Introduction

Solving Multi-Step Problems with Rational Numbers is an important Grade 7 math skill because students are moving from simple answers toward explaining how the math works.

In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with solving multi-step problems with rational numbers.

What Is Solving Multi-Step Problems with Rational Numbers?

Solving Multi-Step Problems with Rational Numbers means using place value, operations, and equations to reason accurately with numbers.

The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.

Understanding Solving Multi-Step Problems with Rational Numbers

Before solving, students should slow down and decide what each number, shape, unit, or label represents.

  • Read the question carefully and identify what is being asked.
  • Choose a model, equation, table, or diagram that matches the situation.
  • Solve one step at a time and keep units or labels attached.
  • Use the answer explanation to check that the result makes sense.

Visual Models

Visual Model 1

Question: Solve: \(2x-5=7\)

Visual Model 1

  • A. \(x=1\)
  • B. \(x=6\)
  • C. \(x=12\)
  • D. \(x=24\)

Why it works: Add \(5\): \(2x=12\). Divide by \(2\): \(x=6\).

Answer: \(x=6\)

Visual Model 2

Question: If \(2(x+3)-5=11\), what is \(x\)? Show steps and verify.

\(x\)\(x+3\)\(2(x+3)\)\(2(x+3)-5\)
1483
25105
36127
  • A. \(x=3\)
  • B. \(x=5\)
  • C. \(x=7\)
  • D. \(x=9\)

Why it works: Add \(5\): \(2(x+3)=16\). Divide by \(2\): \(x+3=8\). Subtract \(3\): \(x=5\). Check: \(2(5+3)-5 = 2(8)-5 = 16-5 = 11\) ✓.

Answer: \(x=5\)

Worked Examples

Example 1

Question: Using the model, what is the original amount?

Example 1

  • A. \(4\)
  • B. \(12\)
  • C. \(20\)
  • D. \(24\)
  1. Original amount = \(12 + 8 = 20\).

Answer: \(20\)

Example 2

Question: What equation did Step 1 start from?

Example 2

  • A. \(4x+5=20\)
  • B. \(4x-5=15\)
  • C. \(4x+8=28\)
  • D. \(4x-4=16\)
  1. If \(4x+8=28\), subtracting \(8\) gives \(4x=20\), then dividing by \(4\) gives \(x=5\).

Answer: \(4x+8=28\)

Example 3

Question: What error, if any, was made in solving the equation?

StepEquation
Original\(3(x-2) = 15\)
Divide by 3\(x-2 = 5\)
Add 2\(x = 7\)
  • A. No error; \(x=7\) is correct
  • B. Wrong operation in Step 2; should subtract 2
  • C. Wrong operation in Step 1; should multiply by 3
  • D. Error in Step 2; \(5+2=8\), not \(7\)
  1. All steps follow correctly: \(3(x-2)=15 \to x-2=5 \to x=7\).

Answer: No error; \(x=7\) is correct

Real-World Word Problems

Problem 1

Question: Marcus earns $12 per hour. After working 8 hours, he spent $15 on lunch. How much money does he have left?

  • A. \($81\)
  • B. \($96\)
  • C. \($111\)
  • D. \($127\)

Why it works: Earnings: \(12 \times 8 = 96\). After spending: \(96 - 15 = 81\).

Answer: \($81\)

Problem 2

Question: A taxi charges $3 per mile plus a $2 fee. If a trip costs $17, how many miles was the trip? Check your answer is reasonable.

  • A. \(3\) miles
  • B. \(5\) miles
  • C. \(6\) miles
  • D. \(8\) miles

Why it works: Set up: \(3m + 2 = 17\). Subtract \(2\): \(3m = 15\). Divide by \(3\): \(m = 5\) miles. Check: \(3(5) + 2 = 15 + 2 = 17\) ✓. Reasonable: ~5 miles at $3/mile plus $2 fee totals $17.

Answer: \(5\) miles

Common Mistakes

  • Rushing before identifying what the numbers represent.
  • Choosing an operation that does not match the situation.
  • Dropping labels, units, or context from the answer.
  • Skipping the estimate or reasonableness check.

Strategy Tips

  • Underline the question being asked.
  • Use a model before jumping to computation.
  • Write an equation that matches the story or picture.
  • Explain the final answer in a sentence.

Practice Questions

Question 1

Solve: \(\frac{1}{2}x-3=5\) and check that your answer is reasonable.

  • A. \(x=6\)
  • B. \(x=10\)
  • C. \(x=16\)
  • D. \(x=24\)

Question 2

Solve \(3x+7=22\) and explain why your answer is reasonable.

  • A. \(x=5\)
  • B. \(x=9.67\)
  • C. \(x=10\)
  • D. \(x=15\)

Question 3

Solve for \(y\): \(2y-8=-14\) and check your answer makes sense.

  • A. \(y=-3\)
  • B. \(y=0\)
  • C. \(y=3\)
  • D. \(y=11\)

Question 4

Which value of \(x\) satisfies \(\frac{x}{4}+2=5\)? Use mental math to verify.

  • A. \(x=6\)
  • B. \(x=12\)
  • C. \(x=20\)
  • D. \(x=28\)

Question 5

Solve: \(5(x-2)=30\) and verify using inverse operations.

  • A. \(x=6\)
  • B. \(x=8\)
  • C. \(x=10\)
  • D. \(x=12\)

Question 6

A savings account has a balance of \(-$15\) (overdraft). After depositing $50, what is the new balance? Write and solve an equation.

  • A. \($25\)
  • B. \($35\)
  • C. \($50\)
  • D. \($65\)
Full Answer Explanations Click to show all answers and explanations

Question 1

Answer: \(x=16\)

Add \(3\) to both sides: \(\frac{1}{2}x=8\). Multiply by \(2\): \(x=16\). Check: \(\frac{1}{2}(16)-3 = 8-3 = 5\) ✓. Reasonableness: Half of 16 is 8; minus 3 gives 5. ✓

Question 2

Answer: \(x=5\)

Subtract \(7\): \(3x=15\). Divide by \(3\): \(x=5\). Check: \(3(5)+7 = 15+7 = 22\) ✓. Reasonable: three times 5 is 15; add 7 gives 22.

Question 3

Answer: \(y=-3\)

Add \(8\): \(2y=-6\). Divide by \(2\): \(y=-3\). Check: \(2(-3)-8 = -6-8 = -14\) ✓. Reasonable: negative answer expected (result is negative).

Question 4

Answer: \(x=12\)

Subtract \(2\): \(\frac{x}{4}=3\). Multiply by \(4\): \(x=12\). Mental math check: \(\frac{12}{4}+2 = 3+2 = 5\) ✓. Reasonable: one-quarter of 12 is 3; add 2 gives 5.

Question 5

Answer: \(x=8\)

Divide by \(5\): \(x-2=6\). Add \(2\): \(x=8\). Check: \(5(8-2) = 5(6) = 30\) ✓. Reasonable: five groups of 6 equals 30.

Question 6

Answer: \($35\)

Equation: \(-15 + d = b\) where \(d = 50\) (deposit). Or: \(-15 + 50 = 35\). Check: Starting with \(-$15\) debt, adding \($50\) leaves \($35\) surplus. Reasonable.

Timed practice quizzes

Related Quizzes

After studying Solving Multi-Step Problems with Rational Numbers, open these two timed quizzes in a new tab for focused extra practice.

Connection to Standards

This lesson supports Grade 7 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.

Summary

Solving Multi-Step Problems with Rational Numbers becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.

GOLDEN RULE

Understand the model before choosing the operation.