Solved Exercises on Real-life Problems with Integers in Grade 7

Master real-life applications of integers: temperature changes, financial transactions, elevation changes, debt calculations, and distance problems through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Temperature Change Problem
Exercise 1
The temperature was -8°C in the morning. It dropped by 5°C in the afternoon and then rose by 12°C in the evening. What was the final temperature?
Definition:

Temperature Change: Positive changes indicate warming, negative changes indicate cooling

Solution method:
  1. Start with the initial temperature
  2. Apply each temperature change sequentially
  3. Use positive numbers for increases and negative numbers for decreases
  4. Perform the addition/subtraction operations
Initial
\(-8°C\)
Change 1
\(-5°C\)
Change 2
\(12°C\)
Final
\(-1°C\)
Step 1: Identify the initial temperature

Starting temperature: -8°C

Step 2: Apply the first change (temperature dropped by 5°C)

"Dropped" means subtract: \(-8 - 5 = -13°C\)

Step 3: Apply the second change (rose by 12°C)

"Rose" means add: \(-13 + 12 = -1°C\)

Step 4: Calculate the final temperature

Combine all changes: \(-8 + (-5) + 12 = -1°C\)

Final temperature: \(-1°C\)
Final answer:

The final temperature was -1°C

Applied rules:

• Positive Changes: Increases in temperature

• Negative Changes: Decreases in temperature

• Sequential Operations: Process changes in order

2 Financial Transaction Problem
Exercise 2
Sarah started with $25 in her bank account. She spent $18 on groceries, received $30 from her grandmother, and then spent $22 on clothes. What is her final balance?
Definition:

Financial Transactions: Positive amounts represent deposits/income, negative amounts represent expenses/spending

Initial
\$25
Expense
\(-\$18\)
Deposit
\$30
Expense
\(-\$22\)
Final
\$15
Step 1: Identify the starting amount

Initial balance: $25

Step 2: Record the grocery expense (negative)

Spent $18: \(25 - 18 = 7\)

Step 3: Record the gift from grandmother (positive)

Received $30: \(7 + 30 = 37\)

Step 4: Record the clothing expense (negative)

Spent $22: \(37 - 22 = 15\)

Step 5: Calculate the final balance

Combine all: \(25 - 18 + 30 - 22 = 15\)

Final balance: \$15
Final answer:

Sarah's final balance is $15

Applied rules:

• Income: Represented by positive numbers

• Expenses: Represented by negative numbers

• Net Change: Sum of all transactions

3 Elevation Change Problem
Exercise 3
A hiker starts at an elevation of 1200 feet above sea level. She climbs up 450 feet, then descends 280 feet, and finally climbs another 150 feet. What is her final elevation?
Definition:

Elevation Changes: Positive changes indicate climbing upward, negative changes indicate descending downward

Start
1200 ft
Climb
450 ft
Descend
-280 ft
Climb
150 ft
Final
1520 ft
Step 1: Identify the starting elevation

Initial elevation: 1200 feet above sea level

Step 2: Apply the first climb (upward movement)

Climb 450 feet: \(1200 + 450 = 1650\) feet

Step 3: Apply the descent (downward movement)

Descend 280 feet: \(1650 - 280 = 1370\) feet

Step 4: Apply the final climb (upward movement)

Climb 150 feet: \(1370 + 150 = 1520\) feet

Step 5: Calculate the final elevation

Combine all: \(1200 + 450 + (-280) + 150 = 1520\) feet

Final elevation: 1520 feet
Final answer:

The hiker's final elevation is 1520 feet above sea level

Applied rules:

• Upward Movement: Represented by positive numbers

• Downward Movement: Represented by negative numbers

• Sea Level Reference: Starting point for elevation measurements

Real-life Applications: Concepts and Methods
Final = Initial + Changes
General Formula for Real-life Problems
Application 1
Temp = Start + Change₁ + Change₂
Temperature Problems
Application 2
Balance = Start + Income - Expenses
Financial Problems
Application 3
Elevation = Base + Up - Down
Elevation Problems
Key definitions:

Integers: Whole numbers including positive, negative, and zero

Positive Numbers: Represent gains, increases, or upward movements

Negative Numbers: Represent losses, decreases, or downward movements

Reference Point: Starting value in real-world contexts

Complete methodology:
  1. Identify the context: Determine what the problem is about (temperature, money, elevation, etc.)
  2. Find the starting value: Locate the initial amount or state
  3. Translate changes: Convert verbal descriptions into positive/negative integers
  4. Apply operations: Add or subtract the changes in sequence
  5. Verify the answer: Check if the result makes sense in context
Tip 1: Keywords like "increase", "gain", "deposit", "rise", "climb" indicate positive numbers.
Tip 2: Keywords like "decrease", "loss", "expense", "drop", "descend" indicate negative numbers.
Tip 3: Always consider the real-world context to verify if your answer makes sense.
Tip 4: Use a number line to visualize the changes in position.
Common errors: Misinterpreting direction words, forgetting to apply negative signs, not considering the reference point.
Exam preparation: Practice translating word problems to mathematical expressions, focus on keyword identification.
Key rules to remember:

• Positive changes increase the value

• Negative changes decrease the value

• Always start with the initial value

• Apply changes in chronological order

• Check if the final answer is reasonable

Solution: Exercises 4 to 5
4 Debt Calculation Problem
Exercise 4
Marcus owed $45 to his friend. He paid back $20, borrowed $35 more, and then paid back $25. What is his current debt?
Definition:

Debt Calculations: Positive amounts represent debt/what is owed, negative amounts represent payments/payments made

Initial Debt
\$45
Payment
\(-\$20\)
New Debt
\$35
Payment
\(-\$25\)
Final Debt
\$40
Step 1: Identify the initial debt

Marcus owes $45 (positive debt)

Step 2: Apply the first payment (reduces debt)

Paid back $20: \(45 - 20 = 25\)

Step 3: Apply the new borrowing (increases debt)

Borrowed $35: \(25 + 35 = 60\)

Step 4: Apply the second payment (reduces debt)

Paid back $25: \(60 - 25 = 35\)

Step 5: Calculate the final debt

Combine all: \(45 - 20 + 35 - 25 = 35\)

Final debt: \$35
Final answer:

Marcus currently owes $35

Applied rules:

• Debt: Represented by positive numbers

• Payments: Represented by negative numbers

• Net Effect: Sum of all transactions determines final debt

5 Distance Problem
Exercise 5
A delivery truck started at the warehouse. It traveled 15 miles north, then 8 miles south, then 12 miles north, and finally 20 miles south. How far is the truck from the warehouse, and in which direction?
Definition:

Directional Distance: North is typically positive, South is negative (or vice versa, depending on the problem)

Start
0 miles
North
15 miles
South
-8 miles
North
12 miles
South
-20 miles
Final
-1 mile
Step 1: Establish the reference point

Warehouse location = 0 miles (origin)

Step 2: Apply the first movement (north)

15 miles north: \(0 + 15 = 15\) miles

Step 3: Apply the second movement (south)

8 miles south: \(15 + (-8) = 7\) miles

Step 4: Apply the third movement (north)

12 miles north: \(7 + 12 = 19\) miles

Step 5: Apply the fourth movement (south)

20 miles south: \(19 + (-20) = -1\) mile

Step 6: Interpret the result

Final position: -1 mile (1 mile south of warehouse)

Truck is 1 mile south of the warehouse
Final answer:

The truck is 1 mile south of the warehouse

Applied rules:

• Directional Convention: Assign positive/negative to directions

• Reference Point: Warehouse is the origin (0)

• Final Position: Distance and direction from origin

Real-life Integer Problems: Comprehensive Guide
Final = Initial + Σ(Changes)
General Formula for Real-life Problems
Key definitions:

Real-life Context: Situations involving measurable quantities that can increase or decrease

Positive Changes: Gains, increases, upward movements, deposits, earnings

Negative Changes: Losses, decreases, downward movements, expenses, debts

Reference Point: Starting value or baseline measurement

Complete methodology:
  1. Analyze the problem: Identify the real-world context (finance, temperature, elevation, etc.)
  2. Identify the initial state: Find the starting value or reference point
  3. Translate actions: Convert each action into a positive or negative integer
  4. Perform calculations: Add or subtract each change in sequence
  5. Interpret results: Express the final answer in the context of the problem
Tip 1: Create a table to track each transaction or change.
Tip 2: Draw a number line to visualize the changes.
Tip 3: Always check if your final answer makes sense in the real-world context.
Tip 4: Pay attention to the specific question being asked (final amount vs. change vs. distance).

Common errors: Misreading direction words, not tracking the sign of each operation, forgetting to consider the context of the final answer.
Exam preparation: Practice identifying keywords, work through multiple examples, focus on interpreting the final result in context.
Key rules to remember:

• Positive = Increase/Gain/Uplift

• Negative = Decrease/Loss/Fall

• Always start with the initial value

• Apply changes in chronological order

• Verify your answer in the problem's context

Exercise with Visualization: Financial Tracking
Exercise 6: Weekly Budget Tracking
Consider a weekly budget starting with $100:
Day 1: +$20 (allowance)
Day 2: -$15 (snacks)
Day 3: -$30 (gift)
Day 4: +$25 (birthday money)
Day 5: -$40 (clothes)

Analysis: The chart shows how positive and negative transactions affect the budget over time.

  • Starting balance: $100
  • After allowance: $120
  • After snacks: $105
  • After gift: $75
  • After birthday money: $100
  • After clothes: $60

Final balance: $60

Questions & Answers

Question: How do I know whether to use positive or negative numbers in real-life problems? I get confused about when to add or subtract.

Answer: Great question! Here are the key guidelines:

  • Positive numbers represent: Gains, increases, additions, deposits, going up, earning money
  • Negative numbers represent: Losses, decreases, subtractions, expenses, going down, spending money

Look for specific keywords in the problem:

  • Positive: increased by, gained, earned, deposited, rose, climbed, added
  • Negative: decreased by, lost, spent, withdrew, dropped, descended, reduced

Think of it this way: positive changes make your situation better (more money, higher temperature, etc.), while negative changes make it worse (less money, lower temperature, etc.).

Question: In the elevation problem, how do I decide which direction is positive and which is negative?

Answer: The standard convention is:

  • Upward movements: Positive (climbing, rising, ascending)
  • Downward movements: Negative (descending, dropping, going down)

However, some problems might specify their own convention. Always read the problem carefully. If it says "above sea level" or "up," those are typically positive. If it says "below sea level" or "down," those are typically negative.

In our example, climbing up was positive and descending was negative. The important thing is to be consistent once you establish your convention.

Question: In the debt problem, why did you say the debt was positive? Isn't debt a negative thing?

Answer: This is a common point of confusion! In the debt problem, we treated the debt amount itself as positive because:

  • We were measuring the amount of debt, which is always positive
  • We treated payments (reducing debt) as negative operations
  • Additional borrowing (increasing debt) as positive operations

Alternatively, you could set up the problem with:

  • Debt as negative (-45 for owing $45)
  • Payments as positive (+20 for paying back $20)

The key is to be consistent with your chosen system throughout the problem. Both approaches will give the same final result when interpreted correctly.

Question: For the distance problem, how did you determine that the truck ended up 1 mile south of the warehouse?

Answer: We established a coordinate system where:

  • North = Positive direction
  • South = Negative direction
  • Warehouse = Origin (0 miles)

Then we calculated the total displacement:

  • Started at 0
  • 15 miles north: 0 + 15 = 15
  • 8 miles south: 15 + (-8) = 7
  • 12 miles north: 7 + 12 = 19
  • 20 miles south: 19 + (-20) = -1

Since the final result is -1, this means 1 unit in the negative direction (south) from the origin (warehouse). Therefore, the truck is 1 mile south of the warehouse.

The absolute value (1) tells us the distance, and the sign (-) tells us the direction.

Question: How can I check if my answer to a real-life integer problem is reasonable?

Answer: Here are several ways to verify your answer:

  • Context check: Does the answer make sense in the real-world scenario? (e.g., can't have negative money in a bank account unless it's debt)
  • Magnitude check: Is the size of the answer reasonable given the inputs?
  • Direction check: Does the sign match what you'd expect? (e.g., if spending more than earning, balance should decrease)
  • Reversibility check: Work backwards from your answer to see if you get the initial conditions

For example, in the temperature problem: we started at -8°C, got colder (-5°C), then warmer (+12°C). Since the warming (+12) was greater than the cooling (-5), we expected the final temperature to be warmer than the start, which it was (-1°C > -8°C).

Always ask yourself: "Does this answer make sense in the real world?"