Question
Unit Rates and Proportion unit test
Answers
GPT-4o mini
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!
### 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!