Solved Exercises on Money Word Problems Algebra in Grade 3

Master money calculations: algebraic thinking, variables, and equations through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Finding Unknown Amount
Exercise 1
Emma has $4.25. She gets some more money from her grandmother. Now she has $7.50. How much did her grandmother give her? Let x = money from grandmother.
Definition:

Variable: A symbol (usually a letter) that represents an unknown number

Algebraic method:
  1. Identify what you know and what you need to find
  2. Set up an equation using a variable for the unknown
  3. Solve the equation by performing the same operation on both sides
  4. Check your answer by substituting it back into the original problem
Known Amount
$4.25
Unknown
x
Total
$7.50

Equation: $4.25 + x = $7.50

Step 1: Write the equation

$4.25 + x = $7.50

Step 2: Subtract $4.25 from both sides

$4.25 + x - $4.25 = $7.50 - $4.25

x = $3.25

Step 3: Check the answer

$4.25 + $3.25 = $7.50 ✓

x = $3.25
Final answer:

Emma's grandmother gave her $3.25

Applied rules:

• Balance Rule: Whatever you do to one side of the equation, do to the other

• Isolation: Get the variable alone on one side

• Verification: Substitute the answer back to check

2 Solving for Missing Value
Exercise 2
Marcus started with some money. He spent $3.75 on lunch and now has $2.45 left. How much money did he start with? Let y = starting money.
Definition:

Subtraction Equation: An equation where the variable is decreased by a known amount

Starting Money
y
Amount Spent
$3.75
Remaining
$2.45

Equation: y - $3.75 = $2.45

Step 1: Write the equation

y - $3.75 = $2.45

Step 2: Add $3.75 to both sides

y - $3.75 + $3.75 = $2.45 + $3.75

y = $6.20

Step 3: Check the answer

$6.20 - $3.75 = $2.45 ✓

y = $6.20
Final answer:

Marcus started with $6.20

Applied rules:

• Inverse Operations: Addition undoes subtraction

• Balance Rule: Perform same operation on both sides

• Verification: Substitute to confirm correctness

3 Multi-step Variable Problem
Exercise 3
Sarah earned some money from chores. She spent $1.50 and then earned $3.25 more. If she now has $8.00, how much did she earn from chores? Let z = money from chores.
Definition:

Multi-step Equation: An equation requiring multiple operations to solve

Started With
z
Spent
-$1.50
Earned More
+$3.25
Final Amount
$8.00

Equation: z - $1.50 + $3.25 = $8.00

Step 1: Write the equation

z - $1.50 + $3.25 = $8.00

Step 2: Simplify the constants

z + ($3.25 - $1.50) = $8.00

z + $1.75 = $8.00

Step 3: Subtract $1.75 from both sides

z + $1.75 - $1.75 = $8.00 - $1.75

z = $6.25

Step 4: Check the answer

$6.25 - $1.50 + $3.25 = $4.75 + $3.25 = $8.00 ✓

z = $6.25
Final answer:

Sarah earned $6.25 from chores

Applied rules:

• Combine Like Terms: Add or subtract constants first

• Balance Rule: Same operation on both sides

• Verification: Substitute to confirm

Money Algebra Methods
\(\text{Starting} + \text{Change} = \text{Final}\)
Basic Money Equation
\(\text{Variable} + \text{Known} = \text{Total}\)
Algebraic Structure
Addition
x + a = b
Solve by subtracting a
Subtraction
x - a = b
Solve by adding a
Multi-step
x + a - b = c
Simplify first, then solve
variable
Variable = Unknown amount
equation
Balance both sides
Solution: Exercises 4 to 5
4 Shopping with Variables
Exercise 4
Tom had some money. He bought a book for $4.25 and a pen for $1.75. If he has $3.50 left, how much money did he start with? Let m = starting money.
Definition:

Multiple Expenses: An equation with multiple subtractions

Starting Money
m
Book Cost
-$4.25
Pen Cost
-$1.75
Remaining
$3.50

Equation: m - $4.25 - $1.75 = $3.50

Step 1: Write the equation

m - $4.25 - $1.75 = $3.50

Step 2: Combine the expenses

$4.25 + $1.75 = $6.00

So: m - $6.00 = $3.50

Step 3: Add $6.00 to both sides

m - $6.00 + $6.00 = $3.50 + $6.00

m = $9.50

Step 4: Check the answer

$9.50 - $4.25 - $1.75 = $9.50 - $6.00 = $3.50 ✓

m = $9.50
Final answer:

Tom started with $9.50

Applied rules:

• Combine Like Terms: Add all expenses together

• Balance Rule: Same operation on both sides

• Verification: Substitute to confirm

5 Savings with Variables
Exercise 5
Lisa saved some money each week for 3 weeks. Then she spent $5.50 on a gift. If she has $8.25 left, how much did she save each week? Let w = weekly savings.
Definition:

Repeated Addition: Multiplication represented as repeated addition

Weekly Savings
w
Weeks Saved
3
Total Saved
3w
Amount Spent
-$5.50
Remaining
$8.25

Equation: 3w - $5.50 = $8.25

Step 1: Write the equation

3w - $5.50 = $8.25

Step 2: Add $5.50 to both sides

3w - $5.50 + $5.50 = $8.25 + $5.50

3w = $13.75

Step 3: Divide both sides by 3

(3w) ÷ 3 = $13.75 ÷ 3

w = $4.58

Step 4: Check the answer

3 × $4.58 - $5.50 = $13.74 - $5.50 ≈ $8.24 (close to $8.25 due to rounding)

w = $4.58 per week
Final answer:

Lisa saved approximately $4.58 each week

Applied rules:

• Repeated Addition: 3 weeks of w = 3w

• Division: To isolate variable multiplied by number

• Verification: Substitute to confirm (allowing for rounding)

Money Algebra Strategies
\(\text{Variable} + \text{Known} = \text{Total} \Rightarrow \text{Variable} = \text{Total} - \text{Known}\)
Solving Addition Equations
Key definitions:

Variable: A symbol representing an unknown value (often x, y, or z)

Equation: A mathematical statement showing two expressions are equal

Balance Rule: Perform the same operation on both sides of the equation

Isolation: Getting the variable alone on one side of the equation

Algebraic problem-solving methodology:
  1. Identify the unknown: Decide what the variable represents
  2. Set up the equation: Write a mathematical sentence
  3. Solve systematically: Use inverse operations to isolate the variable
  4. Check your answer: Substitute back into the original problem
Tip 1: Always define what your variable represents.
Tip 2: Addition equations are solved by subtracting the known amount.
Tip 3: Subtraction equations are solved by adding the known amount.
Tip 4: Always verify your answer by substituting it back.
Common errors: Forgetting to perform the same operation on both sides, misidentifying the unknown, incorrect arithmetic.
Practice strategies: Start with simple equations, work up to complex ones, always check answers.
Algebraic formulas for money problems:

• Addition: x + a = b → x = b - a

• Subtraction: x - a = b → x = b + a

• Multiplication: nx = b → x = b ÷ n

• Balance Rule: Whatever you do to one side, do to the other

Questions & Answers

Question: I don't understand why we use letters like x or y in math problems. Why can't we just use regular numbers?

Answer: We use letters (variables) because sometimes we don't know the answer yet! Think of a variable like a mystery box:

  • Variables represent unknown numbers we need to find
  • We can set up equations that show relationships between known and unknown amounts
  • By solving the equation, we discover what's inside the mystery box

For example: "$5 + x = $12" means we know we started with $5 and ended with $12, but we don't know how much was added. The x holds the place for that unknown amount until we solve for it.

Question: My child is struggling with the concept of keeping both sides of the equation balanced. How can I explain this?

Answer: Use the balance scale analogy to explain this concept:

  • Imagine an old-fashioned balance scale with two pans
  • Whatever you put on one side, you must put the same thing on the other side to keep it balanced
  • If you remove weight from one side, you must remove the same weight from the other side

For example: If x + $3 = $8, imagine $8 on one side and x + $3 on the other. To find x, we subtract $3 from both sides: x = $8 - $3 = $5.

Question: How do I teach students to check their algebraic answers in money word problems?

Answer: Teach the substitution method for checking answers:

  • Solve the equation to find the value of the variable
  • Replace the variable in the original equation with the answer
  • Calculate both sides of the equation
  • Compare to see if both sides are equal

Example: If we solved x + $2.50 = $6.75 and got x = $4.25

• Check: $4.25 + $2.50 = $6.75

• Since $6.75 = $6.75, our answer is correct!

Comprehensive Summary: Money Word Problems Algebra
Key Concepts and Definitions:

Variable: A symbol (usually a letter) that represents an unknown number (e.g., x, y, z)

Equation: A mathematical statement showing two expressions are equal

Balance Rule: Whatever operation you perform on one side of the equation, you must perform on the other side

Isolation: The process of getting the variable alone on one side of the equation

Inverse Operations: Operations that undo each other (addition/subtraction, multiplication/division)

Core Rules and Principles:

• Balance Rule: Maintain equality by performing identical operations on both sides

• Inverse Operations: Use opposite operations to isolate variables

• Verification: Always substitute your answer back to check accuracy

• Define Variables: Clearly state what each variable represents

Step-by-Step Algebraic Method:
  1. Identify Unknown: Decide what the variable represents
  2. Set Up Equation: Write the mathematical relationship
  3. Solve Systematically: Use inverse operations to isolate the variable
  4. Check Answer: Substitute back into original equation
Tip 1: Always define what your variable represents in the context of the problem.
Tip 2: Addition equations are solved by subtracting the known amount from both sides.
Tip 3: Subtraction equations are solved by adding the known amount to both sides.
Tip 4: Always verify your answer by substituting it back into the original problem.
Common Pitfalls: Forgetting to perform operations on both sides, misidentifying the unknown, incorrect arithmetic, skipping verification.
Memorization Aids: Remember "what you do to one side, do to the other" and "opposite operations cancel."
Key Takeaways for Students:

• Variables represent unknown amounts in mathematical equations

• Keep equations balanced by doing the same thing to both sides

• Use opposite operations to isolate variables (addition undoes subtraction)

• Always check your answer by substituting it back into the original problem

• Define what each variable represents in the context of the problem