When a composite index question has one value missing, you do not need a new formula. You write the same weighted-average formula, put a letter where the value is missing, and solve.
This lesson builds on combining weighted indices. The chapter is summarised at Index numbers.
What setup do I use?
Write I = Σ(w × I) ÷ Σw with the unknown inside it. Then multiply both sides by the total weight to clear the fraction before solving.
Worked example 1: a missing weight
Item A has index 120 and weight w. Item B has index 100 and weight 3. The composite index is 108. Find w.
- Write (120w + 100 × 3) ÷ (w + 3) = 108.
- Multiply by (w + 3): 120w + 300 = 108(w + 3) = 108w + 324.
- Collect terms: 12w = 24, so w = 2.
Check: (120 × 2 + 300) ÷ (2 + 3) = 540 ÷ 5 = 108. It matches.
Worked example 2: a missing component index
Three items have weights in the ratio 2 : 3 : 5. Their indices are 115, x and 100. The composite index is 109. Find x.
- Write (2 × 115 + 3x + 5 × 100) ÷ 10 = 109. The total weight is 2 + 3 + 5 = 10.
- Multiply by 10: 230 + 3x + 500 = 1 090.
- Simplify: 3x = 1 090 − 730 = 360, so x = 120.
Check: (230 + 360 + 500) ÷ 10 = 1 090 ÷ 10 = 109.
The mistake that costs marks
The slip in the first type is leaving w out of the denominator. A student writes (120w + 300) ÷ 3 = 108, which gives w = 0.2, a fraction that is too small to be a sensible weight.
In the second type, the slip is dividing by the number of items, three, instead of the total weight, ten. The fix is a line at the start that says “total weight =”, filled in before any other working.
| Check | Wrong | Right |
|---|---|---|
| Denominator in example 1 | 3 | w + 3 |
| Denominator in example 2 | 3 (items) | 10 (total weight) |
Check yourself
Item P has index 110 and weight k. Item Q has index 100 and weight 6. The composite index is 104. Find k.
Answer
(110k + 600) ÷ (k + 6) = 104, so 110k + 600 = 104k + 624.
Then 6k = 24 and k = 4.
Check: (440 + 600) ÷ 10 = 1 040 ÷ 10 = 104.
What to study next
Practise all three lesson types in the index numbers practice set. If you stall on unfamiliar wording, read choosing the right method in an unfamiliar Add Maths question.
For a teacher to read your equation setup, see online one-to-one Additional Mathematics tuition.