Instructions
Welcome to the Variable Detective Challenge! In this worksheet, you will practice translating real-world situations into mathematical equations and solving for the unknown variable. Use the following steps for Sections III and IV:
- Read Carefully: Understand the goal of the problem.
- Define the Variable: Decide what letter (like x or n) represents the unknown quantity.
- Write the Equation: Translate the words into a balanced equation.
- Solve: Use inverse operations to find the value of the variable.
- Check: Make sure your answer makes sense in the original word problem.
Section I: Translating Words into Expressions (Warm-up)
Write the algebraic expression that matches the phrase. Use the variable n for the unknown number.
- Seven more than a number:
- The product of 5 and a number:
- A number decreased by 12:
- Three times a number, minus eight:
Section II: Defining Variables and Setting Up Equations
For the following scenarios, identify the unknown quantity, define the variable, and write the appropriate equation. Do not solve the equation yet.
| Scenario | Define the Variable | Write the Equation |
|---|---|---|
| Example: Alex had some money (m). He spent $10 and now has $45. | Let m = starting money | m - 10 = 45 |
| A concert ticket (t) costs $25, and the total spent was $75. | ||
| A number of shirts (s) tripled is 36. | ||
| 15 less than the total weight (w) is 85 pounds. | ||
| Half of the students (h) attended the field trip, totaling 42 students. |
Section III: Solve the Scenario (Full Problem Solving)
Read each problem. Show your work by defining the variable, writing the equation, solving it, and stating the final answer in a complete sentence.
Problem 1: The Library Mystery
Maria checked out 6 fewer books this month than last month. If she checked out 11 books this month, how many books (b) did she check out last month?
- Define Variable: Let b = __
- Equation:
- Solution:
Answer: __
Problem 2: The Pizza Party
A large pizza costs $15. If a group spent $60 total on pizza, how many large pizzas (p) did they purchase?
- Define Variable: Let p = __
- Equation:
- Solution:
Answer: __
Problem 3: The Arcade Challenge
Noah bought a wristband for $5, which gives him unlimited play. He also spent $2 on special tokens for prize machines. If Noah spent a total of $17, how many additional friends (f) did he buy wristbands for?
Hint: The total cost of wristbands must cover Noah and his friends.
- Define Variable: Let f = __
- Equation:
- Solution:
Answer: __
Section IV: Real-World Application (Challenge Questions)
Challenge 4: Saving for a Game Console
Liam is saving up for a new game console that costs $300. He already has $80 saved. If he earns $25 each week (w) from chores, how many weeks will it take him to save the rest of the money?
- Define Variable: Let w = number of weeks
- Equation:
- Solution:
Answer: __
Challenge 5: Comparing Pets
Sarah has 3 times as many fish (f) as David has dogs (d). If the total number of pets they have is 16, how many fish does Sarah have, and how many dogs does David have?
Hint: This requires two parts. Try expressing both variables in terms of one unknown.
- Define Variables/Relationships: Let d = David's dogs Let 3d = Sarah's fish
- Equation (Total Pets):
- Solution for d:
- Find f:
Answer: David has __ dogs, and Sarah has __ fish.
Answer Key
Section I: Translating Words into Expressions (Warm-up)
- $n + 7$ (or $7 + n$)
- $5n$
- $n - 12$
- $3n - 8$
Section II: Defining Variables and Setting Up Equations
| Scenario | Define the Variable | Write the Equation |
|---|---|---|
| A concert ticket (t) costs $25, and the total spent was $75. | Let t = number of tickets | $25t = 75$ |
| A number of shirts (s) tripled is 36. | Let s = the number of shirts | $3s = 36$ |
| 15 less than the total weight (w) is 85 pounds. | Let w = the total weight | $w - 15 = 85$ |
| Half of the students (h) attended the field trip, totaling 42 students. | Let h = total students | $h / 2 = 42$ (or $0.5h = 42$) |
Section III: Solve the Scenario (Full Problem Solving)
Problem 1: The Library Mystery
- Define Variable: Let b = books checked out last month
- Equation: $b - 6 = 11$
- Solution: $b = 17$ Answer: Maria checked out 17 books last month.
Problem 2: The Pizza Party
- Define Variable: Let p = number of large pizzas purchased
- Equation: $15p = 60$
- Solution: $p = 4$ Answer: They purchased 4 large pizzas.
Problem 3: The Arcade Challenge
Note: The wristband costs $5 per person.
- Define Variable: Let f = number of friends
- Equation: $5(f + 1) + 2 = 17$ OR $5f + 5 + 2 = 17$
- Solution: $5f + 7 = 17$ $5f = 10$ $f = 2$ Answer: Noah bought wristbands for 2 additional friends.
Section IV: Real-World Application (Challenge Questions)
Challenge 4: Saving for a Game Console
- Define Variable: Let w = number of weeks
- Equation: $80 + 25w = 300$
- Solution: $25w = 300 - 80$ $25w = 220$ $w = 220 / 25$ $w = 8.8$ Answer: It will take Liam 8.8 weeks (or 9 weeks, since weeks are usually counted as whole units) to save the money.
Challenge 5: Comparing Pets
- Define Variables/Relationships: Let $d$ = David's dogs Let $3d$ = Sarah's fish
- Equation (Total Pets): $d + 3d = 16$
- Solution for d: $4d = 16$ $d = 4$
- Find f: $f = 3(4) = 12$ Answer: David has 4 dogs, and Sarah has 12 fish.