In a fraction of the remainder problem, the second fraction is taken from what was left, not from the total. With 120 liters and “spend 1/3, then 1/4 of the rest,” the spends are 40 and 20, and 60 is left, which is half. What changes from one step to the next is the base of each fraction. For other kinds of problems, start with fraction word problems, which says when to multiply and when to divide.
| The problem says | Base of the second spend | Result |
|---|---|---|
| "a quarter of the rest" or "a quarter of what is left" | What was left: 80 | Spend 20, 60 left |
| "a quarter of the total" or "a quarter of the original amount" | The total: 120 | Spend 30, 50 left |
The word that decides is the last one in the phrase: “of the rest” shrinks the base, “of the total” leaves it as it was. With the same fraction, the final balance changes from 60 to 50.
Two routes that give the same result
You can chain the amounts, as in steps 1 and 2, or multiply what remains. If a fraction is spent, its complement is left: spending a third leaves two thirds, and spending a quarter leaves three quarters. Because each fraction acts on what the previous one left, the final fraction is the product of those complements. The rule for multiplying fractions, numerator times numerator and denominator times denominator, is on the multiply fractions page. To simplify 6/12, use simplify fractions.
The product is shorter, and the amounts are easier to check. Use the product to find which part is left, and the amounts to see how much that is in liters, dollars or meters. If the wording is confusing, draw a bar with the steps, as the bar model guide explains.
Worked example: spend a third, then a quarter of the rest
A tank holds 120 liters. A third is used, and then a quarter of what is left. How much is left?
- First spend. A third of 120 liters is 40 liters, so 80 liters are left.
- The base changes. The second spend is a quarter of 80 liters, which is 20 liters, and 60 liters are left.
- Look at the final fraction. After the first spend, 2/3 is left, and then 3/4 of that is left. So 2/3 × 3/4 = 6/12 = 1/2 of the total.
- Check with the amounts. Half of 120 liters is 60 liters, the same as in step 2.
Result. 60 liters are left, half of the tank.
Check. 40 + 20 = 60 liters were spent and 60 are left, which together make the original 120.
If both fractions are of the total, 5/12 is left
When both fractions are taken from the starting amount, the spends do add up. With 120 liters, a third is 40 and a quarter is 30, so 70 is spent and 50 is left. In fractions, 12/12 − 4/12 − 3/12 = 5/12, and 5/12 of 120 is 50. Subtract fractions confirms that calculation when the denominators are different.
Finding the total when you only know what is left
If the problem gives the final balance and asks for the total, divide by the fraction that is left. After spending a third and then a quarter of the rest, 60 liters are left, which is half, so the total is 60 ÷ 1/2 = 120 liters. Dividing by one half is the same as multiplying by 2. For other balances, the find the whole from a fraction calculator does that division.
To check, rebuild the chain. A third of 120 is 40, and 80 is left. A quarter of 80 is 20, and 60 is left, the figure given in the problem. If you need to work out each amount separately, fraction of a number does it in one step.
Three spends in a row
With three spends, you multiply three complements. If a quarter is spent, then a third of the rest, and finally half of the new rest, what is left is 3/4 × 2/3 × 1/2 = 6/24 = 1/4 of the total. With 240 units, each step spends 60: from 240 down to 180, to 120 and to 60.
That all three spends are equal is a coincidence of these numbers, not a rule. If you change the fractions, the amounts stop matching, but the product still gives the fraction that is left.
Practice
- From 90 units, a third is spent and then half of the rest. How many are left?
- After spending a quarter and then a third of the rest, 18 units are left. What was the total?
- A budget loses 3/8, and then 2/5 of the remainder goes. Which fraction of the original is still there?
Show the answers
- 30 are left. A third of 90 is 30 and 60 is left; half of 60 is 30, so 30 is left.
- The total was 36. Half is left, because 3/4 × 2/3 = 6/12, and 18 × 2 = 36. Check: a quarter of 36 is 9 and 27 is left, a third of 27 is 9 and 18 is left.
- 3/8 is left. It is 5/8 × 3/5 = 15/40 = 3/8.
Sources
English Wikipedia, article Fraction, section “Multiplying a fraction by another fraction”: to multiply fractions, multiply the numerators and multiply the denominators (accessed October 17, 2026).