2.02 Applying ratios

Proportion

When $2$2 rates or ratio are equal (equivalent) they are in proportion. The following ratios are in proportion:

$3:5=$3:5=$15:25=$15:25=$\frac{1}{5}:\frac{1}{3}$15​:13​

You can use this concept of proportion to find a missing value by making equivalent ratios. If a recipe calls for $2$2 eggs for every $3$3 cups of flour, how many eggs are needed for $15$15 cups of flour? This can be written as:

$2:3$2:3	$=$=	$x:15$x:15	Ratios are equal (in proportion)
$\frac{2}{3}$23​	$=$=	$\frac{x}{15}$x15​	Expressing them as fractions
$\frac{2}{3}\times15$23​×15	$=$=	$x$x	Solving for $x$x
$x$x	$=$=	$10$10	Simplifying

 

You would need $10$10 eggs.

Alternatively, you could use equivalent fractions and make the denominators the same. In this case, multiplying the top and bottom of the left-hand-side fraction by $5$5 would give the same answer.

Practice question

Question 1

 

Dividing quantities using ratio

You may wish to divide a quantity by a given ratio.

If we were dividing a quantity of $4$4 items using a ratio of $1:3$1:3, it could be represented using these blue and green dots.

Here there are $4$4 parts in the ratio and the quantity being divided is $4$4. What happens if we have $40$40 items and we want to divide them in the ratio $1:3$1:3?

First, calculate the total number of parts in the ratio, then use it to divide the quantity into a given ratio.

The total number of parts in the (part-part) ratio is found by adding all the parts. In this case $1+3=4$1+3=4. Then we can divide the total quantity, which is $40$40 in this case by the total number of parts, which is $4$4, to give $10$10. Then using the ratio, you have a blue group of $1\times10=10$1×10=10 and a green group of $3\times10=30$3×10=30. Here we've multiple each term in the ratio by $10$10.

To divide a quantity by a ratio you first identify the number of parts in the ratio.

A ratio of $3:8:1$3:8:1 would have $12$12 parts in total.

Practice questions

Question 2

Question 3

Question 4

 

The unitary method

This is a method of carrying out a calculation to find the value of a number of items by first finding the cost of one of them. This method of solving problems is often handy for solving word problems.

Practice questions

Question 5

Question 6

 

Convert units

Another application of ratios can also be seen in converting units, imperial to metric and vice versa.

Worked Examples

example 1

The ratio of miles to kilometers is $1:1.6$1:1.6. Use this fact to find out how many kilometers are equal to $5$5 miles.

Think:  The given ratio of $1:1.6$1:1.6 means that $1$1 mile is equal to $1.6$1.6 kilometers.  Set up a ratio problem, identifying by what factor each part of the ratio needs to be multiplied to give $5$5 miles.

Do: 

miles	to	kilometers
$1$1	$:$:	$1.6$1.6
to get $5$5 miles, we need $5$5 lots of  the part representing miles in the ratio		so we also need to multiply the number of kilometers by $5$5
$1\times5$1×5	$:$:	$1.6\times5$1.6×5
$5$5	$:$:	$8$8

So $5$5 miles is equivalent to $8$8 kilometers.

 

example 2

$1$1 gallon is around $3.8$3.8 liters.

a)  State this as a ratio of liters to gallons, in the form $a:b$a:b. Give your answer in simplest form.

Think: We want a ratio that compares the number of liters to the numer of gallons. 

Do: Litres:Gallons = $3.8:1$3.8:1

Simplest form is without fractions or decimals:

$3.8$3.8	$:$:	$1$1	multiply both parts by $10$10 to remove the decimal
$38$38	$:$:	$10$10	divide both parts by common factor of $2$2
$19$19	$:$:	$5$5	this is in simplest form

This means that $19$19 litres is equivalent to $5$5 gallons.

 

b) How many liters would a $15$15 gallon vat hold?

Think: Use the conversion ratio.

Do: 

Litres	to	Gallons
$19$19	$:$:	$5$5	for $15$15 gallons we need to multiply the $5$5 "gallons" parts by $3$3.
$19\times3$19×3	$:$:	$5\times3$5×3	so we need to multiply both parts of the ratio by $3$3
$57$57	$:$:	$15$15

So $15$15 Gallons is equivalent to $57$57 Litres. 

 

Practice Questions

Question 7

 

Question 8