Problem

A fraction is a count of unit fractions

Mia paints

\frac{1}{5}

of a poster board every 3 minutes. At that same rate, how long does it take her to paint

\frac{3}{5}

of the board?

Fractions

Your answer

minutes

How to solve

Strategy Look for a Pattern — Seeing

\frac{3}{5}

as three units of

\frac{1}{5}

turns the problem into repeated equal chunks: each

\frac{1}{5}

costs 3 minutes, so the total is just that time pattern repeated three times. A bar drawing makes the equal chunks obvious.

1STEP 1

See $\frac{3}{5}$ as three $\frac{1}{5}$ s

The fraction $\frac{3}{5}$ is three copies of the unit fraction $\frac{1}{5}$ . So painting $\frac{3}{5}$ means painting $\frac{1}{5}$ three separate times.

\frac{3}{5} = \frac{1}{5} + \frac{1}{5} + \frac{1}{5}

2STEP 2

Each fifth costs 3 minutes

Painting one $\frac{1}{5}$ takes 3 minutes, and there are three of them, so multiply 3 minutes by 3.

3 × 3 = 9

Answer

9 minutes

3 × 3 = 9

\frac{3}{5}

is three times as much board as

\frac{1}{5}

, so it should take three times as long as 3 minutes, which is 9 minutes. That is less than the time for the whole board (5 fifths would be 15 minutes), so 9 minutes is reasonable.

Takeaway

A fraction is just a count of unit fractions, so $\frac{3}{5}$ takes three times the time of one $\frac{1}{5}$ !

See $\frac{3}{5}$ as three $\frac{1}{5}$ s
Each fifth costs 3 minutes

Where next?

Level upsuggested

Fractions require equal partitioning of the whole The whole is one; find the remaining fraction

Another one like thissuggested

12 practice problems

Step back and solidifysuggested

Larger dividend gives larger quotient

2 reasoning problems grow out of this type — the bridge toward competition →

▶ Practice — 12 problems

A fraction is a count of unit fractions

See 35\frac{3}{5}53​ as three 15\frac{1}{5}51​s

Each fifth costs 3 minutes

See $\frac{3}{5}$ as three $\frac{1}{5}$ s