A physical fitness evaluation is being undertaken. The final marks are calculated by combining the marks for strength, flexibility, and agility in the ratio 4 : 5 : 6. Rashi has scored 60, 65, and 70. Keerthi has scored 55, 65, and 75 respectively.
- (i) Find out whose total is more without calculating.
- (ii) What are their final marks?
Step-by-step solution
Idea: In a weighted mean, a mark counts in proportion to its weight. Compare where the two girls differ and which weight is bigger there.
(i) Find out whose total is more without calculating.
- Both scored 65 in flexibility, so that part makes no difference. Keerthi is 5 marks behind in strength but 5 marks ahead in agility.½ mark
- Agility has weight 6, strength only 4, so Keerthi’s extra 5 marks count more than her missing 5. Keerthi’s total is more.½ mark
(ii) What are their final marks?
- Rashi: 60 × 4 + 65 × 5 + 70 × 64 + 5 + 6 = 240 + 325 + 42015 = 98515 ≈ 65.67.1 mark
- Keerthi: 55 × 4 + 65 × 5 + 75 × 615 = 220 + 325 + 45015 = 99515 ≈ 66.33. This confirms part (i).1 mark
Check: Difference = (75 − 70) × 6 − (60 − 55) × 415 = 30 − 2015 = 1015 ≈ 0.67, and 66.33 − 65.67 = 0.67 ✓.
Answer to write in the exam
(i)
Flexibility: both 65 (no difference)
Keerthi: −5 in strength (weight 4), +5 in agility (weight 6)
Weight 6 > weight 4
∴ Keerthi’s total is more
(ii)
Rashi = 60 × 4 + 65 × 5 + 70 × 64 + 5 + 6 = 98515 = 65.67
Keerthi = 55 × 4 + 65 × 5 + 75 × 64 + 5 + 6 = 99515 = 66.33
∴ Final marks: Rashi 65.67, Keerthi 66.33
Common mistakes that cost marks
- Saying the totals are equal because 60 + 65 + 70 = 55 + 65 + 75 = 195. The plain totals are equal; the weighted totals are not.
- Using the weights in the wrong order, for example 6 for strength.
- Dividing by 3 instead of by 4 + 5 + 6 = 15.
How this can come in the exam
Two tests are weighted 2 : 3. P scores 50 and 70; Q scores 70 and 50.
Assertion (A): P’s final mark is higher than Q’s.
Reason (R): The test with the larger weight affects the final mark more.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
P = 100 + 2105 = 62, Q = 140 + 1505 = 58. P did better in the heavier test, so R explains A.
Try one yourself
Three rounds are weighted 1 : 2 : 3. Amit scores 80, 70, 60 and Bina scores 60, 70, 80. Who scores more? Find both final marks.
Show answer
Bina, since her higher mark is in the heaviest round. Amit = 80 + 140 + 1806 ≈ 66.67; Bina = 60 + 140 + 2406 ≈ 73.33.
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