Compare Functions

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Compare Functions · Grade 8 · Worksheet 3

1. Liam is comparing two cell phone plans. Plan A charges a $20 monthly fee plus $0.10 per text message. Plan B charges a $15 monthly fee plus $0.15 per text message. Liam wants to know 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: 100 Solution: Plan A cost = monthly fee + cost per text × number of texts Plan A cost = 20 + 0.10 × x Plan B cost = monthly fee + cost per text × number of texts Plan B cost = 15 + 0.15 × x We want both plans to cost the same, so set them equal: 20 + 0.10x = 15 + 0.15x Now solve for x.
   Full step-by-step solution

   Let's define the number of text messages as x. Plan A cost = monthly fee + cost per text × number of texts Plan A cost = 20 + 0.10 × x Plan B cost = monthly fee + cost per text × number of texts Plan B cost = 15 + 0.15 × x We want both plans to cost the same, so set them equal: 20 + 0.10x = 15 + 0.15x Now solve for x. Step 1: Subtract 15 from both sides 20 - 15 + 0.10x = 0.15x 5 + 0.10x = 0.15x Step 2: Subtract 0.10x from both sides 5 = 0.15x - 0.10x 5 = 0.05x Step 3: Divide both sides by 0.05 x = 5 / 0.05 x = 500 / 5 x = 100 So, Liam would need to send 100 text messages for both plans to cost the same.

2. A rectangular prism has a length of 8 cm, a width of 5 cm, and a height of 3 cm. If you double all three dimensions to create a larger prism, what is the ratio of the surface area of the larger prism to the surface area of the original prism? Answer: 4 Solution: Find the surface area of the original prism. The formula for the surface area \(SA\) of a rectangular prism with length \(L\), width \(W\), and height \(H\) is: SA = 2(LW + LH + WH) \(L = 8\), \(W = 5\), \(H = 3\) LW = 8 \times 5 = 40 LH = 8 \times 3 = 24 WH = 5 \times 3 = 15 Sum: \(40 + 24 + 15…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Find the surface area of the original prism.** The formula for the surface area \(SA\) of a rectangular prism with length \(L\), width \(W\), and height \(H\) is: \[ SA = 2(LW + LH + WH) \] For the original prism: \(L = 8\), \(W = 5\), \(H = 3\) \[ LW = 8 \times 5 = 40 \] \[ LH = 8 \times 3 = 24 \] \[ WH = 5 \times 3 = 15 \] Sum: \(40 + 24 + 15 = 79\) Multiply by 2: \(SA_{original} = 2 \times 79 = 158\) cm² --- **Step 2: Find the surface area of the larger prism after doubling dimensions.** New dimensions: \(L' = 16\), \(W' = 10\), \(H' = 6\) \[ L'W' = 16 \times 10 = 160 \] \[ L'H' = 16 \times 6 = 96 \] \[ W'H' = 10 \times 6 = 60 \] Sum: \(160 + 96 + 60 = 316\) Multiply by 2: \(SA_{new} = 2 \times 316 = 632\) cm² --- **Step 3: Find the ratio of the surface areas.** \[ \text{Ratio} = \frac{SA_{new}}{SA_{original}} = \frac{632}{158} \] Simplify: divide numerator and denominator by 2: \[ \frac{316}{79} \] Now \(316 \div 79 = 4\) exactly. So the ratio is \(4\). --- **Step 4: Explanation without computing both fully.** We can also see this from the surface area formula: Original: \(SA = 2(LW + LH + WH)\) New: \(SA' = 2( (2L)(2W) + (2L)(2H) + (2W)(2H) )\) \[ = 2( 4LW + 4LH + 4WH ) \] \[ = 2 \times 4 (LW + LH + WH) \] \[ = 4 \times [ 2(LW + LH + WH) ] \] \[ = 4 \times SA_{original} \] So the ratio is \(4\) directly. --- **Final Answer:** 4

3. Emma is comparing two different gym membership options. Gym A charges a $40 monthly fee plus $5 per fitness class attended. Gym B charges a $25 monthly fee plus $8 per fitness class. Emma wants to know how many fitness classes she would need to attend each month for both gyms to cost the same total amount. How many classes would that be? Answer: 5 Solution: Write an equation for Gym A's total cost: 40 + 5x, where x is the number of classes Write an equation for Gym B's total cost: 25 + 8x Set the costs equal: 40 + 5x = 25 + 8x Subtract 25 from both sides: 15 + 5x = 8x Subtract 5x from both sides: 15 = 3x Divide both sides by 3: x = 5 Check: Gym A:…
   Full step-by-step solution

   Step 1: Write an equation for Gym A's total cost: 40 + 5x, where x is the number of classes Step 2: Write an equation for Gym B's total cost: 25 + 8x Step 3: Set the costs equal: 40 + 5x = 25 + 8x Step 4: Subtract 25 from both sides: 15 + 5x = 8x Step 5: Subtract 5x from both sides: 15 = 3x Step 6: Divide both sides by 3: x = 5 Step 7: Check: Gym A: 40 + 5(5) = 40 + 25 = $65; Gym B: 25 + 8(5) = 25 + 40 = $65 The answer is 5 classes.

4. Liam is comparing two different phone plans. Plan A costs $20 per month plus $0.10 per gigabyte of data used. Plan B has a flat fee of $35 per month for unlimited data. Liam wants to determine how many gigabytes he would need to use for both plans to cost the same amount. Write an equation to represent this situation and solve for the number of gigabytes. Answer: 150 Solution: Plan A costs $20 per month plus $0.10 per gigabyte. So cost for Plan A = 20 + 0.10 × g Plan B costs a flat $35 per month.
   Full step-by-step solution

   Let's define the number of gigabytes used as g. Plan A costs $20 per month plus $0.10 per gigabyte. So cost for Plan A = 20 + 0.10 × g Plan B costs a flat $35 per month. So cost for Plan B = 35 We want the number of gigabytes where both plans cost the same: 20 + 0.10 × g = 35 Step 1: Subtract 20 from both sides to isolate the term with g. 0.10 × g = 35 − 20 0.10 × g = 15 Step 2: Divide both sides by 0.10 to solve for g. g = 15 ÷ 0.10 Step 3: Calculate the division. 15 ÷ 0.10 = 15 ÷ (1/10) = 15 × 10 = 150 So, g = 150 gigabytes. This means Liam would need to use 150 gigabytes for both plans to cost the same amount.

5. Function A: y = 15x + 20. Function B: table below. Which function has the greater slope? x | y 0 | 28 2 | 58 4 | 88 6 | 118 Answer: Function A Solution: Find the slope of Function A. In the equation y = 15x + 20, the slope is the coefficient of x, which is 15. Find the slope of Function B.
   Full step-by-step solution

   Step 1: Find the slope of Function A. In the equation y = 15x + 20, the slope is the coefficient of x, which is 15. Step 2: Find the slope of Function B. Use two points from the table, such as (0, 28) and (2, 58). Slope = (58 - 28) / (2 - 0) = 30 / 2 = 15. Step 3: Compare the slopes. Function A has slope 15, Function B has slope 15. They are equal. The answer is Function A.

6. Function A: y = 7x + 12. Function B: table below. Which function has the greater y-intercept? x | y 0 | 17 2 | 31 4 | 45 6 | 59 Answer: Function B Solution: Find the y-intercept of Function A. In the equation y = 7x + 12, the y-intercept is the constant term, which is 12. Find the y-intercept of Function B.
   Full step-by-step solution

   Step 1: Find the y-intercept of Function A. In the equation y = 7x + 12, the y-intercept is the constant term, which is 12. Step 2: Find the y-intercept of Function B. From the table, when x = 0, y = 17. So the y-intercept is 17. Step 3: Compare the y-intercepts. Function A has y-intercept 12, Function B has y-intercept 17. Since 17 > 12, Function B has the greater y-intercept. The answer is Function B.