Back to Solve multi-step real-life and mathematical problems with positive and negative rational numbers

Solving Multi-Step Problems with Rational Numbers

Common Core Math - Grade 7·standard·7-ee-b-3

Work through problems with immediate feedback

Recall

Which expression gives the product of $-\dfrac{3}{4}$ and $-8$ ?

$-6$

$6$

$-\dfrac{3}{32}$

$\dfrac{3}{32}$

Which decimal is equal to $\dfrac{3}{8}$ ?

$0.38$

$0.375$

$0.35$

$3.75$

Evaluate: $4 + 3 \times (10 - 4)$

Fluency

A store bought 24 items at $1.75 each. Six of the items were damaged and returned for a refund of $1.50 each. Which sequence of operations gives the net cost to the store?

Add $24 + 6 = 30$ total items, then multiply by $1.75.

Multiply $24 \times 1.75$ for the purchase cost, multiply $6 \times 1.50$ for the refund, then subtract the refund from the purchase cost.

Multiply $24 \times 1.50$ , then subtract $6 \times 1.75$ .

Subtract $24 - 6 = 18$ net items, then multiply $18 \times 1.75$ .

Compute $-\dfrac{2}{3} + 1.5 - \dfrac{3}{4}$ using fractions to keep the result exact. Enter your answer as a simplified fraction.

Use a mental-math shortcut to compute $36 \times (2.75 + 0.25)$ . What is the result?

Regroup to simplify: $\left(-\dfrac{5}{6}\right) + \dfrac{2}{3} + \left(-\dfrac{1}{6}\right)$ .

Enter your answer as a simplified fraction.

A student calculated that a plumber earned $146.25 by working 4.5 hours at $31.50 per hour and then paying a $15.75 supply fee. Which estimation best checks whether $146.25 is reasonable?

Round to $30 \times 5 - 16 = 150 - 16 = 134$ , which is close to $146.25.

Round to $30 \times 4 - 15 = 120 - 15 = 105$ , which is far from $146.25, so the answer is wrong.

Round to $35 \times 5 - 15 = 175 - 15 = 160$ , which shows $146.25 is too low.

"About $140" is close enough; no calculation is needed.

Varied Practice

A warehouse has 144 boxes. Three-eighths of the boxes are shipped out on Monday. Then 27 new boxes arrive on Tuesday. In which order should you solve to find how many boxes remain at the end?

Add $144 + 27$ first, then multiply by $\dfrac{3}{8}$ , then subtract.

Multiply $\dfrac{3}{8} \times 144$ to find boxes shipped, subtract that from 144, then add 27.

Subtract 27 from 144 first, then multiply by $\dfrac{3}{8}$ .

Divide 144 by $\dfrac{3}{8}$ to find boxes shipped, then subtract from 144, then add 27.

Subtract $9\dfrac{3}{4}$ from $27\dfrac{1}{2}$ , then divide the result by 2. Express your answer as a mixed number.

$A four-box flowchart showing the problem-solving steps READ, PLAN, COMPUTE, CHECK, with the muffin problem mapped under each step.$

A bakery starts with 240 muffins. They sell $\dfrac{5}{8}$ of them in the morning, then bake 36 more in the afternoon. How many muffins are in the bakery at the end?

Fill in the three blanks to complete the solution plan.

Step 1 (find how many were sold): ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ $\times\ 240 = 150$ muffins sold.

Step 2 (find remaining after sales): $240 - 150 =$ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ muffins.

Step 3 (final count): $90 + 36 =$ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ muffins in the bakery.

Step 1 fraction:

After morning:

Final answer:

A class of 28 students each owes $3.25 for a field trip and $0.75 for supplies. Use the distributive property to find the total amount owed. Enter your answer in dollars.

A student computes $432 as the answer to: "A baker uses 2.5 pounds of flour at $1.80 per pound and spends $3.60 on yeast." Without computing exactly, which estimation shows that $432 is unreasonable?

$2 \times 2 = 4$ , and $4 + 4 = 8$ , which is far less than $432, so the answer cannot be correct.

The computation involves decimals, so a large result like $432 is possible.

$432 \div 2.5 \approx 173$ per pound, which might be a high flour price.

Add the numbers: $2.5 + 1.80 + 3.60 = 7.90 \ne 432$ , confirming an error.

A car travels 215 miles in 3.5 hours. Describe in your own words the plan you would use to find the car's speed in miles per hour. Name the operation you would use and explain why.

Word Problems

$A horizontal number line showing a submarine starting at negative 340 feet, rising 200 feet to negative 140 feet, then descending 60 feet to an unknown final depth marked with a question mark.$

A submarine is at a depth of $-340$ feet. It rises at 25 feet per minute for 8 minutes, then descends 60 feet. What is the submarine's final depth in feet?

(Use a negative number for depth below sea level.)

A towel bar that is $9\dfrac{3}{4}$ inches long is to be mounted and centered on a bathroom door that is $27\dfrac{1}{2}$ inches wide.

Use the given measurements to answer both parts below.

How many total inches of door width are not covered by the towel bar?

Remaining width: ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ inches

remaining width:

Since the bar is centered, how far is each end of the towel bar from the nearest edge of the door?

Distance from each edge: ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ inches

distance from each edge:

A store marks up the price of a jacket by 15%. The original price is $42.80. What is the new price in dollars? Round to the nearest cent.

Maria earns $12.50 per hour. She works 6.5 hours, then spends $8.75 on lunch.

First, write an estimation to predict how much she will have left. Then compute the exact answer. Finally, explain in one or two sentences whether your exact answer is reasonable and why.

Error Analysis

$A two-column contrast card showing the student's incorrect work (5 + 3.25 = 8.25, marked wrong) next to the correct work (5 × 3.25 = 16.25, marked right), with a callout noting that 'each' signals multiplication.$

Problem: Find the total cost of 5 items priced at $3.25 each.

Student's work:
$5 + 3.25 = 8.25$

Student's answer: $8.25

Which response correctly identifies the student's error?

The student added incorrectly; $5 + 3.25$ should equal $8.75$ , not $8.25$ .

The student used addition instead of multiplication. The word "each" with a count and a unit price means multiply: $5 \times 3.25 = 16.25$ .

The student should have divided: $3.25 \div 5 = 0.65$ per item.

The student's equation is set up correctly; only the final answer should be $8.50.

Problem: Jamal mows 4 lawns at $15 each and then pays $12.50 for gas. How much profit does Jamal make?

Student's work:
$4 \times 15 = 60$

Student's answer: "Jamal makes $60."

Which response correctly identifies the student's error?

The student made an arithmetic error; $4 \times 15 \ne 60$ .

The student stopped at an intermediate step. The question asks for profit after paying $12.50 for gas, so one more step is needed: $60 - 12.50 = 47.50$ .

The student should have subtracted $12.50 from $15 first, then multiplied by 4.

The student's answer is correct because $60 is what Jamal earned.

Challenge

A pool holds 5,000 gallons at full capacity. On Monday it loses $\dfrac{3}{4}$ of its water due to a drain malfunction. On Tuesday, 0.35 times the pool's full capacity is pumped back in. On Wednesday, $37\dfrac{1}{2}$ gallons evaporate. How many gallons are in the pool at the end of Wednesday?

A jacket originally costs $180. A store offers a 35% discount.

A student estimates the discount as "about $50 off."

(a) Use a better estimation strategy to find a closer estimate of the discount. Show your reasoning.

(b) Compute the exact discount and the sale price.

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