1. PRESENTATION 9 Ratios and Proportions

2. RATIOS A ratio is the comparison of two like quantities
The terms of a ratio must be compared in the order in which they are given
Terms must be expressed in the same units
The first term is the numerator of a fraction, and the second term is the denominator
A ratio should be expressed in lowest fractional terms

3. RATIOS Ratios are expressed in two ways:
With a colon between the terms, such as 4 : 9
This is read as “4 to 9”
With a division sign separating the two numbers, such as 4 ÷ 9 or

4. RATIOS Example: Express 5 to 15 as a ratio in lowest terms
Write the ratio as a fraction and reduce
The ratio is 1 : 3

5. RATIOS Example: Express 10 to as a ratio in lowest terms Divide
The ratio is 12 : 1

6. PROPORTIONS
A proportion is an expression that states the equality of two ratios
Proportions are expressed in two ways
As 3 : 4 = 6 : 8, which is read as “3 is to 4 as 6 is to 8”
As , which is the equation form

7. PROPORTIONS A proportion consists of four terms
The first and fourth terms are called extremes
The second and third terms are called means
In the proportion 3 : 4 = 6 : 8, 3 and 8 are the extremes and 4 and 6 are the means
The product of the means equals the product of the extremes (if the terms are cross-multiplied, their products are equal)

8. PROPORTIONS Example: Solve the proportion below for F:
Cross multiply: F = 6.2(9.8)
Divide both sides by 21.7:
Therefore F = 2.8

9. DIRECT PROPORTIONS
Two quantities are directly proportional if a change in one produces a change in the other in the same direction
When setting up a direct proportion in fractional form:
Numerator of the first ratio must correspond to the numerator of the second ratio
Denominator of the first ratio must correspond to the denominator of the second ratio

10. DIRECT PROPORTIONS
Example: A machine produces 280 pieces in 3.5 hours. How long does it take to produce 720 pieces?
Analyze: An increase in the number of pieces produced (from 280 to 720) requires an increase in time
Time increases as production increases; therefore, the proportion is direct

11. DIRECT PROPORTIONS
Set up the proportion and let t represent the time required to produce 720 pieces
The numerator of the first ratio corresponds to the numerator of the second ratio (280 pieces to 3.5 hours)
The denominator of the first ratio corresponds to the denominator of the second ratio (720 pieces to t)

12. DIRECT PROPORTIONS Solve for t:
It will take 9 hours to produce 720 pieces

13. INVERSE PROPORTIONS
Two quantities are inversely or indirectly proportional if a change in one produces a change in the other in the opposite direction
Two quantities are inversely proportional if
An increase in one produces a decrease in the other
A decrease in one produces an increase in the other

14. INVERSE PROPORTIONS
When setting up an inverse proportion in fractional form:
The numerator of the first ratio must correspond to the denominator of the second ratio
The denominator of the first ratio must correspond to the numerator of the second ratio

15. INVERSE PROPORTIONS
Example: Five identical machines produce the same parts at the same rate. The 5 machines complete the required number of parts in 1.8 hours. How many hours does it take 3 machines to produce the same number of parts?
Analyze: A decrease in the number of machines (from 5 to 3) requires an increase in time
Time increases as the number of machines decrease and this is an inverse proportion

16. INVERSE PROPORTIONS
Let x represent the time required by 3 machines to produce the parts
The numerator of the first ratio corresponds to the denominator of the second ratio; 5 machines corresponds to 1.8 hours
The denominator of the first ratio corresponds to the numerator of the second ratio; 3 machines corresponds to x

17. INVERSE PROPORTIONS
Solve for x:
It will take 3 hours

18. PRACTICAL PROBLEMS
A piece of lumber 2.8 meters long weighs 24.5 kilograms
A piece 0.8 meters long is cut from the 2.8-meter length
Determine the weight of the 0.8- meter piece

19. PRACTICAL PROBLEMS
Analyze: Since the weight of 0.8 meters is less than the total weight of the piece of lumber, this is a direct proportion
Set up the proportion and let x represent the weight of the 0.8-meter piece

20. PRACTICAL PROBLEMS
Solve for x:
The piece of lumber weighs 7 kilograms