1. SOLVING AND APPLYING PROPORTIONS
Chapter 4

2. Introduction We will learn about ratios and proportions
We will discuss how the ratio is related to percentages.

3. Ratio and Proportion (4.1)
Ratio: A comparison of two numbers by division.
A ratio can take two forms (for any real number a and b and b ≠0):
Rate: If a and b above represent quantities measured in different units then the ratio is known as a rate.
Unit rate: A rate with a denominator of 1.
a : b
𝑎 𝑏
Velocity is a rate: 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 (𝑚) 𝑡𝑖𝑚𝑒 (𝑠)

4. Ratio and Proportion (4.1)
Sample Problem (Calculating for the unit rate) The table below gives the prices for different sizes of the same brand of apple juice. Find the unit rate (cost per ounce) for the 16-oz size.
Price
Volume
$0.72
16 oz
$1.20
32 oz
$1.60
64 oz

5. Ratio and Proportion (4.1)
Dimensional analysis (aka: unit analysis): The process of converting from one unit to another.
This process is a very powerful tool for solving word problems as well.
We can also use it to confirm our answers are correct and that the formulas we are using are valid.
This process involves examining units as variables and multiplying or dividing these units.
Units: These are descriptors of physical quantities (i.e. inches, centimeters) .
The key to doing dimensional analysis is choosing and using the correct conversion factors.
Conversion factors: Rates that are equal to one.
Mathematically, dimensional analysis involves multiplying or dividing ratios.

6. Ratio and Proportion (4.1)
Sample Problem A cheetah ran 300 feet in 2.92 seconds. What was the cheetah’s speed in miles per hour. 1 mile = 5280 ft 60 s = 1 minute 60 min = 1 hour

7. Ratio and Proportion (4.1)
Proportion: An equation that states that two ratios are equal.
The proportion is read as “a is to b as c is to d.”
The above proportion can also be written:
a:b = c:d.
We can use the Multiplication Property of Equality to solve a proportion for a variable.
𝑎 𝑏 = 𝑐 𝑑 𝑓𝑜𝑟 𝑏 ≠0 𝑎𝑛𝑑 𝑑 ≠0

8. Ratio and Proportion (4.1)
Sample Problem Solve 𝑡 9 = 5 6 .

9. Ratio and Proportion (4.1)
Using the Multiplication Property of Equality, we can prove an important property of proportions.
Cross Products of a Proportions
If 𝑎 𝑏 = 𝑐 𝑑 , then ad=bc.
Deriving the cross products.

10. Ratio and Proportion (4.1)
Sample Problem Us e the cross product to solve the proportion 𝑦 2.5 =− 3 4 .

11. Ratio and Proportion (4.1)
We can also use the cross product along with the other properties that we have learned to solve for proportions that involve multiple steps.
Sample Problem
Solve the proportion 𝑥+4 5 = 𝑥 −2 7 .

12. Proportions and Percent Equations (4.3)
A percent is a ratio that compares some number to 100.
We can solve for a percent problem by writing and solving a proportion.
Percentages can be greater
than 100 and also less than 1.
Example
23% =
𝑛 100 = 𝑝𝑎𝑟𝑡 𝑤ℎ𝑜𝑙𝑒

13. Proportions and Percent Equations (4.3)
Sample Problem What percent of 80 is 18?

14. Proportions and Percent Equations (4.3)
Sample Problem Find 75% of 320.

15. Proportions and Percent Equations (4.3)
Sample Problem According to the U.S. Geological Survey, surface water accounts for 77.6% of our country’s water supply. The surface water supply is about billion gallons per day. Find the total water supply.

16. SOLVING AND APPLYING PROPORTIONS
Chapter 1
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