PRESENTATION 9 Ratios and Proportions
Jan 03, 2016
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