Proportions Using Equivalent Ratios

What is a proportion?  A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal.  = 8 is an example of a proportion.

Read each proportion. Why is each one a proportion?  a) = 2  "5 is to 1 as 10 is to 2." 5 is five times 1, as 10 is five times 2.  b) = 24  "2 is to 8 as 6 is to 24." 2 is a fourth of 8, as 6 is a fourth of 24.  c) = 4  "75 is to 100 as 3 is to 4." 75 is three fourths of 100, as 3 is three fourths of 4.

Proportion problem from the 2003 TAKS If the ratio of boys to girls in the sixth-grade is 2 to 3, which of these show the possible numbers of the boys and girls in the chorus? A. 20 boys, 35 girlsC. 35 boys, 20 girls B.24 boys, 36 girlsD. 36 boys, 24 girls

Solving a Proportion.  When one of the four numbers in a proportion is unknown, setting up equivalent ratios may be used to find the unknown number.  This is called solving the proportion. Question marks or letters are frequently used in place of the unknown number. Example:  Solve for n: =  Solve for n: = 1 2n4

Solve each proportion.. 4 3n9 = = n 278n = n = 9218 n = 18 4n2 =

Conversion Ratios  A conversion ratio is a fraction where the numerator and denominator express the same quantity using different units.  1 hr 1 yd 1 T 60 min 3 ft2000 lb 60 min 3 ft2000 lb  Note that since each of these examples has the same quantity in both the numerator and the denominator.  We can use conversion ratios in a proportion to solve conversion problems.

Use Proportions to Solve Conversion Problems  Setting up a proportion to solve a conversion problem needs to be written carefully.  Example: 12yd = ____ft 12yd = ____ft 1. Write a conversion ratio for the units given in the problem. 2. Then write the information in the problem as a ratio. 3. Solve the proportion (Hint: Label both ratios in your proportion) 1 yd 3 ft = 12 yd n ft

Rate A rate is a ratio that expresses how long it takes to do something, such as traveling a certain distance. To walk 3 kilometers in one hour is to walk at the rate of 3 km/h. The fraction expressing a rate has units of distance in the numerator and units of time in the denominator. Problems involving rates typically involve setting two ratios equal to each other and solving for an unknown quantity, that is, solving a proportion.

Use Proportions to Solve Rate Problems Juan runs 4 km in 30 minutes. At that rate, how far could he run in 45 minutes?  Give the unknown quantity the name n. In this case, n is the number of km Juan could run in 45 minutes at the given rate.  We know that running 4 km in 30 minutes is the same as running n km in 45 minutes; that is, the rates are the same.  So we have the proportion 4km = n km 30min 45min

Two Rate Problems  The Browns averaged 55 miles per hour on their vacation. How many miles did they travel in four days?  Sharon rode her bicycle a total of 36 km at a rate of 9 km per hour. Hour long did she ride?

Proportion problem from the 2003 TAKS Corinne’s group was responsible for painting windows on the set of the school play. The group painted 18 windows in 90 minutes. If they continued painting at this rate, how many windows would they paint in 3 hours?