Modeling with Equations Examples: 47 Problems with Answers
Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Modeling with Equations.
This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.
Concept Recap
Translating a real-world situation into one or more equations that capture its mathematical relationships and constraints.
Turn a word problem into math: identify what's unknown, write relationships as equations.
Read the full concept explanation →How to Use These Examples
- Read the first worked example with the solution open so the structure is clear.
- Try the practice problems before revealing each solution.
- Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea: Modeling with equations turns a word problem into equations you can actually solve for the unknowns.
Common stuck point: The procedure for modeling with equations is the easy part; the trap is flipping subtraction order. Asking "Am I turning a described situation into an equation I can then solve for the unknown?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint: Ask: Am I turning a described situation into an equation I can then solve for the unknown?
Worked Examples
Example 1
easyA phone plan costs $15 per month plus $0.10 per text. Write an equation for the monthly cost if you send texts.
Answer
First step
1
Fixed cost: $15 per month.
Full solution
- 2 Variable cost: $0.10 per text, so for texts.
- 3 Total: .
Modeling translates a real situation into an equation. Identify the fixed part (constant) and the changing part (variable term) to build the equation.
Example 2
mediumTwo trains leave the same station traveling in opposite directions at 50 mph and 70 mph. After how many hours are they 360 miles apart?
Example 3
mediumA gym charges a $30 sign-up fee plus $25 per month. Write an equation for the total cost after months and find when .
Example 4
mediumAt a bake sale, brownies cost $2 and cookies $1. Selling brownies and cookies brought in $50 with items total. Find and .
Example 5
hardA store discount of plus a coupon of $5 off brings a shirt to $25. Find the original price .
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easyWrite an equation: 'A number doubled and increased by 7 equals 25.'
Example 2
hardThe sum of three consecutive integers is 72. Find them.
Example 3
easyA number plus 7 equals 12. Write and solve an equation.
Example 4
easyTwice a number is 18. Find the number.
Example 5
easy3 apples cost $1.50. Write an equation for the price of one apple.
Example 6
easyA number decreased by 4 is 10. Find it.
Example 7
easyThe perimeter of a square with side is 20. Find .
Example 8
easyHalf of a number is 7. Find the number.
Example 9
easySara has dollars; after earning $5 she has $20. Find .
Example 10
easyA rectangle's length is 3 and area is 15. Write an equation for width .
Example 11
mediumTom is 4 years older than Jen. Together they are 28. Find Jen's age.
Example 12
mediumA phone plan costs $20 plus $0.10 per minute. For what minutes is the bill $35?
Example 13
mediumTwo consecutive integers sum to 45. Find them.
Example 14
mediumA 30% discount on price gives $42. Find .
Example 15
mediumA boat travels 30 miles downstream in 2 hours. If current is 3 mph, find boat speed in still water.
Example 16
mediumThe sum of three consecutive even integers is 48. Find the smallest.
Example 17
mediumA rectangle's length is twice its width and its perimeter is 36. Find the width.
Example 18
challengeA 40-liter mixture is 25% acid. How many liters of pure acid are added to make it 40% acid?
Example 19
challengeTwo trains start 300 miles apart heading toward each other at 40 and 60 mph. When do they meet?
Example 20
challengeA father is 4 times as old as his son now; in 20 years he will be twice as old. Find the son's current age.
Example 21
mediumA shirt costs . Buying 5 shirts and a 43. Find .
Example 22
mediumThree times a number minus 5 equals 16. Find the number.
Example 23
easyA number plus equals . Write and solve an equation.
Example 24
easyTriple a number is . Find the number.
Example 25
easyFive less than a number equals . Find the number.
Example 26
easyFour apples cost $3.20. Find the cost per apple.
Example 27
easyMike has $12. After buying a book for dollars, he has $5. Find .
Example 28
easyA rectangle has area and width . Write and solve for length .
Example 29
mediumSarah is years older than Tom. Their ages sum to . Find Tom's age.
Example 30
mediumThe sum of three consecutive integers is . Find them.
Example 31
mediumA tip on a bill gives $9. Find .
Example 32
mediumA car travels miles at mph for hours. Find .
Example 33
mediumA rectangular field has perimeter m. Length is m more than width. Find dimensions.
Example 34
mediumA taxi charges $2.50 base plus $1.20 per mile. For what mileage is the fare $10.90?
Example 35
mediumA boat travels miles upstream in hours. Current is mph. Find boat speed in still water.
Example 36
mediumThe sum of three consecutive even integers is . Find the largest.
Example 37
hardA -liter solution is acid. How many liters of pure acid must be added to reach acid?
Example 38
hardTwo pipes fill a tank in and hours respectively. How long together?
Example 39
hardA father is his son's age now. In years he will be twice as old. Find the son's age.
Example 40
hardTwo cars leave the same point traveling in the same direction at and mph. How long until they are miles apart?
Example 41
challengeA coin jar has nickels and dimes worth $4.30 with coins total. How many of each?
Example 42
challengeA rectangular garden has perimeter ft and area sq ft. Find the dimensions.
Related Concepts
Background Knowledge
These ideas may be useful before you work through the harder examples.
equationsalgebraic representation