Solved Exercises on Real-World Applications of Numbers in Grade 7

Master real-world applications of numbers: temperature changes, financial calculations, distances, measurements, and more through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Temperature Change
Exercise 1
The temperature was -5°C in the morning. It increased by 12°C during the day, then decreased by 8°C in the evening. What is the final temperature?
Definition:

Temperature change: Using positive and negative integers to represent increases and decreases in temperature

Calculation method:
  1. Start with the initial temperature
  2. Add increases (positive numbers)
  3. Subtract decreases (negative numbers)
  4. Combine all changes step by step
Initial
-5°C
+12°C
+7°C
-8°C
-1°C
Step 1: Start with initial temperature

Initial temperature = -5°C

Step 2: Add the increase of 12°C

-5 + 12 = 7°C

Step 3: Subtract the decrease of 8°C

7 - 8 = -1°C

Final temperature = -1°C
Final answer:

The final temperature is -1°C

Applied rules:

• Adding positive to negative: -5 + 12 = 7 (take the sign of the larger absolute value)

• Subtracting positive from positive: 7 - 8 = -1 (result takes sign of larger absolute value)

• Real-world context: Temperature changes follow integer arithmetic rules

2 Bank Account Balance
Exercise 2
Sarah started with $150 in her bank account. She withdrew $45, then deposited $75, and finally withdrew $30. What is her final balance?
Definition:

Bank account operations: Positive amounts represent deposits, negative amounts represent withdrawals

Start
$150
Withdraw
$105
Deposit
$180
Final
$150
Step 1: Start with initial amount

Initial balance = $150

Step 2: Subtract withdrawal of $45

$150 - $45 = $105

Step 3: Add deposit of $75

$105 + $75 = $180

Step 4: Subtract final withdrawal of $30

$180 - $30 = $150

Final balance = $150
Final answer:

Sarah's final balance is $150

Applied rules:

• Deposits: Represented by positive numbers

• Withdrawals: Represented by negative numbers

• Balance calculation: Add/subtract each transaction sequentially

3 Distance Traveled
Exercise 3
A car travels 85 km north, then 32 km south, then 48 km north again. What is the car's final displacement from its starting point?
Definition:

Displacement: Distance from starting point considering direction (north = positive, south = negative)

North
+85 km
South
+53 km
North
+101 km
Step 1: Assign directions

North = positive direction, South = negative direction

Step 2: Convert movements to signed numbers

85 km north = +85 km, 32 km south = -32 km, 48 km north = +48 km

Step 3: Calculate total displacement

+85 + (-32) + 48 = 85 - 32 + 48 = 101 km

Step 4: Interpret the result

Positive result means 101 km north of starting point

Final displacement = 101 km north
Final answer:

The car is 101 km north of its starting point

Applied rules:

• Directional quantities: Assign positive/negative based on reference direction

• Vector addition: Combine movements considering direction

• Displacement vs distance: Displacement considers direction, distance is total path length

Real-World Applications Overview
Final Value = Initial Value + Changes
General Formula
Temperature
°C = °C + Δ°C
Changes in degrees Celsius
Finance
Balance = Start + Deposits - Withdrawals
Bank account calculations
Distance
Displacement = Σ(Direction × Distance)
Vector-based movement
Key definitions:

Integers: Whole numbers including positive, negative, and zero

Real-world context: Numbers representing actual measurable quantities

Operations: Addition, subtraction, multiplication, division in practical scenarios

Problem-solving approach:
  1. Identify the context: Determine what the numbers represent (temperature, money, distance, etc.)
  2. Assign signs: Decide which direction or change is positive/negative
  3. Perform calculations: Apply arithmetic operations correctly
  4. Interpret results: Connect mathematical answer back to real-world meaning
Tip 1: Always define your reference direction or starting point before calculating.
Tip 2: Check if your answer makes sense in the real-world context.
Tip 3: Use number lines to visualize problems involving direction.
Common applications: Temperature, elevation, financial transactions, sports scores, inventory management.
Key insight: Negative numbers represent decreases, losses, or opposite directions in real-world contexts.
Solution: Exercises 4 to 5
4 Stock Market Change
Exercise 4
A stock price was $42.50 at the beginning of the week. It rose by $3.25 on Monday, fell by $1.80 on Tuesday, rose by $2.40 on Wednesday, and fell by $4.10 on Thursday. What is the final price?
Definition:

Stock price changes: Using decimals to track financial fluctuations over time

Start
$42.50
+ Rise
$45.75
- Fall
$43.95
+ Rise
$46.35
- Fall
$42.25
Step 1: Start with initial price

Initial price = $42.50

Step 2: Add Monday's rise of $3.25

$42.50 + $3.25 = $45.75

Step 3: Subtract Tuesday's fall of $1.80

$45.75 - $1.80 = $43.95

Step 4: Add Wednesday's rise of $2.40

$43.95 + $2.40 = $46.35

Step 5: Subtract Thursday's fall of $4.10

$46.35 - $4.10 = $42.25

Final price = $42.25
Final answer:

The final stock price is $42.25

Applied rules:

• Decimal arithmetic: Align decimal points when adding/subtracting

• Financial precision: Maintain decimal places for accurate monetary calculations

• Sequential operations: Process changes in chronological order

5 Elevation Changes
Exercise 5
A hiker starts at an elevation of 2,450 meters. She descends 850 meters, then ascends 1,200 meters, then descends 450 meters. What is her final elevation?
Definition:

Elevation changes: Using positive numbers for ascending and negative for descending

Start
2,450m
Descend
1,600m
Ascend
2,800m
Final
2,350m
Step 1: Start with initial elevation

Starting elevation = 2,450 meters

Step 2: Subtract descent of 850 meters

2,450 - 850 = 1,600 meters

Step 3: Add ascent of 1,200 meters

1,600 + 1,200 = 2,800 meters

Step 4: Subtract final descent of 450 meters

2,800 - 450 = 2,350 meters

Final elevation = 2,350 meters
Final answer:

The hiker's final elevation is 2,350 meters

Applied rules:

• Altitude calculations: Positive changes indicate climbing, negative indicate descending

• Large number handling: Maintain place value accuracy with commas

• Contextual interpretation: Result represents actual physical height above sea level

Real-World Applications Guide
Net Change = Sum of All Changes
Net Change Formula
Key definitions:

Real-world numbers: Quantities that represent actual measurable phenomena

Contextual meaning: Numbers with specific units and practical significance

Directional values: Positive/negative representing opposite directions or states

Application methodology:
  1. Identify the scenario: Recognize the real-world context (finance, temperature, etc.)
  2. Define variables: Assign mathematical representations to real quantities
  3. Set up operations: Translate words into mathematical expressions
  4. Calculate: Perform arithmetic with attention to signs and units
  5. Verify: Check if the result makes logical sense in the context
Tip 1: Always include units in your calculations and final answer.
Tip 2: Draw a number line to visualize changes in direction or state.
Tip 3: Round appropriately based on the context (dollars to cents, temperature to degree).
Tip 4: Practice with various real-world contexts to build familiarity.
Common applications: Banking, weather, geography, sports, inventory, physics, engineering.
Problem types: Change over time, comparisons, tracking, budgeting, measuring.
Essential formulas to remember:

• Temperature change: Final = Initial + Rise - Fall

• Bank balance: Final = Initial + Deposits - Withdrawals

• Distance/displacement: Final = Start + Σ(Direction × Distance)

• Net change: Total = Σ(All individual changes)

• Percentage change: New Value = Old Value × (1 + %Change)

Real-World Applications Visualization
Exercise 6: Multiple Contexts
Compare different real-world applications of numbers:
Temperature changes: -5°C to +7°C
Financial transactions: $100 to $150
Elevation changes: 200m to 350m

Analysis: The chart shows how numbers apply across different real-world contexts.

  • Temperature: Uses negative values, measured in degrees
  • Finance: Only positive values typically, measured in currency
  • Elevation: Can use negative (below sea level) or positive values

Questions & Answers

Question: In the temperature exercise, why did we get -1°C as the final temperature when we added 12°C and then subtracted 8°C? Shouldn't it still be positive?

Answer: Great observation! Let's break it down step by step:

  • We started at -5°C (very cold)
  • We added 12°C: -5 + 12 = 7°C (this brings us to a positive temperature)
  • Then we subtracted 8°C: 7 - 8 = -1°C (subtracting 8 from 7 gives us -1)

Even though we had a positive temperature of 7°C after the first change, subtracting 8°C took us back below zero. Think of it like this: starting at -5°C, we went up 12 degrees (+12) and then down 8 degrees (-8). The net change was +12 - 8 = +4°C. So -5°C + 4°C = -1°C.

This demonstrates how multiple changes can result in a final value that's different from what you might expect based on just the first change.

Question: In the bank account exercise, Sarah ended with the same amount she started with ($150). Is this just a coincidence?

Answer: Yes, this is a coincidence created for the exercise! Let's verify:

  • Started with: $150
  • Withdrew $45: $150 - $45 = $105
  • Deposited $75: $105 + $75 = $180
  • Withdrew $30: $180 - $30 = $150

In the real world, ending with the same balance would happen when total deposits equal total withdrawals. In this case, total withdrawals were $45 + $30 = $75, and total deposits were $75. Since deposits equal withdrawals, the balance returned to the starting amount.

This demonstrates the principle that in accounting, net changes determine the final outcome. When net change is zero, the final value equals the initial value.

Question: In the hiking example, why do we call it "displacement" instead of just "distance"? Aren't they the same thing?

Answer: Excellent question! There's an important distinction between displacement and distance:

  • Distance: The total length of the path traveled, regardless of direction. Always positive.
  • Displacement: The straight-line distance from start to end point, considering direction. Can be positive or negative.

In our hiking example: - Total distance traveled: 850m + 1,200m + 450m = 2,500m - Displacement: Final position (2,350m) - Starting position (2,450m) = -100m

Actually, let me recalculate: Starting at 2,450m and ending at 2,350m means displacement is 2,350 - 2,450 = -100m (100m lower than start). However, in our example we said final elevation is 2,350m, which is 100m lower than the starting 2,450m.

The negative sign indicates the hiker ended up at a lower elevation than where she started, even though she climbed partway up during the journey.

Question: How do I know whether to use positive or negative numbers in real-world problems? It seems confusing.

Answer: The key is to establish a consistent reference system. Here are standard conventions:

  • Temperature: Above/below zero (positive/negative)
  • Finance: Credit/debit, gain/loss (positive/negative)
  • Direction: North/East = positive, South/West = negative (or vice versa if specified)
  • Elevation: Above sea level = positive, below sea level = negative
  • Time: Future = positive, past = negative

Always identify what represents "more" or "up" in your context and assign that as positive. Then "less" or "down" becomes negative. Write down your convention at the beginning of the problem to stay consistent.

For example, in banking: deposits increase your balance (positive) and withdrawals decrease it (negative).

Question: Are there any shortcuts or memory tricks to solve these real-world problems faster?

Answer: Yes, here are some efficient strategies:

  1. Group operations: Instead of calculating step-by-step, group all additions and subtractions: -5 + 12 - 8 = -5 + (12 - 8) = -5 + 4 = -1
  2. Track net change: Calculate total increases minus total decreases, then add to initial value
  3. Use number lines: Visualize the problem by drawing movements along a line
  4. Estimate first: Get a rough idea of the answer before calculating precisely

For financial problems: Final = Initial + (All deposits) - (All withdrawals)

For temperature: Final = Initial + (All rises) - (All falls)

These formulas let you calculate everything in one step rather than sequentially!