Algebraic Representation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Algebraic Representation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Using algebraic expressions and equations to represent and analyze mathematical relationships and real-world situations.

Translating 'the cost is $5 plus $2 per item' into C=5+2nC = 5 + 2n.

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: Algebraic representation captures a verbal relationship as a reusable expression or equation.

Common stuck point: The procedure for algebraic representation is the easy part; the trap is using the same letter for two different quantities. Asking "Can I name the varying quantity with a letter and write the relationship as an equation?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can I name the varying quantity with a letter and write the relationship as an equation?

Worked Examples

Example 1

easy
Represent this situation algebraically: 'Sarah has 5 more apples than Tom. Together they have 23 apples.'

Answer

t+(t+5)=23t + (t+5) = 23; Tom: 9, Sarah: 14.

First step

1
Let tt = Tom's apples. Then Sarah has t+5t + 5 apples.

Full solution

  1. 2
    Together: t+(t+5)=23t + (t + 5) = 23.
  2. 3
    Simplify: 2t+5=232t + 5 = 23, so 2t=182t = 18 and t=9t = 9.
  3. 4
    Tom has 9 apples; Sarah has 14 apples.
Algebraic representation translates words into symbols. Choose a variable for an unknown quantity, express other quantities in terms of it, and write an equation from the given relationship.

Example 2

medium
Write an algebraic formula for the perimeter of a rectangle with length ℓ\ell and width ww.

Example 3

medium
A phone plan costs $40 plus $0.10 per text. Write total cost CC for tt texts and find CC when t=50t = 50.

Example 4

medium
A class of ss students raises $d\$d each. Write the total raised, then their average per student.

Example 5

hard
Three consecutive integers sum to 84. Represent them and solve.

Practice Problems

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

Example 1

easy
Write an expression for the total cost of nn books at $8 each.

Example 2

medium
A taxi charges $3 base fare plus $2 per mile. Write a formula for the fare FF in terms of miles mm.

Example 3

easy
Write '5 more than a number nn' as an expression.

Example 4

easy
Write '3 times a number nn' algebraically.

Example 5

easy
Write '7 less than xx' as an expression.

Example 6

easy
A pencil costs $2. Write the cost of nn pencils.

Example 7

easy
Write 'the sum of xx and yy' as an expression.

Example 8

easy
Write 'twice a number decreased by 4' for number nn.

Example 9

easy
A taxi charges $3 to start plus $2 per mile. Write the cost for mm miles.

Example 10

easy
Write 'a number nn divided by 5' as an expression.

Example 11

medium
A number nn is increased by 3, then doubled. Write the expression and define the variable.

Example 12

medium
A rectangle's length is 3 more than its width ww. Write its perimeter.

Example 13

medium
Sara is 5 years older than twice Tom's age tt. Write Sara's age, then her age in 4 years.

Example 14

medium
Translate 'the product of a number and 4 is 5 less than the number' into an equation (number nn).

Example 15

medium
A store sells shirts at $15 each with a one-time $5 shipping fee. Write total cost for ss shirts and define ss.

Example 16

medium
Translate 'half of a number, increased by 6, equals 10' into an equation and solve (number nn).

Example 17

challenge
A two-digit number has tens digit tt and units digit uu. Write its value, and the value when the digits are reversed.

Example 18

challenge
Two consecutive even integers sum to 38. Represent them algebraically and find them.

Example 19

challenge
A worker earns $12/hour for the first 40 hours and $18/hour beyond. Write total pay PP for h>40h>40 hours.

Example 20

medium
Translate '8 less than three times a number nn' into an expression.

Example 21

medium
A gym charges $25 to join plus $10 per month. Write total cost for mm months and define mm.

Example 22

medium
Translate 'the sum of a number and twice another equals 12' into an equation (numbers a,ba,b).

Example 23

easy
Write an expression for '6 less than a number nn'.

Example 24

easy
Write an expression for the total length of kk ribbons that are each 4 feet long.

Example 25

easy
An apple costs pp dollars. Write the cost of 12 apples.

Example 26

easy
Write the expression for 'a number nn squared, decreased by 5'.

Example 27

easy
A square has side ss. Write its area and perimeter.

Example 28

medium
A rectangle has length twice its width. If width is ww, write length, perimeter, and area.

Example 29

medium
A number nn is multiplied by 3 then 4 is subtracted. The result equals 17. Write the equation and solve.

Example 30

medium
Anna has $30. She spends $x\$x on a book and $5 on a sandwich. Write the amount left.

Example 31

medium
A bus travels at vv mph for tt hours. Write the distance dd. If v=55v = 55 and t=3t = 3, compute dd.

Example 32

medium
Translate 'twice a number is 8 more than the number' into an equation and solve.

Example 33

medium
A coffee shop sells small cups at $3 and large at $5. If xx small and yy large cups are sold, write total revenue.

Example 34

hard
A printer charges $0.10/page for the first 50 pages and $0.05/page after. Write total cost CC when p>50p > 50 pages are printed.

Example 35

hard
A boat travels at bb mph in still water; the river current is cc mph. Write the boat's speed downstream and upstream.

Example 36

hard
Translate 'the square of a number is 6 more than 5 times the number' and find the positive integer solution.

Example 37

hard
A movie ticket costs $aa for adults and $cc for children. A family of 2 adults and 3 children pays $54. Write the equation; if a=12a = 12, find cc.

Example 38

hard
A water tank holds VV gallons. It is filled at rr gallons/min. Write the time tt needed to fill it.

Example 39

hard
A worker plants pp trees on day 1, then 4 more trees each subsequent day. Write the total planted after dd days.

Example 40

medium
A car rental is $25/day plus $0.20/mile. Write total cost CC for dd days and mm miles.

Example 41

challenge
A sum of $1000 is split between two accounts: one earning 4% simple interest, one earning 6%. If xx is in the 4% account, write the total interest after 1 year, and find xx if total interest is $50.

Example 42

challenge
A three-digit number has hundreds digit aa, tens bb, units cc. Write its value, and the new value if you swap the hundreds and units digits.
← Back to Algebraic Representation Practice Mode

Background Knowledge

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

expressionsequations