Fraction of a number: divide then multiply

3.NF.A.13.OA.A.2 · take · grade 3

Archetype: Multiplicative Comparison and Unit Rate · step in a 7-type progression

A ribbon $30\ \text{m}$ long is cut by Liam, Mia, and Noah, each taking the length they need to wrap a gift. Liam cut $\frac{1}{6}$ of the whole ribbon, Mia cut $\frac{4}{15}$ of the whole ribbon, and Noah cut $\frac{3}{10}$ of the whole ribbon. Who cut the longest piece of ribbon?

Show solution

Understand

A 30 m ribbon is shared. Liam takes 1/6 of the whole, Mia takes 4/15 of the whole, and Noah takes 3/10 of the whole. I need to find each person's length and decide who cut the longest piece.

Givens

The whole ribbon is 30 m.
Liam cut 1/6 of 30 m.
Mia cut 4/15 of 30 m.
Noah cut 3/10 of 30 m.

Unknowns

Who cut the longest piece (and the lengths to compare).

Constraints

Each fraction is of the same whole, 30 m.
A fraction of 30 m is found by dividing 30 into the denominator's equal parts, then taking the numerator's worth.

Plan

#7 Identify Subproblems · also uses: #2 Make a Systematic List

Each person's length is one fraction-of-a-number subproblem (divide 30 by the denominator, multiply by the numerator). Listing the three results side by side makes the comparison straightforward.

Execute

#7 Identify Subproblems 3.NF.A.1

Divide 30 into 6 equal parts, then take 1 part.

30 \div 6 = 5,\quad 5 \times 1 = 5

A fraction of a length means split into equal parts and take that many.

#7 Identify Subproblems 3.NF.A.1

Divide 30 into 15 equal parts, then take 4 parts.

30 \div 15 = 2,\quad 2 \times 4 = 8

Same split-and-take rule with denominator 15 and numerator 4.

#7 Identify Subproblems 3.NF.A.1

Divide 30 into 10 equal parts, then take 3 parts.

30 \div 10 = 3,\quad 3 \times 3 = 9

Same rule with denominator 10 and numerator 3.

#2 Make a Systematic List 3.OA.A.2

List them: Liam 5 m, Mia 8 m, Noah 9 m. The largest is 9 m, so Noah cut the longest piece.

5 < 8 < 9 \Rightarrow \text{Noah}

Once all three are plain meter amounts, comparing is just ordering whole numbers.

Answer: Noah (his piece is 9 m, the longest)

Review

The three pieces total 5 + 8 + 9 = 22 m, which is less than the 30 m ribbon, so the shares are sensible (some ribbon is left over). Noah's 9 m is clearly the largest of 5, 8, 9.

Compare the fractions directly (tool 15) by a common denominator of 30: Liam 5/30, Mia 8/30, Noah 9/30; the largest fraction of the same whole belongs to Noah.

Standards · min grade 3

3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Computing each person's length as a fraction of the 30 m whole.
3.OA.A.2 Interpret whole-number quotients of whole numbers — Dividing 30 by each denominator and comparing the resulting lengths.

💡 This only needs Grade 3 fraction sense: split the 30 m, take your parts, then see whose pile is biggest!