Ratio and Proportion
Ratios A ratio is the comparison of two numbers written as a fraction. For example: Your school’s basketball team has won 7 games and lost 3 games. What is the ratio of wins to losses? Because we are comparing wins to losses the first number in our ratio should be the number of wins and the second number is the number of losses. The ratio is ___________ games won _______ 7 games __ 7 = = games lost 3 games 3
Rates In a ratio, if the numerator and denominator are measured in different units then the ratio is called a rate. A unit rate is a rate per one given unit, like 60 miles per 1 hour. Example: You can travel 120 miles on 60 gallons of gas. What is your fuel efficiency in miles per gallon? 120 miles ________ ________ 2 miles Rate = = 60 gallons 1 gallon Your fuel efficiency is 2 miles per gallon.
Proportion An equation in which two ratios are equal is called a proportion. A proportion can be written using colon notation like this a:b=c:d or as the more recognizable (and useable) equivalence of two fractions. ___ ___ a c = b d
Proportion When Ratios are written in this order, a and d are the extremes, or outside values, of the proportion, and b and c are the means, or middle values, of the proportion. ___ ___ a c a:b::c:d = b d Extremes Means
Proportion To solve problems which require the use of a proportion we can use one of two properties. The reciprocal property of proportions. If two ratios are equal, then their reciprocals are equal. The cross product property of proportions. The product of the extremes equals the product of the means
Proportion Example: Write the original proportion. Use the cross product property. Divide both sides by 6 to isolate the variable, then simplify.
Three Kinds of Proportion • Direct Proportion - a proportion when both antecedent and the consequent increase or decrease. Example: For every 15 boys who attended the program, there were 10 girls. How many girls were there if there were 60 boys who attended the program?
Solution: Proportion: 15 boys to 10 girls = 60 boys : x girls 15 : 10 = 60 : x 15x = 600 15 15 x = 40 girls
2. Indirect or Inversed Proportion - A proportion when the antecedent increases as the consequent decreases or vice versa. Example: If 6 men can paint our house in 18 days, how long will it take 4 men to do the same job?
Solution men hours 6 x 4 18 6 : 4 = x : 18 4x = 108 4 4 x = 27 hours
3. Partitive Proportion - used to divide a number into parts proportional to the given. Example: Four children decided to give their grandmother a gift for her 70th birthday. They decided to divide the expenses in the ratio 2:3:4:5. How much will each one pay if the cost of the gift is P1,400 only?
Solution 2 : 3 : 4 : 5 = 1,400 2n + 3n + 4n + 5n = 1,400 14n = 1,400 14 14 n = 100 P200 will be given by the first child P300 will be given by the second child P400 will be given by the third child P500 will be given by the fourth child
Board Drill Find the value of x: 4: 8 = x : 10
Board Drill Find the value of x: 24: 48 = x : 8
Board Drill Find the value of x: 15: x = 45 : 27
Board Drill There are 24 flowers in 3 vases. If there are 6 vases, how many flowers are there?
Board Drill Two numbers are in the ratio 5:7. If the sum of the two numbers is 120, what are the two numbers?
Board Drill If 8 men can build the extension of our house in 6 days, how many men are needed to finish it in 4 days?