Systems Word Problems

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Systems Word Problems · Grade 8 · Worksheet 2

1. A science class is growing bacteria in two different petri dishes. Dish A starts with 8,000 bacteria and grows at a rate of 400 bacteria per hour. Dish B starts with 2,000 bacteria and grows at a rate of 800 bacteria per hour. After how many hours will both dishes contain the same number of bacteria? Answer: 15 Solution: Let \( t \) = number of hours after the start. Dish A starts with 8,000 bacteria and grows at 400 bacteria per hour.
   Full step-by-step solution

   Let's solve this step by step. --- **Step 1: Define variables** Let \( t \) = number of hours after the start. Dish A starts with 8,000 bacteria and grows at 400 bacteria per hour. So, number of bacteria in Dish A after \( t \) hours is: \[ A(t) = 8000 + 400t \] Dish B starts with 2,000 bacteria and grows at 800 bacteria per hour. So, number of bacteria in Dish B after \( t \) hours is: \[ B(t) = 2000 + 800t \] --- **Step 2: Set up the equation** We want the time \( t \) when both dishes have the same number of bacteria: \[ 8000 + 400t = 2000 + 800t \] --- **Step 3: Solve for \( t \)** Subtract 2000 from both sides: \[ 8000 - 2000 + 400t = 800t \] \[ 6000 + 400t = 800t \] Subtract 400t from both sides: \[ 6000 = 800t - 400t \] \[ 6000 = 400t \] Divide both sides by 400: \[ t = 6000 / 400 \] \[ t = 15 \] --- **Step 4: Interpret the result** After 15 hours, both dishes will have the same number of bacteria. --- **Step 5: Check the answer** Dish A after 15 hours: \( 8000 + 400 \times 15 = 8000 + 6000 = 14000 \) Dish B after 15 hours: \( 2000 + 800 \times 15 = 2000 + 12000 = 14000 \) They match. --- **Final answer:** 15

2. Hana is helping organize a school art exhibition. She needs to order two types of display boards: large boards that cost $14 each and small boards that cost $9 each. Her total budget for boards is $410, and she needs exactly 35 boards in total to display all the artwork. How many large boards and how many small boards should Hana order to use her entire budget and get exactly 35 boards? Answer: Large boards: 19, Small boards: 16 Solution: Let x = number of large boards and y = number of small boards.
   Full step-by-step solution

   Let x = number of large boards and y = number of small boards. Step 1: Write the equation for total number of boards: x + y = 35 Step 2: Write the equation for total cost: 14x + 9y = 410 Step 3: Solve the first equation for x: x = 35 - y Step 4: Substitute into the second equation: 14(35 - y) + 9y = 410 Step 5: Simplify: 490 - 14y + 9y = 410 Step 6: Combine like terms: 490 - 5y = 410 Step 7: Subtract 490 from both sides: -5y = -80 Step 8: Divide both sides by -5: y = 16 Step 9: Substitute back to find x: x = 35 - 16 = 19 Hana should order 19 large boards and 16 small boards. The answer is 19 large boards and 16 small boards.

3. Liam is planning a school fundraiser and needs to buy supplies. He can purchase markers for $2 each and notebooks for $3 each. He has a budget of $120 and needs exactly 50 items total. How many markers and how many notebooks should Liam buy to spend exactly his budget and get exactly 50 items? Answer: 30 markers and 20 notebooks Solution: Let m = number of markers Let n = number of notebooks 1. Each marker costs $2, each notebook costs $3 2. Total budget is $120 3.
   Full step-by-step solution

   Let's define variables: Let m = number of markers Let n = number of notebooks We know: 1. Each marker costs $2, each notebook costs $3 2. Total budget is $120 3. Total items needed is 50 Step 1: Write the equations from the problem. From the total items: m + n = 50 From the total cost: 2m + 3n = 120 Step 2: Solve the first equation for one variable. From m + n = 50, we get m = 50 - n Step 3: Substitute this into the second equation. 2(50 - n) + 3n = 120 Step 4: Simplify and solve for n. 100 - 2n + 3n = 120 100 + n = 120 n = 120 - 100 n = 20 Step 5: Find m using m = 50 - n m = 50 - 20 m = 30 Step 6: Check the solution. Total items: 30 + 20 = 50 ✓ Total cost: 2×30 + 3×20 = 60 + 60 = 120 ✓ Therefore, Liam should buy 30 markers and 20 notebooks.

4. Liam is planning a school fundraiser and needs to order t-shirts. The printing company charges a $25 setup fee plus $8 per shirt. Liam has budgeted $185 for the t-shirts. How many t-shirts can he order within his budget? Answer: 20 Solution: Write down the total cost equation. - A fixed setup fee: $25 - Plus $8 per shirt \text{Total cost} = 25 + 8x Set up the budget equation. Liam’s budget is $185, so: 25 + 8x = 185 Isolate the term with \( x \).
   Full step-by-step solution

   Let's solve this step by step. --- **Step 1: Write down the total cost equation.** The printing company charges: - A fixed setup fee: $25 - Plus $8 per shirt If Liam orders \( x \) shirts, the total cost is: \[ \text{Total cost} = 25 + 8x \] --- **Step 2: Set up the budget equation.** Liam’s budget is $185, so: \[ 25 + 8x = 185 \] --- **Step 3: Isolate the term with \( x \).** Subtract 25 from both sides: \[ 8x = 185 - 25 \] \[ 8x = 160 \] --- **Step 4: Solve for \( x \).** Divide both sides by 8: \[ x = 160 / 8 \] \[ x = 20 \] --- **Step 5: Interpret the result.** \( x = 20 \) means Liam can order **20 shirts** within his budget. --- **Final Answer:** 20

5. Hana is selling two types of raffle tickets. Type A costs $4 each and Type B costs $6 each. She sells 40 tickets in total and collects $180. How many of each type did she sell? Answer: Type A: 30, Type B: 10 Solution: Define variables. Let x = number of Type A tickets, y = number of Type B tickets. Write equations.
   Full step-by-step solution

   Step 1: Define variables. Let x = number of Type A tickets, y = number of Type B tickets. Step 2: Write equations. Total tickets: x + y = 40. Total money: 4x + 6y = 180. Step 3: Solve by elimination. Multiply the first equation by 4: 4x + 4y = 160. Step 4: Subtract from the second equation: (4x + 6y) - (4x + 4y) = 180 - 160, so 2y = 20, y = 10. Step 5: Substitute y = 10 into x + y = 40: x + 10 = 40, so x = 30. Step 6: Check: 30 + 10 = 40 tickets, and 4(30) + 6(10) = 120 + 60 = 180 dollars. Correct. The answer is Type A: 30, Type B: 10.

6. Matiu and Mere are collecting donations for a charity. Matiu collected 4 times as much money as Mere. Together they collected $150. How much did each person collect? Answer: Matiu collected $120, Mere collected $30 Solution: Let m = amount Mere collected (in dollars). Matiu collected 4 times as much, so Matiu collected 4m dollars.
   Full step-by-step solution

   Let m = amount Mere collected (in dollars). Matiu collected 4 times as much, so Matiu collected 4m dollars. Together they collected $150, so the equation is: m + 4m = 150 Combine like terms: 5m = 150 Divide both sides by 5: m = 30 So Mere collected $30. Matiu collected 4 × $30 = $120. Check: $30 + $120 = $150. Correct. The answer is Matiu collected $120, Mere collected $30.

7. Liam is planning a school fundraiser and needs to determine how many chocolate bars and bags of popcorn to sell. He knows that chocolate bars sell for $2 each and popcorn bags sell for $3 each. His goal is to raise exactly $120. Additionally, he has a limited number of boxes and can only fit a total of 50 items. How many chocolate bars and how many bags of popcorn must Liam sell to meet his fundraising goal and item limit? Answer: 30 chocolate bars and 20 popcorn bags Solution: Let c = number of chocolate bars Let p = number of popcorn bags Each chocolate bar sells for $2 → money from chocolate = 2c Each popcorn bag sells for $3 → money from popcorn = 3p Goal: total money = 120 So: 2c + 3p = 120 ...
   Full step-by-step solution

   Let's define variables: Let c = number of chocolate bars Let p = number of popcorn bags We know: Each chocolate bar sells for $2 → money from chocolate = 2c Each popcorn bag sells for $3 → money from popcorn = 3p Goal: total money = 120 So: 2c + 3p = 120 ... (1) Also: total items = 50 So: c + p = 50 ... (2) --- **Step 1: Solve for one variable from equation (2)** From (2): c = 50 - p --- **Step 2: Substitute into equation (1)** 2(50 - p) + 3p = 120 100 - 2p + 3p = 120 100 + p = 120 --- **Step 3: Solve for p** p = 120 - 100 p = 20 --- **Step 4: Solve for c** c = 50 - p = 50 - 20 = 30 --- **Step 5: Check** Money: 2*30 + 3*20 = 60 + 60 = 120 ✔ Total items: 30 + 20 = 50 ✔ --- **Final answer:** Liam must sell 30 chocolate bars and 20 popcorn bags.