Solved Exercises on Applications with Negative Numbers in Grade 7

Master applications with negative numbers: temperature, finance, elevation, physics through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Temperature Changes
Exercise 1
The temperature was 12°C in the morning and dropped by 18°C by night. What was the nighttime temperature?
Definition:

Temperature change: A decrease in temperature is represented by a negative number

Temperature change method:
  1. Identify the starting temperature
  2. Identify the change (drop = negative)
  3. Add the change to the starting temperature
Starting Temperature
\(12°C\)
Temperature Change
\(-18°C\)
Final Temperature
\(12 + (-18) = -6°C\)
Step 1: Identify the starting temperature

Initial temperature = \(12°C\)

Step 2: Identify the temperature change

Temperature dropped by \(18°C\), so change = \(-18°C\)

Step 3: Calculate the final temperature

Final temperature = Starting temperature + Change

\(12°C + (-18°C) = 12°C - 18°C = -6°C\)

Step 4: Interpret the result

The nighttime temperature was \(-6°C\)

Nighttime temperature: \(-6°C\)
Final answer:

The nighttime temperature was \(-6°C\)

Applied rules:

• Temperature drop: Represented as a negative number

• Addition: Starting value + change = final value

• Integer arithmetic: \(12 + (-18) = -6\)

2 Financial Loss
Exercise 2
Sarah had $45 in her bank account. She spent $72. What is her account balance now?
Definition:

Financial loss: Spending more than you have creates a negative balance

Starting Balance
\$45
Amount Spent
\$72
Final Balance
\$45 - \$72 = -\$27
Step 1: Identify the starting balance

Initial balance = \$45

Step 2: Identify the amount spent

Amount spent = \$72

Step 3: Calculate the final balance

Final balance = Starting balance - Amount spent

\$45 - \$72 = -\$27

Step 4: Interpret the result

A negative balance means Sarah owes \$27

Account balance: \(-\$27\)
Final answer:

Sarah's account balance is \(-\$27\) (she owes \$27)

Applied rules:

• Financial loss: Spending more than available creates negative balance

• Subtraction: Starting amount - spending = final amount

• Integer arithmetic: \$45 - \$72 = -\$27

3 Elevation Change
Exercise 3
A hiker starts at 250 meters above sea level and descends 380 meters. What is her new elevation?
Definition:

Elevation change: Descending means going below sea level

Starting Elevation
\(250m\)
Descent
\(380m\)
Final Elevation
\(250m - 380m = -130m\)
Step 1: Identify the starting elevation

Starting elevation = \(250m\) above sea level

Step 2: Identify the descent

Descended = \(380m\) (going down)

Step 3: Calculate the final elevation

Final elevation = Starting elevation - Descent

\(250m - 380m = -130m\)

Step 4: Interpret the result

A negative elevation means \(130m\) below sea level

New elevation: \(-130m\) (130m below sea level)
Final answer:

The hiker is now \(130m\) below sea level

Applied rules:

• Elevation: Above sea level = positive, below sea level = negative

• Subtraction: Starting elevation - descent = final elevation

• Integer arithmetic: \(250 - 380 = -130\)

Rules and methods, laws,...
\( \text{Final} = \text{Initial} + \text{Change} \)
Change Formula
Temperature
\( T_{final} = T_{initial} + \Delta T \)
Drop = negative change, Rise = positive change
Finance
\( B_{final} = B_{initial} + \text{Income} - \text{Expenses} \)
Income = positive, Expenses = negative
Elevation
\( E_{final} = E_{initial} + \text{Change} \)
Ascent = positive, Descent = negative
Key definitions:

Negative number: A number less than zero, indicated by a minus sign

Positive number: A number greater than zero

Zero: Neither positive nor negative, represents equilibrium

Application: Real-world situations where negative numbers represent opposite directions or states

Change: Increase or decrease in a quantity

Direction: Negative numbers indicate opposite direction from positive

Application methods:
  1. Identify the context: Determine what the numbers represent
  2. Assign signs: Determine what positive and negative mean in context
  3. Perform operations: Apply integer arithmetic rules
  4. Interpret results: Make sure answer makes sense in context
Tip 1: Always define what positive and negative represent in the context
Tip 2: Use number lines to visualize changes
Tip 3: Check if your answer makes sense in the real-world context
Tip 4: Remember: opposite operations can be represented by opposite signs
Common errors: Misinterpreting what positive/negative means, forgetting to apply sign rules correctly, not checking if answer makes sense in context.
Exam preparation: Practice various contexts, memorize common sign assignments, use number lines for visualization.
Formulas to know by heart:

• General change: \(\text{Final} = \text{Initial} + \text{Change}\)

• Temperature: \(T_{final} = T_{initial} + \Delta T\)

• Finance: \(B_{final} = B_{initial} + \text{Income} - \text{Expenses}\)

• Elevation: \(E_{final} = E_{initial} + \text{Change}\)

• Profit/Loss: \(\text{Net} = \text{Revenue} - \text{Costs}\)

Solution: Exercises 4 to 5
4 Weight Change
Exercise 4
Maria weighed 65 kg and lost 8 kg during a diet program. Then she gained 3 kg back. What is her final weight?
Definition:

Weight change: Loss = negative change, Gain = positive change

Starting Weight
\(65kg\)
First Change
\(-8kg\)
Second Change
\(+3kg\)
Final Weight
\(65 + (-8) + 3 = 60kg\)
Step 1: Identify the starting weight

Initial weight = \(65kg\)

Step 2: Apply the first change (weight loss)

Lost \(8kg\) = change of \(-8kg\)

Weight after loss = \(65kg + (-8kg) = 57kg\)

Step 3: Apply the second change (weight gain)

Gained \(3kg\) = change of \(+3kg\)

Final weight = \(57kg + 3kg = 60kg\)

Step 4: Alternative calculation

Net change = \(-8kg + 3kg = -5kg\)

Final weight = \(65kg + (-5kg) = 60kg\)

Final weight: \(60kg\)
Final answer:

Maria's final weight is \(60kg\)

Applied rules:

• Weight loss: Represented as a negative number

• Weight gain: Represented as a positive number

• Sequential changes: Apply changes in order

5 Speed Change
Exercise 5
A car traveling at 60 km/h slows down by 25 km/h, then speeds up by 15 km/h. What is its final speed?
Definition:

Speed change: Slowing down = negative change, Speeding up = positive change

Starting Speed
\(60 km/h\)
First Change
\(-25 km/h\)
Second Change
\(+15 km/h\)
Final Speed
\(60 + (-25) + 15 = 50 km/h\)
Step 1: Identify the starting speed

Initial speed = \(60 km/h\)

Step 2: Apply the first change (slowing down)

Slowed down by \(25 km/h\) = change of \(-25 km/h\)

Speed after slowing = \(60 km/h + (-25 km/h) = 35 km/h\)

Step 3: Apply the second change (speeding up)

Sped up by \(15 km/h\) = change of \(+15 km/h\)

Final speed = \(35 km/h + 15 km/h = 50 km/h\)

Step 4: Verify the result

Net change = \(-25 km/h + 15 km/h = -10 km/h\)

Final speed = \(60 km/h + (-10 km/h) = 50 km/h\) ✓

Final speed: \(50 km/h\)
Final answer:

The car's final speed is \(50 km/h\)

Applied rules:

• Slowing down: Represented as a negative change

• Speeding up: Represented as a positive change

• Sequential changes: Apply changes in chronological order

Key Concepts: Laws, Methods, Rules, Definitions
\( \text{Final Value} = \text{Initial Value} + \text{Total Change} \)
Change Accumulation Rule
Key definitions:

Negative number: A number less than zero, representing the opposite of a positive quantity

Positive number: A number greater than zero, representing a standard quantity

Integer: A whole number that can be positive, negative, or zero

Absolute value: The distance of a number from zero, always non-negative

Opposite numbers: Numbers that are the same distance from zero but in opposite directions

Real-world applications: Situations where negative numbers represent opposite directions or states

Change: The difference between initial and final states

Complete application methodology:
  1. Context identification: Determine what the situation represents
  2. Sign assignment: Define what positive and negative mean in the context
  3. Data collection: Identify initial values and changes
  4. Operation planning: Determine which operations to perform
  5. Calculation: Apply integer arithmetic rules
  6. Verification: Check if answer makes sense in context
  7. Interpretation: Explain the result in real-world terms
Tip 1: Create a number line to visualize the problem
Tip 2: Always define what positive and negative represent in the specific context
Tip 3: For multiple changes, calculate net change or apply sequentially
Tip 4: Always verify that your answer makes sense in the real-world context
Common errors: Misinterpreting positive/negative meanings, incorrect sign application, not considering the context of the result.
Exam preparation: Practice various contexts, master sign interpretation, use visual aids like number lines.
Formulas to know by heart:

• General change: \(\text{Final} = \text{Initial} + \text{Change}\)

• Sequential changes: \(\text{Final} = \text{Initial} + C_1 + C_2 + ... + C_n\)

• Net change: \(\text{Net Change} = C_1 + C_2 + ... + C_n\)

• Temperature: \(T_{final} = T_{initial} + \Delta T\)

• Finance: \(B_{final} = B_{initial} + \text{Income} - \text{Expenses}\)

• Elevation: \(E_{final} = E_{initial} + \text{Change}\)

Exercise with Visualization: Negative Number Applications
Exercise 6: Multi-Scenario Applications
Compare different negative number applications:
Temperature: \(10°C \to -5°C\)
Bank balance: \(\$50 \to -\$20\)
Elevation: \(100m \to -30m\)
Speed: \(40 km/h \to 15 km/h\)

Analysis: The chart shows how negative numbers apply in different contexts.

  • Temperature: From \(10°C\) to \(-5°C\) (decrease of \(15°C\))
  • Bank balance: From \(\$50\) to \(-\$20\) (loss of \(\$70\))
  • Elevation: From \(100m\) to \(-30m\) (descent of \(130m\))
  • Speed: From \(40 km/h\) to \(15 km/h\) (decrease of \(25 km/h\))

Questions & Answers

Question: How do I know when to use negative numbers in real-world problems? It's confusing!

Answer: Negative numbers typically represent:

  • Opposite direction: Below zero, left of center, downward
  • Loss or decrease: Money spent, temperature dropped, elevation descended
  • Debt or deficit: Owing money, below sea level
  • Opposite state: Reverse of a positive action

Common contexts:

  • Temperature: Below freezing = negative
  • Finance: Debt, loss, withdrawal = negative
  • Elevation: Below sea level = negative
  • Time: Before reference point = negative

Always define what positive and negative mean in your specific context!

Question: What's the easiest way to remember which financial transactions are positive and which are negative?

Answer: Use this simple memory aid:

  • Money coming IN: Deposits, income, gains = POSITIVE
  • Money going OUT: Expenses, withdrawals, losses = NEGATIVE
  • Money owed: Debt, loans = NEGATIVE (owe others)
  • Money owned: Assets, savings = POSITIVE

Think of your bank account:

  • Getting paid = positive
  • Buying something = negative
  • Earning interest = positive
  • Paying bills = negative

The key is to think about the direction of money flow!

Question: How do I check if my answer makes sense in real-world applications?

Answer: Here are several ways to verify your answer:

  1. Reasonableness check: Does the number make sense in context?
  2. Sign check: Does the positive/negative sign match the expected outcome?
  3. Unit check: Are the units correct?
  4. Back-calculation: Can you work backwards to verify?
  5. Estimation: Is the answer close to what you estimated?

Example verification: For a temperature problem where it dropped from 12°C to -6°C:

  • Reasonable: Temperature can indeed be negative ✓
  • Sign: Drop means negative, final result is negative ✓
  • Back-check: 12 - 18 = -6 ✓
  • Estimation: Close to expected drop ✓

Always interpret your answer in the context of the original problem!