Question 1

In a class, the number of boys exceeds the number of girls by 12% of the total class strength. What is the ratio of boys to girls?

Accepted Answer

Introduction / Context:This problem assesses understanding of how to convert a verbal comparison involving a percentage of the total into a concrete ratio between two parts (boys and girls). Many learners mix up “percent of total” with “percent of one part,” so translating the statement into algebra relative to total strength is the key. Given Data / Assumptions: Boys are more than girls by 12% of the total class strength. Total strength is the sum of boys and girls only. We must express boys : girls in simplest whole-number form. Concept / Approach:Let total = T, boys = B, girls = G. We are told B − G = 0.12*T and B + G = T. Solving these two linear relations gives B and G in terms of T. Finally, convert B:G to a whole-number ratio. Step-by-Step Solution:B − G = 0.12TB + G = TAdd both: 2B = 1.12T ⇒ B = 0.56TThen G = T − B = T − 0.56T = 0.44TRatio B : G = 0.56T : 0.44T = 56 : 44 = 14 : 11 Verification / Alternative check:Check the difference if B:G = 14:11 out of total 25 parts. Difference = 3 parts; 3/25 of total = 12%, which matches the given condition. Why Other Options Are Wrong:11 : 14 and 25 : 28 imply boys fewer or the same proportion as girls; 28 : 25 and 7 : 6 give a difference larger than 12% of total. Common Pitfalls:Taking 12% of girls (or boys) instead of 12% of total; skipping the system B − G and B + G and thereby miscomputing the ratio. Final Answer:14 : 11

Question 2

Three friends divide Rs. 624 in the ratio 1/2 : 1/3 : 1/4. What is the share of the third friend?

Accepted Answer

Introduction / Context:Ratios can be given as fractions. The amounts received are proportional to these fractional numbers. The process is to convert the fractional ratio into proportional parts, total those parts, and then distribute the total sum accordingly. Given Data / Assumptions: Total amount = Rs. 624 Ratio among three friends = 1/2 : 1/3 : 1/4 “Third friend” corresponds to the 1/4 part in the given ratio order. Concept / Approach:If the ratio is a : b : c, then shares are (a/total of ratio) * sum, etc. Here, a = 1/2, b = 1/3, c = 1/4. First compute a + b + c, then the third share = (c / (a + b + c)) * 624. Step-by-Step Solution:Sum of ratio terms = 1/2 + 1/3 + 1/4 = 6/12 + 4/12 + 3/12 = 13/12Third share fraction of total = (1/4) / (13/12) = (1/4) * (12/13) = 12/52 = 3/13Third share = 624 * (3/13) = 48 * 3 = Rs. 144 Verification / Alternative check:Compute first and second shares similarly and confirm that all three add to Rs. 624. Why Other Options Are Wrong:Rs. 288 and Rs. 192 correspond to other parts or doubled values; Rs. 148 and Rs. 156 are rounding guesses not matching exact proportional division. Common Pitfalls:Adding the numerators directly without common denominators; treating 1/2 : 1/3 : 1/4 as 1 : 1 : 1 inadvertently. Final Answer:Rs. 144

Question 3

Two numbers are in the ratio 3 : 4 and their sum is 420. What is the greater of the two numbers?

Accepted Answer

Introduction / Context:When a sum and a ratio are known, each part equals the sum divided by the total number of parts. Multiply the unit part by the appropriate ratio component to obtain each number. Given Data / Assumptions: Ratio = 3 : 4 Sum = 420 We want the larger number (the “4” part). Concept / Approach:Total parts = 3 + 4 = 7. Each part (unit) = 420 / 7. Then greater number = 4 * (unit part). Step-by-Step Solution:Unit part = 420 / 7 = 60Greater number = 4 * 60 = 240 Verification / Alternative check:Smaller number = 3 * 60 = 180. Sum check: 180 + 240 = 420 (correct ratio and total). Why Other Options Are Wrong:175, 200, and 315 do not align with 3:4 and total 420; 180 is the smaller number, not the greater. Common Pitfalls:Using 420/4 directly and ignoring the ratio; swapping the 3-part and 4-part accidentally. Final Answer:240

Question 4

A carton contains a dozen (12) mirrors and is dropped. Which of the following cannot be the ratio of broken mirrors to unbroken mirrors?

Accepted Answer

Introduction / Context:For a ratio to be feasible, broken + unbroken must equal the fixed total (12 mirrors). Thus, the sum of the ratio terms times a common multiplier must be 12. This divisibility check quickly rules out impossible options. Given Data / Assumptions: Total mirrors = 12 Broken : Unbroken must sum to 12 when scaled by some integer k. We test each option by checking if (a + b) * k = 12 has an integer k. Concept / Approach:If the ratio is a : b, then numbers are ak and bk for some integer k. Total = (a + b)k must be 12. Therefore, 12 must be divisible by (a + b). Step-by-Step Solution:2 : 1 ⇒ a + b = 3; 12/3 = 4 ⇒ possible3 : 1 ⇒ a + b = 4; 12/4 = 3 ⇒ possible3 : 2 ⇒ a + b = 5; 12/5 is not an integer ⇒ impossible7 : 5 ⇒ a + b = 12; 12/12 = 1 ⇒ possible Verification / Alternative check:Try to construct numbers for 3 : 2; you cannot reach exactly 12 total with integer counts, confirming impossibility. Why Other Options Are Wrong:They are feasible because their combined parts divide 12 exactly. Common Pitfalls:Forgetting that counts of items must be whole numbers; using non-integer k makes the ratio unrealizable. Final Answer:3 : 2

Question 5

Two whole numbers have sum 64. Which of the following ratios cannot be their ratio?

Accepted Answer

Introduction / Context:If two whole numbers are in ratio a : b, then the actual numbers are ak and bk for some positive integer k, and their sum is (a + b)k. This sum must equal 64; hence 64 must be divisible by (a + b). We use this divisibility condition to find the impossible ratio. Given Data / Assumptions: a and b are positive integers representing the ratio. Sum (a + b)k = 64 for some integer k. Both numbers are whole (nonnegative integers). Concept / Approach:Compute (a + b) for each option, then check if 64/(a + b) is an integer. If not, that ratio cannot produce whole-number addends summing to 64. Step-by-Step Solution:5 : 3 ⇒ a + b = 8 ⇒ 64/8 = 8 (OK)7 : 1 ⇒ a + b = 8 ⇒ 64/8 = 8 (OK)3 : 4 ⇒ a + b = 7 ⇒ 64/7 not an integer (Not possible)9 : 7 ⇒ a + b = 16 ⇒ 64/16 = 4 (OK) Verification / Alternative check:Construct numbers for possible cases (e.g., 5k and 3k with k = 8 give 40 and 24) to confirm. For 3:4, you cannot reach sum 64 with an integer k. Why Other Options Are Wrong:They are actually possible; only 3 : 4 fails the divisibility test. Common Pitfalls:Confusing the sum 64 with a single term; forgetting k must be an integer to keep numbers whole. Final Answer:3 : 4

Question 6

A 13 m long iron rod weighs 23.4 kg. What is the weight of a 6 m length of the same rod (assume uniform density)?

Accepted Answer

Introduction / Context:If a rod is uniform, its weight is directly proportional to its length. Therefore, a simple proportion can be used to find the weight of any other length of the same rod. Given Data / Assumptions: 13 m length has weight 23.4 kg. Rod has uniform cross-section and density (weight ∝ length). Find weight for 6 m. Concept / Approach:Use the proportion Weight/Length = constant. Hence W2 = W1 * (L2/L1). This keeps units consistent and leverages direct proportionality. Step-by-Step Solution:W2 = 23.4 * (6 / 13)Compute numerator: 23.4 * 6 = 140.4Divide by 13: 140.4 / 13 = 10.8 kg Verification / Alternative check:Per-meter weight = 23.4 / 13 = 1.8 kg/m. For 6 m, weight = 1.8 * 6 = 10.8 kg, confirming the result. Why Other Options Are Wrong:7.2 kg or 9.6 kg correspond to other incorrect fractions; 12.4 kg and 18 kg exceed or mismatch the proportional outcome. Common Pitfalls:Using inverse proportion accidentally; rounding early and introducing errors; mixing meters and kilograms carelessly. Final Answer:10.8 kg

Question 7

Find the third proportional to 0.8 and 0.2. (That is, 0.8 : 0.2 = 0.2 : x. Find x.)

Accepted Answer

Introduction / Context:The third proportional to a and b is a number c such that a : b = b : c. This leads to the relation c = b^2 / a for positive a, b. Knowing this formula is a standard ratio-and-proportion skill. Given Data / Assumptions: a = 0.8 b = 0.2 Find c such that 0.8 : 0.2 = 0.2 : c Concept / Approach:Using a : b = b : c ⇒ c = b^2 / a. Substitute a = 0.8 and b = 0.2 and compute carefully to avoid decimal mistakes. Step-by-Step Solution:c = b^2 / a = (0.2 * 0.2) / 0.8= 0.04 / 0.8= 0.05 Verification / Alternative check:Check the proportion: 0.8 : 0.2 = 4 : 1. With c = 0.05, 0.2 : 0.05 = 4 : 1 as well. Hence the proportion holds. Why Other Options Are Wrong:0.4 and 0.8 are too large; 0.032 and 0.02 are too small and do not satisfy 0.2 : c = 4 : 1. Common Pitfalls:Using c = a^2 / b (reversing the formula) or mishandling decimals during division. Final Answer:0.05

Question 8

Find the mean proportional (geometric mean) between 0.32 and 0.02.

Accepted Answer

Introduction / Context:The mean proportional (geometric mean) m between two positive numbers a and b satisfies a : m = m : b, or m^2 = a * b. Hence m = sqrt(a * b). Compute accurately with decimals. Given Data / Assumptions: a = 0.32 b = 0.02 m^2 = a * b ⇒ m = sqrt(0.32 * 0.02) Concept / Approach:Multiply the two decimals and take the square root. Keep precision to avoid rounding errors that might push you toward a nearby option. Step-by-Step Solution:Product = 0.32 * 0.02 = 0.0064m = sqrt(0.0064) = 0.08 Verification / Alternative check:Check proportion: 0.32 : 0.08 = 4 : 1 and 0.08 : 0.02 = 4 : 1; proportions match, confirming m. Why Other Options Are Wrong:0.34 and 0.30 are far too large; 0.16 would be m^2, not m; 0.06 does not satisfy the proportion check. Common Pitfalls:Confusing arithmetic mean with geometric mean; forgetting to take the square root after multiplying. Final Answer:0.08

Question 9

Find the fourth proportional to 0.2, 0.12, and 0.3. (That is, 0.2 : 0.12 = 0.3 : x. Find x.)

Accepted Answer

Introduction / Context:The fourth proportional x to a, b, c is defined by a : b = c : x. This yields x = (b * c) / a. With decimal values, careful multiplication and division are enough to get the exact result. Given Data / Assumptions: a = 0.2 b = 0.12 c = 0.3 Concept / Approach:Use x = (b * c) / a directly. No further transformation is required; just compute cleanly. Step-by-Step Solution:x = (0.12 * 0.3) / 0.2= 0.036 / 0.2= 0.18 Verification / Alternative check:Check the proportion numerically: 0.2 : 0.12 = 5 : 3; with x = 0.18, 0.3 : 0.18 = 5 : 3 as well. Why Other Options Are Wrong:0.13 and 0.15 understate the correct division; 0.80 is unrelatedly large; 0.12 only repeats b and breaks the proportion. Common Pitfalls:Inverting the formula to x = (a * c)/b (incorrect), or losing decimal places during computation. Final Answer:0.18

Question 10

What number should be subtracted from each of 54, 71, 75, and 99 so that the resulting numbers are in proportion?

Accepted Answer

Introduction / Context:Numbers a, b, c, d are in proportion if a : b = c : d. Here, we seek x such that (54 − x) : (71 − x) = (75 − x) : (99 − x). This leads to a simple cross-multiplication and linear equation in x. Given Data / Assumptions: Numbers: 54, 71, 75, 99 Subtract the same x from each. Condition: (54 − x)/(71 − x) = (75 − x)/(99 − x) Concept / Approach:Use cross-multiplication to eliminate denominators, expand, and solve for x. Quadratic terms cancel, yielding a first-degree equation. Step-by-Step Solution:(54 − x)(99 − x) = (71 − x)(75 − x)54*99 − (54 + 99)x + x^2 = 71*75 − (71 + 75)x + x^25346 − 153x = 5325 − 146x5346 − 5325 = 153x − 146x ⇒ 21 = 7x ⇒ x = 3 Verification / Alternative check:Compute pairs after subtracting 3: 51:68 and 72:96. Reduce: 51/68 = 3/4, 72/96 = 3/4. Proportion holds. Why Other Options Are Wrong:1, 2, 4, or 6 do not produce equal ratios for the two pairs when checked. Common Pitfalls:Arithmetic slips in products 54*99 and 71*75; forgetting that x^2 cancels on both sides. Final Answer:3

Question 11

What number should be added to each of 6, 14, 18, and 38 so that they become proportionate (i.e., 6+x : 14+x = 18+x : 38+x)?

Accepted Answer

Introduction / Context:We want one constant x that, when added to each term, creates two equal ratios. This is a classic “bring into proportion” problem based on cross-multiplication and solving a linear equation. Given Data / Assumptions: Numbers: 6, 14, 18, 38 We need (6 + x)/(14 + x) = (18 + x)/(38 + x) x is the same for all four numbers. Concept / Approach:Set up the proportion and cross-multiply to clear denominators. Expand, simplify, and solve the resulting linear equation for x. Step-by-Step Solution:(6 + x)(38 + x) = (14 + x)(18 + x)Left: 6*38 + 44x + x^2 = 228 + 44x + x^2Right: 14*18 + 32x + x^2 = 252 + 32x + x^2Equate and cancel x^2: 228 + 44x = 252 + 32x ⇒ 12x = 24 ⇒ x = 2 Verification / Alternative check:Substitute x = 2: Left ratio (8:16) = 1:2; Right ratio (20:40) = 1:2. Proportion satisfied. Why Other Options Are Wrong:Adding 1, 3, 4, or 5 yields ratios that do not match upon reduction. Common Pitfalls:Errors in expansion; forgetting to cancel x^2 on both sides; miscomputing 6*38 or 14*18. Final Answer:2

Question 12

A fraction bears the same ratio to 1/27 as 3/7 bears to 5/9. Find the fraction.

Accepted Answer

Introduction / Context:We are given a proportion involving a fraction x: x : (1/27) = (3/7) : (5/9). Solving for x uses the property that a : b = c : d implies a/b = c/d, hence a = b * (c/d). Given Data / Assumptions: x / (1/27) = (3/7) / (5/9) All values are positive rational numbers. Concept / Approach:Compute the right-hand ratio exactly, then multiply by 1/27 to get x. Be careful when dividing by a fraction: (3/7)/(5/9) = (3/7)*(9/5). Step-by-Step Solution:(3/7) / (5/9) = (3/7) * (9/5) = 27/35x = (1/27) * (27/35) = 1/35 Verification / Alternative check:Compute the left ratio with x = 1/35: (1/35)/(1/27) = 27/35, matching the right side. Why Other Options Are Wrong:7/45 and 5/21 are unrelated ratios; 45/7 is far too large; 3/35 is close but not equal to the required 1/35. Common Pitfalls:Inverting the division incorrectly; canceling 27s prematurely; arithmetic mistakes with complex fractions. Final Answer:1 / 35

Question 13

What number must be added to each term of the ratio 7 : 13 so that the ratio becomes 2 : 3?

Accepted Answer

Introduction / Context:To transform one ratio into another by adding the same number k to both terms, set up an equation: (7 + k)/(13 + k) = 2/3, then solve for k. This is a direct proportion manipulation. Given Data / Assumptions: Original ratio: 7 : 13 Target ratio: 2 : 3 Same k added to both terms. Concept / Approach:Cross-multiply and solve the resulting linear equation in k. Ensure integer arithmetic is clean to avoid mistakes. Step-by-Step Solution:(7 + k)/(13 + k) = 2/33(7 + k) = 2(13 + k)21 + 3k = 26 + 2k ⇒ k = 5 Verification / Alternative check:New ratio = (7 + 5) : (13 + 5) = 12 : 18, which reduces to 2 : 3. Works perfectly. Why Other Options Are Wrong:1, 2, 3, or 6 do not convert 7 : 13 to 2 : 3 when added to both terms. Common Pitfalls:Setting 7/13 = 2/3 directly (ignoring the added k); small arithmetic slips in cross-multiplication. Final Answer:5

Question 14

A ratio equals 5 : 8. If the antecedent (first term) is 40, what is the consequent (second term)?

Accepted Answer

Introduction / Context:Scaling a ratio to actual numbers requires multiplying both terms by the same factor. If the first term becomes 40, compute the common multiplier and apply it to the second term to find the consequent. Given Data / Assumptions: Ratio = 5 : 8 Antecedent (first term) after scaling = 40 Concept / Approach:Find k such that 5k = 40 ⇒ k = 8. Then consequent = 8k = 8 * 8 = 64. Step-by-Step Solution:5k = 40 ⇒ k = 8Consequent = 8k = 8 * 8 = 64 Verification / Alternative check:40 : 64 simplifies by 8 to 5 : 8, agreeing with the original ratio. Why Other Options Are Wrong:25 and 48 are inconsistent with 5k = 40; “None” is unnecessary since 64 is correct; 56 corresponds to k = 7, not matching 40. Common Pitfalls:Adding 3 to get 8 from 5 and then doing the same for 40 (incorrect reasoning). Ratios scale multiplicatively, not additively. Final Answer:64

Question 15

Among the ratios 7 : 15, 15 : 23, 17 : 25, and 21 : 29, which is the smallest?

Accepted Answer

Introduction / Context:To compare fractions a:b, compute a/b (or cross-multiply pairwise). The smallest fraction corresponds to the smallest decimal value. We compare the four given ratios by converting to decimals or by quick cross-multiplications. Given Data / Assumptions: Ratios: 7:15, 15:23, 17:25, 21:29 All positive, so direct comparison of decimals is valid. Concept / Approach:Compute approximate decimals: 7/15 ≈ 0.4667; 15/23 ≈ 0.6522; 17/25 = 0.68; 21/29 ≈ 0.724. The smallest is the least decimal value. Step-by-Step Solution:7/15 ≈ 0.466715/23 ≈ 0.652217/25 = 0.6821/29 ≈ 0.724Smallest = 7 : 15 Verification / Alternative check:Cross-multiply 7/15 vs. each other ratio to confirm 7/15 is smaller in each comparison. Why Other Options Are Wrong:Each corresponds to a larger decimal value and hence a larger ratio. Common Pitfalls:Assuming larger denominators always mean smaller fractions without considering numerators; rounding too coarsely. Final Answer:7 : 15

Question 16

If 1/5 : 1/N = 1/N : 1/1.25, find the value of N.

Accepted Answer

Introduction / Context:This is a proportion of the form a : b = b : c. When a, b, c are reciprocals or decimals, it is best to translate to an equation in b and solve. Here, a = 1/5, b = 1/N, c = 1/1.25. Given Data / Assumptions: (1/5) : (1/N) = (1/N) : (1/1.25) N > 0 Concept / Approach:For a : b = b : c, we have b^2 = a*c. Hence (1/N)^2 = (1/5) * (1/1.25). Compute the right-hand side and then invert to find N. Step-by-Step Solution:(1/N)^2 = (1/5) * (1/1.25) = (1/5) * 0.8 = 0.161/N^2 = 0.16 ⇒ N^2 = 1 / 0.16 = 6.25N = √6.25 = 2.5 Verification / Alternative check:Plug back: 1/5 : 1/2.5 = 0.2 : 0.4 = 1 : 2; and 1/2.5 : 1/1.25 = 0.4 : 0.8 = 1 : 2. Matches. Why Other Options Are Wrong:1.25, 1.5, 2.25, 2.0 do not satisfy b^2 = a*c when substituted. Common Pitfalls:Squaring the wrong term or inverting the ratio incorrectly; mishandling 1/1.25 = 0.8. Final Answer:2.5

Question 17

If one-third of A, one-fourth of B, and one-fifth of C are equal, find the ratio A : B : C.

Accepted Answer

Introduction / Context:When scaled parts of quantities are equal, express each original quantity in terms of a common variable. If A/3 = B/4 = C/5 = k, then A, B, C can be written in terms of k and their ratio found directly. Given Data / Assumptions: A/3 = B/4 = C/5 All quantities are positive. Concept / Approach:Let the common value be k. Then A = 3k, B = 4k, C = 5k. Their ratio is simply 3 : 4 : 5 after canceling k. Step-by-Step Solution:Assume A/3 = B/4 = C/5 = kA = 3k, B = 4k, C = 5kA : B : C = 3 : 4 : 5 Verification / Alternative check:Compute A/3, B/4, C/5 with A:B:C = 3:4:5: each gives k, confirming consistency. Why Other Options Are Wrong:They invert or permute the order incorrectly; 1/3 : 1/4 : 1/5 restates the given scaled parts, not A:B:C. Common Pitfalls:Writing A = k/3 instead of A = 3k; mixing up which number multiplies k. Final Answer:3 : 4 : 5

Question 18

Find the ratio that (1/3 of Rs. 9.30) bears to (0.6 of Rs. 1.55).

Accepted Answer

Introduction / Context:Ratios with monetary amounts work like any numbers. Compute each expression exactly, then form the ratio and reduce to simplest terms to avoid decimal clutter. Given Data / Assumptions: (1/3) of Rs. 9.30 0.6 of Rs. 1.55 Find their ratio in simplest form. Concept / Approach:Compute both quantities: A = 9.30/3, B = 0.6 * 1.55. Then A : B = (A/B) : 1 or simplify by dividing both by a common factor. Step-by-Step Solution:A = 9.30 / 3 = 3.10B = 0.6 * 1.55 = 0.93Ratio = 3.10 : 0.93Divide both by 0.31 ⇒ 10 : 3 Verification / Alternative check:Compute 3.10/0.93 ≈ 3.333..., which equals 10/3. Hence 10 : 3 is correct. Why Other Options Are Wrong:1 : 3 and 3 : 10 invert or shrink the correct ratio; 3 : 1 is too large; 5 : 3 is 1.666..., not matching. Common Pitfalls:Rounding early; miscomputing 0.6 * 1.55; forgetting to fully reduce the ratio. Final Answer:10 : 3

Question 19

If a : b = 2 : 3 and 2 : a = 1 : 2, find the value of b.

Accepted Answer

Introduction / Context:Two ratio statements allow solving for a and then b concretely. The second ratio fixes a numerically, and the first then gives b from a standard proportion. Given Data / Assumptions: a : b = 2 : 3 2 : a = 1 : 2 Concept / Approach:From 2 : a = 1 : 2, write 2/a = 1/2 ⇒ a = 4. Substitute into a : b = 2 : 3 to find b. Step-by-Step Solution:2/a = 1/2 ⇒ a = 4a : b = 2 : 3 ⇒ 4 : b = 2 : 3Cross-multiply: 4*3 = 2*b ⇒ b = 12/2 = 6 Verification / Alternative check:Check both ratios with a = 4, b = 6: a : b = 4 : 6 = 2 : 3; 2 : a = 2 : 4 = 1 : 2. Both satisfied. Why Other Options Are Wrong:4, 1/3, 3/2, 8 contradict one or both given ratios when substituted. Common Pitfalls:Interpreting 2 : a = 1 : 2 as 2/a = 2/1 (reversing); skipping the final consistency check. Final Answer:6

Question 20

If 2A = 3B and 4B = 5C, find the ratio A : C.

Accepted Answer

Introduction / Context:Linked proportional equations can be combined to relate two variables that do not appear together initially. Express A and C in terms of B (or a common parameter) and then form their ratio. Given Data / Assumptions: 2A = 3B ⇒ A = (3/2)B 4B = 5C ⇒ C = (4/5)B All variables are positive. Concept / Approach:Write A and C each as a multiple of B. Then A : C = [(3/2)B] : [(4/5)B] = (3/2) : (4/5) = (3/2) * (5/4) = 15/8. Step-by-Step Solution:A = (3/2)BC = (4/5)BA : C = (3/2) : (4/5) = (3/2)*(5/4) = 15/8 Verification / Alternative check:Choose B = 40 (LCM approach). Then A = (3/2)*40 = 60; C = (4/5)*40 = 32; A : C = 60 : 32 = 15 : 8. Why Other Options Are Wrong:3 : 4 and 4 : 3 invert or mismatch; 8 : 15 is the reciprocal; 5 : 6 does not satisfy the derived relation. Common Pitfalls:Dividing instead of multiplying by reciprocals when forming the ratio; losing the common factor B too early. Final Answer:15 : 8

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