Problem
Measure a target in fewest pourings
Only a 10-gallon and a 7-gallon jug are used to leave exactly 36 gallons in the tank. One use pours a full jug in or dips a full jug out. The tank may never go below empty. Find the fewest uses.
Reasoning · Grade 3-2 Measuring Capacity
Your answer
Watch out Pouring alone never hits 36 — you have to dip water back out.
How to solve
Strategy Guess and Check — We need whole counts a (10-gallon pours in) and b (7-gallon dips out, or pours in) so that the net is 36. Guess small counts and check the total 10a minus 7b (or plus), keeping the count as low as possible. A short systematic list of the possibilities makes sure we do not miss a smaller answer.
1STEP 1
Try adding only
Sums of 10s and 7s never reach 36.
10a + 7b = 36 has no whole-number solution
Listing the totals you can build by only adding shows 36 is skipped, so we must also be allowed to pour some water back out.
3.OA.D.9Make A Systematic List2STEP 2
Allow pouring water back out
Allowing dips, five pours of 10 and two dips of 7 works.
10 × 5 - 7 × 2 = 50 - 14 = 36
Going a little over with the big jug and then trimming back with the small jug is a natural way to land exactly on 36.
3.OA.D.8Guess And CheckGoing a little past the target with the big jug and then pouring some back with the small one lands exactly on 36.
Why?
Pouring out reverses pouring in, so an overshoot can be undone as precisely as it was made.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
The water in the tub is every jugful poured in less every jugful poured out, so the target is one plus-and-minus sum.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
Count the uses and confirm it is fewest
No shorter combination exists, so 7 uses is fewest.
5 + 2 = 7 uses
Once you find the smallest pair of counts that nets 36, adding them gives the fewest number of times you touch the jugs.
3.OA.D.8Make A Systematic ListAnswer
7 uses
10 × 5 − 7 × 2 = 36
Check the net: 5 pours of 10 gallons add 50, and 2 dips of 7 gallons remove 14, leaving 50 - 14 = 36 gallons exactly. The count 5 + 2 = 7 is small and no combination with 6 or fewer uses produces 36, so the answer is reasonable.
Takeaway
Pouring a little extra in with the big jug and dipping some back out with the small jug lets you hit the target in just 7 moves!
- Try adding only
- Allow pouring water back out
- Count the uses and confirm it is fewest