We are asked to explain why estimating products of fractions is useful. Imagine this situation.

Jack is wrapping gifts for his family. Each gift is a drawing he made. He wants to know how many square meters of wrapping paper he will need.

Each drawing is shaped like a rectangle. Their dimensions are fractions, expressed in meters.

Gifts: a dragon with 3/7 times 3/7 written below, a dolphin with 1/9 times 2/9 written below and a tulip with 2/3 times 3/4 written below

Jack does not need to know the exact amount of paper he will need to use in order to buy the paper. An estimate will be good enough. This means we should estimate the size of each drawing. The length and width of the first drawing are

\frac{3}{7}

meters. The area of the drawing is the length multiplied by the width.

A = \frac{3}{7} \times \frac{3}{7}

When we estimate products, we usually round fractions to

0,

\frac{1}{2},

or

1 .

Notice that

\frac{1}{2}

is the closest to

\frac{3}{7} .

Let's substitute

\frac{1}{2}

for

\frac{3}{7}

in the expression and simplify.

A = \frac{3}{7} \times \frac{3}{7}

$\frac{3}{7} \approx \frac{1}{2}$

A \approx (\frac{1}{2}) (\frac{1}{2})

MultFrac

Multiply fractions

A = \frac{1 \times 1}{2 \times 2}

Multiply

A = \frac{1}{4}

The estimated area of the first drawing is

\frac{1}{4}

square meters. Below are the results if we repeat these steps for the other drawings.

We found the areas for all three drawings, but notice that Jack will need to wrap both the front and back of each image. Let's double each area. Then we can add these areas to find how much wrapping paper Jack needs.

2 (\frac{1}{4}) + 2 (0) + 2 (\frac{1}{2}) = \frac{3}{2}

Jack will need about

\frac{3}{2}

or

1.5

square meters of wrapping paper. One roll of wrapping paper should do.

Summary

In real life, we mostly care about how fast we calculate things, not how precisely. Since we cannot buy a precise amount of wrapping paper, we only have to estimate the area we are wrapping. This is why we round numbers, especially fractions.

Exercise