Ratios and Rates

Understanding and simplifying ratios and rates

Ratios and Rates

Ratio

A ratio compares two quantities by division.

Three ways to write:

  • Using colon: 3:53:5
  • Using "to": 3 to 53 \text{ to } 5
  • As a fraction: 35\frac{3}{5}

Example: If there are 3 cats and 5 dogs, the ratio of cats to dogs is 3:53:5.

Simplifying Ratios

Divide both numbers by their GCF (like simplifying fractions).

Example: 12:18=1218=23=2:312:18 = \frac{12}{18} = \frac{2}{3} = 2:3

Rate

A rate is a ratio that compares two different units.

Examples:

  • 60 miles per hour: 60 miles1 hour\frac{60 \text{ miles}}{1 \text{ hour}}
  • \3perpound:per pound:\frac{$3}{1 \text{ lb}}$
  • 25 miles per gallon: 25 miles1 gallon\frac{25 \text{ miles}}{1 \text{ gallon}}

Unit Rate

A rate with a denominator of 1.

To find unit rate: Divide numerator by denominator.

Example: If you travel 150 miles in 3 hours: 150 miles3 hours=50 miles per hour\frac{150 \text{ miles}}{3 \text{ hours}} = 50 \text{ miles per hour}

Comparing Rates

Find the unit rate for each, then compare.

Example: Which is cheaper: 3 lb for \6or5lbforor 5 lb for$9$?

  • First: \frac{\6}{3} = $2$ per lb
  • Second: \frac{\9}{5} = $1.80$ per lb

The second option is cheaper!

📚 Practice Problems

1Problem 1easy

❓ Question:

Simplify the ratio 15:2515:25.

💡 Show Solution

Find the GCF of 15 and 25: GCF=5\text{GCF} = 5

Divide both by 5: 15÷5=315 \div 5 = 3 25÷5=525 \div 5 = 5

Answer: 3:53:5

2Problem 2medium

❓ Question:

A car travels 240 miles in 4 hours. Find the unit rate (miles per hour).

💡 Show Solution

Divide distance by time:

Rate=240 miles4 hours\text{Rate} = \frac{240 \text{ miles}}{4 \text{ hours}}

=60 miles per hour= 60 \text{ miles per hour}

Answer: 60 mph

3Problem 3hard

❓ Question:

Which is the better buy: 8 oz for \2.40or12ozforor 12 oz for$3.36$?

💡 Show Solution

Find the unit price for each:

First option: $2.408 oz=$0.30 per oz\frac{\$2.40}{8 \text{ oz}} = \$0.30 \text{ per oz}

Second option: $3.3612 oz=$0.28 per oz\frac{\$3.36}{12 \text{ oz}} = \$0.28 \text{ per oz}

\0.28 < $0.30$

Answer: 12 oz for \3.36$ is the better buy

4Problem 4medium

❓ Question:

A recipe calls for 3 cups of flour for every 2 cups of sugar. If you use 9 cups of flour, how much sugar do you need?

💡 Show Solution

Step 1: Write the ratio. Flour : Sugar = 3 : 2

Step 2: Set up a proportion. 3/2 = 9/x

Step 3: Cross multiply. 3x = 2 × 9 3x = 18

Step 4: Solve for x. x = 18/3 x = 6

Step 5: Check the ratio. 9:6 simplifies to 3:2 ✓

Answer: 6 cups of sugar

5Problem 5hard

❓ Question:

A car uses 12 gallons of gas to travel 300 miles. At this rate, how many gallons are needed to travel 450 miles? What is the unit rate in miles per gallon?

💡 Show Solution

Part 1: Gallons for 450 miles Step 1: Set up proportion. 12/300 = x/450

Step 2: Cross multiply. 300x = 12 × 450 300x = 5,400

Step 3: Solve. x = 5,400/300 = 18 gallons

Part 2: Unit rate (miles per gallon) Step 1: Divide miles by gallons. 300 miles ÷ 12 gallons = 25 miles per gallon

Or: 450 ÷ 18 = 25 mpg ✓

Answer: 18 gallons needed; Unit rate: 25 miles per gallon

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