7-4 Comparing and Scaling - Concepts and Explanation
A comparison of two quantities
Ratios can be written in several forms. You can write the ratio of 3 cups of water to 2 cups of lemonade concentrate as 2 to 3, 2 : 3, or 2/3
A proportion is a statement of equality between two ratios.
Kendra takes 70 steps on the treadmill to run 0.1 mile. When her workout is done, she has run 3 miles. How many steps has she taken?
Proportion: 70 steps/0.1 mile = x steps/3 miles
70 steps x 30/0.1 miles x 30 = 21.00 steps/3 miles Solution of the proportion
Two Types of Ratios
Ratios can be part-to-part or part-to-whole comparisons. Part-to-whole comparisons can be written as fractions or percents. Part-to-part comparisons can be written in fraction form, but do not represent a fraction.
The ratio of concentrate to water in a mix for lemonade is 3 cups concentrate to 16 cups water. What fraction of the mix is concentrate?
3/16 is the part-to-part comparison. This does not mean that the fraction of mix that is concentrate is 3/16. Find the total, 19 cups, to write the fraction of the mix that is concentrate. Write a part-to-whole comparison using a fraction, 3/19, or a percent, 3 ÷ 19 = 0.15789 ≈ 15.8%, to describe the part that is concentrate.
A comparison of measures with two different units
Examples of rates: miles to gallons, sandwiches to people, dollars to hours, calories to ounces, kilometers to hours
A rate in which the second quantity is 1 unit
Students sometimes find unit rates difficult because they have two options when dividing the two numbers of a rate. Tracking the units helps students think through such situations. The goal is to build flexibility in using either set of unit rates to compare the quantities.
Sascha rides 6 miles in 20 minutes during the first leg of his bike ride. During the second leg, he rides 8 miles in 24 minutes. During which leg is Sascha faster?
6 miles/20 minutes = 0.3 miles per minute
8 miles/24 minutes = 0.333 miles per minute
The times, 1 minute, are the same, so 8 miles in 24 minutes is faster.
You can divide the other way as well:
20 minutes/6 miles = 3.333 minutes per mile
24 minutes/8 miles = 3 minutes per mile
The distances, 1 mile, are the same, and 3 minutes per mile is faster.
Scaling Ratios (and Rates)
Finding a common denominator or common numerator to make comparisons easier
Which is cheaper, 3 roses for $5 or 7 roses for $9 ?
Scale the costs to be the same by finding a common denominator. Use a common multiple of 5 and 9:
3 roses/$5 = 3 roses x 9/ $5 x 9 = 27 roses/$45; 7 roses/$9 = 7 roses x 5/$9 x 5 = 35 roses/$45
7 roses for $9 gives more roses for the same amount of money. Or, scale the numerators to be the same:
3 roses/$5 = 3 roses x 7/$5 x 7 = 21 roses/$35; 7 roses/$9 = 7 roses x 3/$9 x 3 = 21 roses/$27
21 roses for $27 is cheaper than 21 roses for $35.
A relationship in which you multiply one variable by a constant number to find the value of another variable
The price of one pizza is $13.
To find the cost C of any number of pizzas n, multiply the number of pizzas by 13. The unit rate 13 is also called the constant of proportionality, k. The relationship appears as a straight line on a graph. The equation can be written as y = kx. In this case, C = 13n.