Question

Evaluate the following expression and name the property used in each step:

$3\frac{2}{3}+4+5\frac{1}{3}$

Short answer

The value of the expression is: $\mathbf{3}\frac{\mathbf{2}}{\mathbf{3}}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{5}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{=}\mathbf{13}$.

Step-by-step solution

Step 1. Given information.

The given expression is:

$3\frac{2}{3}+4+5\frac{1}{3}$

Step 2. Write the concept.

The substitution property for multiplication, addition, and subtraction is:

$\begin{array}{c}a\cdot b=c\\ a÷b=c\\ a\pm b=c\end{array}$

The Associative property is:

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

Also, to add or subtract the fraction, the denominator must be the same.

Step 3. Write mixed fraction to fraction.

$\begin{array}{c}3\frac{2}{3}=\frac{11}{3}\\ 5\frac{1}{3}=\frac{16}{3}\end{array}$

Step 4. Evaluate the expression.

The expression $3\frac{2}{3}+4+5\frac{1}{3}$ can be written as in below:

$3\frac{2}{3}+4+5\frac{1}{3}=\frac{11}{3}+4+\frac{16}{3}$

Then,

$\begin{array}{l}\frac{11}{3}+4+\frac{16}{3}=\left(\frac{11}{3}+\frac{16}{3}\right)+4\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}}\left(\text{Associative\hspace{0.17em}\hspace{0.17em}Property}\right)\\ =\left(\frac{27}{3}\right)+4\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}}\left(\text{Substitution\hspace{0.17em}\hspace{0.17em}Property}\right)\end{array}$

Since, $\left(\frac{27}{3}\right)+4=\frac{27}{3}+\frac{4}{1}$. So,

$\begin{array}{l}\frac{11}{3}+4+\frac{16}{3}=\frac{27\times 1}{3\times 1}+\frac{4\times 3}{1\times 3}\\ =\frac{27}{3}+\frac{12}{3}\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}}\left(\text{Substitution\hspace{0.17em}\hspace{0.17em}Property}\right)\\ =\frac{39}{3}\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}}\left(\text{Substitution\hspace{0.17em}\hspace{0.17em}Property}\right)\\ =\frac{3\times 13}{3}\\ =13\end{array}$