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Solving Real-World Problems with Linear Equations

Turn everyday situations about money, distance, and measurements into a linear equation, solve it, and answer the question that was asked.

Practice this skill 25 min Real-world linear equations Leer en español
Key idea

What you'll learn

Most of the equations on the GED do not arrive already written for you. They come inside a short story: a repair bill, a phone plan, a road trip, a garden fence, a sale price. In this lesson you will learn a four-step plan for turning a story into an equation and solving it:

  1. Name the unknown with a letter.
  2. Write an equation that says the same thing as the story.
  3. Solve the equation.
  4. Answer the question that was actually asked — with units — and check that it makes sense.

You will practice the problem types the GED uses again and again: a fixed fee plus a rate, comparing two plans, consecutive integers, perimeter, distance–rate–time, percent discounts and taxes, and "two kinds of items" problems.

Why it matters on the GED: Word problems are a large part of the Mathematical Reasoning test, and the test is designed to see whether you can use algebra, not just push symbols around. Real life: comparing cell phone plans, figuring out how many hours of overtime you need, or working out what an item cost before tax are all linear-equation problems.

Words to know

  • unknown — the quantity the question asks for; you give it a letter (for "how many hours?", let h = hours).
  • fixed cost (flat fee) — an amount you pay once no matter what (a $65 service call).
  • rate — an amount per one unit ($45 per hour, 55 miles per hour, 2.5 gallons per minute).
  • total — the result of adding everything up (fixed cost + rate × number of units).
  • consecutive integers — whole numbers in a row, each 1 more than the last (n, n + 1, n + 2, e.g. 28, 29, 30).
  • consecutive even (or odd) integers — each is 2 more than the last (n, n + 2, n + 4, e.g. 44, 46, 48).
  • perimeter — the distance around a figure (a 19-ft by 8-ft rectangle has perimeter 2(19) + 2(8) = 54 ft).
  • discount — an amount taken off a price (20% off means you pay 80%).
  • break-even point — the amount at which two options cost the same.
  • "is", "equals", "results in", "the total is" — words that become the = sign.

All Math words in the glossary

The four-step plan and translating words

Step 1 — Name the unknown. Write a sentence like "Let h = the number of hours." Choose the quantity the question asks for whenever you can. If there are two related unknowns, name the smaller or simpler one and write the other in terms of it ("Let w = width; then length = 2w + 3").

Step 2 — Write the equation. Translate the story piece by piece. These phrases appear constantly:

WordsMath
more than, increased by, plus, total of+
less than, decreased by, fewer, minus-
times, of, per (with a quantity), twice\times
divided by, split equally, per (as an average)\div
is, equals, will be, results in, costs=
"7 less than x"x - 7 (order flips!)
"twice the sum of x and 3"2(x + 3)

Step 3 — Solve using the method you already know: distribute, clear fractions/decimals, combine like terms, collect x on one side, divide.

Step 4 — Answer and check. Re-read the question. Does it ask for the width or the length? The smallest integer or the largest? Hours of overtime or total hours? Then check your answer in the story, not just in your equation — if your equation was set up wrong, checking it in the equation will not catch the mistake.

Is the answer reasonable? A number of people or tickets must be a whole number. A time or a length cannot be negative. A sale price must be less than the original price. If your answer breaks the story, go back and look for a set-up error.

The problem types the GED loves

1. Fixed fee + rate. Total = fixed + rate \times quantity. A plumber charges $65 plus $45 per hour, and the bill is $245: 65 + 45h = 245, so 45h = 180 and h = 4 hours.

Bar model for the plumber bill. The total bar, 245, dollars is made of a 65 dollars fee block followed by four 45 dollars blocks, one for each hour. Taking away the 65 dollars fee leaves 180, dollars and 180 ÷ 45 = 4 hours.total bill $245$65$45$45$45$45fee (once)4 hours × $45 = $180 → 180 ÷ 45 = 4 hours65 + 45h = 245 → 45h = 180 → h = 4

2. Comparing two plans (break-even). Write an expression for each plan and set them equal. Plan A: 20 + 0.08m; Plan B: 35 + 0.05m. They cost the same when 20 + 0.08m = 35 + 0.05m, so 0.03m = 15 and m = 500 minutes.

Graph comparing two phone plans. Horizontal axis: minutes from 0 to 1,000. Vertical axis: cost in dollars from 0 to 100. Plan A, cost = 20 + 0.08m, starts at 20 dollars and rises faster. Plan B, cost = 35 + 0.05m, starts at 35 dollars and rises more slowly. The lines cross at the break-even point (500, 60): at 500 minutes both plans cost 60 dollars.02004006008001000$0$20$40$60$80$100Minutes (m)Cost$20$35break-even (500, 60)Plan A: 20 + 0.08mPlan B: 35 + 0.05m

3. Consecutive integers. Three consecutive integers: n, n + 1, n + 2. Three consecutive even (or odd) integers: n, n + 2, n + 4. Add them and set equal to the total.

4. Perimeter and geometry. Label every side in terms of one letter, then use the perimeter (or angle-sum) fact. A rectangle with width w and length 2w + 3 and perimeter 54 ft: 2w + 2(2w + 3) = 54, so 6w + 6 = 54, w = 8 ft, and length = 19 ft.

Rectangle drawn to scale with width w = 8 ft on the short sides and length 2w + 3 = 19 ft on the long sides. Perimeter: 2w + 2(2w + 3) = 54 ft.length = 2w + 3 = 19 ft2w + 3 = 19 ftw = 8 ftwP = 2w + 2(2w + 3) = 54 ft6w + 6 = 54 → w = 8, length = 2(8) + 3 = 19

5. Distance = rate × time. Draw a quick sketch.

  • Moving apart (opposite directions): add the speeds. Cars at 55 mph and 65 mph separate at 55 + 65 = 120 mph, so they are 300 miles apart after 300 \div 120 = 2.5 hours.
  • Catching up (same direction, later start): set the two distances equal.

6. Percent of an unknown original. "After a 20% discount the jacket costs $68." You paid 80% of the original price p: 0.80p = 68, so p = 85. Do not just add 20% of $68 back on — 20% of the new price is a different amount.

7. Two kinds of items, one total. "150 tickets, adults $12 and children $7, total $1,465." Let a = adult tickets; then children = 150 - a. Money: 12a + 7(150 - a) = 1465.

Formula

Formulas you will use

These three formulas are on the GED formula sheet:

d = rt \qquad\qquad \text{Total cost} = (\text{number of units}) \times (\text{price per unit}) \qquad\qquad I = prt
  • d = distance, r = rate (speed), t = time. Units must match: miles per hour needs time in hours.
  • In simple interest I = prt: I = interest, p = principal (starting amount), r = yearly rate as a decimal, t = time in years.

This pattern is not on the sheet, but you should know it by heart:

\text{Total} = \text{fixed amount} + (\text{rate}) \times (\text{number of units})

Example: $30 to join + $25 per month for m months: \text{Total} = 30 + 25m.

Percent change of a price (not on the sheet): after a discount of r\% you pay (100 - r)\%; after a tax of r\% you pay (100 + r)\%. As decimals: 20% off → \times 0.80; 6% tax → \times 1.06.

Worked example

Example 1 (easy): the repair bill

An electrician charges a $75 visit fee plus $50 per hour. A bill came to $325. How many hours did the electrician work?

  1. Let h = hours worked. Total = 75 + 50h.
  2. Equation: 75 + 50h = 325.
  3. Subtract 75 from both sides: 50h = 250.
  4. Divide by 50: h = 5.
Bar model for the electrician bill. The total bar, 325, dollars is made of a 75 dollars visit-fee block followed by five 50 dollars blocks, one for each hour. Taking away the 75 dollars fee leaves 250, dollars and 250 ÷ 50 = 5 hours.total bill $325$75$50$50$50$50$50fee (once)5 hours × $50 = $250 → 250 ÷ 50 = 5 hours75 + 50h = 325 → 50h = 250 → h = 5

Answer: 5 hours

Check: 75 + 50(5) = 75 + 250 = \$325 ✓

Worked example

Example 2: consecutive even integers

The sum of three consecutive even integers is 138. What are the integers?

  1. Let n = the smallest. The next two even integers are n + 2 and n + 4.
  2. Equation: n + (n + 2) + (n + 4) = 138.
  3. Combine like terms: 3n + 6 = 138.
  4. Subtract 6: 3n = 132. Divide by 3: n = 44.
  5. The integers are 44, 46, and 48.
Bar model for three consecutive even integers with sum 138. Three rows: the first bar is n; the second is n plus a block of 2; the third is n plus two blocks of 2. Together they are 3n + 6 = 138, so 3n = 132 and n = 44, giving 44, 46 and 48.nn = 4444n + 2n = 44246n + 4n = 4422483 blue bars = 3n3 red blocks = 63n + 6 = 138 → 3n = 132 → n = 44

Answer: 44, 46, 48

Check: 44 + 46 + 48 = 138 ✓, and all three are even ✓

Worked example

Example 3: a triangle's perimeter

The sides of a triangle are x, x + 4, and 2x - 1 inches. The perimeter is 39 inches. How long is the longest side?

  1. Perimeter = sum of the sides: x + (x + 4) + (2x - 1) = 39.
  2. Combine like terms: 4x + 3 = 39.
  3. Subtract 3: 4x = 36. Divide by 4: x = 9.
  4. The sides are 9, 9 + 4 = 13, and 2(9) - 1 = 17. The longest is 17 inches.
Triangle drawn to scale with sides x = 9 inches, x + 4 = 13 inches, and 2x − 1 = 17 inches. The perimeter is 9 + 13 + 17 = 39 inches. The longest side, 17 inches, is the base.2x − 1 = 17 in (longest)x = 9 inx + 4 = 13 inx + (x + 4) + (2x − 1) = 39 → 4x + 3 = 39 → x = 9

Answer: 17 inches (not 9 — the question asked for the longest side)

Check: 9 + 13 + 17 = 39 ✓

Worked example

Example 4 (real world): catching up

Rosa leaves home driving 50 mph. Two hours later her brother leaves from the same place on the same road driving 70 mph. How long after he leaves does he catch up to her?

  1. Let t = hours the brother drives. Rosa has been driving 2 hours longer: t + 2 hours.
  2. When he catches up, they have gone the same distance. Using d = rt: 50(t + 2) = 70t.
  3. Distribute: 50t + 100 = 70t.
  4. Subtract 50t: 100 = 20t. Divide by 20: t = 5.
Distance diagram for the catch-up problem, drawn on a scale from 0 to 350 miles. Rosa's bar: a 100-mile head start (2 hours at 50 mph) followed by five 50-mile blocks, ending at 350 miles. Her brother's bar: five 70-mile blocks, also ending at 350 miles. They meet at 350 miles, 5 hours after he leaves.050100150200250300350miles from homeRosahead start 10050505050502 h before50(t + 2) with t = 5 hBrother707070707070t with t = 5 hmeet350 mi

Answer: 5 hours after he leaves

Check: Rosa drives 50 \times 7 = 350 miles; her brother drives 70 \times 5 = 350 miles ✓

Worked example

Example 5 (GED-hard, work backwards): discount, then tax

A TV was on sale for 25% off. Then 6% sales tax was added to the sale price. The final cost was $381.60. What was the original price?

  1. Let p = original price. After 25% off you pay 75%: sale price = 0.75p.
  2. Tax adds 6%, so you pay 106% of the sale price: 1.06(0.75p) = 381.60.
  3. Multiply the decimals: 0.795p = 381.60.
  4. Divide: p = 381.60 \div 0.795 = 480.
Bar model for the TV price. Top bar: the original price p = 480, dollars split into four quarters of 120 dollars; one quarter (25%, 120 dollars) is crossed off as the discount, leaving the sale price 360 dollars. Bottom bar: the sale price 360 dollars plus 6% tax of 21.60 dollars gives the final cost 381.60 dollars.original price p = $48025% = $12025% = $12025% = $12025% off = $120sale price = 0.75p = $360$3606% tax= $21.60final cost = 1.06(0.75p) = $381.600.795p = 381.60 → p = 381.60 ÷ 0.795 = 480

Answer: $480

Check: 480 \times 0.75 = \$360; 360 \times 1.06 = \$381.60 ✓

Calculator tip

Calculator tip (TI-30XS MultiView)

Word problems on linear equations are in the calculator part of the GED, and they often have messy decimals. Use the calculator for the arithmetic, but write the equation on your noteboard first.

  • Divide decimals carefully: for 0.795p = 381.60, type 381.6 ÷ 0.795 enter → 480.
  • Use parentheses for multi-step checks: 1.06 ( 0.75 × 480 ) enter → 381.6.
  • Fraction answers: if a problem uses fractions, enter them with the n/d key; press toggle to see the decimal.
  • Check the story, not just the equation: compute each piece (sale price, tax, total) and compare to the given numbers.

Don't let the calculator decide how to set up the problem. If you are unsure which operation to use, try a simple guess number (like 10 hours) to see how the total is built, then replace it with the variable.

Common mistake

Common mistakes

  • Answering the wrong question. You solve for the width (8 ft) but the question asked for the length (19 ft). Wrong: stop at x. Right: re-read the question and finish.
  • Forgetting the fixed fee. For "$65 plus $45 per hour = $245", wrong: 245 \div 45 \approx 5.4 hours. Right: subtract the fee first, 180 \div 45 = 4 hours.
  • Taking a percent of the wrong amount. For "after 20% off, the price is $68", wrong: 68 + 0.20(68) = \$81.60. Right: 0.80p = 68, so p = \$85.
  • Mixing up consecutive and consecutive even/odd. Consecutive integers go up by 1 (n, n + 1, n + 2); consecutive even or odd integers go up by 2 (n, n + 2, n + 4).
  • Subtracting speeds for objects moving apart. Cars going in opposite directions separate at the sum of their speeds, not the difference.
  • "Less than" order. "7 less than 3x" is 3x - 7, not 7 - 3x.
Test strategy

GED test strategy

  • Read twice. First for the story, then for the exact question and the units. Underline what you need to find.
  • Work backwards from the answers. On multiple choice you can test each option in the story. For the plumber problem, try 4 hours: 65 + 45(4) = 245 ✓. This is often faster than writing an equation.
  • Eliminate unreasonable options. A sale's original price must be more than the sale price; a number of people must be a whole number; a time can't be negative.
  • Estimate first. "Two cars, 120 mph apart-speed, 300 miles" — about 2 to 3 hours. An option of 30 hours is clearly wrong.
  • Watch for "trap" options that are an in-between number from the work (the overtime hours instead of total hours, or the width instead of the length).
  • Units: convert minutes to hours (divide by 60) or cents to dollars before you set up d = rt or a cost equation.

Check yourself

Real-world linear equations

A plumber charges a $65 service fee plus $45 per hour of work. Mr. Ortiz's bill was $245. How many hours did the plumber work?

Answer choices
Hint

The $65 fee is paid once, no matter how long the job takes. Remove it before dividing by the hourly rate.

Check yourself

Real-world linear equationsFill in

A gym charges a one-time $30 joining fee plus $25 per month. Kendra has paid $230 in total so far. For how many months has she been a member?

months

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Write the total as the joining fee plus $25 times the number of months.

Summary

  • Use the four steps: name the unknown, write the equation, solve, answer the real question with units.
  • Total = fixed amount + rate × number of units. Two plans are equal when their expressions are set equal.
  • Consecutive integers: n, n + 1, n + 2; consecutive even/odd: n, n + 2, n + 4.
  • d = rt, total cost, and I = prt are on the formula sheet; percent-of-original problems use 0.80p, 1.06p, etc.
  • Check your answer in the story and make sure it is reasonable.
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