A proportional split divides a total into parts that keep the same relationship as a set of weights. If 120 is split between two groups with weights 3 and 5, one gets 45 and the other 75. The method has three steps: add the weights, find each share as total × weight ÷ sum, and check that the shares add up to the total.
Weights are relative, and they do not have to add up to 1 or 100. When the items being shared are whole objects, or when the weights are fractions, two decisions come up, and this guide explains them with worked cases. To calculate directly, use the proportional sharing calculator.
| Situation | How to solve it | Example |
|---|---|---|
| Whole-number weights, total divides evenly | Total × weight ÷ sum of weights | 120 in 3:5 gives 45 and 75 |
| Fractional weights | Add the fractions and divide each weight by the sum | 60 with 1/4 and 1/6 gives 36 and 24 |
| Items that can't be split | Whole part of each share, and the leftover item to the largest remainder | 11 items in 3:2 give 7 and 4 |
| More weight, smaller part | Use the reciprocals as weights | 70 inversely to 2 and 5 gives 50 and 20 |
The table follows the guide in order: each row adds one difficulty to the basic case in the first row.
There are two checks, and you need both. Giving 60 to each group also adds up to 120, but the ratio would be 1:1. Giving 3 and 5 keeps the ratio, but leaves 112 unshared.
Weights that are fractions
You do not need to convert the weights to whole numbers or percentages. Add them as fractions and divide each weight by that sum. To split 60 with weights 1/4 and 1/6, the common denominator is 12, with 1/4 = 3/12 and 1/6 = 2/12, which add up to 5/12.
The first group’s fraction of the total is (3/12) ÷ (5/12) = 3/5 and the second group’s is 2/5. The shares are 60 × 3/5 = 36 and 60 × 2/5 = 24, and 36 + 24 = 60. Multiplying both weights by 12 gives the whole-number ratio 3:2, which leads to the same split.
If the shares do not come out whole, it is best to keep them as fractions until the end. The value 100/3, for example, is written 33.3̅ and not 33.33; rounding is applied only once, to the final result.
Worked example: 120 in the ratio 3:5
Two groups receive 120 in the ratio 3:5. How much does each one get?
- Add the weights. 3 + 5 = 8.
- Turn each weight into a fraction of the total. The first group gets 3/8 and the second gets 5/8.
- Multiply by the total. 120 × 3/8 = 45 and 120 × 5/8 = 75.
The result is 45 and 75.
Check. 45 + 75 = 120. Also, 45/3 = 15 and 75/5 = 15, so each unit of weight is worth the same in both groups.
Items that can’t be split
With 11 items and weights 3:2, the sum is 5 and the shares are 33/5 = 6.6 and 22/5 = 4.4. Neither is a whole number. The largest-remainder method first assigns the whole part, 6 and 4, which uses 10 items. The remaining item goes to the group with the larger remainder: 0.6 against 0.4, so it goes to the first.
| Step | Group A (weight 3) | Group B (weight 2) |
|---|---|---|
| Share | 33/5 = 6.6 | 22/5 = 4.4 |
| Whole part | 6 | 4 |
| Remainder | 0.6 | 0.4 |
| Leftover item | 1 | 0 |
| Final split | 7 | 4 |
All 11 items are used, but 7:4 only resembles 3:2, because no whole-number split of 11 can be exact in that ratio. Rounding each share separately would give 7 and 4 here by chance; with other numbers the total can come out higher or lower than it should, which is why the leftover item is handed out by a fixed rule. If two remainders tie, the rule has to say who wins.
Inverse splitting
Sometimes the person with the smaller weight should get more, as in a split that is inversely proportional to ages. It works the same way, with reciprocals as the weights. To split 70 inversely to 2 and 5, the weights are 1/2 and 1/5, which add up to 7/10.
The fractions of the total are (1/2) ÷ (7/10) = 5/7 and (1/5) ÷ (7/10) = 2/7. The shares are 70 × 5/7 = 50 and 70 × 2/7 = 20. The one with weight 2 gets more than the one with weight 5, as intended, and 50 + 20 = 70.
Practice
- Split 90 in the ratio 1:2:3.
- Split 50 with weights 2/3 and 1/3.
- Split 8 items in the ratio 1:2 using largest remainders.
- Split 84 inversely to 3 and 4.
See the answers
- 15, 30, and 45 (the sum of the weights is 6 and each unit is worth 15).
- 100/3 and 50/3, that is, 33.3̅ and 16.6̅; they add up to 50.
- 3 and 5, because the shares 8/3 and 16/3 have whole parts 2 and 5, and the leftover item goes to the remainder 2/3.
- 48 and 36, because the weights 1/3 and 1/4 add up to 7/12, the fractions of the total are 4/7 and 3/7, and 84 × 4/7 = 48.
Sources and tools
The Wikipedia article Proportionality (mathematics) covers direct and inverse proportionality between quantities (accessed October 17, 2026); it does not describe splitting a total, which this page checks with the worked calculations. To split among several groups, use proportional sharing. The link between ratio, proportion, and fraction is in the guide to ratio, proportion, and fraction, and for a single part, fraction of a number works. The 17 camels puzzle uses a split in which the fractions do not add up to one. To reduce weights, see simplify ratios.