Unit Rates and Proportion unit test

Answer 1

Sure! I can help you prepare for a unit test on unit rates and proportions. Below are some key concepts, example questions, and practice problems.

Key Concepts

1. Unit Rate: A unit rate is a comparison of two quantities with different units, expressed as a ratio where the second quantity is 1. For example, if you travel 150 miles in 3 hours, your unit rate (speed) is 50 miles per hour (150 miles ÷ 3 hours).

2. Proportion: A proportion is an equation that states two ratios are equal. For example, if 1/2 = 2/4, then the two ratios form a proportion.

3. Solving Proportions: To solve a proportion, you can use cross multiplication. For example, if \( a/b = c/d \), then cross-multiplying gives you \( ad = bc \).

Example Questions

1. Find the Unit Rate:

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

   Solution: Unit rate = 300 miles ÷ 10 gallons = 30 miles per gallon.

2. Determine if the Proportions are Equivalent:

   • Are the ratios 3/4 and 6/8 equivalent?

   Solution: Cross multiply: \( 3 \times 8 = 24 \) and \( 4 \times 6 = 24 \). Since both products are equal, the ratios are equivalent.

3. Solve for x:

   • If \( \frac{5}{x} = \frac{15}{9} \), what is the value of x?

   Solution: Cross multiply to get \( 5 \times 9 = 15 \times x \). This simplifies to \( 45 = 15x \) which leads to \( x = 3 \).

Practice Problems

1. A worker earns $240 for 20 hours of work. What is the hourly rate?

2. If 3 apples cost $1.50, how much do 12 apples cost?

3. Solve for x: \( \frac{2}{5} = \frac{x}{20} \).

4. A recipe requires 2 cups of flour for 3 cups of sugar. What is the unit rate of flour per cup of sugar?

5. Is the ratio 8:12 equivalent to the ratio 2:3?

6. A car consumes 12 gallons of fuel over 300 miles. What is the car's fuel efficiency in miles per gallon?

Answers to Practice Problems

1. Hourly Rate: $240 ÷ 20 hours = $12 per hour.

2. Cost of 12 Apples: If 3 apples cost $1.50, then 12 apples cost \( 12 ÷ 3 \times 1.50 = 4 \times 1.50 = $6.00 \).

3. Solve for x: Cross-multiplying gives \( 2 \times 20 = 5 \times x \), leading to \( 40 = 5x \), so \( x = 8 \).

4. Unit Rate of Flour per Cup of Sugar: \( 2 cups flour ÷ 3 cups sugar = \frac{2}{3} \) cups of flour per cup of sugar.

5. Equivalent Ratios: \( 8:12 = \frac{8}{12} = \frac{2}{3} \) and \( 2:3 \) are equivalent.

6. Fuel Efficiency: \( 300 miles ÷ 12 gallons = 25 miles per gallon. \)

Feel free to ask if you need more specific questions or further clarifications on any of these concepts! Good luck with your test!