Problem
The whole is one; find the remaining fraction
Sea spends 5/12 of her money on snacks, then spends 2/7 of what is left on school supplies. Write, as a fraction of the money she started with, how much is left.
Fractions
Your answer
Watch out It's tempting to take 2/7 of the original whole, but the problem says 2/7 comes out of what was left after snacks (7/12), not the starting amount.
How to solve
Strategy Identify Subproblems — Solve it in stages: first the money left after snacks (whole minus 5/12), then the fraction of that remainder still left after spending 2/7 of it. A bar model makes the 'fraction of what is left' step clear.
1STEP 1
Money left after snacks
The whole is 1 = 12/12. Spending 5/12 leaves 12/12 - 5/12 = 7/12 of the original money.
1 - 5/12 = 12/12 - 5/12 = 7/12
Since the whole is 1, what is left is simply 1 minus the part spent.
3.NF.A.3Identify Subproblems2STEP 2
Fraction of the remainder still left
She spends 2/7 of what was left, so 7/7 - 2/7 = 5/7 of the remainder stays. The 5/7 applies to the 7/12 she had after snacks.
1 - 2/7 = 5/7 of the 7/12 left
Spending 2 of every 7 equal parts of the remainder leaves 5 of those 7 parts.
3.NF.A.1Draw A Diagram3STEP 3
Combine to a fraction of the whole
Take 5/7 of 7/12. Split the 7/12 into 7 equal parts of 1/12 each; 5 of them are left, which is 5/12 of the original money.
5/7 of 7/12 = (5 × 7)/(7 × 12) = 5/12
Seven twelfths split into 7 equal pieces gives pieces of 1/12, and keeping 5 of them is 5/12.
3.NF.A.1Identify SubproblemsTaking 5/7 of the 7/12 that is left lands on a whole number of twelfth-sized pieces, so it is again a plain fraction of the money she started with.
Why?
The 7/12 left after snacks is just seven equal pieces, each one 1/12 of the whole.
Why?
A twelfth comes from cutting the whole into 12 equal pieces with nothing left over, and 7/12 simply collects seven of those single 1/12 pieces.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Taking 5/7 of a batch of seven equal pieces keeps exactly five of the pieces, a whole count.
Why?
The seven equal shares that sevenths cut the batch into match the seven pieces one for one, so each share is one piece and five shares are five pieces.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
Five kept pieces, each 1/12 of the whole, together make five twelfths of the whole.
Why?
Five copies of 1/12 is 1/12 added over and over five times, which counts up to that many twelfths.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Answer
5/12
5/7 of 7/12 = (5 × 7)/(7 × 12) = 5/12
After snacks 7/12 remains; she then spends a bit more, so the final amount must be less than 7/12. The answer 5/12 is less than 7/12 and still positive, which fits. Check: spent 5/12 on snacks plus 2/12 on supplies (2/7 of 7/12) = 7/12 spent, leaving 5/12.
Takeaway
The whole is just 1, so subtract each part spent in turn to find the fraction that is left!
- Money left after snacks
- Fraction of the remainder still left
- Combine to a fraction of the whole
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested