Solved Exercises on Evaluating Expressions Word Problems in Grade 9

Master evaluating expressions in real-world contexts: translating word problems into algebraic expressions and evaluating them through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Shopping Problem
Exercise 1
Sarah buys notebooks at $2.50 each and pens at $1.25 each. If she buys x notebooks and y pens, the total cost is 2.50x + 1.25y dollars. Find the total cost if she buys 4 notebooks and 6 pens.
Definition:

Word problem: Real-world situation expressed in words requiring mathematical solution

Algebraic expression: Mathematical phrase with variables, constants, and operations

Substitution: Replacing variables with given values

Word problem method:
  1. Identify the variables and what they represent
  2. Identify the expression to evaluate
  3. Substitute the given values for variables
  4. Apply order of operations to evaluate
  5. Interpret the result in context
Given
x=4 notebooks, y=6 pens
Expression
2.50x + 1.25y
Substitute
2.50(4) + 1.25(6)
Step 1: Identify variables and their values

x = number of notebooks = 4

y = number of pens = 6

Step 2: Identify the expression

Total cost = 2.50x + 1.25y

Step 3: Substitute values

Total cost = 2.50(4) + 1.25(6)

Step 4: Apply order of operations (multiplication first)

Total cost = 10.00 + 7.50

Step 5: Complete the calculation

Total cost = $17.50

Total cost = $17.50
Final answer:

The total cost for 4 notebooks and 6 pens is $17.50

Applied rules:

• Variable identification: Determine what each variable represents

• Substitution: Replace variables with given values

• Order of operations: Multiplication before addition

2 Distance Problem
Exercise 2
The distance traveled by a car moving at a constant speed is given by the expression d = rt, where r is the rate in mph and t is the time in hours. Find the distance if the car travels at 65 mph for 3.5 hours.
Definition:

Distance formula: d = rt (distance = rate × time)

Rate: Speed at which something moves (mph, km/h)

Time: Duration of travel

Given
r = 65 mph, t = 3.5 hours
Formula
d = rt
Substitute
d = 65 × 3.5
Step 1: Identify the variables and their values

r = rate = 65 mph

t = time = 3.5 hours

Step 2: Identify the formula

d = rt (distance = rate × time)

Step 3: Substitute the values

d = 65 × 3.5

Step 4: Perform the multiplication

d = 227.5 miles

Step 5: Interpret the result

The car travels 227.5 miles

Distance = 227.5 miles
Final answer:

The car travels 227.5 miles

Applied rules:

• Distance formula: d = rt

• Unit consistency: Ensure units match (mph × hours = miles)

• Substitution: Replace variables with values

3 Temperature Conversion
Exercise 3
The formula to convert Celsius to Fahrenheit is F = (9/5)C + 32. What is the Fahrenheit temperature when Celsius is 25°C?
Definition:

Temperature conversion: Mathematical relationship between temperature scales

Fahrenheit: Temperature scale where water freezes at 32°F and boils at 212°F

Celsius: Temperature scale where water freezes at 0°C and boils at 100°C

Given
C = 25°C
Formula
F = (9/5)C + 32
Substitute
F = (9/5)(25) + 32
Step 1: Identify the variable and its value

C = 25°C

Step 2: Identify the conversion formula

F = (9/5)C + 32

Step 3: Substitute the value

F = (9/5)(25) + 32

Step 4: Perform the multiplication

F = (9 × 25)/5 + 32 = 225/5 + 32 = 45 + 32

Step 5: Complete the calculation

F = 77°F

F = 77°F
Final answer:

25°C is equivalent to 77°F

Applied rules:

• Temperature conversion: F = (9/5)C + 32

• Order of operations: Multiplication before addition

• Fraction multiplication: (a/b) × c = (a × c)/b

Word Problem Solving Guide
\(\text{Distance: } d = rt, \text{ Cost: } C = px, \text{ Temperature: } F = \frac{9}{5}C + 32\)
Common Formulas
Distance
d = rt
Distance = Rate × Time
Cost
C = px
Cost = Price × Quantity
Area
A = lw
Area = Length × Width
Key definitions:

Word Problem: A mathematical problem presented in narrative form describing a real-world situation

Variable: A symbol representing an unknown quantity

Expression: A mathematical phrase combining numbers, variables, and operations

Word problem methodology:
  1. Read carefully: Understand the situation described
  2. Identify variables: Determine what quantities are unknown
  3. Locate expression: Find the formula or expression to evaluate
  4. Substitute values: Replace variables with given values
  5. Evaluate: Apply order of operations to calculate
  6. Interpret: State the answer in the context of the problem
Tip 1: Underline or highlight important numbers and units in the problem.
Tip 2: Always include units in your final answer.
Tip 3: Check if your answer makes sense in the real-world context.
Tip 4: Write out the substitution step clearly to avoid errors.
Common errors: Misidentifying variables, calculation mistakes, incorrect order of operations, wrong units.
Exam preparation: Practice common formulas, memorize key relationships, work with real-world scenarios.
Essential rules:

• Variable identification: Understand what each variable represents

• Substitution: Replace variables with given values

• Order of operations: Follow PEMDAS/BODMAS rules

• Unit consistency: Ensure units match in calculations

Solution: Exercises 4 to 5
4 Rectangle Area
Exercise 4
A rectangular garden has length l feet and width w feet. The area is given by A = lw. If the length is 12 feet more than the width, and the width is 8 feet, what is the area of the garden?
Definition:

Rectangle area: Product of length and width (A = lw)

Algebraic relationship: l = w + 12 (length is 12 more than width)

Substitution: Replace variables with their values or expressions

Given
w = 8, l = w + 12
Area formula
A = lw
Substitute
A = (8+12)(8)
Step 1: Identify the given information

Width (w) = 8 feet

Length (l) = width + 12 = w + 12

Step 2: Find the length

l = w + 12 = 8 + 12 = 20 feet

Step 3: Identify the area formula

A = lw (Area = length × width)

Step 4: Substitute the values

A = (20)(8)

Step 5: Calculate the area

A = 160 square feet

Area = 160 sq ft
Final answer:

The area of the garden is 160 square feet

Applied rules:

• Rectangle area: A = lw

• Algebraic substitution: Replace variables with their expressions

• Arithmetic: Perform multiplication accurately

5 Electrical Power
Exercise 5
The power P (in watts) dissipated by an electrical resistor is given by P = I²R, where I is the current in amperes and R is the resistance in ohms. Find the power when the current is 3 amperes and the resistance is 10 ohms.
Definition:

Electrical power: Energy consumed per unit time (P = I²R)

Current (I): Flow of electric charge measured in amperes (A)

Resistance (R): Opposition to current flow measured in ohms (Ω)

Given
I = 3A, R = 10Ω
Power formula
P = I²R
Substitute
P = (3)²(10)
Step 1: Identify the variables and their values

Current (I) = 3 amperes

Resistance (R) = 10 ohms

Step 2: Identify the power formula

P = I²R (Power = Current² × Resistance)

Step 3: Substitute the values

P = (3)²(10)

Step 4: Apply order of operations (exponent first)

P = 9 × 10

Step 5: Complete the calculation

P = 90 watts

Power = 90 W
Final answer:

The power dissipated by the resistor is 90 watts

Applied rules:

• Power formula: P = I²R

• Order of operations: Exponents before multiplication

• Units: Power in watts, current in amperes, resistance in ohms

Detailed Summary: Evaluating Expressions Word Problems
\(\text{Word problem: Identify} \rightarrow \text{Substitute} \rightarrow \text{Evaluate} \rightarrow \text{Interpret}\)
Word Problem Solving Process
Comprehensive definitions:

Word Problem: A mathematical problem presented in narrative form that describes a real-world situation

Algebraic Expression: A mathematical phrase that combines numbers, variables, and operations

Substitution: The process of replacing variables with their given numerical values

Evaluation: Calculating the numerical value of an expression after substitution

Variable: A symbol (usually a letter) that represents an unknown or changing quantity

Complete word problem methodology:
  1. Comprehension: Read the problem carefully to understand the situation
  2. Identification: Identify what is given and what needs to be found
  3. Variable recognition: Determine which quantities are represented by variables
  4. Expression identification: Locate the formula or expression to evaluate
  5. Substitution: Replace variables with their given values
  6. Evaluation: Apply order of operations to calculate the result
  7. Interpretation: State the answer in the context of the problem
Tip 1: Always read the problem twice - first for understanding, then to identify key information.
Tip 2: Draw a diagram when possible to visualize the problem.
Tip 3: Pay attention to units - they must be consistent in calculations.
Tip 4: Check if your answer makes sense in the real-world context of the problem.
Common applications: Finance, physics, engineering, geometry, business, science.
Key skills: Reading comprehension, variable identification, substitution, arithmetic, interpretation.
Essential rules and procedures:

• Problem reading: Understand the context and what is being asked

• Variable identification: Determine what each variable represents

• Expression recognition: Identify the formula or expression to use

• Substitution: Replace variables with their values accurately

• Order of operations: Follow PEMDAS/BODMAS when evaluating

• Unit consistency: Ensure units are compatible in calculations

Visualization: Word Problem Relationships
Real-World Applications
Visualizing common word problem relationships:
Distance, area, cost, temperature conversions
How variables relate in real-world scenarios

Analysis: The chart shows how different variables relate in word problems.

  • Linear relationships: Distance = rate × time
  • Quadratic relationships: Area = length × width
  • Exponential relationships: Growth and decay problems

Questions & Answers

Question: How do I know which variable represents what in a word problem?

Answer: Look for clues in the problem:

  • Explicit definitions: Problems sometimes directly state "let x = number of items"
  • Contextual clues: "rate" usually means speed, "area" relates to dimensions
  • Common formulas: Distance = rate × time, Area = length × width
  • Units: mph suggests rate, hours suggests time, miles suggests distance

When variables aren't explicitly defined, assign letters to unknown quantities based on their meaning (d for distance, t for time, etc.)

Always clearly define your variables before starting calculations.

Question: What should I do if the word problem doesn't give me a formula?

Answer: When no formula is provided, you need to construct one based on the problem context:

  • Draw a diagram: Visualize the situation
  • Identify relationships: How do quantities relate to each other?
  • Recall common formulas: Area, perimeter, distance, cost, etc.
  • Build the expression: Translate the verbal description into mathematical terms

For example, if a problem mentions "the sum of two numbers is 15," you can write x + y = 15.

Practice recognizing common patterns in word problems to build your formula-creation skills.

Question: How do I handle word problems with multiple unknowns?

Answer: For problems with multiple unknowns:

  • Define all variables: Assign a letter to each unknown quantity
  • Look for relationships: Find connections between the unknowns
  • Express in terms of one variable: If possible, express some unknowns in terms of others
  • Use substitution: Replace expressions to reduce the number of variables

For example, if "one number is 5 more than another," you can say x = first number and y = x + 5.

Sometimes you'll need multiple equations to solve problems with multiple unknowns.

Question: How can I check if my answer makes sense?

Answer: Use these verification strategies:

  • Reality check: Does the answer make sense in the real-world context?
  • Unit check: Do the units match what's expected?
  • Magnitude check: Is the answer reasonable given the input values?
  • Substitution check: Plug your answer back into the original expression

For example, if calculating distance traveled, a negative answer doesn't make sense. If calculating time, a negative answer is also illogical.

Always consider the context of the problem when evaluating your answer.

Question: What are common mistakes to avoid in word problems?

Answer: Common mistakes include:

  • Rushing to solve: Not reading the problem carefully enough
  • Wrong variable assignment: Misidentifying what variables represent
  • Unit inconsistency: Mixing different units without conversion
  • Order of operations: Not following PEMDAS/BODMAS rules
  • Calculation errors: Arithmetic mistakes in substitution
  • Ignoring context: Not checking if the answer makes real-world sense

Take your time, write out each step clearly, and always verify your final answer against the problem requirements.

Practice regularly to build confidence and reduce common errors.