Neft Teacher · Learn It

Unit 9 · Standard 6.AT.11

Apply Two-Variable Relationships to Solve Problems

Lesson 9-4 · Learn It — how the math works

Why we're learning this

An automobile plant's assembly line runs in 8-hour shifts and produces 2.5 cars per hour. An order arrives: 1,000 cars in 3 weeks. Should the manufacturer accept? Later, the student council plans a fundraiser car wash — last year they washed 34 cars and raised $408, and this year the goal is $500.

✱ Math Workbench

Learn It

Use the equation to answer a real question

Writing the equation is only half the job. Applying it means substituting what you know, solving for what you don't, and checking whether the answer makes sense in the situation.

Picture itSee what the math looks like before you read about it

An assembly line at an automobile manufacturing plant runs in 8-hour shifts and produces 2.5 cars per hour. Next to it, an order form reads: 1,000 cars needed in 3 weeks. A table shows: 1 hour, 2.5 cars. 8 hours, 20 cars. 16 hours, 40 cars. 24 hours, 60 cars.
Where you see this in real lifeAn assembly line at an automobile manufacturing plant runs in 8-hour shifts and produces 2.5 cars per hour. Next to it, an order form reads: 1,000 cars needed in 3 weeks. A table shows: 1 hour, 2.5 cars. 8 hours, 20 cars. 16 hours, 40 cars. 24 hours, 60 cars.

Watch me solve itI do — read one step at a time

The problem looks like this
Table: 8 hours make 20 cars, and 400 hours make 1,000 cars; the rule is c = 2.5h.Hours (h)8400hCars (c)201,0002.5h
  1. The number of cars produced depends on the number of hours of operation. Every hour, the plant produces 2.5 cars.
  2. Let h = hours and c = cars. The is c = 2.5h. Check with the table: 8 hours gives 2.5 × 8 = 20 cars per shift.2.5 × 8 = 20
  3. The order needs 1,000 cars, so : 2.5h = 1,000.
  4. Divide both sides by 2.5:h = 1,000 ÷ 2.5 = 400 hours
  5. Interpret it: 400 hours running 24 hours a day is about 17 days — about 2.5 weeks. That fits inside 3 weeks only if the plant runs 24 hours a day, 7 days a week.

Watch out — mistakes students make here

  • 40 hours would produce only 2.5 × 40 = 100 cars — one-tenth of the order.
  • 4 days would be only 4 × 24 = 96 hours, far short of 400.
  • 50 comes from dividing by 8 (one shift per day) — this question says the plant runs 24 hours a day.

Try it with meWe do — you answer first, then check

Same steps, new numbers.

  1. Last year the council washed 34 cars and raised $408. What did they charge per car?
    Check this step
    Divide:408 ÷ 34 = $12
  2. At $12 per car, let n = cars washed and t = total raised: t = 12n.
  3. The goal is $500, so 12n = 500 means n = 500 ÷ 12, which is about 41.7 — so they must wash 42 cars, because 41 cars raise only 41 × 12 = $492.
  4. Or fix the cars instead: if they 35 cars, then 35p = 500 gives p = 500 ÷ 35, 14.285714… — so the minimum whole-cent price is $14.29 per car.

On your ownYou do — solve this one by yourself

Before you start, remember:

  • For each problem below, write the , what you know, and solve.
  • Then say what the means in the situation — and check whether it actually meets the goal.

Solve: The order requires 1,000 cars. Using 2.5h = 1,000, how many hours will the plant need?

My answer

Stuck? Open one hint at a time.

Hint 1

h is multiplied by 2.5 — undo it with division.

Hint 2

Divide 1,000 by 2.5.

Hint 3

Check: 2.5 × your answer should equal 1,000.

Check my answer
✅ Answer: 400 hours
Divide both sides by 2.5: h = 1,000 ÷ 2.5 = 400 hours.

Turn & TalkSay it out loud with a partner

The plant produces 2.5 cars per hour, and the order needs 1,000 cars. Solving 2.5h = 1,000 gives h = 400 hours. Why do you divide instead of multiply to find h?

Try starting with:

  • I divide 1,000 by 2.5 because multiplication and division ___ each other.Divido 1,000 entre 2.5 porque la multiplicación y la división se ___ entre sí.
  • I know h = 400 is correct because 2.5 × ___ = 1,000.Sé que h = 400 es correcto porque 2.5 × ___ = 1,000.
Use these wordssolutionsubstituteat least