PRESENTATION 9 Ratios and Proportions. RATIOS A ratio is the comparison of two like quantities The terms of a ratio must be compared in the order in which.

PRESENTATION 9Ratios and Proportions

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

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

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

RATIOS

•Example: Express 10 to as a ratio in lowest terms

•Divide

•The ratio is 12 : 1

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

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)

PROPORTIONS•Example: Solve the proportion below for F:

• Cross multiply: 21.7F = 6.2(9.8)

• Divide both sides by 21.7:

• Therefore F = 2.8

9.86.2 21.7F

21.7 60.7621.7 21.7

F

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

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

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)

280pieces 3.5hours720pieces hourst

DIRECT PROPORTIONS•Solve for t:

• It will take 9 hours to produce 720 pieces

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

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

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

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

5machines hours3machines 1.8hours

x

INVERSE PROPORTIONS•Solve for x:

• It will take 3 hours

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

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

PRACTICAL PROBLEMS•Solve for x:

•The piece of lumber weighs 7 kilograms