Algebraic Representation Examples: 47 Problems with Answers
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 .
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
easyRepresent this situation algebraically: 'Sarah has 5 more apples than Tom. Together they have 23 apples.'
Answer
; Tom: 9, Sarah: 14.
First step
1
Let = Tom's apples. Then Sarah has apples.
Full solution
- 2 Together: .
- 3 Simplify: , so and .
- 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
mediumWrite an algebraic formula for the perimeter of a rectangle with length and width .
Example 3
mediumA phone plan costs $40 plus $0.10 per text. Write total cost for texts and find when .
Example 4
mediumA class of students raises each. Write the total raised, then their average per student.
Example 5
hardThree 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
easyWrite an expression for the total cost of books at $8 each.
Example 2
mediumA taxi charges $3 base fare plus $2 per mile. Write a formula for the fare in terms of miles .
Example 3
easyWrite '5 more than a number ' as an expression.
Example 4
easyWrite '3 times a number ' algebraically.
Example 5
easyWrite '7 less than ' as an expression.
Example 6
easyA pencil costs $2. Write the cost of pencils.
Example 7
easyWrite 'the sum of and ' as an expression.
Example 8
easyWrite 'twice a number decreased by 4' for number .
Example 9
easyA taxi charges $3 to start plus $2 per mile. Write the cost for miles.
Example 10
easyWrite 'a number divided by 5' as an expression.
Example 11
mediumA number is increased by 3, then doubled. Write the expression and define the variable.
Example 12
mediumA rectangle's length is 3 more than its width . Write its perimeter.
Example 13
mediumSara is 5 years older than twice Tom's age . Write Sara's age, then her age in 4 years.
Example 14
mediumTranslate 'the product of a number and 4 is 5 less than the number' into an equation (number ).
Example 15
mediumA store sells shirts at $15 each with a one-time $5 shipping fee. Write total cost for shirts and define .
Example 16
mediumTranslate 'half of a number, increased by 6, equals 10' into an equation and solve (number ).
Example 17
challengeA two-digit number has tens digit and units digit . Write its value, and the value when the digits are reversed.
Example 18
challengeTwo consecutive even integers sum to 38. Represent them algebraically and find them.
Example 19
challengeA worker earns $12/hour for the first 40 hours and $18/hour beyond. Write total pay for hours.
Example 20
mediumTranslate '8 less than three times a number ' into an expression.
Example 21
mediumA gym charges $25 to join plus $10 per month. Write total cost for months and define .
Example 22
mediumTranslate 'the sum of a number and twice another equals 12' into an equation (numbers ).
Example 23
easyWrite an expression for '6 less than a number '.
Example 24
easyWrite an expression for the total length of ribbons that are each 4 feet long.
Example 25
easyAn apple costs dollars. Write the cost of 12 apples.
Example 26
easyWrite the expression for 'a number squared, decreased by 5'.
Example 27
easyA square has side . Write its area and perimeter.
Example 28
mediumA rectangle has length twice its width. If width is , write length, perimeter, and area.
Example 29
mediumA number is multiplied by 3 then 4 is subtracted. The result equals 17. Write the equation and solve.
Example 30
mediumAnna has $30. She spends on a book and $5 on a sandwich. Write the amount left.
Example 31
mediumA bus travels at mph for hours. Write the distance . If and , compute .
Example 32
mediumTranslate 'twice a number is 8 more than the number' into an equation and solve.
Example 33
mediumA coffee shop sells small cups at $3 and large at $5. If small and large cups are sold, write total revenue.
Example 34
hardA printer charges $0.10/page for the first 50 pages and $0.05/page after. Write total cost when pages are printed.
Example 35
hardA boat travels at mph in still water; the river current is mph. Write the boat's speed downstream and upstream.
Example 36
hardTranslate 'the square of a number is 6 more than 5 times the number' and find the positive integer solution.
Example 37
hardA movie ticket costs $ for adults and $ for children. A family of 2 adults and 3 children pays $54. Write the equation; if , find .
Example 38
hardA water tank holds gallons. It is filled at gallons/min. Write the time needed to fill it.
Example 39
hardA worker plants trees on day 1, then 4 more trees each subsequent day. Write the total planted after days.
Example 40
mediumA car rental is $25/day plus $0.20/mile. Write total cost for days and miles.
Example 41
challengeA sum of $1000 is split between two accounts: one earning 4% simple interest, one earning 6%. If is in the 4% account, write the total interest after 1 year, and find if total interest is $50.
Example 42
challengeA three-digit number has hundreds digit , tens , units . Write its value, and the new value if you swap the hundreds and units digits.
Related Concepts
Background Knowledge
These ideas may be useful before you work through the harder examples.
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