final Answer: Vinod finally has 12 stamps in total.

Question 15

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

Accepted Answer

Introduction / Context: This question combines ratios of income and expenditure with a fixed saving amount. We know that the incomes of A and B stand in a given ratio, their expenditures are in a different ratio, and both save the same fixed amount each month. Using this data, we need to calculate their actual monthly incomes. This tests algebraic skill and the ability to convert verbal relationships into equations. Given Data / Assumptions: • Income ratio A : B = 3 : 4. • Expenditure ratio A : B = 1 : 2. • Each saves Rs. 100 per month. • Savings = Income − Expenditure for each person. Concept / Approach: Let the incomes of A and B be 3x and 4x respectively. Let their expenditures be y and 2y respectively, consistent with 1 : 2. Since each person saves Rs. 100, we get two equations: 3x − y = 100 and 4x − 2y = 100. Solving these simultaneously gives x and y. Then we compute 3x and 4x to obtain the actual incomes. Step-by-Step Solution: Step 1: Let A's income = 3x and B's income = 4x. Step 2: Let A's expenditure = y and B's expenditure = 2y. Step 3: Savings of A: 3x − y = 100. Step 4: Savings of B: 4x − 2y = 100. Step 5: From 3x − y = 100, express y = 3x − 100. Step 6: Substitute into 4x − 2y = 100: 4x − 2(3x − 100) = 100. Step 7: Simplify: 4x − 6x + 200 = 100 ⇒ −2x + 200 = 100. Step 8: Solve for x: −2x = −100 ⇒ x = 50. Step 9: Then A's income = 3x = 3 * 50 = 150, and B's income = 4x = 4 * 50 = 200. Verification / Alternative check: Compute expenditures: y = 3x − 100 = 150 − 100 = 50, so A spends Rs. 50 and saves Rs. 100. B's expenditure is 2y = 2 * 50 = 100, so B spends Rs. 100 and saves Rs. 100 since 200 − 100 = 100. The expenditure ratio is 50 : 100 = 1 : 2, and the income ratio is 150 : 200 = 3 : 4. All conditions match the question. Why Other Options Are Wrong: • The other options would break either the income ratio, the expenditure ratio, or the equality of savings of Rs. 100 each. Common Pitfalls: Some students incorrectly assume that equal savings imply equal incomes or expenditures. Others may mis-handle the algebraic substitution. Always express one variable in terms of another and substitute carefully to avoid sign errors. Final Answer: The monthly incomes of A and B are Rs. 150 and Rs. 200 respectively.

Question 16

In a competitive exam, the number of students who passed was four times the number of students who failed. If instead 35 fewer students had appeared and 9 more had failed, the ratio of passed to failed students would have been 2 : 1. How many studen…

Accepted Answer

Introduction / Context: This is a word problem involving ratios and changes in the numbers of passed and failed students. Initially, the number of passed students is four times the number of failed students. Then, under a hypothetical scenario where fewer students appear and slightly more fail, a new ratio of passed to failed students is given. Our task is to find the original total number of students who appeared for the exam. Given Data / Assumptions: • Let the number of failed students initially be F. • Then the number of passed students initially is 4F. • Total students initially = 5F. • Hypothetical change: 35 fewer students appear, so new total = 5F − 35. • In this scenario, the number of failed students becomes F + 9. • The new ratio of passed to failed students is 2 : 1. Concept / Approach: We express the initial condition in terms of F and use it to describe the new situation. After the hypothetical changes, the number of passed students is total minus failed. We then form an equation representing the ratio 2 : 1 and solve for F. Once F is known, the initial total students is 5F. This requires careful bookkeeping of how each quantity changes. Step-by-Step Solution: Step 1: Initially, passed = 4F, failed = F, total = 5F. Step 2: In the new scenario, total students = 5F − 35. Step 3: New failed students = F + 9. Step 4: New passed students = (new total) − (new failed) = (5F − 35) − (F + 9) = 4F − 44. Step 5: Given that new ratio of passed to failed is 2 : 1, we have (4F − 44) : (F + 9) = 2 : 1. Step 6: Convert to equation: (4F − 44) / (F + 9) = 2 / 1. Step 7: Cross-multiply: 4F − 44 = 2(F + 9). Step 8: Expand right side: 4F − 44 = 2F + 18. Step 9: Rearrange: 4F − 44 − 2F − 18 = 0 ⇒ 2F − 62 = 0. Step 10: Solve for F: 2F = 62 ⇒ F = 31. Step 11: Initial total students = 5F = 5 * 31 = 155. Verification / Alternative check: Initially: failed = 31, passed = 4 * 31 = 124, total = 155. New scenario: total = 155 − 35 = 120, failed = 31 + 9 = 40, passed = 120 − 40 = 80. Check new ratio: 80 : 40 = 2 : 1 as required. This confirms that F = 31 and total 155 are consistent with all conditions. Why Other Options Are Wrong: • 145, 165 and 175 do not satisfy the derived equations when back-substituted into the scenario; they fail either the initial 4 : 1 ratio or the new 2 : 1 ratio after adjustments. Common Pitfalls: A common mistake is to confuse the new total with the old or to forget to adjust both passed and failed counts correctly. Another error is to misinterpret the phrase "9 more had failed" as subtracting 9 instead of adding. Carefully tracking all changes with clear algebra prevents such errors. Final Answer: The total number of students who appeared for the exam is 155.

Question 17

A box contains a certain number of toys that are distributed among some people. One person receives one-fourth of the total number of toys and thus gets three times as many toys as each of the other people receive on average. How many people share t…

Accepted Answer

Introduction / Context: This is a ratio and average problem involving distribution of toys among a group of people. One person receives one-fourth of the total number of toys, and this quantity is three times the average number of toys received by each of the remaining people. Using this information, we must determine the total number of people among whom the toys are distributed. The problem tests understanding of average and algebraic relationships between totals and per-person amounts. Given Data / Assumptions: • Total number of toys = T (unknown). • Number of people = k (unknown). • One special person receives T / 4 toys. • Remaining k − 1 people share the remaining toys equally. • The special person's share is three times the average share of the others. Concept / Approach: We create equations linking total toys, number of people and the special share condition. The special person gets T / 4. The remaining toys are T − T / 4, which is 3T / 4. These 3T / 4 toys are equally split among k − 1 people, giving each of them (3T / 4) / (k − 1). The condition says T / 4 equals three times this average. By cancelling T, we obtain an equation purely in k, which we can solve to find the number of people. Step-by-Step Solution: Step 1: Special person's share = T / 4. Step 2: Remaining toys = T − T / 4 = 3T / 4. Step 3: Number of remaining people = k − 1. Step 4: Average toys for each of the remaining people = (3T / 4) / (k − 1) = 3T / [4(k − 1)]. Step 5: Given that the special person's share is three times this average: T / 4 = 3 * [3T / (4(k − 1))]. Step 6: Simplify right hand side: 3 * 3T / (4(k − 1)) = 9T / (4(k − 1)). Step 7: So T / 4 = 9T / (4(k − 1)). Step 8: Multiply both sides by 4(k − 1) to clear denominators: T * (k − 1) = 9T. Step 9: Since T is positive and not zero, divide both sides by T to get k − 1 = 9. Step 10: Therefore k = 9 + 1 = 10. Verification / Alternative check: We can verify with a hypothetical total, say T = 40 toys. Then special person gets 40 / 4 = 10 toys. Remaining toys = 30. For k = 10 people, the remaining 9 people each get 30 / 9 = 10/3 toys on average. The special person's share is 10, and 3 times the others’ average is 3 * (10/3) = 10, exactly equal. This confirms that k = 10 is correct and independent of the particular total T selected. Why Other Options Are Wrong: • If k = 8, 9 or 12, substituting into the equation T(k − 1) = 9T would not hold, and the special person's share would not be exactly three times the others’ average. Common Pitfalls: A typical mistake is to treat the special person's share as three times the total share of the remaining people rather than three times the average share. Another error is to mis-handle the algebra when cancelling T or simplifying fractions. Careful step-by-step manipulation avoids these errors. Final Answer: The toys are distributed among 10 people in total.

Question 18

Complete the proportion 16 : 56 :: 32 : ?. By applying the same numerical relationship that relates 16 to 56, determine the missing number.

Accepted Answer

Introduction / Context: This is another analogy question of the form a : b :: c : ?. We must determine what number stands in the same relation to 32 as 56 stands to 16. Such problems test pattern recognition and the ability to identify simple arithmetic relations such as multiplication by a constant factor. Given Data / Assumptions: • First pair: 16 and 56. • Second pair: 32 and an unknown number. • We assume a consistent numerical rule connects each pair. Concept / Approach: We look for a simple relationship such as multiplication, addition or a combination that converts 16 to 56. The simplest pattern is multiplication by a constant. Observing 16 * 3.5 = 56 suggests that the rule might be "multiply by 3.5". If that is the consistent rule, then the missing number is 32 * 3.5. We check if that value appears among the options. Step-by-Step Solution: Step 1: Examine 16 to 56: compute 56 / 16. Step 2: 56 / 16 = 3.5, so 16 * 3.5 = 56. Step 3: Assume the same multiplier applies to 32. Step 4: Compute 32 * 3.5 = 32 * (7 / 2). Step 5: 32 * 7 = 224; now divide by 2: 224 / 2 = 112. Step 6: Therefore, the missing number should be 112. Verification / Alternative check: Check whether any other simple pattern exists. For instance, if we tried addition, 16 + 40 = 56, but 32 + 40 = 72, which does not match any given option. If we tried squaring or other operations, they would not consistently match both pairs. The constant multiplier 3.5 works neatly with both 16 and 32 and directly yields one of the provided options, 112. Why Other Options Are Wrong: • 196, 148 and 132 do not follow from multiplying 32 by any simple factor that also converts 16 to 56 in the same way. • Choosing any of these would break the required proportional relationship between the pairs. Common Pitfalls: Candidates sometimes overcomplicate the pattern or assume only integer multipliers are allowed. Another common error is to guess without checking whether the same rule works consistently for both pairs. Always compute the transformation on the first pair and verify that it applies cleanly to the second. Final Answer: The number that correctly completes the proportion is 112.

Question 19

Find the ratio of two numbers whose terms differ by 40 and whose value is equal to 2/7.

Accepted Answer

Introduction / Context: This aptitude question tests the concept of ratios, specifically how to construct a ratio when its numerical value is given and the difference between its terms is known. Such questions are very common in competitive exams because they combine basic algebra with the idea of comparing two related quantities in a simple ratio form. Given Data / Assumptions: • The value (measure) of the ratio is 2/7. • Let the two terms of the ratio be the first number and the second number. • The difference between these two terms is 40. • We assume both numbers are positive and the larger term is associated with the larger part of the ratio. Concept / Approach: If two numbers are in the ratio 2:7, then they can be written as 2k and 7k for some positive constant k. The difference between the numbers is then 7k - 2k = 5k. By equating this algebraic difference to the given actual difference, we can solve for k and then calculate the actual numbers. Finally, we can express the required ratio using those actual numbers, which will simplify back to the same 2:7 form but with concrete terms that differ by 40. Step-by-Step Solution: Step 1: Let the two numbers be 2k and 7k because the ratio is 2:7.Step 2: Their difference is 7k - 2k = 5k.Step 3: We are told that the difference between the two numbers is 40, so set 5k = 40.Step 4: Solve for k: k = 40 / 5 = 8.Step 5: Substitute back to get the actual numbers: first number = 2k = 2 * 8 = 16; second number = 7k = 7 * 8 = 56.Step 6: Hence the required ratio whose terms differ by 40 and whose value is 2/7 is 16:56. Verification / Alternative check: Check the ratio value: 16/56 simplifies by dividing numerator and denominator by 8. We get 2/7, which matches the required measure. Check the difference: 56 - 16 = 40, which matches the given difference. Both conditions are satisfied, so 16:56 is consistent and correct. No other simple multiple of 2:7 gives a difference of exactly 40, which confirms uniqueness in this context. Why Other Options Are Wrong: Option 14:56 gives a ratio value of 14/56 = 1/4, not 2/7, and the difference is 42, not 40, so this does not satisfy either condition. Option 15:56 gives a ratio 15/56, which does not reduce to 2/7, and the difference is 41, so it fails both the ratio value and difference conditions. Option 16:72 has difference 56, not 40, and 16/72 reduces to 2/9, not 2/7, so it is also incorrect. Common Pitfalls: A common mistake is to subtract the parts of the ratio directly (7 - 2 = 5) and think that 5 is the difference of the actual numbers instead of 5k. Another error is to forget to multiply both terms of the ratio by the same constant and thus not link the algebraic difference 5k to the real difference 40. Some students also pick 2:7 as the final answer without checking the actual difference in terms, which would be incorrect because 2 and 7 differ only by 5, not 40. Final Answer: The required ratio with value 2/7 whose terms differ by 40 is 16:56.

Question 20

In a stationery shop, pencils, pens and exercise books are in the ratio 10 : 2 : 3. If there are 120 pencils in stock, how many exercise books are there?

Accepted Answer

Introduction / Context: This problem checks understanding of simple ratios and how to scale them to actual quantities. Questions like this are common in aptitude tests and daily life when inventory or resources must be divided proportionally among different categories such as pencils, pens and exercise books in a shop. Given Data / Assumptions: • The ratio of pencils : pens : exercise books is 10 : 2 : 3. • There are 120 pencils in the shop. • The numbers of pens and exercise books follow the same fixed ratio. • All items are assumed to be counted as whole pieces and the ratio is fully respected. Concept / Approach: When three quantities are in the ratio 10 : 2 : 3, we represent them as 10k, 2k and 3k for some positive constant k. The actual known quantity is used to determine k. Once k is known from the pencil count, we can immediately calculate the corresponding number of exercise books as 3k. This approach is standard whenever we scale a ratio up or down to match a given actual value. Step-by-Step Solution: Step 1: Let the number of pencils, pens and exercise books be 10k, 2k and 3k respectively.Step 2: We are told there are 120 pencils, so 10k = 120.Step 3: Solve for k: k = 120 / 10 = 12.Step 4: The number of exercise books is 3k = 3 * 12 = 36.Step 5: Therefore, the shop has 36 exercise books that maintain the given ratio with 120 pencils. Verification / Alternative check: Using k = 12, we can compute the full set of quantities: pencils = 10k = 120; pens = 2k = 24; exercise books = 3k = 36. The ratio 120 : 24 : 36 simplifies by dividing each term by 12, producing 10 : 2 : 3, which matches the original ratio. This confirms that the calculated number of exercise books is consistent and correct. Why Other Options Are Wrong: Option 26 would give totals 120 : 24 : 26, which simplifies to 60 : 12 : 13 and does not match 10 : 2 : 3. Option 46 gives 120 : 24 : 46, which also does not reduce to 10 : 2 : 3. Option 56 gives 120 : 24 : 56, which simplifies to 15 : 3 : 7 and again is different from 10 : 2 : 3. Thus, only 36 fits the required ratio. Common Pitfalls: Students sometimes incorrectly divide 120 directly by the third term of the ratio (3) instead of first finding the scale factor k from the matching term, which is 10 in this case. Another mistake is to misinterpret the ratio as percentages instead of fixed proportional parts, leading to wrong arithmetic. Carefully mapping each ratio term to its corresponding variable expression avoids these errors. Final Answer: The number of exercise books in the shop is 36.

Ratio and Proportion Questions