Fractions & Decimals

Problem

Find the whole from a fractional part

A tank starts with some unknown amount of water. First, 2/7 of all the water is used. Then 4/5 of whatever is left after that is poured out. After both steps, exactly 12 liters remain. We need the starting amount.
Fractions
Your answer
How to solve
Strategy Work Backwards — The end-state (12 L left) is given and we want the start, so working backwards fits perfectly. We split the journey into two stages (use 2/7, then pour 4/5) and undo each one, using fraction-of-an-amount reasoning to recover the original whole.
1STEP 1

What fraction is left after the first step

Using 2/7 leaves 7/7 - 2/7 = 5/7 of the original water.

1 - 2/7 = 5/7
2STEP 2

What fraction is left after the second step

Pouring out 4/5 of that leaves one fifth of 5/7, which is 1/7 of the original.

1/5 × 5/7 = 5/35 = 1/7
3STEP 3

Match the leftover to 12 liters and work backwards

If one seventh is 12 L, the whole tank held 12 x 7 = 84 L.

1/7 × W = 12 → W = 12 × 7 = 84
Answer
84 liters
Check forward: start 84 L, use 2/7 - > 24 L used, 60 L left; pour 4/5 of 60 - > 48 L out, 12 L left. That matches the given 12 L, and 84 L is comfortably larger than 12 L as expected after two removals.
Takeaway

If you know the value of just one equal part, multiply back up to find the whole.

  • What fraction is left after the first step
  • What fraction is left after the second step
  • Match the leftover to 12 liters and work backwards
Where next?
Another one like thissuggested
Same look, different logicsuggested

▶ Practice — 12 problems