Writing Equations from Context Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Writing Equations from Context.

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 real-world situations and word problems into algebraic equations by identifying the unknown, choosing a variable, and expressing relationships mathematically.

Word problems are stories in disguise. Your job is to find the main character (the unknown—call it xx), figure out what's happening to it (the operations), and write down the punchline (the equation). 'Five more than twice a number is 17' becomes 2x+5=172x + 5 = 17.

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: Writing an equation from context means naming the unknown and making two descriptions of the same quantity equal.

Common stuck point: The procedure for writing equations from context is the easy part; the trap is skipping the variable definition. Asking "Can I name the unknown and write two expressions that must be equal in this situation?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can I name the unknown and write two expressions that must be equal in this situation?

Worked Examples

Example 1

easy
Write an equation: 'Three times a number decreased by 4 is 17.'

Answer

3n−4=17;n=73n - 4 = 17; \quad n = 7

First step

1
Let nn = the unknown number.

Full solution

  1. 2
    'Three times a number' = 3n3n.
  2. 3
    'Decreased by 4' = 3n−43n - 4.
  3. 4
    'Is 17' = =17= 17. Equation: 3n−4=173n - 4 = 17.
  4. 5
    Solve: 3n=213n = 21, n=7n = 7.
Translating word problems: assign a variable, convert each phrase to math, and connect with an equals sign.

Example 2

medium
A plumber charges \$50 for a visit plus \$30 per hour. If the bill was \$170, how many hours did the job take?

Example 3

medium
A gym membership has a $30\$30 start-up fee and costs $25\$25/month. The total spent is $255\$255. Write and solve an equation for the number of months mm.

Example 4

medium
Tank A has 200200 L and drains at 55 L/min. Tank B has 5050 L and fills at 1010 L/min. After how many minutes tt are they equal?

Example 5

hard
A boat travels 2020 km at speed b+4b+4 km/h with the current, then 2020 km back at b−4b-4 km/h against the current. The trip totals 33 hours. Write an equation in bb.

Example 6

hard
A worker earns $15\$15/h for the first 4040 hours and $22.50\$22.50/h for overtime. He earned $735\$735 working hh hours, where h>40h>40. Write an equation in hh.

Example 7

challenge
A ball is thrown upward from a 55-m roof with initial velocity 1010 m/s. Its height in meters after tt seconds is h(t)=5+10t−5t2h(t)=5+10t-5t^2. Write the equation for when the ball hits the ground.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Write an equation: 'Half of a number plus 6 equals 10.'

Example 2

hard
The perimeter of a rectangle is 34 cm. The length is 5 cm more than the width. Find the dimensions.

Example 3

easy
A number increased by 7 equals 12. Write an equation using xx for the number.

Example 4

easy
Twice a number is 18. Write an equation for the number nn.

Example 5

easy
Five less than a number is 9. Write the equation using xx.

Example 6

easy
A pencil costs $3\$3. Write an equation for the cost CC of pp pencils.

Example 7

easy
The sum of a number and 4 is 10. Write an equation using xx.

Example 8

easy
A number divided by 6 equals 5. Write the equation using yy.

Example 9

easy
Maria has xx dollars. She earns $15\$15 more. Now she has $40\$40. Write the equation.

Example 10

easy
Three times a number, plus 2, is 17. Write the equation using xx.

Example 11

medium
The length of a rectangle is 3 more than its width ww. Its perimeter is 26. Write an equation in ww.

Example 12

medium
Tom is twice as old as Sara. In 5 years their ages sum to 40. Let Sara's age be ss; write the equation.

Example 13

medium
A phone plan costs $20\$20 plus $0.10\$0.10 per minute. The bill is $32\$32. Write an equation for minutes mm.

Example 14

medium
A number is 4 less than three times another number yy. The number is 11. Write an equation in yy.

Example 15

medium
Two consecutive integers sum to 47. Let the smaller be nn; write the equation.

Example 16

medium
A jar has nickels and dimes. There are 3 more dimes than nickels, and 12 coins total. Let nickels be nn; write the count equation.

Example 17

medium
A car travels at 60 mph for tt hours and covers 150 miles. Write an equation for tt.

Example 18

medium
Admission is $8\$8 for adults and $5\$5 for children. A group of 6 people pays $39\$39. Let adults be aa; write the cost equation (children =6−a=6-a).

Example 19

medium
A rectangle's length is twice its width ww, and its perimeter is 30. Write an equation in ww.

Example 20

challenge
A rectangle's length is 2 more than twice its width ww, and its area is 60. Write an equation in ww, and state why this is not linear.

Example 21

challenge
A boat goes 12 miles downstream and back upstream. Downstream speed is b+3b+3, upstream is b−3b-3 (mph), where bb is boat speed in still water. The trip takes 2 hours. Write an equation in bb.

Example 22

challenge
Define a variable and write an equation: 'The price of a shirt after a 20% discount is $36\$36.' Explain the choice of variable.

Example 23

easy
A taxi charges $3\$3 to start and $2\$2 per km. Let dd be distance in km and CC be total cost. Write the equation for CC in terms of dd.

Example 24

easy
Let hh be hours worked. A worker earns $15\$15/hour plus a $20\$20 bonus. Write an equation for total pay PP.

Example 25

medium
A rectangle's length is 44 more than its width ww. The perimeter is 4040. Write an equation in ww.

Example 26

medium
Maya is 66 years older than her sister Lina. In 44 years, their ages will sum to 3030. Let Lina's current age be ℓ\ell; write an equation in ℓ\ell.

Example 27

medium
Adult admission is $10\$10 and child admission is $6\$6. A group of 55 people pays $38\$38. Let aa be adults; write a cost equation.

Example 28

medium
A printer prints 2424 pages/minute. A job takes tt minutes and produces 480480 pages. Write an equation for tt.

Example 29

medium
A $1200\$1200 phone is paid off in equal monthly installments of $x\$x over 2424 months with a $60\$60 down payment. Write an equation in xx.

Example 30

medium
A vendor sells lemonade for $3\$3/cup. Costs are $50\$50 flat per day plus $0.50\$0.50/cup. Let cc be cups sold and PP be profit. Write PP in terms of cc.

Example 31

medium
A bus holds 4040 students. There are bb buses and 55 extra students who go in a van. The total is 245245 students. Write an equation in bb.

Example 32

hard
The sum of three consecutive odd integers is 9393. Let the smallest be nn; write the equation.

Example 33

hard
A rectangle's length is 33 more than its width ww. Its area is 108108 cm². Write an equation in ww and state why it is not linear.

Example 34

hard
A chemistry experiment mixes xx liters of a 20%20\% acid solution with (10−x)(10-x) liters of a 50%50\% acid solution to get a 30%30\% acid mixture. Write the equation.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

equationsvariablesexpressions