Divide 378 in the Ratio 2:7: Two Shares and a Check

Read a part-to-part ratio correctly

The ratio 2:7 compares the first share with the second share. It does not say that the first share is 2/7 of the total. Add the ratio terms: 2 + 7 = 9 equal parts. This is the denominator needed to split a combined amount.

The first share is 2/9 of the whole; the second is 7/9. Those fractions add to one. The order matters: reversing the names attached to the shares reverses who receives each amount.

Calculate the value of one part

For a total of 378, one ratio part is 378 ÷ 9 = 42. The first share is 2 × 42 = 84; the second is 7 × 42 = 294. Both use the same value per part.

This method works with money, lengths or counts as long as the units match. When the objects cannot be divided, a fractional result needs an allocation rule. The present example is chosen so that both shares are whole numbers.

Check the total and the proportion

Total check: 84 + 294 = 378. Ratio check: divide both shares by 42; the result is 2:7. A correct total alone is not enough, because many different pairs of values add to 378.

Cross-multiplication is another check: 84 × 7 = 294 × 2 = 588. Equal cross-products confirm the intended part-to-part ratio.

Scale the same ratio

If the total doubles to 756, one part doubles to 84 and the shares become 168 and 588. The ratio does not change. If the total instead falls to 9, the shares are simply 2 and 7.

A fixed additional amount is different from proportional scaling. Adding the same number to both shares generally changes the ratio; multiplying both shares by the same positive number preserves it.

A common wrong answer and a practice question

Dividing 378 by 7 treats the second ratio term as the entire total. That confuses part-to-part with part-to-whole. Write “total parts = 9” before doing the division to prevent this mistake.

Practice: split 423 using the same ratio. One part is 47, so the answers are 94 and 329. Check their sum and reduce their ratio before moving on.

Check the mathematical definitions

For fractions, percentages and worked arithmetic, see OpenStax Prealgebra. For number definitions, see OpenStax on prime and composite numbers and Wolfram MathWorld on digital roots. Worked examples on this page are calculated for this guide.