A student multiplied a number by 3/5 instead of 5/3 . What is the percentage error in the calculation ?

Correct — option (1), 64%.

Take the number as 15, which both 3 and 5 divide.

Step 1 — the correct result.
15 × 5/3 = 25

Step 2 — the student's result.
15 × 3/5 = 9

Step 3 — the error.
25 − 9 = 16

Step 4 — the error as a percentage of the correct result.
16 ÷ 25 × 100 = 64%

Check for any non-zero number N.
Correct result = 5N/3; obtained result = 3N/5
Obtained ÷ correct = (3/5) ÷ (5/3) = 9/25 = 36%
Error = 100% − 36% = 64%

The number cancels, so every non-zero starting number gives the same 64%.

The idea to remember: percentage error is measured against the correct value, not against the wrong result or the original number.

Why the others are wrong

• (2)
  54% — Option (2) would put the student's result at 46% of the correct value. On the number 15, that is 46% of 25 = 11.5.

  But 15 × 3/5 = 9, which is 36% of 25. The result falls short by 64%, not 54%.

• (3)
  44% — Option (3) would make the student's result 56% of the correct value: 14 out of 25 on the number 15.

  The actual result is 9. Since (3/5) ÷ (5/3) = 9/25, the wrong result is 36% of the right one, an error of 64%.

• (4)
  34% — Option (4) would leave the student's result at 66% of the correct value: 16.5 out of 25 on the number 15.

  The factor 3/5 is much smaller than 5/3; their ratio is 9/25. The result falls to 36% of what it should be, so the error is 64%.

Percentage error = (difference between the correct and obtained values) ÷ correct value × 100.

When a non-zero number is multiplied by a wrong factor in place of the right one, the number cancels: the error depends only on the two factors. Here the obtained result is (3/5) ÷ (5/3) = 9/25 of the correct one.

Picking a convenient number, a multiple of every denominator, turns the fractions into whole numbers and gives the same percentage.

Key facts

• Percentage error = (difference between correct and obtained values) ÷ correct value × 100.
• For the number 15: correct result 15 × 5/3 = 25; obtained result 15 × 3/5 = 9.
• The error of 16 on 25 is 64%; the obtained result is 36% of the correct one.
• Multiplying by a/b in place of b/a gives (a/b)² of the correct result.
• A percentage change is taken on the starting value; a percentage error on the correct value.

Common traps

• Dividing by the wrong base. 16 on 9, the obtained value, is about 177.8%, and 16 on 15, the original number, is about 106.7%; the error is taken on the correct 25.
• Reporting the obtained share as the error. 9 is 36% of 25; the error is the remaining 64%.
• Subtracting the fractions and stopping. 5/3 − 3/5 = 16/15 is the error as a fraction of the original number, not of the correct result.

UnlockIAS will link similar questions from RAS Pre 2018 here once that paper is published on this site.

• practice — not a real PYQ

  A student divided a number by 4 instead of multiplying it by 4. What is the percentage error in the result ?

  • (a)93.75%
  • (b)75%
  • (c)15%
  • (d)6.25%
  Answer(1) — Take the number 4: correct result 4 × 4 = 16, obtained result 4 ÷ 4 = 1. The error of 15 on 16 is 93.75%.

  Option (2) would need an obtained result of 4, option (3) takes the error of 15 as a percentage, and option (4) is the obtained result as a share of the correct one (1 on 16).

• practice — not a real PYQ

  A student multiplied a number by 4/5 instead of 5/4. What is the percentage error in the calculation ?

  • (a)45%
  • (b)36%
  • (c)64%
  • (d)56.25%
  Answer(2) — Take 20: correct result 20 × 5/4 = 25, obtained result 20 × 4/5 = 16. The error of 9 on 25 is 36%.

  Option (1) takes the error on the original number (9 on 20), option (3) is the obtained result as a share of the correct one (16 on 25), and option (4) takes the error on the obtained result (9 on 16).