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Writing Linear Expressions from Words

Translate phrases and real-life situations into algebraic expressions.

Practice this skill 20 min Write linear expressions from context Leer en español
Key idea

What you'll learn

Word problems become easy once you can turn the words into math. In this lesson you will learn to write linear expressions from phrases and real situations:

  • translate key words such as sum, less than, product and quotient into symbols,
  • get the order right for subtraction and division,
  • use parentheses when words group a quantity together,
  • build the very common "fixed amount + rate × quantity" model for costs, pay and savings,
  • describe related quantities ("twice as many", "the rest") with one variable,
  • write expressions for perimeters and for situations with a threshold, like overtime pay.

Why it matters on the GED: the test often asks, "Which expression represents …?" You do not have to compute anything — you have to recognize the right model. Writing the expression is also the first step in every equation and inequality word problem, so this skill feeds into much of the algebra section.

Words to know

  • Expression — a math phrase without an equals sign (22m + 35).
  • Variable — a letter for an unknown or changing amount; always say what it stands for ("let m = number of months").
  • Sum / more than / increased by / total — addition ("5 more than x" is x + 5).
  • Difference / less than / decreased by / fewer than — subtraction ("5 less than x" is x - 5).
  • Product / times / of / twice — multiplication ("twice n" is 2n; "half of n" is \tfrac{1}{2}n).
  • Quotient / divided by / per / ratio — division ("the quotient of x and 4" is \tfrac{x}{4}).
  • Rate — an amount per one unit ($22 per month, 0.79 dollars per mile); it multiplies the variable.
  • Fixed amount (constant) — a one-time amount that does not change (a $35 sign-up fee).
  • Consecutive integers — integers in a row (n, n + 1, n + 2); consecutive even or odd integers go up by 2 (n, n + 2, n + 4).

All Math words in the glossary

Translating words into symbols

WordsOperationExampleExpression
sum, plus, more than, increased by, added to+8 more than a numbern + 8
difference, minus, less than, fewer than, decreased by-8 less than a numbern - 8
product, times, of, twice, triple\timesthe product of 8 and a number8n
quotient, divided by, per, ratio\diva number divided by 8\tfrac{n}{8}

Watch the order. Addition and multiplication can be written in either order (n + 8 = 8 + n), but subtraction and division cannot.

  • "8 less than n" means start with n and take 8 away: n - 8. The order is reversed from the words.
  • "8 minus n" or "the difference of 8 and n" keeps the order: 8 - n.
  • "The quotient of x and 4" means x \div 4 = \tfrac{x}{4}, in the order given.
Word-to-expression mapping. Top: “8 less than a number” — the arrows cross, sending “a number” to n at the front and 8 to the end, giving n − 8 (the order reverses). Bottom: “8 minus a number” — the arrows go straight down, giving 8 − n (the order stays).“8 less than a number”88less than−a numbernorder reverses“8 minus a number”88minus−a numbernorder stays

Watch the grouping. Words like the sum of or the difference of tell you to group with parentheses when another operation is applied to them.

  • "3 times a number, plus 5" → 3n + 5
  • "3 times the sum of a number and 5" → 3(n + 5)
  • "the quotient of a number and 4, increased by 9" → \tfrac{n}{4} + 9
  • "the sum of a number and 9, divided by 4" → \tfrac{n + 9}{4}
Two phrases mapped to expressions. “3 times a number, plus 5”: 3 times a number gives 3n, then plus 5, so 3n + 5. “3 times the sum of a number and 5”: the sum of a number and 5 is one group (n + 5), and 3 multiplies the whole group, so 3(n + 5).“3 times a number, plus 5”3 times3na numberplus 5+ 5= 3n + 5“3 times the sum of a number and 5”3 timesthe sum of a number and 53(n + 5)= 3(n + 5)the sum is one group

Quick check: pick a number for the variable, follow the words in order with that number, and see if your expression gives the same result.

Real situations: fixed amount + rate × quantity

Many GED word problems follow one pattern:

\text{total} = \text{fixed amount} + (\text{rate}) \times (\text{quantity})
  • A gym charges a $35 sign-up fee plus $22 per month: 22m + 35.
  • A plumber charges $60 per visit plus $45 per hour: 45h + 60.
  • Sam has $500 and spends $35 per week: 500 - 35w (spending is a negative rate).
Graph and table for the gym cost 22m + 35. The line starts at 35 dollars when m = 0 (the fixed sign-up fee) and rises 22 dollars for each month (the rate): 35, 57, 79, 101, 123 and 145 dollars for 0 to 5 months.Gym cost: 22m + 3504080120160012345months, m$start: $35 fee+22tablemcost035157279310141235145rate 22 = +22per month

How to spot the parts: the rate comes with per, each, every, or an hour/a mile/a month. It multiplies the variable. The fixed amount happens once (a fee, a starting balance, a down payment). It stands alone.

Several quantities, one variable. When the problem links two amounts, write both with the same variable.

  • "twice as many pens as notebooks": notebooks = n, pens = 2n.
  • "50 tickets, x of them adult": adult = x, child = 50 - x (the rest is total minus the part).
  • "three consecutive even integers, the smallest is n": n, n + 2, n + 4.

Then multiply each amount by its own price and add. For 50 tickets at $12 (adult) and $8 (child): 12x + 8(50 - x).

Geometry and thresholds

Geometry. Label each side with an expression, then use the formula.

Rectangle with width w and length 2w + 3 (length is 3 more than twice the width).2w + 3wLength is 3 more than twice the width w

If the width is w and the length is "3 more than twice the width", the length is 2w + 3. The perimeter adds all four sides:

P = 2(\text{length}) + 2(\text{width}) = 2(2w + 3) + 2w = 4w + 6 + 2w = 6w + 6

Thresholds ("after the first …", "over 200", "beyond 40 hours"). Only the part above the threshold gets the extra rate. If a phone plan costs $40 and includes 200 texts, with $0.10 for each text over 200, then for t texts (t > 200), the extra texts number t - 200:

40 + 0.10(t - 200) = 0.10t + 20

Overtime works the same way: the first 40 hours are paid at the normal rate, and only the hours over 40, that is h - 40, are paid at the overtime rate.

Formula

Patterns to know (not on the formula sheet)

\text{Total} = b + mx
  • b = the fixed amount (one-time fee, starting value). If the amount goes down over time, the rate m is negative.
  • m = the rate: how much the total changes for each one unit of x.
  • x = the quantity (hours, months, miles, items).

Other patterns: "the rest" = \text{total} - x; amount over a threshold = x - \text{threshold}; consecutive integers n, n+1, n+2; consecutive even or odd integers n, n+2, n+4.

These patterns are not on the GED formula sheet. The perimeter and area formulas you combine them with are on the sheet: perimeter of a rectangle P = 2l + 2w, area A = lw.

Worked example

Example 1 (easy): Translate a phrase

Write an expression for "7 more than half of a number n".

  1. "Half of a number": \tfrac{1}{2}n (or \tfrac{n}{2}).
  2. "7 more than" that amount: add 7.

The figure sums it up:

Word-to-expression mapping for “7 more than half of a number n”: “half of” maps to ½, “a number n” maps to n, “more than” maps to +, and 7 goes at the end, giving ½n + 7. Check with n = 10: ½(10) + 7 = 12.“7 more than half of a number n”77more than+half of½a number nn= ½n + 7Check n = 10: half of 10 is 5, and 7 more is 12; ½(10) + 7 = 12

Answer: \tfrac{1}{2}n + 7

Check: if n = 10, the words say half of 10 is 5, and 7 more is 12. The expression gives \tfrac{1}{2}(10) + 7 = 12 ✓

Worked example

Example 2 (word problem): Profit from a car wash

A youth club spends $45 on supplies for a car wash and charges $8 per car. Write an expression for the club's profit if it washes c cars.

  1. Money in: $8 for each car, so 8c.
  2. Money out: the one-time $45 for supplies.
  3. Profit = money in − money out.

The figure sums it up:

Mapping for the car wash: “8 dollars per car” becomes 8c (rate times quantity) and the one-time 45 dollars for supplies becomes − 45, so profit is 8c − 45. A bar for c = 10 shows money in of 80 split into 45 for supplies and 35 profit.$8 per car, $45 supplies once$8 per car8c$45 supplies (once)− 45rate × quantityfixed amount outCheck c = 10: money in 8(10) = 80supplies 45profit 358c − 45

Answer: 8c - 45 dollars

Check: with 10 cars, the club takes in $80 and keeps 80 - 45 = 35 dollars. The expression gives 8(10) - 45 = 35 ✓

Worked example

Example 3 (word problem): Two kinds of tickets

A school group buys 30 museum tickets. Student tickets cost $6 and adult tickets cost $10. If s tickets are student tickets, write and simplify an expression for the total cost.

  1. Student tickets: s. Adult tickets are the rest: 30 - s.
  2. Cost of student tickets: 6s. Cost of adult tickets: 10(30 - s).
  3. Total: 6s + 10(30 - s) = 6s + 300 - 10s = 300 - 4s.

The figure sums it up:

Tape diagram of 30 tickets split into s student tickets and 30 − s adult tickets (the rest). Student tickets cost 6 dollars each, giving 6s; adult tickets cost 10 dollars each, giving 10(30 − s). Total: 6s + 10(30 − s) = 300 − 4s.30 tickets in alls student30 − s adult (the rest)× $6× $106s10(30 − s)Total: 6s + 10(30 − s) = 6s + 300 − 10s = 300 − 4s

Answer: 300 - 4s dollars (or -4s + 300)

Check: if all 30 are adults (s = 0), the cost is 30 \times 10 = 300 dollars ✓. Each student ticket saves $4, which matches the -4s.

Worked example

Example 4: Perimeter of a triangle

The sides of a triangle are: the shortest side is x inches, the second side is twice the shortest, and the third side is 5 inches longer than the shortest. Write a simplified expression for the perimeter.

  1. Sides: x, 2x, and x + 5.
  2. Perimeter = sum of the sides: x + 2x + (x + 5).
  3. Combine like terms: 4x + 5.

The figure sums it up:

Triangle with sides labelled x (shortest), 2x and x + 5, drawn to scale for x = 3 so the sides are 3, 6 and 8 inches. Perimeter: x + 2x + (x + 5) = 4x + 5.x + 52xxDrawn for x = 3: sides 6, 3 and 8 inchesP = x + 2x + (x + 5) = 4x + 5

Answer: 4x + 5 inches

Check: if x = 3, the sides are 3, 6 and 8, for a total of 17. The expression gives 4(3) + 5 = 17 ✓

Worked example

Example 5 (GED-hard): A threshold

A parking garage charges $5 for the first 2 hours and $3 for each additional hour. Write a simplified expression for the cost of parking h hours, where h is a whole number greater than 2.

  1. The first 2 hours cost a flat $5.
  2. The additional hours are the hours beyond 2: h - 2.
  3. Cost of the additional hours: 3(h - 2).
  4. Total: 5 + 3(h - 2) = 5 + 3h - 6 = 3h - 1.

The figure sums it up:

Timeline for the parking garage with h = 5 hours: the first 2 hours form one block costing a flat 5, dollars and each of the h − 2 = 3 extra hours is a 3 dollars block. Cost = 5 + 3(h − 2) = 3h − 1; for h = 5 this is 14 dollars.Parking h hours (example: h = 5)first 2 h: $5$3$3$3012345hoursextra hours: h − 2Cost = 5 + 3(h − 2) = 3h − 1h = 5: 5 + 3(3) = 14 and 3(5) − 1 = 14

Answer: 3h - 1 dollars

Check: for 5 hours, the garage charges $5 plus 3 extra hours at $3, which is 5 + 9 = 14 dollars. The expression gives 3(5) - 1 = 14 ✓

Calculator tip

Calculator tip (TI-30XS MultiView)

Writing an expression is a reading skill, so the calculator cannot write it for you. But it is an excellent way to test answer choices.

  1. Pick a simple number for the variable that fits the story (for overtime, choose h = 45 so it is over 40).
  2. Work out the real answer from the words, step by step (for 45 hours at $18, with time and a half over 40: 18 \times 40 = 720 and 27 \times 5 = 135, total 855).
  3. Type each answer choice with that number, for example 2 7 × 4 5 − 3 6 0 enter → 855. The choice that matches is correct.

For decimals such as 0.10(t - 200), use the parentheses keys so the calculator subtracts before it multiplies.

When not to use it: for short translations like "5 less than 3 times a number", reasoning about the words is faster than any keystrokes.

Common mistake

Common mistakes

  • Reversing "less than". Wrong: "5 less than 3n" → 5 - 3n. Right: start with 3n and take away 5: 3n - 5.
  • Missing parentheses. Wrong: "3 times the sum of n and 5" → 3n + 5. Right: 3(n + 5), because the sum is found first.
  • Swapping the rate and the fixed fee. Wrong: $35 sign-up plus $22 a month → 35m + 22. Right: the monthly amount multiplies the months: 22m + 35.
  • Using "the rest" incorrectly. Wrong: with 20 people and a adults, children = a - 20. Right: children = 20 - a.
  • Applying the extra rate to all units. Wrong: texts over 200 at $0.10 → 40 + 0.10t. Right: only t - 200 texts are extra: 40 + 0.10(t - 200).
  • Consecutive even numbers going up by 1. Wrong: n, n + 1, n + 2. Right: n, n + 2, n + 4.
Test strategy

GED test strategy

  • Define the variable first. Write "m = months" on your erasable note board. Then ask what gets multiplied by m.
  • Test with a number. Choose a value, compute the answer from the words, and check each choice. This catches every reversed subtraction and missing parenthesis.
  • Use extreme values. In "50 tickets, x adult", try x = 0 (all child tickets) or x = 50 (all adult) — the answer is easy to see from the story.
  • Eliminate by the fixed amount. A one-time fee is never multiplied by the variable. Cross out any choice where it is.
  • Expect equivalent forms. The right answer may be simplified (6w + 6) or not (2(2w + 3) + 2w). If no choice looks like yours, simplify yours and compare again.
  • Read the last sentence twice. Does it ask for the cost, the profit, or the amount left?

Check yourself

Write linear expressions from context

Which expression represents "5 less than 3 times a number n"?

Answer choices
Hint

Find "3 times a number" first. Then "5 less than" that means you take 5 away from it.

Check yourself

Write linear expressions from context

A gym charges a one-time sign-up fee of $35 plus $22 per month. Which expression gives the total cost, in dollars, of a membership for m months?

Answer choices
Hint

Which amount is paid every month? That amount is multiplied by m.

Summary

  • Translate key words: more than +, less than - (reversed order), product/of \times, quotient/per \div.
  • Use parentheses when a sum or difference is multiplied or divided as a group.
  • Costs and pay usually follow fixed amount + rate × quantity; a decreasing amount has a negative rate.
  • Write related quantities with one variable: "twice as many" = 2n, "the rest" = \text{total} - x.
  • For thresholds, only the amount over the limit, x - \text{limit}, gets the extra rate.
  • Always test your expression with a simple number.
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