Question

Find the value of x in each of the following proportions. a. 6 : 9 = x : 72 b. 8⁄3 = 40⁄x c. x : 55 = 3 : 5 d. 33⁄x = 11⁄5 A recipe calls for 5 cups of flour for every 6 teaspoons of salt. If you wanted to make a larger batch of the same recipe with 15 cups of flour, how much salt would you need? A 20-foot flagpole casts a 6-foot shadow. How tall is a nearby building that casts a 30-foot shadow (assume the shadows are measured at the same time of the day and both the flagpole and the building face the sun at approximately the same angle)? For every 15 columns of news stories in a newspaper, 4 columns of ads are required. If a newspaper contains 300 columns of news stories, how many columns of ads must it have? A certain color paint should have a mix of blue and green in a ratio of 5 : 2. If a house painter needs to use 8 gallons of green paint, how many gallons of blue paint should he mix with it in order to obtain the right color?

Answer 1

QUESTION 1a

We want to find the value of $x$ in the proportion,

$6:9=x:72$

We rewrite the ratio on each side of the equation as fractions to obtain,

$\frac{6}{9}=\frac{x}{72}$

We now multiply both sides of the equation by $72$ to get,

$\frac{6}{9}\times 72=\frac{x}{72}\times 72$

This simplifies to  

$\frac{6}{1}\times 8=\frac{x}{1}\times 1$

$\therefore 48=x$

$x=48$

QUESTION 1b

The given proportion is,

$\frac{8}{3}=\frac{40}{x}$

We cross multiply to obtain,

$8x=3\times 40$

We divide both sides by 8 to get,

$x=\frac{3\times 40}{8}$

$\Rightarrow x=\frac{3\times 40}{8}$

$\Rightarrow x=\frac{3\times 5}{1}$

$\therefore x=15$

QUESTION 1c

The given proportion is

$\frac{x}{55}=\frac{3}{5}$

We multiply both sides by 55 to get,

$\frac{x}{55}\times 55=\frac{3}{5}\times 55$

This simplifies to  

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

$\therefore x=33$

QUESTION 1d

The given proportion is,

$\frac{33}{x}=\frac{11}{5}$

We cross multiply to get,

$33\times 5=11x$

We now divide both sides by 11 to get,

$\frac{33\times 5}{11}=x$

$\Rightarrow x=15$

QUESTION 2

Since the recipe calls for 5 cups of flour for every 6 teaspoons of salt, we can write the ratio,

$5:6$

Let a larger recipe of 15 cups of flour correspond to $x$ teaspoons of salt,

Then the proportion becomes,

$5:6=15:x$

This implies that,

$\frac{5}{6}=\frac{15}{x}$.

We cross multiply to get,

$5x=6\times 15$

We divide through by 5 to get,

$x=\frac{6\times 15}{5}$

$\therefore x=18$

Hence you will need 18 teaspoons of salt.

QUESTION 3

We were given that a 20-foot flagpole casts a 6-foot shadow.

We write this as a ratio to get,

$20:6$

Let the nearby 30-foot building cast an x-foot shadow.

We can also write this as a ratio to get,

$30:x$

The proportion now becomes,

$20:6=30:x$

We cross multiply to get,

$20x=6\times 30$  

This implies that

$x=\frac{6\times 30}{20}$  

$\Rightarrow x=9$  

QUESTION 4

If 15 corresponds to 4, we write the ratio,

$15:4$

We want to find what number of ads corresponds to 300 columns. Let that number be x.

Then we have the ratio,

$300:x$

The proportion is

$15:4=300:x$

This implies that,

$15x=300\times 4$

$x=\frac{300\times 4}{15}$

$x=80$

It must have 80 ads.

QUESTION 5

The given ratio is $5:2$,

Let the gallons of blue paint be $x$

We can write the ratio $x:8$

The required proportion is

$5:2=x:8$

This implies that,

$\frac{5}{2}=\frac{x}{8}$

We multiply both sides by 8

$x=\frac{5}{2}\times 8$

$x=20$

8 gallons of blue paint is needed