TACHS: Fractions of what remains: multiply what is kept
Track an amount through two fraction changes, decide when to multiply and when to subtract, and work back to the starting amount.
When a question takes a fraction away and then a fraction of what is left, how do you find what remains?
Short answer
Write the fraction kept at each step and multiply those fractions. Subtract from 1 only when every fraction is taken of the full amount, and divide by the fraction kept to find a starting amount.
Know first: Multiplying fractions, Subtracting a fraction from 1
Step 1 of 6
Each fraction is taken of the amount in front of it
When 2/5 of a pitcher is poured out, 3/5 is kept. That kept fraction matters more than the fraction used, because it is what the next step starts from.
The words of what remains or of the rest mean the next fraction is taken of a smaller amount. When both steps say of what remains, multiply the kept fractions. When a step says of the full tank, that fraction is taken of the starting amount, so you subtract it from 1.
| The question says | What the fraction is taken of | What to do |
|---|---|---|
| "then 1/3 of what remains" | The smaller amount left | Multiply the kept fractions |
| "then 1/4 of the full tank" | The starting amount | Subtract both fractions from 1 |
| "12 cups are left. How much at the start?" | Unknown start | Divide 12 by the fraction kept |
| "what fraction of the water there just before" | The amount after step one | Divide the part taken by that amount |
The same two fractions give different answers depending on what each one is taken of.
Step 2 of 6
Multiply the kept fractions, then test with real numbers
Worked example: A pitcher poured twice
A full pitcher of lemonade is served. First 2/5 of it is poured. Then 1/3 of what is left is poured. What fraction of the full pitcher remains?
- A4/15
- B2/15
- C2/5
- D3/5
- 1
Write the fraction kept at step one
Pouring 2/5 keeps 3/5.
- 2
Write the fraction kept at step two
Pouring 1/3 of the rest keeps 2/3 of that 3/5.
- 3
Multiply
3/5 × 2/3 = 6/15 = 2/5.
- 4
Test with 30 cups
First pour: 2/5 × 30 = 12 cups, leaving 18. Second pour: 1/3 × 18 = 6 cups, leaving 12. And 12 out of 30 is 2/5.
- 5
See why the other choices are there
4/15 is 1 − 2/5 − 1/3, which treats the second pour as 1/3 of the full pitcher. 2/15 multiplies the fractions poured instead of the fractions kept. 3/5 stops after the first pour.
Answer
2/5 of the pitcher remains.
Wrong: 1 − 2/5 − 1/3 = 4/15
Right: 3/5 × 2/3 = 2/5
The second pour was 1/3 of the 18 cups left, which is 6 cups. Subtracting 1/3 from 1 acts as if it were 1/3 of all 30 cups, or 10 cups.
Step 3 of 6
Change the base or run it backward
Some questions say the second amount is a fraction of the full tank. Then both fractions share the same base, and you subtract. A full tank loses 1/6, then water equal to 1/4 of the full tank: 1 − 1/6 − 1/4 = 7/12 remains. With 24 liters, that is 24 − 4 − 6 = 14 liters, and 14/24 = 7/12.
Other questions give the amount left and ask for the start. The amount left is a fraction of the start, so the start is bigger. Divide by the fraction kept.
Working back to the start
A tank loses 1/4 of its water, then 2/3 of what remains. 15 liters are left.
Fraction kept: 3/4 × 1/3 = 1/4
Start: 15 ÷ 1/4 = 15 × 4 = 60 liters
Check: 60 → 45 after step one → 15 after step two
Step 4 of 6
Decide the base before you calculate
Check yourself · Question 1
Eli spends 1/5 of his savings on a game. Then he spends 1/3 of the money that is left on a jacket. What fraction of his savings does he have now?
Answer: C
Spending 1/5 keeps 4/5. Spending 1/3 of the rest keeps 2/3 of that. Multiply: 4/5 × 2/3 = 8/15.
Trap answer: 7/15
Wrong base for the second step
- Why it looks right
- Both fractions appear in the question, so subtracting both from 1 feels like accounting for everything.
- Why it's wrong
- The jacket cost 1/3 of what was left, and only 4/5 was left. A third of 4/5 is 4/15 of his savings, less than a third of all of it. So he keeps 4/5 − 4/15 = 8/15.
- The right answer
- 8/15, because 4/5 × 2/3 = 8/15.
Check yourself · Question 2
A jar of beads loses 1/5 of its beads. After a second scoop, 3/5 of the full jar is left. What fraction of the beads in the jar just before the second scoop was taken?
Answer: B
After the first scoop, 4/5 of the jar is left. The second scoop takes 4/5 − 3/5 = 1/5 of the full jar. As a fraction of the 4/5 that was there: (1/5) ÷ (4/5) = 1/4. With 40 beads: 32 before the scoop, 24 after, and 8 out of 32 is 1/4.
Check yourself · Question 3
A tank loses 1/3 of its water. Then it loses 3/4 of the water that remains. Now 10 gallons are left. How many gallons were in the tank at the start?
Answer: D
Fraction kept: 2/3 × 1/4 = 1/6. So 1/6 of the start is 10 gallons, and the start is 10 ÷ 1/6 = 60. Check: 60 → 40 → 10.
Common mistake
"10 gallons are left and 1/6 is kept, so I multiply: 10 × 1/6."
- Why it's tempting
- Multiplying by the kept fraction is exactly what you did going forward, so it feels like the same move.
- Do this instead
- Going forward, start × 1/6 = amount left. Going backward, you undo that, so divide: amount left ÷ 1/6. Ask whether the start should be bigger than what is left. It always is.
Step 5 of 6
What to remember
Remember
- Write the fraction kept at each step. Multiply kept fractions when each step acts on what remains.
- Subtract from 1 only when every fraction is taken of the full amount.
- To find the start, divide the amount left by the total fraction kept, then check forward.
Step 6 of 6
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, what is the second fraction taken of, and which wrong choice comes from subtracting both fractions from 1?