Question

Examples 2 and 3 Evaluate each expression using properties of numbers. Name the property used in each step.

28.$\mathbf{2}\frac{\mathbf{3}}{\mathbf{4}}\mathbf{\cdot }\mathbf{1}\frac{\mathbf{1}}{\mathbf{8}}\mathbf{\cdot }\mathbf{32}$

Short answer

The value of expression is 99.

Step-by-step solution

– Write the formulae used

Associative property of multiplication:

$\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)$

Inverse property of multiplication:

$a\cdot \frac{1}{a}=\frac{1}{a}\cdot a=1$

Commutative property of multiplication:

$a\cdot b=b\cdot a$

– Write the given information 

The expression given is

$2\frac{3}{4}\cdot 1\frac{1}{8}\cdot 32$

- DESCRIPTION OF STEP 

To solve the given expression first simplify the mixed fraction(s) in the given expression as:

$\begin{array}{c}2\frac{3}{4}\cdot 1\frac{1}{8}\cdot 32=\frac{2\times 4+3}{4}\cdot \frac{1\times 8+1}{8}\cdot 32\\ =\frac{11}{4}\cdot \frac{9}{8}\cdot 32\end{array}$

Now use the associative property to simplify the above expression further as:

$\begin{array}{c}\frac{11}{4}\cdot \frac{9}{8}\cdot 32=\frac{11}{4}\cdot \left(\frac{9}{8}\cdot 32\right)\\ =\frac{11}{4}\cdot \left(\frac{9}{\cancel{8}}\cdot \cancel{8}\cdot 4\right)\\ =\frac{11}{4}\cdot 9\cdot 4\\ =\frac{11}{\cancel{4}}\cdot \cancel{4}\cdot 9\left(\text{commutativerule}\right)\\ =11\cdot 9\\ =99\end{array}$

Thus,$2\frac{3}{4}\cdot 1\frac{1}{8}\cdot 32=99$