Algebra in Real-World Contexts: Modeling and Solving Simple Problems

A Repeatable Modeling Process

In real-world algebra, the goal is not just to “get x,” but to model a situation accurately and interpret the solution with units. Use this repeatable process every time:

• Understand the situation: What is happening? What is being asked?
• Define a variable: Choose one unknown and state what it represents with units.
• Write an equation: Translate the relationships in the story into math.
• Solve: Use algebra to find the value of the variable.
• Interpret with units: Answer the question in a sentence and include units (dollars, miles, years, feet, etc.).
• Check if it makes sense: Substitute back or reason about whether the result is reasonable.

Quick Checklist Before You Solve

• Did you define the variable clearly (including units)?
• Does each number in your equation match something in the story?
• Does the equation reflect comparison language correctly (especially “less than”)?
• Will your answer be positive? If not, does a negative value make sense in context?

Problem Set 1: Budgeting (Total Cost)

Budgeting problems often involve a fixed fee plus a per-item cost, or a total budget that must cover several costs.

Guided Model Example

Context: A movie theater charges $12 for a ticket and $5 for a snack. You spend $32 total. How many snacks did you buy?

1) Understand: Total cost is ticket + snack cost. Ticket cost happens once; snack cost depends on how many snacks.

2) Define variable: Let s = number of snacks (snacks).

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3) Write equation: 12 + 5s = 32

4) Solve:

12 + 5s = 32  (dollars)
5s = 32 - 12
5s = 20
s = 20 / 5
s = 4

5) Interpret with units: You bought 4 snacks.

6) Check: Ticket $12 + 4 snacks at $5 each is $12 + $20 = $32. Matches.

Practice Problems

For each problem: (a) define a variable with units, (b) write an equation, (c) solve, (d) interpret with units, (e) check.

• A streaming service charges a $9 monthly fee plus $3 per movie rental. Your bill is $24. How many movies did you rent?
• You have $50. You buy a notebook for $8 and spend the rest on pens that cost $2 each. How many pens can you buy?
• A taxi charges a $6 pickup fee plus $2 per mile. The total fare is $22. How many miles did you travel?
• You buy 3 identical T-shirts and pay $45 total. There is also a one-time shipping fee of $6 included in the total. What is the price of one T-shirt?

Common Pitfalls (Budgeting)

• Missing units: Writing s = 4 without stating “snacks” can cause confusion later. Always attach meaning and units.
• Wrong variable choice: If the question asks “How many snacks?” define the variable as snacks, not dollars.
• Equation doesn’t match the story: A one-time fee should not be multiplied by the variable (don’t write 5(12 + s) here).

Problem Set 2: Distance–Rate–Time (Simple Numbers)

Many travel problems use the relationship:

distance = rate × time

Be consistent with units (miles with miles per hour, minutes with miles per minute, etc.).

Guided Model Example

Context: A cyclist rides at 12 miles per hour for t hours and travels 30 miles. How long did the ride take?

1) Understand: Distance depends on rate and time.

2) Define variable: Let t = time (hours).

3) Write equation: 12t = 30

4) Solve:

12t = 30
t = 30 / 12
t = 2.5

5) Interpret with units: The ride took 2.5 hours (2 hours 30 minutes).

6) Check: 12 miles/hour × 2.5 hours = 30 miles. Works.

Practice Problems

• A car travels 180 miles at 60 miles per hour. Let t be the time in hours. Write and solve an equation for t. Interpret your answer with units.
• You walk at 3 miles per hour for 2.5 hours. Let d be the distance in miles. Write and solve an equation for d. Interpret with units.
• A train travels 240 miles in 4 hours. Let r be the rate in miles per hour. Write and solve an equation for r. Interpret with units.
• A runner completes 6 miles in 48 minutes. Let r be the rate in miles per minute. Write and solve an equation for r. Interpret with units.

Common Pitfalls (Distance–Rate–Time)

• Unit mismatch: If time is in minutes but rate is miles per hour, convert one so they match before solving.
• Choosing the wrong variable: If the question asks for time, define the variable as time, not distance.
• Forgetting what the number means: A result like t = 2.5 is hours, not minutes. State the unit and, if helpful, convert.

Problem Set 3: Ages

Age problems often compare ages now, in the past, or in the future. The key is to keep the time reference consistent (all “now,” all “in 5 years,” etc.).

Guided Model Example

Context: Maya is 7 years older than Jordan. Together, they are 41 years old. How old is Jordan?

1) Understand: One person’s age is described in terms of the other’s, and the sum is given.

2) Define variable: Let j = Jordan’s age (years).

3) Write equation: Maya’s age is j + 7. Together: j + (j + 7) = 41

4) Solve:

j + (j + 7) = 41
2j + 7 = 41
2j = 34
j = 17

5) Interpret with units: Jordan is 17 years old.

6) Check: Maya is 24; 17 + 24 = 41. Works.

Practice Problems

• Sam is 4 years younger than Lee. The sum of their ages is 30. Define a variable for Lee’s age (years), write an equation, solve, and interpret.
• In 6 years, Ava will be 2 times as old as she is now. Ava is currently a years old. Write and solve an equation for a. Interpret with units.
• Ben is 3 times as old as his sister. Together they are 28 years old. Define a variable for the sister’s age (years), write an equation, solve, and interpret.
• Three years ago, Noah was 12. Let n be Noah’s current age (years). Write and solve an equation for n. Interpret with units.

Common Pitfalls (Ages)

• Mixing time frames: Don’t add “in 5 years” to one person but keep the other person “now.” If one age is “in 5 years,” make all ages “in 5 years.”
• Wrong variable definition: If you define j as Maya’s age but later treat it like Jordan’s age, the equation will not match the story.
• Interpreting the wrong quantity: Sometimes you solve for one person but the question asks for the other. Use your variable definition to answer correctly.

Problem Set 4: Perimeter Problems

Perimeter is the distance around a shape. For rectangles, P = 2L + 2W. For other polygons, perimeter is the sum of side lengths.

Guided Model Example

Context: A rectangular garden has a perimeter of 50 feet. The length is 5 feet more than the width. Find the width.

1) Understand: Perimeter uses both length and width. One is described using the other.

2) Define variable: Let w = width (feet).

3) Write equation: Length is w + 5. Perimeter: 2(w + 5) + 2w = 50

4) Solve:

2(w + 5) + 2w = 50
2w + 10 + 2w = 50
4w + 10 = 50
4w = 40
w = 10

5) Interpret with units: The width is 10 feet.

6) Check: Length is 15 feet. Perimeter = 2(15) + 2(10) = 30 + 20 = 50 feet. Works.

Practice Problems

• A rectangle has perimeter 64 cm. The length is 4 cm less than twice the width. Define a variable for width (cm), write an equation, solve, and interpret with units.
• A triangle has side lengths 8 in, 11 in, and x in. The perimeter is 30 in. Write and solve an equation for x. Interpret with units.
• A square has perimeter 36 m. Let s be the side length (m). Write and solve an equation for s. Interpret with units.
• A rectangle has width w feet and length w + 9 feet. The perimeter is 78 feet. Write and solve an equation for w. Interpret with units.

Common Pitfalls (Perimeter)

• Forgetting to double both dimensions in a rectangle: Perimeter is 2L + 2W, not L + W.
• Not summing all sides: For polygons, include every side exactly once.
• Units missing: If perimeter is in cm, your side lengths should be in cm too, and your final answer should state cm.

Comparison Language and “Less Than” Traps

Many modeling errors happen when the equation does not match the story’s comparison language. Use these reminders:

Mini-check: If you see “less than,” pause and ask: “Less than what?” The order in the sentence can be different from the order in the expression.

Mixed Review: Translate, Simplify, Solve, Interpret, Check

For each problem: define a variable with units, write an equation that matches the story, solve, interpret with units, and check for reasonableness.

• You have $80. After buying a $14 book, you buy g games that cost $11 each. You spend all your money. Find g.
• A plumber charges a $35 service fee plus $20 per hour. The total bill is $115. How many hours did the plumber work?
• A car travels at 55 miles per hour for t hours and goes 165 miles. Find t and interpret the unit.
• A hiker walks 12 miles in 4 hours. At the same rate, how long will it take to walk 21 miles? (Define the variable as time in hours.)
• Maria is 9 years less than twice Devon’s age. Maria is 27 years old. How old is Devon?
• The sum of two ages is 52. One person is 6 years older than the other. Find both ages.
• A rectangle’s perimeter is 90 inches. The length is 3 inches more than the width. Find the width and length.
• A fence surrounds a triangular yard with sides x, x + 4, and 2x feet. The perimeter is 64 feet. Find x and interpret what each side length is.
• “Seven less than a number is 18.” Write an equation, solve, and check by substitution.
• A gym membership costs $25 to join and $15 per month. Your total cost after some months is $130. How many months did you pay for?