Zoey made 5 1/2 cups of trail mix for a camping trip. She wants to divide the trail mix into 3/4 cup servings. How many serviings can she make?

Nov 19, 2017

Zoey can divide the $5 \frac{1}{2}$ cups of trail mix into $7$ sets of cups that are$\frac{3}{4}$ full, with $\frac{1}{4}$ of a 100% full cup remaining.

Explanation:

we can do this in two ways, we can do it with a diagram, showing the different cups, or we can use simple division.

$\textcolor{w h i t e}{c}$

$\underline{\textcolor{b l a c k}{\text{Method 1, diagramming:}}}$

original amount of trail mix: $5 \frac{1}{2}$ cups

$\textcolor{red}{\text{cup " 1: {3/4 " cup}}$

Amount of trail mix left: 5 1/2 - 3/4 = color(blue)(4 3/4 " cups remaining"

$\textcolor{red}{\text{cup " 2: {3/4 " cup}}$

Amount of trail mix left: 4 3/4 - 3/4 = color(blue)(4 " cups remaining"

$\textcolor{red}{\text{cup " 3: {3/4 " cup}}$

Amount of trail mix left: 4 - 3/4 = color(blue)(3 1/4 " cups remaining"

$\textcolor{red}{\text{cup " 4: {3/4 " cup}}$

Amount of trail mix left: 3 1/4 - 3/4 = color(blue)(2 1/2 " cups remaining"

$\textcolor{red}{\text{cup " 5: {3/4 " cup}}$

Amount of trail mix left: 2 1/2 - 3/4 = color(blue)(1 3/4 " cups remaining"

$\textcolor{red}{\text{cup " 6: {3/4 " cup}}$

Amount of trail mix left: 1 3/4 - 3/4 = color(blue)(1 " cups remaining"

$\textcolor{red}{\text{cup " 7: {3/4 " cup}}$

Amount of trail mix left: 1 - 3/4 = color(blue)(1/4 " cups remaining"

From this, we can see that after $7$ cups, there is only $\frac{1}{4}$ of a cup left, not enough to fill another $\frac{3}{4}$ cup. So Zoey can divide the $5 \frac{1}{2}$ cups of trail mix into $7$ sets of $\frac{3}{4}$ full cups with $\frac{1}{4}$ of a cup remaining.

$\textcolor{w h i t e}{c}$
$\textcolor{w h i t e}{c}$

$\underline{\textcolor{b l a c k}{\text{Method 2, simple division:}}}$

splitting $5 \frac{1}{2}$ cups of trail mix into $x$ sets of $\frac{3}{4}$ cups can be written algebraically as $x \times \frac{3}{4} = 5 \frac{1}{2}$

$x \times \frac{3}{4} = 5 \frac{1}{2}$

In this, we need to isolate $x$, to find it's value.

$\frac{x \times \textcolor{red}{\cancel{\textcolor{b l a c k}{\frac{3}{4}}}}}{\textcolor{red}{\cancel{\frac{3}{4}}}} = \frac{5 \frac{1}{2}}{\textcolor{red}{\frac{3}{4}}}$

$x = 5 \frac{1}{2} \div \frac{3}{4}$

$x = \frac{11}{2} \div \frac{3}{4}$

Finding the reciprocal of the second fraction and replacing the $\div$ with $\times$

$x = \frac{11}{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}} 1}} \times \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{4}} 2}}{3}$

$x = \frac{11}{1} \times \frac{2}{3}$

$x = \frac{22}{3}$

$x = 7 \frac{1}{3}$

This is represented as $7 \frac{1}{3}$ sets of $\frac{3}{4}$ cups, $\frac{1}{3} \textcolor{b l u e}{\text{(remaining amount of " 3/4 " cup)}}$ of 3/4 color(green)("(Serving size of cup)" is $\frac{1}{4}$, so there is $\frac{1}{4}$ of a full cup remaining and $\frac{1}{3}$ of a $\frac{3}{4}$ cup remaining.

$\textcolor{w h i t e}{c}$

Zoey can divide the $5 \frac{1}{2}$ cups of trail mix into $7$ sets of cups that are$\frac{3}{4}$ full, with $\frac{1}{4}$ of a 100% full cup remaining.