  ### Home > MC2 > Chapter 10 > Lesson 10.1.4 > Problem10-46

10-46.

Maria and Jorge were trying to simplify the expression $1 \frac { 2 } { 5 } \cdot ( - \frac { 3 } { 4 } ) \cdot ( - \frac { 4 } { 3 } )$. Maria started by rewriting $1 \frac { 2 } { 5 }$ as $\frac { 7 } { 5 }$. Her work is below.

$\left. \begin{array} { c } { 1 \frac { 2 } { 5 } \cdot ( - \frac { 3 } { 4 } ) \cdot ( - \frac { 4 } { 3 } ) } \\ { \frac { 7 } { 5 } \cdot ( - \frac { 3 } { 4 } ) \cdot ( - \frac { 4 } { 3 } ) } \\ { ( - \frac { 21 } { 20 } ) \cdot ( - \frac { 4 } { 3 } ) } \end{array} \right.$
$\frac { 84 } { 60 } = 1 \frac { 24 } { 60 } = 1 \frac { 2 } { 5 }$

Jorge had a different idea. He multiplied $( - \frac { 3 } { 4 } ) \cdot ( - \frac { 4 } { 3 } )$ first.

1. Simplify the expression $1 \frac { 2 } { 5 } \cdot ( - \frac { 3 } { 4 } ) \cdot ( - \frac { 4 } { 3 } )$ using Jorge's method. Is your answer equal to Maria's?

What do you get when you multiply $\left(-\frac{3}{4}\right) \cdot \left(-\frac{4}{3}\right)$?

Since you are multiplying by reciprocal fractions, the fractions reduce to $1$.

$\text{Yes, }1\frac{2}{5} \cdot \left(-\frac{3}{4}\right) \cdot \left(-\frac{4}{3}\right) = 1\frac{2}{5}$

2. Why might Jorge have decided to multiply $( - \frac { 3 } { 4 } ) \cdot ( - \frac { 4 } { 3 } )$ first?

As you found in part (a), $\left(-\frac{3}{4}\right)\left(-\frac{4}{3}\right)$ is $1$, due to the multiplicative property.

3. In a multiplication problem, the factors can be grouped together in different ways. This is called the Associative Property of Multiplication. Read the Math Notes box for this lesson, then show that $( \frac { 3 } { 4 } \cdot \frac { 2 } { 5 } ) \cdot ( - 2 ) = \frac { 3 } { 4 } \cdot ( \frac { 2 } { 5 } \cdot ( - 2 ) )$.

Solve each equation. What do you get?

$-\frac{12}{20} = -\frac{12}{20}$

4. Simplify each expression. First, decide if you want to group some factors together.

1. $\left(-9\right)\cdot\frac{1}{9}\cdot\frac{3}{8}$

1. $-\frac{5}{12}\cdot\frac{3}{7}\cdot\frac{4}{9}$

1. $-8.1·5·2$

i. Like numbers in the numerator and denominator cancel out.
ii. Multiply the last two values. Do you see anything you can cancel out?
iii. Multiply two terms together to get an easier number to deal with.

$ii. -\frac{5}{60}$