Compare Functions

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Answer Key & Explanations

Compare Functions · Grade 8 · Worksheet 2

1. Liam is comparing two different streaming services. Service A charges a $5 monthly fee plus $0.50 per movie watched. Service B charges a $12 monthly fee plus $0.25 per movie watched. Liam wants to know how many movies he would need to watch for both services to cost the same amount. Write an equation to represent this situation and solve for the number of movies. Answer: 28 Solution: Service A cost = monthly fee + cost per movie × number of movies Service A cost = 5 + 0.50 × m Service B cost = monthly fee + cost per movie × number of movies Service B cost = 12 + 0.25 × m 5 + 0.50m = 12 + 0.25m Subtract 0.25m from both sides to get m terms on one side.
   Full step-by-step solution

   Let's define the number of movies watched as m. Service A cost = monthly fee + cost per movie × number of movies Service A cost = 5 + 0.50 × m Service B cost = monthly fee + cost per movie × number of movies Service B cost = 12 + 0.25 × m We want the costs to be equal: 5 + 0.50m = 12 + 0.25m Step 1: Subtract 0.25m from both sides to get m terms on one side. 5 + 0.50m - 0.25m = 12 + 0.25m - 0.25m 5 + 0.25m = 12 Step 2: Subtract 5 from both sides to isolate the m term. 5 + 0.25m - 5 = 12 - 5 0.25m = 7 Step 3: Divide both sides by 0.25 to solve for m. m = 7 / 0.25 Step 4: Calculate 7 divided by 0.25. Dividing by 0.25 is the same as multiplying by 4. m = 7 × 4 m = 28 So, Liam would need to watch 28 movies for both services to cost the same.

2. (3 × 10⁴) ÷ (6 × 10²) = ? Answer: 50 Solution: Write the expression as (3 ÷ 6) × (10⁴ ÷ 10²) Simplify the coefficients: 3 ÷ 6 = 0.5 Simplify the exponents: 10⁴ ÷ 10² = 10^(4-2) = 10² = 100 Multiply the results: 0.5 × 100 = 50 The answer is 50.
   Full step-by-step solution

   Step 1: Write the expression as (3 ÷ 6) × (10⁴ ÷ 10²) Step 2: Simplify the coefficients: 3 ÷ 6 = 0.5 Step 3: Simplify the exponents: 10⁴ ÷ 10² = 10^(4-2) = 10² = 100 Step 4: Multiply the results: 0.5 × 100 = 50 The answer is 50.

3. Liam is comparing two different cell phone plans. Plan A charges a $25 monthly fee plus $0.10 per minute of talk time. Plan B charges a $15 monthly fee plus $0.15 per minute of talk time. Liam wants to know how many minutes of talk time would make both plans cost the same amount. Write an equation to represent this situation and solve for the number of minutes. Answer: 200 minutes Solution: Write the cost equation for Plan A. Plan A has a $25 monthly fee plus $0.10 per minute. So, Cost_A = 25 + 0.10 * m Write the cost equation for Plan B.
   Full step-by-step solution

   Let's define the number of minutes of talk time as m. Step 1: Write the cost equation for Plan A. Plan A has a $25 monthly fee plus $0.10 per minute. So, Cost_A = 25 + 0.10 * m Step 2: Write the cost equation for Plan B. Plan B has a $15 monthly fee plus $0.15 per minute. So, Cost_B = 15 + 0.15 * m Step 3: Set the costs equal to each other. We want to find when Cost_A = Cost_B. So, 25 + 0.10 * m = 15 + 0.15 * m Step 4: Solve for m. First, subtract 15 from both sides: 25 - 15 + 0.10 * m = 0.15 * m 10 + 0.10 * m = 0.15 * m Step 5: Subtract 0.10 * m from both sides: 10 = 0.15 * m - 0.10 * m 10 = 0.05 * m Step 6: Divide both sides by 0.05 to isolate m: m = 10 / 0.05 m = 1000 / 5 m = 200 Step 7: Interpret the result. The number of minutes that makes both plans cost the same is 200 minutes. Final Answer: 200 minutes

4. Noah is comparing two different car rental companies. DriveEasy charges a fixed fee of $36 plus $0.16 per mile driven. SpeedyRent's total cost for different numbers of miles is shown in the table below: | Miles Driven (x) | Total Cost (y) | |------------------|----------------| | 50 | $41 | | 100 | $46 | | 150 | $51 | | 200 | $56 | Noah wants to rent a car for a trip. He wants to know which company has the lower cost per mile (the smaller rate). What is the cost per mile for SpeedyRent, and which company has the lower cost per mile? Answer: SpeedyRent has a cost per mile of $0.10, which is lower than DriveEasy's $0.16 per mile. Solution: Identify the rate (cost per mile) for DriveEasy from the equation y = 0.16x + 36. The coefficient of x is 0.16, so DriveEasy charges $0.16 per mile. Find the rate for SpeedyRent using two points from the table.
   Full step-by-step solution

   Step 1: Identify the rate (cost per mile) for DriveEasy from the equation y = 0.16x + 36. The coefficient of x is 0.16, so DriveEasy charges $0.16 per mile. Step 2: Find the rate for SpeedyRent using two points from the table. Use the points (50, 41) and (100, 46). Rate = (46 - 41) / (100 - 50) = 5 / 50 = 0.10. So SpeedyRent charges $0.10 per mile. Step 3: Verify with another pair, (100, 46) and (150, 51): Rate = (51 - 46) / (150 - 100) = 5 / 50 = 0.10. This confirms the rate is constant. Step 4: Compare the rates: DriveEasy = $0.16 per mile, SpeedyRent = $0.10 per mile. Since 0.10 < 0.16, SpeedyRent has the lower cost per mile. The answer is SpeedyRent has a cost per mile of $0.10, which is lower than DriveEasy's $0.16 per mile.

5. Function A: y = 6x + 11. Function B: table below. Which function has the greater slope? x | y 1 | 16 3 | 26 5 | 36 7 | 46 Answer: Function B Solution: Find the slope of Function A. In the equation y = 6x + 11, the slope is the coefficient of x, which is 6. Find the slope of Function B.
   Full step-by-step solution

   Step 1: Find the slope of Function A. In the equation y = 6x + 11, the slope is the coefficient of x, which is 6. Step 2: Find the slope of Function B. Use two points from the table, for example (1, 16) and (3, 26). Slope = (26 - 16) / (3 - 1) = 10 / 2 = 5. Step 3: Compare the slopes. Function A has slope 6, Function B has slope 5. Since 6 > 5, Function A has the greater slope. The answer is Function A.

6. Liam is comparing two phone plans. Plan A costs $25 per month plus $0.10 per text message. Plan B has a fixed cost of $40 per month for unlimited texting. Liam wants to determine how many text messages he would need to send for both plans to cost the same amount. Write an equation to represent this situation and solve for the number of text messages. Answer: 150 Solution: Plan A cost = 25 + 0.10 * x Plan B cost = 40 25 + 0.10 * x = 40 Subtract 25 from both sides to isolate the term with x. 0.10 * x = 40 - 25 0.10 * x = 15 Divide both sides by 0.10 to solve for x.
   Full step-by-step solution

   Let's define the number of text messages as x. Plan A cost = 25 + 0.10 * x Plan B cost = 40 We want the costs to be equal: 25 + 0.10 * x = 40 Step 1: Subtract 25 from both sides to isolate the term with x. 0.10 * x = 40 - 25 0.10 * x = 15 Step 2: Divide both sides by 0.10 to solve for x. x = 15 / 0.10 Step 3: Calculate the division. 15 divided by 0.10 is the same as 15 divided by (1/10), which is 15 * 10 = 150. So, x = 150. This means Liam would need to send 150 text messages for both plans to cost the same.

7. 2³ × 3² = ? Answer: 72 Solution: 2³ × 3² 2³ means 2 raised to the power of 3, which is 2 × 2 × 2. 3² means 3 raised to the power of 2, which is 3 × 3. Calculate 2³ 2 × 2 = 4 4 × 2 = 8 So, 2³ = 8.
   Full step-by-step solution

   Let's solve the problem step by step. We are given: 2³ × 3² **Step 1: Understand the notation** 2³ means 2 raised to the power of 3, which is 2 × 2 × 2. 3² means 3 raised to the power of 2, which is 3 × 3. **Step 2: Calculate 2³** 2 × 2 = 4 4 × 2 = 8 So, 2³ = 8. **Step 3: Calculate 3²** 3 × 3 = 9 So, 3² = 9. **Step 4: Multiply the results** 8 × 9 = 72. **Step 5: Final answer** Therefore, 2³ × 3² = 72.