# For a certain kind of plasterwork, 1.3 cu yd of sand are needed for every 100 sq yd of surface....

## Question:

For a certain kind of plasterwork, 1.3 cu yd of sand are needed for every 100 sq yd of surface. How much sand will be needed for 380 sq yd of surface?

## Ratio and Proportion

A ratio is an expression comparing two quantities, for example a and b. This can be written using a colon, for example {eq}a:b {/eq}, and read as "a is to b". Ratios can also be written as fractions, where the first quantity is the numerator and the second quantity is the denominator, {eq}\dfrac{a}{b} {/eq}

A proportion, on the other hand is an equation comparing two ratios. When two ratios are proportional, they are equal. For example two ratios {eq}a:b = c:d {/eq} and can also be written as {eq}\dfrac{a}{b} = \dfrac{c}{d} {/eq}

Given that {eq}1.3\ \mathrm{yd^3} {/eq} of sand are needed for every {eq}100\ \mathrm{yd^2} {/eq} of surface, we can write it as a ratio as follows.

$$1.3 \,\mathrm{yd^3} : 100 \,\mathrm{yd^2}$$

This ratio can be expressed as a fraction and it can be rewritten as follows.

$$\dfrac{1.3 \,\mathrm{yd^3}}{100 \,\mathrm{yd^2}}$$

For the second ratio, we have an unknown amount of sand {eq}n {/eq} needed for {eq}380\ \mathrm{yd^2} {/eq} of surface. We can do the same as we did for the previous ratio in writing it as a fraction.

$$n \,\mathrm{yd^3}: 380 \,\mathrm{yd^2}\\ \dfrac{n \,\mathrm{yd^3}}{380 \,\mathrm{yd^2}}$$

We can compare the two ratios by making a proportion out of them. We do this by equating the two fractions we obtained.

$$\dfrac{1.3 \,\mathrm{yd^3}}{100 \,\mathrm{yd^2}} = \dfrac{n \,\mathrm{yd^3}}{380 \,\mathrm{yd^2}}$$

We now have an equation with an unknown value, {eq}n {/eq}. We can solve for this value by using cross multiplication.

\begin{align*} (380 \,\mathrm{yd^2}) \dfrac{1.3 \,\mathrm{yd^3}}{100 \,\mathrm{yd^2}} &= n \,\mathrm{yd^3} \\ n &= 4.94 \,\mathrm{yd^3} \end{align*}

Therefore, 4.94 cubic yards of sand is needed for 380 square yards of surface. 