Savitri’s marks in Kashmiri are as follows: 35 out of 50 in internal tests, 44 out of 60 in the project, and 80 out of 100 in the final exam. The annual percentage score is calculated by combining the internals, project, and final exam in the ratio 3 : 4 : 5. Which of the following expression(s) gives her annual score (as a percentage) in Kashmiri?
(i) 35 × 3 + 44 × 4 + 80 × 53 + 4 + 5
(ii) (35/100) × 3 + (44/100) × 4 + (80/100) × 53 + 4 + 5
(iii) (35/50) × 3 + (44/60) × 4 + (80/100) × 53 + 4 + 5 × 100
(iv) ((35/50) × 100) × 3 + ((44/60) × 100) × 4 + ((80/100) × 100) × 53 + 4 + 5

Step-by-step solution

Idea: Marks out of different maximums cannot be combined directly. First change each mark to a percentage (or fraction) of its own maximum, then take the weighted mean.

1. Percentages: internals 3550 × 100 = 70%, project 4460 × 100 ≈ 73.33%, final 80100 × 100 = 80%.½ mark
2. (i) uses the raw marks 35, 44, 80, which are out of 50, 60 and 100. It treats 35 out of 50 as if it were 35%, so it is wrong (it gives 56.75).½ mark
3. (ii) divides every mark by 100, again treating 35 out of 50 and 44 out of 60 as out of 100; it is wrong (and gives a fraction, 0.5675, not a percentage).½ mark
4. (iii) takes the weighted mean of the correct fractions 0.7, 0.733…, 0.8 and then multiplies by 100: correct. (iv) takes the weighted mean of the correct percentages 70, 73.33, 80: correct. Multiplying by 100 before or after the averaging gives the same number.1 mark
5. Value: 70 × 3 + 73.33 × 4 + 80 × 512 = 210 + 293.33 + 40012 = 903.3312 ≈ 75.28%.½ mark

Check: (iii): 2.1 + 2.933 + 412 × 100 = 9.03312 × 100 ≈ 75.28 ✓, the same as (iv). 75.28% lies between 70% and 80% ✓.

Answer to write in the exam

Common mistakes that cost marks

• Choosing (i) because it looks like the weighted-mean formula. The values must first be made comparable (all out of 100).
• Choosing (ii), which divides 35 and 44 by 100 although they are out of 50 and 60.
• Picking only one of (iii) and (iv). They are the same expression; one multiplies by 100 at the end, the other at the start.

How this can come in the exam

Try one yourself

A student scores 12 out of 20 in a quiz and 45 out of 50 in an exam. They are combined in the ratio 1 : 4. Write a correct expression and find the final percentage.

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