## Ratios, rates and proportions

### Ratio

A ratio is a comparison of two or more numerical values of the same kind (or same units).

There are 3 apples and 5 pears in a bowl. The ratio of apples to pears would be \dfrac{3}{5}.

As \dfrac35=0.6, we can say that there's 0.6 apple for each pear.

A ratio can be written in different forms. However, when spoken aloud, they are read "... to ..." (excluding decimal form).

The following are all equivalent:

- \dfrac{1}{4}
- 1:4
- 0.25
- 1 to 4

### Rate

A rate is a ratio where different units are compared.

Frank drives 60 miles for every 2 hours. This is a rate because it compares distance units (mile) to time units (hour).

Mary paid $10 for 3 gallons of fuel. This is a rate because it compares money ($) to volume (gallon).

80 candies per bag is a ratio. This is a rate because it compares units of candies to units of bags.

### Unit Rate

A unit rates is a rate that is expressed to a quantity of 1.

5 miles per hour is a unit rate since it compares to one hour.

Reading 16 pages per hour is a unit rate since it compares to one hour.

$3.25 per gallon is a unit rate since it compares to one gallon.

Given a rate \dfrac{a }{b}, the unit rate is found by a\div b.

Harry drives 120 miles in 3 hours. The unit rate is:

120 \div 3 =60 \text { miles per hour }

Jane can read 32 pages in 2 hours. The unit rate is:

32 \div 2 = 16 \text{ pages per hour }

Molly paid $7.30 for two gallons of gas. The unit rate is:

\$ 7.30 \div 2 =$3.65 \text{ per gallon }

### Proportion

A proportion is a type of ratio relating a part to a whole.

Assuming a room has 3 boys and 4 girls:

- The ratio of boys to girls is \dfrac{3}{4}.
- As there are 7 students in the room (3 boys and 4 girls), the proportion of boys in the room to total number of students is \dfrac{3}{7}.

A proportion can be written as a fraction, decimal, or percent.

The following proportions are the same:

- \dfrac{7}{10}
- 7:10

## Percents

### Percent

A percent is a ratio whose second term is 100:

** x\%=\dfrac{x}{100} **

25% means all of the following:

- \dfrac{25}{100}
- 25 per 100
- 0.25

Percents are used to express the quantity of a unit from a total of 100.

In a group of 100 people, 36 people wear glasses. That means that 36% of the group wears glasses.

In a class of 25 people, 14 people are blonde.

\dfrac{14}{25}=\dfrac{56}{100}

That means that 56% of the people in the class are blonde.

### Percent change

The percent change from a to b is given by :

** \dfrac{b-a}{a} \times 100 **

The number of bears in a zoo goes from 8 to 14 in two years. The percent change is :

\dfrac{14-8}{8} \times 100 = 75\%

The value of a real number *a* after a change of b\% is given by:

** a \times \dfrac{\left(100+b\right)}{100} **

There are 24 students in a class. After a percent change of 25%, the number of students in the class is:

24 \times \dfrac{\left(100+25\right)}{100} =30

The price of a yoga class is $40. The new price, after a percent change of 10%, is:

40 \times \dfrac{\left(100+10\right)}{100}=44

A shirt sells for $24. There is a 30% off sale at the store, therefore there is a change in price of -30%.

24 \times \dfrac{\left(100-30\right)}{100} = 16{,}8

The new price of the shirt is 16,80$.