Ratio and Proportion: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 1)

A compound ratio combines two or more ratios into one. To find it, multiply the corresponding terms together. For a:b and c:d, that gives (a x c):(b x d).

Inverse proportionality means x and y move in opposite directions. Their product stays fixed: xy = k, where k is a constant. As y increases, x decreases so the product stays the same. For example, doubling y halves x.

Direct proportionality means x and y grow together at a fixed rate. This is written as x = ky, where k is a constant. As y increases, x increases by the same fixed multiple. For example, doubling y doubles x.

The mean proportional m satisfies a:m = m:b. Cross-multiplying gives m x m = a x b, so m² = ab. Taking the square root gives m = √(ab). This is why it is also called the geometric mean of a and b.

When a:b = b:c, the middle term b is repeated in the proportion. Here c is called the third proportional to a and b. Cross-multiplying gives a x c = b x b, so c = b²/a.

When a:b = c:d, the four terms a, b, c, d form a proportion. Here d is called the fourth proportional to a, b, and c. It is found by cross-multiplying: a x d = b x c, so d = (b x c)/a.

More workers finish the task faster, so workers x days stays constant. 4 workers take 27 days, so workers x days = 4 x 27 = 108. For 6 workers: days = 108/6 = 18. Check: 6 x 18 = 108, which equals 4 x 27 = 108.

More workers finish the task faster, so workers x days stays constant. 16 workers take 22 days, so workers x days = 16 x 22 = 352. For 11 workers: days = 352/11 = 32. Check: 11 x 32 = 352, which equals 16 x 22 = 352.

More workers finish the task faster, so workers x days stays constant. 11 workers take 6 days, so workers x days = 11 x 6 = 66. For 6 workers: days = 66/6 = 11. Check: 6 x 11 = 66, which equals 11 x 6 = 66.

More workers finish the task faster, so workers x days stays constant. 16 workers take 18 days, so workers x days = 16 x 18 = 288. For 3 workers: days = 288/3 = 96. Check: 3 x 96 = 288, which equals 16 x 18 = 288.

More workers finish the task faster, so workers x days stays constant. 16 workers take 12 days, so workers x days = 16 x 12 = 192. For 12 workers: days = 192/12 = 16. Check: 12 x 16 = 192, which equals 16 x 12 = 192.

More workers finish the task faster, so workers x days stays constant. 7 workers take 24 days, so workers x days = 7 x 24 = 168. For 6 workers: days = 168/6 = 28. Check: 6 x 28 = 168, which equals 7 x 24 = 168.

More workers finish the task faster, so workers x days stays constant. 16 workers take 3 days, so workers x days = 16 x 3 = 48. For 8 workers: days = 48/8 = 6. Check: 8 x 6 = 48, which equals 16 x 3 = 48.

More workers finish the task faster, so workers x days stays constant. 12 workers take 22 days, so workers x days = 12 x 22 = 264. For 8 workers: days = 264/8 = 33. Check: 8 x 33 = 264, which equals 12 x 22 = 264.

Output is directly proportional to workers, so output/workers stays constant. 9 workers produce 333 units, a rate of 333/9 units per worker. For 13 workers: units = (13 x 333)/9 = 4329/9 = 481. Check: 9 x 481 = 4329 and 13 x 333 = 4329, which match.

Output is directly proportional to workers, so output/workers stays constant. 4 workers produce 108 units, a rate of 108/4 units per worker. For 11 workers: units = (11 x 108)/4 = 1188/4 = 297. Check: 4 x 297 = 1188 and 11 x 108 = 1188, which match.

Output is directly proportional to workers, so output/workers stays constant. 3 workers produce 483 units, a rate of 483/3 units per worker. For 14 workers: units = (14 x 483)/3 = 6762/3 = 2254. Check: 3 x 2254 = 6762 and 14 x 483 = 6762, which match.

Output is directly proportional to workers, so output/workers stays constant. 15 workers produce 219 units, a rate of 219/15 units per worker. For 10 workers: units = (10 x 219)/15 = 2190/15 = 146. Check: 15 x 146 = 2190 and 10 x 219 = 2190, which match.

Output is directly proportional to workers, so output/workers stays constant. 6 workers produce 184 units, a rate of 184/6 units per worker. For 3 workers: units = (3 x 184)/6 = 552/6 = 92. Check: 6 x 92 = 552 and 3 x 184 = 552, which match.

Output is directly proportional to workers, so output/workers stays constant. 12 workers produce 372 units, a rate of 372/12 units per worker. For 7 workers: units = (7 x 372)/12 = 2604/12 = 217. Check: 12 x 217 = 2604 and 7 x 372 = 2604, which match.

Output is directly proportional to workers, so output/workers stays constant. 3 workers produce 234 units, a rate of 234/3 units per worker. For 17 workers: units = (17 x 234)/3 = 3978/3 = 1326. Check: 3 x 1326 = 3978 and 17 x 234 = 3978, which match.

Output is directly proportional to workers, so output/workers stays constant. 12 workers produce 96 units, a rate of 96/12 units per worker. For 18 workers: units = (18 x 96)/12 = 1728/12 = 144. Check: 12 x 144 = 1728 and 18 x 96 = 1728, which match.

10% of A equals 38% of B, so (10/100) x A = (38/100) x B. Multiplying both sides by 100: 10 x A = 38 x B. So A/B = 38/10. Their H.C.F. is 2. Dividing both terms by 2 gives A:B = 19:5. Check: 10 x 19 = 190 and 38 x 5 = 190, which are equal.

27% of A equals 66% of B, so (27/100) x A = (66/100) x B. Multiplying both sides by 100: 27 x A = 66 x B. So A/B = 66/27. Their H.C.F. is 3. Dividing both terms by 3 gives A:B = 22:9. Check: 27 x 22 = 594 and 66 x 9 = 594, which are equal.

51% of A equals 86% of B, so (51/100) x A = (86/100) x B. Multiplying both sides by 100: 51 x A = 86 x B. So A/B = 86/51. Their H.C.F. is 1. Dividing both terms by 1 gives A:B = 86:51. Check: 51 x 86 = 4386 and 86 x 51 = 4386, which are equal.