Section C: Practice Problems The Size of Products

Section Summary

Details

Number line. 2 tick marks, labeled 0, 4 sevenths. Point between the tick marks, labeled 3 fifths times 4 sevenths.

In this section, we learned how to compare the size of a product to the size of the factors. To compare $\frac{3}{5} \times \frac{4}{7}$ with $\frac{4}{7}$ , for example, we can put them on a number line.
Since $\frac{3}{5}$ is 3 equal parts with 5 parts in the whole, it is to the left of $\frac{4}{7}$ , only part of the way there. We can also see this by writing $\frac{3}{5}$ as $1 - \frac{2}{5}$ .

$(1 - \frac{2}{5}) \times \frac{4}{7} = \frac{4}{7} - (\frac{2}{5} \times \frac{4}{7})$

The product is less than $\frac{4}{7}$ because it is $\frac{4}{7}$ minus a fraction.

Problem 1 (Lesson 16)

Andre ran $\frac{4}{5}$ of a 7 mile trail. Did Andre run more or less than 7 miles? Explain or show your reasoning.
Clare ran $\frac{}{10}$ of a 7 mile trail. She ran more than 7 miles. Choose a number that could go in the box. Explain or show your reasoning.

Problem 2 (Lesson 17)

The point J on the number line shows how many miles Jada ran. Label the points on the number line to show how far each of these students ran.

Clare ran $\frac{8}{5}$ as far as Jada.
Tyler ran $\frac{4}{3}$ as far as Jada.
Lin ran $\frac{1}{2}$ as far as Jada.

Problem 3 (Lesson 18)

The point A is labeled on the number line.

Label each of these points on the number line.

$\frac{2}{5} \times A$
$\frac{13}{10} \times A$
$\frac{7}{4} \times A$

Number line. Tick mark, labeled 0. Point, labeled A to the right.

Problem 4 (Lesson 19)

Use the equation $\frac{5}{7} = (1 - \frac{2}{7})$ to explain why $\frac{5}{7} \times \frac{11}{3} < \frac{11}{3}$ .

Problem 5 (Lesson 20)

Explain why multiplying a fraction by a number less than 1 makes the fraction smaller.

Problem 6 (Exploration)

A point P is labeled on the number line.

P is $\frac{3}{4}$ of a number A. Plot A on the number line. Explain or show your reasoning.
P is $\frac{5}{9}$ of a number B. Plot B on the number line. Explain or show your reasoning.

Number line. Tick mark, labeled 0. Point to the right of 0. Labeled P.

Problem 7 (Exploration)

About $10^{6}$ people live in Michigan. About $10^{4}$ of the people in Michigan live in Flint.
- How many times as many people live in Michigan as in Flint?
- How many times as many people live in Flint as in Michigan?
There are about $10^{11}$ stars in the Milky Way. There are about $10^{21}$ stars in the universe.
- How many times as many stars are there in the universe than in the Milky Way?
- How many times as many stars are there in the Milky Way than in the universe?