Unit 5: Proportional Reasoning

Ratios, Rates, Proportions, and Percents

Topic 1: Ratios & Rates

A ratio is just a way to compare two things. A rate is a special ratio we use all the time, like comparing miles driven to the hours it took.

What Are They? (Examples)

What is a Ratio?

A ratio compares any two quantities.

"In a classroom, there are 12 girls and 10 boys. The ratio of girls to boys is 12 to 10."

What is a Rate?

A rate is a ratio that compares two quantities with different units.

"A car travels 150 miles in 3 hours. The rate is 150 miles per 3 hours."

How to Write a Ratio

You can write the ratio of "a" to "b" in three ways:

$ a:b \quad \quad \frac{a}{b} \quad \quad \text{a to b} $

Step 1: Identify the two things you are comparing. The order matters!

Step 2: Write the numbers in the same order they are asked for. "Apples to oranges" is different from "oranges to apples".

Step 3: Simplify the ratio if possible. Just like a fraction, find the simplest way to say the same comparison.
For example, a ratio of 6 to 8 is the same as saying 3 to 4.

Practice: Ratios

A basket has 6 apples and 8 oranges. (That's 14 total pieces of fruit!)

I Do

Ratio of apples to oranges:

$ \frac{6}{8} = \frac{3}{4} $

We Do

Ratio of oranges to total fruit:

$ \frac{8}{14} = \frac{4}{7} $

You Do

Ratio of apples to total fruit:

$ \frac{6}{14} = \frac{3}{7} $

A basket has 6 apples and 8 oranges (14 total fruit).

I Do: Ratio of apples to oranges: 3/4

We Do: Ratio of oranges to total fruit:

You Do: Ratio of apples to total fruit:

Independent Practice: Ratios

Level 1:

Write the ratio of 5 dogs to 7 cats in two different ways.

Level 2:

A car travels 150 miles on 5 gallons of gas. What is the rate of miles to gallons?

Level 3:

In a class of 30 students, 18 are girls. What is the ratio of boys to girls? (Simplify your answer)

Independent Practice

1. Write the ratio of 5 dogs to 7 cats in two different ways.

2. A car travels 150 miles on 5 gallons of gas. What is the rate of miles to gallons?

3. In a class of 30 students, 18 are girls. What is the ratio of boys to girls?

Exit Ticket: Ratios & Rates

A store sells a pack of 12 sodas for $9. Write the rate of dollars to sodas and simplify it.

$ \frac{\$9}{12 \text{ sodas}} = \frac{\$3}{4 \text{ sodas}} $

Answer:

Topic 2: Unit Rate

A unit rate tells you "how much for ONE". Think about the price for one item at the grocery store. This helps you find the best deal!

How to Find a Unit Rate

Step 1: Write the comparison as a rate (fraction). Make sure the quantity you want to be "1" is in the denominator (bottom).

Step 2: Divide the top number (numerator) by the bottom number (denominator).

Step 3: Write your answer with the correct units. This is very important!

Key Idea: The word "per" (like in miles PER hour) is a clue that you're looking for a unit rate.

Practice: Unit Rate

I Do

Fly 600 miles in 3 hours:

$ \frac{600 \text{ miles}}{3 \text{ hours}} $

200 mph

We Do

$12 for 4 pounds:

$ \frac{\$12}{4 \text{ pounds}} $

$3 per pound

You Do

Type 240 words in 5 minutes:

$ \frac{240 \text{ words}}{5 \text{ minutes}} $

48 wpm

I Do: Fly 600 miles in 3 hours: 200 mph

We Do: $12 for 4 pounds:

You Do: Type 240 words in 5 minutes:

Independent Practice: Unit Rate

Level 1:

A car travels 300 miles on 10 gallons of gas. What is the unit rate in miles per gallon?

Level 2:

Store A sells a 16 oz box of cereal for $4.80. Store B sells a 20 oz box for $5.60. Which is the better buy? (Hint: Find the cost per ounce for each).

Level 3:

If you earn $120 for an 8-hour shift, how much will you earn in a 5-hour shift at the same rate? (Hint: First find the hourly rate).

Independent Practice

1. A car travels 300 miles on 10 gallons of gas. Unit rate?

2. 16 oz for $4.80 vs 20 oz for $5.60. Better buy?

3. If you earn $120 for 8 hours, how much for 5 hours?

Exit Ticket: Unit Rate

A 5-pack of pencils costs $1.50. What is the cost per pencil?

$ \frac{\$1.50}{5 \text{ pencils}} = \$0.30 $ per pencil

Answer:

Topic 3: Solving Proportions

A proportion is an equation with two equal fractions. We use them to solve problems where two things are changing at the same rate.

How to Solve a Proportion

Step 1: Write down the proportion. Make sure the units match up across the top and across the bottom.

Step 2: Cross-multiply. Draw a "butterfly" or "X" to connect the numbers diagonally. Multiply the connected numbers.

$ \frac{a}{b} $$\searrow$$ \frac{c}{d} $ and $ \frac{a}{b} $$\nwarrow$$ \frac{c}{d} \rightarrow a \cdot d = b \cdot c $

Step 3: Solve the simple one-step equation that you just created.

Practice: Proportions

I Do

$ \frac{x}{12} = \frac{2}{3} $

$ 3 \cdot x = 12 \cdot 2 $

$ 3x = 24 $

x = 8

We Do

$ \frac{5}{y} = \frac{10}{6} $

$ 5 \cdot 6 = y \cdot 10 $

$ 30 = 10y $

y = 3

You Do

$ \frac{7}{3} = \frac{z}{9} $

$ 7 \cdot 9 = 3 \cdot z $

$ 63 = 3z $

z = 21

I Do: $ \frac{x}{12} = \frac{2}{3} $. x = 8

We Do: $ \frac{5}{y} = \frac{10}{6} $. y =

You Do: $ \frac{7}{3} = \frac{z}{9} $. z =

Topic 4: Percent Applications

Percents are just special fractions out of 100. We use them every day for shopping, eating out, and saving money!

How to Find Percent of a Number

Step 1: Convert the percent to a decimal. To do this, divide by 100 (or just move the decimal point two places to the left).

Step 2: Multiply the decimal by the number you are finding the percent of.

Key Idea: The word "of" in math problems almost always means you need to multiply.

Example: What is 40% of 200?

Step 1: 40% $\rightarrow$ 40. $\rightarrow$ 0.40

Step 2: $ 0.40 \times 200 = 80 $

How to Calculate Discounts & Tax/Tip

Discounts (Subtract)

1. Find the discount amount:
$ \text{Original Price} \times \text{Percent (as decimal)} $

2. Find the sale price:
$ \text{Original Price} - \text{Discount Amount} $

Think: A discount makes the price go down, so we subtract!

Tax & Tip (Add)

1. Find the tax/tip amount:
$ \text{Original Price} \times \text{Percent (as decimal)} $

2. Find the total cost:
$ \text{Original Price} + \text{Tax/Tip Amount} $

Think: Tax and tip make the price go up, so we add!

Practice: Percent Applications

I Do

Find 15% of 60.

$ 0.15 \times 60 $

9

We Do

20% discount on $50 item. What is the sale price?

Discount: $0.2 \times 50 = \$10$

$ \$50 - \$10 = \$40 $

You Do

7% tax on a $30 meal. What is the total cost?

Tax: $0.07 \times 30 = \$2.10$

$ \$30 + \$2.10 = \$32.10 $

I Do: Find 15% of 60: 9

We Do: 20% discount on $50. Sale price?

You Do: 7% tax on $30. Total cost?

Independent Practice: Percents

Level 1:

What is 50% of 90?

Level 2:

A video game costs $60. It is on sale for 25% off. What is the final price?

Level 3:

Your dinner bill is $75. You want to leave a 20% tip, and there is an 8% sales tax. What is your total cost? (Hint: Find tip and tax separately, then add both to the bill).

Independent Practice

1. What is 50% of 90?

2. A $60 game is 25% off. Final price?

3. Bill is $75. Calculate a 20% tip and 8% tax. Total cost?

Exit Ticket: Percents

A pair of shoes costs $80. You have a coupon for 30% off. How much do the shoes cost after the discount?

Discount: $0.30 \times \$80 = \$24$

Final Price: $ \$80 - \$24 = \$56 $

Answer:

Congratulations!

You have completed the notes for Unit 5.