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 ^{5}⁄_{7}, 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 ^{6}⁄_{8} or ^{3}⁄_{4}.
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 : 24You 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. ^{3}⁄_{4} = ^{9}⁄_{12} is an example of a proportion.
Example: Solve y : 3 = ^{6}⁄_{9}We can first write this problem as ^{y}⁄_{3} = ^{6}⁄_{9}
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 = ^{D}⁄_{T},
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 |
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 = 2^{1}⁄_{2} hours, or we can say 2.5 hours.
He then runs for 2 ÷ 2 = 1 hour.
The total time for the journey is 2^{1}⁄_{2} + 1 = 3^{1}⁄_{2} hours.
The average rate of speed for his journey is total distance ÷ total time, which equals
12 km ÷ 3^{1}⁄_{2} hours.
= 12 ÷ ^{7}⁄_{2}
= 12 × ^{2}⁄_{7}
= ^{24}⁄_{7}
3^{3}⁄_{7} km per hour.
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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