Solved Exercises on Word Problems with Rational Numbers in Grade 7

Master word problems with rational numbers: real-life applications and problem-solving strategies through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Temperature Change Problem
Exercise 1
The temperature was -12.5°F at midnight. It rose by 8.7°F in the morning and then dropped by 5.3°F in the afternoon. What was the final temperature?
Definition:

Rational Numbers: Numbers that can be expressed as fractions \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \neq 0\), including decimals

Problem-solving method:
  1. Read the problem carefully and identify key information
  2. Identify the starting value and all changes
  3. Set up the equation with rational numbers
  4. Solve step by step
  5. Check if the answer makes sense in the context
Initial
\(-12.5°F\)
Change 1
\(8.7°F\)
Change 2
\(-5.3°F\)
Final
\(-9.1°F\)
Step 1: Identify the starting temperature

Initial temperature: -12.5°F

Step 2: Apply the first change (temperature rose)

Temperature rose by 8.7°F: \(-12.5 + 8.7 = -3.8°F\)

Step 3: Apply the second change (temperature dropped)

Temperature dropped by 5.3°F: \(-3.8 + (-5.3) = -9.1°F\)

Step 4: Combine all operations

\(-12.5 + 8.7 + (-5.3) = -9.1°F\)

Final temperature: \(-9.1°F\)
Final answer:

The final temperature was -9.1°F

Applied rules:

• Positive Change: Represents increase in temperature

• Negative Change: Represents decrease in temperature

• Sequential Operations: Process changes in chronological order

2 Financial Transaction Problem
Exercise 2
Sarah started with $45.75 in her account. She spent $18.25 on groceries, received $32.50 from her grandmother, and then spent $27.80 on clothes. What is her final balance?
Definition:

Financial Transactions: Money-related activities where positive numbers represent income/gains and negative numbers represent expenses/losses

Initial
\$45.75
Expense
\(-\$18.25\)
Deposit
\$32.50
Expense
\(-\$27.80\)
Final
\$32.20
Step 1: Identify the starting amount

Initial balance: $45.75

Step 2: Apply the grocery expense (negative)

Spent $18.25: \(45.75 - 18.25 = 27.50\)

Step 3: Apply the gift from grandmother (positive)

Received $32.50: \(27.50 + 32.50 = 60.00\)

Step 4: Apply the clothing expense (negative)

Spent $27.80: \(60.00 - 27.80 = 32.20\)

Step 5: Combine all operations

\(45.75 - 18.25 + 32.50 - 27.80 = 32.20\)

Final balance: \$32.20
Final answer:

Sarah's final balance is $32.20

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 2,450.75 feet above sea level. She climbs up 1,230.5 feet, then descends 895.25 feet, and finally climbs another 675.75 feet. What is her final elevation?
Definition:

Elevation Changes: Vertical distance measurements where positive changes indicate climbing upward and negative changes indicate descending downward

Start
2,450.75 ft
Climb
1,230.5 ft
Descend
-895.25 ft
Climb
675.75 ft
Final
3,461.75 ft
Step 1: Identify the starting elevation

Initial elevation: 2,450.75 feet above sea level

Step 2: Apply the first climb (upward movement)

Climb 1,230.5 feet: \(2,450.75 + 1,230.5 = 3,681.25\) feet

Step 3: Apply the descent (downward movement)

Descend 895.25 feet: \(3,681.25 - 895.25 = 2,786.00\) feet

Step 4: Apply the final climb (upward movement)

Climb 675.75 feet: \(2,786.00 + 675.75 = 3,461.75\) feet

Step 5: Combine all operations

\(2,450.75 + 1,230.5 + (-895.25) + 675.75 = 3,461.75\) feet

Final elevation: 3,461.75 feet
Final answer:

The hiker's final elevation is 3,461.75 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

Word Problems with Rational Numbers: Concepts and Methods
Final = Initial + Σ(Changes)
General Formula for Sequential Changes
Application 1
Temp = Start + Rise - Drop
Temperature Problems
Application 2
Balance = Start + Deposits - Withdrawals
Financial Problems
Application 3
Elevation = Base + Up - Down
Elevation Problems
Key definitions:

Rational Numbers: Numbers that can be expressed as fractions \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \neq 0\), including terminating and repeating decimals

Positive Numbers: Represent gains, increases, or upward movements

Negative Numbers: Represent losses, decreases, or downward movements

Reference Point: Starting value in real-world contexts

Word Problems: Mathematical problems presented in narrative form describing real-world situations

Complete methodology:
  1. Read Carefully: Understand the situation described in the problem
  2. Identify Information: Find the starting value and all changes mentioned
  3. Assign Signs: Determine which values are positive and which are negative
  4. Set Up Equation: Write the mathematical expression representing the situation
  5. Solve: Perform the operations in the correct order
  6. Check: Verify that the answer makes sense in the context
Tip 1: Look for keywords that indicate positive or negative changes ("increase", "decrease", "gain", "loss", "rise", "fall").
Tip 2: Always consider the units of measurement and ensure consistency throughout the problem.
Tip 3: Draw a diagram or number line to visualize the problem if it helps.
Tip 4: Estimate your answer first to check if the calculated result is reasonable.
Common errors: Misinterpreting direction words, making sign errors, not following the order of operations, failing to check if the answer is reasonable.
Exam preparation: Practice identifying keywords, work through multiple examples, focus on setting up the correct equation.
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 Distance and Speed Problem
Exercise 4
A car travels 45.25 miles east, then 28.75 miles west, then 32.5 miles east, and finally 18.25 miles west. What is the car's final position relative to its starting point?
Definition:

Directional Distance: Distance measurements with direction, where positive values indicate one direction and negative values indicate the opposite direction

Start
0 miles
East
45.25 miles
West
-28.75 miles
East
32.5 miles
West
-18.25 miles
Final
30.75 miles
Step 1: Establish the reference point

Starting position = 0 miles (origin)

Step 2: Apply the first movement (east)

45.25 miles east: \(0 + 45.25 = 45.25\) miles

Step 3: Apply the second movement (west)

28.75 miles west: \(45.25 + (-28.75) = 16.5\) miles

Step 4: Apply the third movement (east)

32.5 miles east: \(16.5 + 32.5 = 49.0\) miles

Step 5: Apply the fourth movement (west)

18.25 miles west: \(49.0 + (-18.25) = 30.75\) miles

Step 6: Combine all operations

\(0 + 45.25 + (-28.75) + 32.5 + (-18.25) = 30.75\) miles

Car is 30.75 miles east of the starting point
Final answer:

The car is 30.75 miles east of its starting point

Applied rules:

• Directional Convention: Assign positive/negative to directions

• Reference Point: Origin is the starting position

• Final Position: Distance and direction from origin

5 Fractional Parts Problem
Exercise 5
A recipe calls for 2.5 cups of flour. Maria already added \(\frac{3}{4}\) cup, then added another \(\frac{1}{2}\) cup, and accidentally spilled \(\frac{1}{3}\) cup. How much more flour does she need to add to reach the required amount?
Definition:

Fractional Parts: Parts of a whole represented as fractions, which can be added or subtracted from the total requirement

Required
2.5 cups
Added 1
\(\frac{3}{4}\) cup
Added 2
\(\frac{1}{2}\) cup
Lost
\(-\frac{1}{3}\) cup
Total Added
1.083... cups
Remaining
1.416... cups
Step 1: Convert all values to the same form

Convert fractions to decimals: \(\frac{3}{4} = 0.75\), \(\frac{1}{2} = 0.5\), \(\frac{1}{3} ≈ 0.333\)

Step 2: Calculate the total amount added

Amount added: \(0.75 + 0.5 - 0.333 = 0.917\) cups

Step 3: Calculate how much more is needed

Required: 2.5 cups

Available: 0.917 cups

Needed: \(2.5 - 0.917 = 1.583\) cups

Step 4: Alternative method using fractions

Convert 2.5 to fraction: \(2.5 = \frac{5}{2}\)

Find LCD for \(\frac{3}{4} + \frac{1}{2} - \frac{1}{3}\)

\(\frac{9}{12} + \frac{6}{12} - \frac{4}{12} = \frac{11}{12}\)

Step 5: Calculate remaining flour needed

\(\frac{5}{2} - \frac{11}{12} = \frac{30}{12} - \frac{11}{12} = \frac{19}{12} = 1\frac{7}{12}\) cups

Maria needs to add \(1\frac{7}{12}\) cups (≈1.583 cups) more flour
Final answer:

Maria needs to add \(1\frac{7}{12}\) cups more flour to reach the required amount

Applied rules:

• Unit Conversion: Convert all values to same form (fractions or decimals)

• Spilled Amount: Treated as negative in the calculation

• Remaining Calculation: Required minus available

Word Problems with Rational Numbers: Comprehensive Guide
Net Change = Final Value - Initial Value
Net Change Formula
Key definitions:

Rational Numbers: Numbers expressible as \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\), including fractions, decimals, and integers

Word Problems: Mathematical problems presented in narrative form that describe real-world situations

Key Information: The important numbers and relationships in the problem

Context: The real-world situation that gives meaning to the mathematical operations

Verification: Checking if the mathematical answer makes sense in the context of the problem

Complete methodology:
  1. Analyze the problem: Read carefully and understand what is being asked
  2. Identify knowns and unknowns: List all given information and what needs to be found
  3. Assign variables: Represent unknowns with variables if necessary
  4. Translate to math: Convert the words into mathematical expressions or equations
  5. Solve: Perform the necessary calculations
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