# Ratio and Proportion

Objectives

1. Students will become familiar with the concepts of ratio.
2. Students will understand how to determine ratio.
3. Students will become familar with the concepts of proportion.
4. Students will understand the concepts of cross multiplication.

A ratio is an expression that compare quantities relative to each other. We generally separate the numbers in the ratio with a colon (:). Suppose we want to write the ratio of 5 and 7. We can write this as 5:7 or as a fraction 57, and we say the ratio is five to seven.

Example: Shevon has a bag with 6 pens, 8 pencils and 2 erasers. What is the ratio of pens to pencils?

(i) Expressed as a ratio we write 6:8 which can be simplified as 3:4.

(ii) Expressed as a fraction, with the numerator equal to the first quantity and the denominator equal to the second.

The answer would be 68 or 34.

Simplfying Ratio

To simplify a ratio you look for common factors in both sides then divide both sides by those common factors.

Example: Simplify 18 : 24

You can first divide by 2. When divided by 2,

18 : 24 = 9 : 12. The next step is to divide by 3.

When divided by 3, 9 : 12 = 3 : 4. So our final answer is 3 : 4.

Proportion

A proportion is an equation with a ratio on each side. It is an equation that states that two things are equal. 34 = 912 is an example of a proportion.

Example: Solve y : 3 = 69
We can first write this problem as y3 = 69
Using cross-multiplication we see that 9 × y = 3 × 6
So we have 9y = 18
Then y = 18 ÷ 9
The answer is y = 2.

Average Rate of Speed

A rate is a ratio that expresses how long it takes to do something, such as traveling a certain distance.
Rate is distance given in units such as miles, feet, kilometers, meters, etc., divided by time.
The average rate of speed is the total distance traveled divided by the total time taken.

The general formula is A.S = DT,
where A.S stands for average speed, D stands for distance and T stands for time.

 For you to remember: Averege speed = distance ÷ time Time = distance ÷ average speed Distance = average speed × time

Example: Jonny walks 10 km at 4 km per hour, then runs for 2 km at 2 km per hour.
What is Jonny's average speed for the distance he traveled?

Solution

The total distance traveled is 10 + 2 = 12 km.

Now we must find the total time he was traveling.

For the first part of the journey, he walked for 10 ÷ 4 = 212 hours, or we can say 2.5 hours.

He then runs for 2 ÷ 2 = 1 hour.

The total time for the journey is 212 + 1 = 312 hours.

The average rate of speed for his journey is total distance ÷ total time, which equals

12 km ÷ 312 hours.

= 12 ÷ 72

= 12 × 27

= 247

337 km per hour.

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More Problems -

Problem 1. Three T-shirts cost 15 dollars. How much do 5 T- shirts cost?

Method of Unitary Analysis,

Number of T-shirts    Cost

3                            \$15

1                             15 ÷ 3 = 5

5                              5 × 5 = 25

Thus 5 T-shirts costs \$25.00

Problem 2. Making 5 apple pies requires 2 pounds of apples. How many pounds of apples are needed to make 8 pies?

Number of pie                                Weight/pounds

5                                                 2

1                                                 2 ÷ 5 = 0.4

8                                                8× 0.4 = 3.2

Thus, 8 pie requires 3.2 pounds of apples.

Problem 3. Jack and Jill went up the hill to pick apples and pears. Jack picked 10 apples 15 pears and Jill picked 20 apples and some pears. The ratio of apples to pears picked by both Jack and Jill were the same. Determine how many pears Jill picked.

Solution                Apples        Pears

Jack      10              15

To obtain the 20 apples, Jill picked , we need simply double the 10 apples Jack picked. So we multiply the chart by two.

Apples        Pears

Jack      10              15

Jill       20              30

Thus, Jill picked 30 pears.

Let's redo this problem with less nice numbers:

Problem 4. Jack picked 12 apples 15 pears and Jill picked 16 apples and some pears. The ratio of apples to pears picked by Jack and Jill were the same. Determine how many pears Jill picked.

Solution                Apples        Pears

Jack      12              15

To obtain the 16 apples, Jill picked , we need to find what number multiplied by 12 will yield 16. This is one meaning of division: 16/ 12 = 4/3. So multiply the one-line chart by 4/3:

Apples        Pears

Jack       12              15

Jill       16              20

Thus, Jill picked 20 pears.

Next, we present another solution in the spirit of unitary analysis. First divide the one-line chart by 12, then multiply by 16:

Alternate Solution           Apples        Pears

Jack       12              15

1             15/12

Jill         16            16 x (15 ÷ 12) = 20

Thus, Jill picked 20 pears.

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