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}

Answer and Explanation:

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.


Learn more about this topic:

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Ratios and Proportions: Definition and Examples

from Geometry: High School

Chapter 7 / Lesson 1
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