Integer Word Problems with Step-by-Step Solutions

Grade: 6-7 | Topic: Arithmetic

What You Will Learn

After working through this page you will be able to translate everyday situations -- temperature changes, bank transactions, elevation shifts, and game scores -- into integer expressions, solve them step by step, and check that your answer makes sense in context. These skills connect integer arithmetic to the real world and are tested frequently in grades 6 and 7.

Theory

A four-step strategy for integer word problems

Word problems become manageable when you follow a consistent approach:

Step 1 -- Identify the integers. Translate the words into positive and negative numbers. Use negative for "below," "loss," "debt," "drop," "withdrawal," and similar words. Use positive for "above," "gain," "credit," "rise," "deposit."

Step 2 -- Choose the operation. Decide whether to add, subtract, multiply, or divide based on what the problem asks.

Step 3 -- Write and solve the expression. Apply the integer rules you already know.

Step 4 -- Interpret and check. State your answer in the context of the problem (with units), and ask: "Does this make sense?"

Common real-world contexts for integers

Context	Positive means	Negative means
Temperature	Above zero	Below zero
Money / Banking	Deposit, credit, profit	Withdrawal, debt, loss
Elevation	Above sea level	Below sea level
Sports / Games	Points scored, yards gained	Points lost, yards lost
Time zones	Hours ahead of UTC	Hours behind UTC

Translating words into expressions

Here are key phrases and their integer translations:

• "The temperature dropped 8 degrees from $-3°\text{C}$" $\to$ $-3 + (-8)$
• "She deposited $50 then withdrew $120" $\to$ $50 + (-120)$
• "A submarine at $-200$ m rose 75 m" $\to$ $-200 + 75$
• "He lost 4 points three times in a row" $\to$ 3 \times (-4)$

The hardest part is the translation. Once you have the expression, the arithmetic rules do the rest.

Worked Examples

Example 1: Temperature change (easy)

Problem: At midnight the temperature in Fargo is $-12°\text{C}$. By noon it rises $19°\text{C}$. What is the noon temperature?

Step 1: Starting temperature: $-12$. Change: $+19$ (it rises).

Step 2: Add the change to the starting temperature. $-12 + 19$

Step 3: Different signs -- subtract absolute values: $19 - 12 = 7$. The larger absolute value belongs to $+19$ (positive), so the result is positive.

Step 4: Check -- a 19-degree rise from $-12$ should pass through zero and go positive. $7°\text{C}$ makes sense.

Answer: The noon temperature is $\mathbf{7°\text{C}}$.

Example 2: Bank balance (easy)

Problem: Mia's checking account has a balance of \85$. She writes checks for$$120$and$$45$, then deposits$$60$. What is her final balance?

Step 1: Translate each transaction. $85 + (-120) + (-45) + 60$

Step 2: Group positives and negatives. $\text{Positives: } 85 + 60 = 145$ $\text{Negatives: } (-120) + (-45) = -165$

Step 3: Combine. $145 + (-165) = -(165 - 145) = -20$

Step 4: Check -- she spent more than she had and deposited, so a negative balance (overdrawn) makes sense.

Answer: Mia's final balance is \mathbf{-\20} (overdrawn by \20).

Example 3: Elevation change (medium)

Problem: A hiker starts at an elevation of $350$ m above sea level. She descends $480$ m into a canyon, then climbs $200$ m. What is her final elevation?

Step 1: Translate. $350 + (-480) + 200$

Step 2: Group. $\text{Positives: } 350 + 200 = 550$ $\text{Negatives: } -480$

Step 3: Combine. $550 + (-480) = +(550 - 480) = 70$

Step 4: Check -- she descended more than her starting height but then climbed, ending slightly above sea level. $70$ m makes sense.

Answer: Her final elevation is $\mathbf{70}$ m above sea level.

Example 4: Football yardage (medium)

Problem: A football team records the following plays: gain of 8 yards, loss of 3 yards, loss of 12 yards, gain of 15 yards. What is the team's net yardage?

Step 1: Translate each play. $8 + (-3) + (-12) + 15$

Step 2: Group. $\text{Positives: } 8 + 15 = 23$ $\text{Negatives: } (-3) + (-12) = -15$

Step 3: Combine. $23 + (-15) = +(23 - 15) = 8$

Step 4: Check -- gains exceeded losses, so a positive net yardage is correct.

Answer: The team's net yardage is $\mathbf{8}$ yards gained.

Example 5: Multi-day temperature tracking (challenging)

Problem: Over four days, a city's temperature changed as follows from each day to the next: dropped $6°\text{C}$, dropped $4°\text{C}$, rose $11°\text{C}$, dropped $3°\text{C}$. If the temperature on Day 1 was $2°\text{C}$, what was the temperature at the end of Day 5?

Step 1: Translate the changes. $\text{Day 1 to 2: } -6 \qquad \text{Day 2 to 3: } -4 \qquad \text{Day 3 to 4: } +11 \qquad \text{Day 4 to 5: } -3$

Step 2: Find the total change. $(-6) + (-4) + 11 + (-3)$

Step 3: Group. $\text{Positives: } 11$ $\text{Negatives: } (-6) + (-4) + (-3) = -13$

Step 4: Total change. $11 + (-13) = -(13 - 11) = -2$

Step 5: Apply to the starting temperature. $2 + (-2) = 0$

Step 6: Check -- a net drop of 2 degrees from $2°\text{C}$ should give $0°\text{C}$. Correct.

Answer: The temperature at the end of Day 5 is $\mathbf{0°\text{C}}$.

Common Mistakes

Mistake 1: Using the wrong sign for a "drop" or "loss"

❌ "The temperature dropped 7 degrees" $\to$ $+7$

✅ "Dropped" means a decrease $\to$ $-7$ (or add $-7$)

Why this matters: Misreading the direction word ("rose" vs "dropped," "gained" vs "lost") flips the sign and produces an answer that is completely wrong. Always underline the direction word before translating.

Mistake 2: Forgetting to include the starting value

❌ Changes are $-5$, $+3$, $-2$. Answer: $-5 + 3 + (-2) = -4$.

✅ If the starting value was $10$, the correct answer is $10 + (-5) + 3 + (-2) = 6$.

Why this matters: The changes alone give you the net change, not the final value. You must add (or apply) the changes to the starting point.

Mistake 3: Giving an answer without units or context

❌ Answer: $-20$

✅ Answer: $-20°\text{C}$ (or -\20$, or$-20$ metres, etc.)

Why this matters: An integer by itself does not answer a word problem. Always include the unit and explain what the number means (e.g., "overdrawn by $20" or "20 metres below sea level").

Practice Problems

Try these on your own before checking the answers:

1. The temperature at dawn is $-9°\text{C}$. It rises $15°\text{C}$ by noon, then drops $7°\text{C}$ by evening. What is the evening temperature?
2. A scuba diver is at $-18$ metres. She descends another $12$ metres, then ascends $25$ metres. What is her final depth?
3. Tom's game score starts at $0$. He scores $+25$, loses $-40$, scores $+30$, and loses $-10$. What is his final score?
4. A company's profit over four quarters was: +$5,000, −$3,000, −$7,000, +$8,000. What was the total annual profit?
5. An elevator starts on floor $-2$ (basement level 2). It goes up 7 floors, then down 4 floors, then down 3 floors. What floor is it on now?

Click to see answers

1. $-9 + 15 = 6$; $6 + (-7) = \mathbf{-1°\text{C}}$
2. $-18 + (-12) = -30$; $-30 + 25 = \mathbf{-5}$ metres (still 5 m below the surface)
3. $0 + 25 + (-40) + 30 + (-10)$. Positives: $55$. Negatives: $-50$. Total: $\mathbf{5}$ points
4. $5000 + (-3000) + (-7000) + 8000$. Positives: $13{,}000$. Negatives: $-10{,}000$. Total: +$3,000 profit
5. $-2 + 7 = 5$; $5 + (-4) = 1$; $1 + (-3) = \mathbf{-2}$ (back at basement level 2)

Summary

• Translate direction words into signs: "drop," "loss," "below" $=$ negative; "rise," "gain," "above" $=$ positive.
• Write the expression using the starting value and all changes, then apply integer addition/subtraction rules.
• Always include units (degrees, dollars, metres, points) and interpret the answer in context.
• Check reasonableness: if losses exceed gains, the result should be negative; if gains exceed losses, the result should be positive.
• Use the four-step strategy (identify, choose operation, solve, check) for every problem.

Related Topics

• Integers — Operations, Number Line, and Word Problems
• Adding and Subtracting Integers — Rules and Examples
• Multiplying and Dividing Integers — Sign Rules Explained
• Comparing and Ordering Integers on a Number Line

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