6.1 RATIOS
6.1
RATIOS
INTRODUCTION TO RATIOS
A ratio is used to compare two or more quantities.
For example, if there are 4 males and 5 females, the ratio of males to females can be expressed as
4 : 5, where a colon is used to separate the two numbers. It is read as “4 is to 5”. 4 and 5 are the terms
4
of the ratio. This can also be expressed as a fraction: .
5
When comparing more than two quantities, we use a colon to represent a ratio.
For example, if you have US$10, C$5, and £2, then the ratio of the different amounts you have is:
US$ : C$ : £ = 10 : 5 : 2
SIMPLIFYING RATIOS
Just as we can simplify fractions, we can simplify ratios to their lowest terms.
For example, if Andrew has completed 5 courses out of 35 courses, the ratio of the courses completed
to the total number of courses is 5 : 35.
This ratio can be simplified to its lowest terms as follows:
5 : 35
Divide each term by 5
1:7
Therefore, 5 : 35 = 1 : 7
1 : 7 is called an equivalent ratio of 5 : 35 as they are equal when reduced to the lowest terms.
Note • A ratio is generally expressed in its lowest terms (but not always).
• A ratio has no units.
• When the terms of a ratio are multiplied or divided by the same number, the resulting
ratio is called an equivalent ratio.
• Changing the order of the terms of a ratio results in a different ratio. 1 : 2 ≠ 2 : 1
USING RATIOS TO SHARE QUANTITIES
Ratios can be used to share quantities as illustrated in the following examples:
EXAMPLE 6.1A - USING RATIOS TO SHARE QUANTITIES BETWEEN 2 PEOPLE
Ann and Ben invested $10,000 in the ratio of 2 : 3. How much did they each invest?
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CHAPTER 6 | RATIOS AND PROPORTIONS
SOLUTION
They distributed this investment in 2 + 3 = 5 parts.
Ann’s investment =
2
× $10, 000 = $4, 000
5
Ben’s investment =
3
× $10, 000 = $6, 000
5
Therefore, Ann invested $4,000 and Ben invested $6,000.
EXAMPLE 6.1B - USING RATIOS TO SHARE QUANTITIES AMONG 3 PEOPLE
Pam, Quincy, and Ron invested $50,000 in a ratio of 1 : 2 : 3 to start a business. What was the amount
each person invested?
SOLUTION
They distributed this investment in 1 + 2 + 3 = 6 parts.
Pam’s investment =
Quincy’s investment =
Ron’s investment =
1
× $50, 000 = $8, 333.33
6
2
× $50, 000 = $16, 666.67
6
3
× $50, 000 = $25, 000
6
Therefore, Pam, Quincy, and Ron invested $8,333.33, $16,666.67, and $25,000, respectively.
EXAMPLE 6.1C - Using a ratio as a “rate” or “unit rate”
John worked 5 hours and earned $80. Calculate his hourly rate of pay
SOLUTION
5hrs : $80 = 1hr : x
1
5
= x
80
5x = 80
x = 80
5
x = 16
1
Therefore, John’s rate of pay is $16/hr.
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Cross multiplying
Simplifying
Another Example:
The speed of a car per hour is
a “unit rate” because there is
a 1 in the denominator.
6.1 RATIOS
EXERCISE 6.1
Answers to the odd numbered problems are available online
Reduce the following ratios in Problems 1 and 2 to their lowest terms:
1.
(a) 2 : 4
(b) 6 : 18
(c) 12 : 22
(d) 16 : 24
(e) 21 : 24
(f) 200 : 125
(h) 2 : 4 : 8
(i) 50 : 30 : 25
(a) 3 : 9
(b) 5 : 35
(c) 14 : 16
(d) 15 : 20
(e) 9 : 18
(f) 300 : 180
(h) 3 : 9 : 18
(i) 70 : 40 : 80
(g)
2.
(g)
3.
4
:2
3
8
:4
5
Write the following as ratios and simplify to the lowest terms:
(a) A college hired 20 professors for 600 students.
(b) A student has worked an average of 6 hours per day this year, whereas he worked 4 hours
per day last year.
(c) A college graduate earned three times as much per hour as his brother who did not have a
college diploma.
(d) A textbook for English cost twice as much as the one for math.
(e) Mary works 40 hours per week, Julie works 35 hours per week, and Trish works 30 hours
per week.
(f) Taylor earns $15 per hour, Kenny earns $12 per hour and their little brother Ryan earns $8
per hour.
4.
Write the following as ratios and simplify to lowest terms:
(a) A restaurant hired 5 chefs and 25 waiters.
(b) A financial consultant worked an average of 40 hours per week this year whereas he worked
35 hours per week last year.
(c) A senior manager earns four times as much per hour as a junior manager.
(d) A large coffee costs five-fourths as much as a medium one.
(e) Vern, Hugo and Lena study 12, 16 and 20 hours per week, respectively.
(f) Ingrid, Hebe, and Roger earn $2,700, $3,300 and $3,900 every month.
5.
Amy and Kevin invested $24,000 in their bakery. If the ratio of their investment was 2 : 3, how
much did each invest?
6.
Samuel and Henry shared the profit of $16,000 from their company last month. If they shared the
profit in the ratio of 5 : 8, how much did each receive?
7.
Carson and Hannah invested $5,000 in their business making lamps. If the ratio of their
investment was 3 : 5 respectively, how much did each invest?
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CHAPTER 6 | RATIOS AND PROPORTIONS
8.
Louise and Sylvain invested $3,000 in furniture for their business. If the ratio of their investment
was 2 : 5 respectively, how much did each invest?
9.
Tanya and Matilde invested $4,000 in their ice cream specialty store. If the ratio of their investment
was 3 : 7 respectively, how much did each invest?
10. Don, Bill, and Nolan invested $20,000 to upgrade their machinery for their business in the ratio
of 2 : 3 : 4 respectively. How much did each invest?
11. Chris, Lacey, and Nadia invested a total of $30,000 for their new timeshare property in the ratio
of 3 : 4 : 5 respectively. How much did each invest?
12. Kim, Tim, and Tracey bought a condo in Florida for $100.000 and invested in the ratio of
1 : 2 : 7 respectively. How much did each invest?
13. In your groups create and solve 4 more problems using ratios to share investments similar to the
above problems. Have another group check your work.
14. If John worked 12 hours and earned $200, was was his hourly wage?
15. If Tim drove 150 km in 2 hours, what was his speed in km/hr?
16. A recipe for 3.5 litres (one 4L punch bowl) of peach punch calls for the following quantities: 375
mL peach nectar, 500 mL water, 250 mL sugar, 1375 mL can of pineapple juice, 1 litre (1000 mL)
1
bottle of Ginger Ale, bag of ice – as needed, and 2 lemons sliced – garnish, as needed.
2
(a) With the exception of the ice and lemon slices, what is the ratio of the ingredients so that the
catering company can calculate for large functions?
(b) If you have 14 cans of pineapple juice, then how much of the other ingredients do you need
and how many litres of punch will that make?
6.2
PROPORTIONS
When two ratios are equal, we may write it as a proportion.
For example, the ratios 4 : 6 and 20 : 30 are proportionate to each other.
We can verify this as follows:
4 : 6 = 20 : 30
4 20
=
6 30
Write as fractions,
Simplify to lowest terms,
2 2
=
3 3
As the ratios are equal when reduced to lowest terms, they are proportionate.
SOLVING FOR AN UNKNOWN TERM IN A PROPORTION
If any term in a proportion is unknown, we can cross-multiply to calculate the unknown term.
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