Ratios and ProportionsREVIEW CONCEPTS

What is a ratio?• A ratio is a comparison of two numbers.• Ratios can be written in three different ways:

a to ba : b

ab

NOTE: Ratios must be expressed in lowest terms.

3 to 43 : 4

34

How to simplify ratios. Simplify ratios the same way you simplify fractions: Find a common factor.

16:4 = 4:118:24 = 3:4

125:25 = 5:1

Example 1: Part to Whole In a 100g sample of copper alloy, there is 55g of copper and 45g of tungsten.

What is the ratio of copper alloy to tungsten?

The ratio of copper alloy to tungsten is 20:9.

Tungsten : Copper Alloy45 : 100

9 : 20

Example 2: Part to Part In a 100g sample of copper alloy, there is 55g of copper and 45g of tungsten.

What is the ratio of copper to tungsten?

The ratio of copper to tungsten is 11:9.

Copper : Tungsten55:4511:9

Proportions A proportion is an equation that equates two ratios.

d

c

b

a

16

12

4

3

Cross-Product Property

4

6

2

3

2x6=12 3x4 = 12

d

c

b

a

ad = bc

Therefore:

Example 3:

Step 1: Cross Multiply

Step 2: Multiply

Step 3: Divide

750

=𝑥110

7×110=50𝑥

770=50 𝑥

77050

=50𝑐  50

15.4=𝑥

Example 4: Direct Proportion In a 100g sample of copper alloy, there is 55g of copper and 45g of tungsten.

How much copper is in a 120g sample of copper alloy?

Step 1: Set up the proportion

55100

=?120

Original Copper

Original Alloy

New Copper

New Alloy

Example 4: Continued…

Step 2: Cross Multiply

Step 3: Solve for the unknown

Therefore, there are 66g of copper in a 120 sample of copper alloy.

55100

=𝑐120

55×120=100𝑐

660 0=100𝑐

6600100

=100𝑐  100

66=𝑐

Example 5: Direct Proportion The ratio of tungsten to copper in a sample of copper alloy is 9:20.

If you have a sample of copper alloy that has is 220g of copper, how many grams of tungsten is in the sample?

Step 1: Set up the proportion

920

=?220

Ratio of Tungsten

Ratio of Copper

Amount of Tungsten

Amount of Copper

Example 5: Continued…

Step 2: Cross Multiply

Step 3: Solve for the unknown

Therefore, there are 99g of tungsten in the sample of copper alloy.

920

=𝑡220

t

t

198020

=20𝑐  20

99=𝑐

Example 6: Direct Proportion If 450 nails are used to build a 2300m fence, how many nails will be used to build a 5000m fence?

Indirect Proportion An indirect proportion is a comparison between two ratios that are INVERSELY proportional

This means that an increase in one quantity leads to a decrease in the other quantity

When we solve for inverse proportions, we need to invert one of our ratios

Example 1: Indirect Proportion If it takes 3 carpenters 30 days to build one house, how many days would it take 5 carpenters to build the same house?

Let’s think about this to see if it is a direct or indirect proportion. If we INCREASE the number of carpenters, will the amount of time it takes to build the house decrease?

Example 1: Indirect Proportion (Cont)

If it takes 3 carpenters 30 days to build one house, how many days would it take 5 carpenters to build the same house?

Setting up the proportion if it was direct:

330

=5𝑋

But because it is indirect:

3𝑋

=530

We switch the two values on the bottom

Example 1: Indirect Proportion (Cont)

If it takes 3 carpenters 30 days to build one house, how many days would it take 5 carpenters to build the same house?

3𝑋

=530

Solving for X yields 18 days

Example 2: Indirect Proportion If it takes 8 carpenters 26 days to complete a project, how many days would it take 11 carpenters to complete the same project?

826

=11𝑋

ORIGINAL: INVERTING:

8𝑋

=1126

X = 18.9 Days

Other Examples A journeyman takes 2 hours to install a door. An apprentice takes 3.5 hours to do the same job. How long would it take them do install a door if they were working together?

Let’s try to understand what this question is asking

The first thing we need to do is to figure out how much of the door each of them can complete PER hour

The journeyman takes 2 hours for 1 door. So in 1 hour he completes 0.5 of the door (1/2)

The apprentice takes 3.5 hours for 1 door. So in 1 hour he completes 0.2857 of the door (1/3.5)

Example Cont Combining the two,

Journeyman Apprentice

0.5 + 0.2857 = 0.7857

This means they complete 0.7857 of the door in 1 hour (60 minutes). Setting up the proportion:

0.7857𝑑𝑜𝑜𝑟𝑠60𝑚𝑖𝑛𝑢𝑡𝑒𝑠

=1𝑑𝑜𝑜𝑟𝑋 X = 76.365 minutes

Example Cont0.7857𝑑𝑜𝑜𝑟𝑠60𝑚𝑖𝑛𝑢𝑡𝑒𝑠

=1𝑑𝑜𝑜𝑟𝑋

X = 76.365 minutes

We don’t want to leave our answer in minutes so we will use a proportion to convert to hours.

60𝑚𝑖𝑛𝑢𝑡𝑒𝑠1h𝑜𝑢𝑟

=76.35𝑚𝑖𝑛𝑢𝑡𝑒𝑠

𝑋X = 1.27275 hours

0.27275 * 60 minutes = 16.365 minutes

Therefore, it would take the apprentice and the journeyman 1 hour and 16 minutes to install a

door if they are working together

Trying another one A journeyperson takes 3 hours to install two doors. An apprentice takes 6.75 hours to install 2 doors. If they are working together to install two doors, how long will it take them?

Try it yourself!

Trying another one A journeyperson takes 3 hours to install two doors. An apprentice takes 6.75 hours to install 2 doors. If they are working together to install two doors, how long will it take them?

Journeyperson: 2/3 = 0.667 doors per hour

Apprentice = 2/6.75 = 0.296 doors per hour

0.667 + 0.296 = 0.963 doors per hour

0.963𝑑𝑜𝑜𝑟60𝑚𝑖𝑛𝑢𝑡𝑒𝑠

=2𝑑𝑜𝑜𝑟𝑠

𝑋

0.963X = 120 X = 124.6 minutes

601

=124.6𝑋

X = 2.077 hours

0.077*60 = 4.62 minutes

Therefore, it would take them 2 hours and 5 minutes to install two doors

Your turn!!!