5. Ratio & Proportion

Lesson

Recall that a ratio uses division to compare two numbers. There are three ways to write a ratio of two numbers. In a sport's game, we might refer to a team's record as a ratio of wins to losses. That same ratio might be represented as a fraction or as a ratio written with the ":" symbol between two numbers. All three of these statements have the same meaning.

Representing ratios

Ratios may be represented in a variety of ways. Here are some examples:

Words | Numbers | Algebra |
---|---|---|

wins to losses | $5$5 to $4$4 | $a$a to $b$b |

$\frac{\text{wins}}{\text{losses}}$winslosses |
$\frac{5}{4}$54 | $\frac{a}{b}$ab |

wins : losses | $5:4$5:4 | $a:b$a:b |

Note that the denominator in a ratio cannot be zero.

Have you ever had to double a recipe to make enough for everyone? If you forget to double one of the ingredients, the recipe doesn't quite turn out. That's because the ingredients weren't in proportion. We use proportions a lot in everyday life.

A proportion is a statement of equality between two ratios. A proportion is true if both sides of the proportion simplify to be equivalent ratios.

Proportion

A proportion is a statement fo equality between two ratios.

It can be represented in words, or as an equation.

Words | Equations |
---|---|

$a$a is to $b$b as $c$c is to $d$d |
$\frac{a}{b}=\frac{c}{d}$ab=cd or $a:b=c:d$a:b=c:d |

$2$2 is to $5$5 as $4$4 is to $10$10 | $\frac{2}{5}=\frac{4}{10}$25=410 or $2:5=4:10$2:5=4:10 |

Is the following proportion true or false?

$3$3 gallons is to $4$4 square feet as $12$12 gallons is to $16$16 square feet

True

AFalse

B

The two quantities are in proportion. Find the missing value.

- $\frac{\editable{}}{10}:\frac{35}{50}$10:3550

The number of students and teachers competing in a charity race is in the ratio $10:3$10:3. If $70$70 students take part in the race, how many teachers are there?

We say that two quantities have a proportional relationship if the values always maintain the same ratio. When two quantities are proportional, we can use a ratio table to determine unknown values.

Complete the patterns of equivalent ratios by filling in the gaps.

$18$18 : $27$27 $\editable{}$ : $21$21 $10$10 : $15$15 $6$6 : $\editable{}$ $2$2 : $\editable{}$

Fill in the ratio table below, then use it to answer the following questions.

US dollar ($) $1$1 $2$2 $3$3 $4$4 $5$5 British pound (£) $0.60$0.60 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ How many pounds will you be able to buy with $$11$11 US?

Justin has just returned from holiday with £$15.00$15.00. How many US dollars can he exchange this for?

Fill in the ratio table below, then use it to answer the following questions.

pens $10$10 $20$20 $30$30 $40$40 $50$50 $\editable{}$ cost (dollars) $\editable{}$ $11.60$11.60 $\editable{}$ $\editable{}$ $29.00$29.00 $58.00$58.00 Calculate the cost of buying $90$90 pens.

How much would you expect to pay for $5$5 pens?

Analyze proportional relationships and use them to model and solve real-world and mathematical problems.

Represent proportional relationships by equations. Example: If total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.