The whole is one; find the remaining fraction

3.NF.A.13.NF.A.3 · take · grade 3

Archetype: Part-Whole Fraction Reasoning · step in a 5-type progression

Sea spent $\dfrac{5}{12}$ of the money she had on snacks, then spent $\dfrac{2}{7}$ of what was left on school supplies. Write, as a fraction, what part of the money she started with is left.

Show solution

Understand

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.

Givens

The starting amount of money counts as the whole, 1
She spends 5/12 of the whole on snacks
Then she spends 2/7 of what remains on school supplies

Unknowns

The fraction of the original money that is left at the end

Constraints

The whole is 1
The 2/7 is taken from the remainder after snacks, not from the original whole

Plan

#7 Identify Subproblems · also uses: #1 Draw a Diagram

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.

Execute

#7 Identify Subproblems 3.NF.A.3

The whole is 1 = 12/12. Spending 5/12 leaves 12/12 - 5/12 = 7/12 of the original money.

1 - \dfrac{5}{12} = \dfrac{12}{12} - \dfrac{5}{12} = \dfrac{7}{12}

Since the whole is 1, what is left is simply 1 minus the part spent.

#1 Draw a Diagram 3.NF.A.1

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 - \dfrac{2}{7} = \dfrac{5}{7} \text{ of the } \dfrac{7}{12} \text{ left}

Spending 2 of every 7 equal parts of the remainder leaves 5 of those 7 parts.

#7 Identify Subproblems 3.NF.A.1

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.

\dfrac{5}{7} \text{ of } \dfrac{7}{12} = \dfrac{5 \times 7}{7 \times 12} = \dfrac{5}{12}

Seven twelfths split into 7 equal pieces gives pieces of 1/12, and keeping 5 of them is 5/12.

Answer: 5/12

Review

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.

Work it as parts of twelfths (tool 15): the 7/12 left is 7 twelfths; spending 2/7 of those 7 twelfths means spending exactly 2 twelfths, so 7/12 - 2/12 = 5/12 remains.

Standards · min grade 3

3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Taking 5/7 of the 7/12 remainder as 5 equal parts of 1/12
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Computing 1 - 5/12 = 7/12 with the whole written as 12/12

💡 The whole is just 1, so subtract each part spent in turn to find the fraction that is left!