Problem
Recover the whole from a fractional part
A mystery number ★ has the property that 7/12 of it equals 21. I first need to find ★, then report 1/9 of ★.
FractionsOperations
Your answer
Watch out It's tempting to stop once you find the whole ★ = 36 and report 36 as the answer — but the question asks for 1/9 of ★, not ★ itself.
How to solve
Strategy Work Backwards — I work backwards from the part to the whole: if 7 twelfths equal 21, then one twelfth equals 21/7, and the whole is 12 of those. Then taking 1/9 of the recovered whole is a second small subproblem.
1STEP 1
Find one twelfth of ★
7 equal twelfths make 21, so one twelfth is 21 divided by 7.
21 ÷ 7 = 3
If 7 equal parts total 21, each part is found by sharing 21 into 7 equal pieces.
3.OA.A.4Work BackwardsSince 7 equal twelfths of star add up to 21, one twelfth of star must be 21 divided by 7.
Why?
The 21 is 7 equal twelfths piled together, and 7 equal piles of one twelfth is the same as 7 times one twelfth.
Why?
Making 7 same-size groups of one twelfth is just adding that twelfth over and over, and repeated adding of equal groups is what multiplying means.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
If 7 times one twelfth is 21, then one twelfth is found by sharing 21 back into 7 equal parts, which is 21 divided by 7.
Why?
Dividing undoes multiplying, so splitting 21 into 7 equal parts reverses the 'times 7' and returns the single twelfth it started from.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
2STEP 2
Build the whole ★
★ is made of 12 such twelfths, so multiply one twelfth (3) by 12.
3 × 12 = 36
The whole is just all 12 of its equal parts put back together.
3.NF.A.1Work Backwards3STEP 3
Take 1/9 of ★
Now find 1/9 of 36 by dividing 36 into 9 equal parts.
36 ÷ 9 = 4
A unit fraction 1/9 of a number is that number shared into 9 equal parts.
3.NF.A.1Identify SubproblemsAnswer
4
36 ÷ 9 = 4
Check: 7/12 of 36 is 36 ÷ 12 × 7 = 3 × 7 = 21, which matches the given. And 1/9 of 36 is 4. Both come out as clean whole numbers, so 36 is the right whole and 4 is the answer.
Takeaway
This only needs Grade 3 fraction sense: find one equal part, build the whole, then split again!
- Find one twelfth of ★
- Build the whole ★
- Take 1/9 of ★
Where next?
Another one like thissuggested
Step back and solidifysuggested