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Writing Rational Expressions from Situations

Write fraction expressions for rates, averages, shared costs, mixtures and work problems.

Practice this skill 25 min Write rational expressions from context Leer en español
Key idea

What you'll learn

Whenever a word problem says "per," "each," "average," or "how long," there is usually a division hiding in it. When the thing you divide by contains a variable, the result is a rational expression, such as \dfrac{150}{r} hours or \dfrac{1200}{n} dollars per person.

In this lesson you will learn to write rational expressions for:

  • Rates and time: time = distance ÷ speed, including speeds that change by a current or wind.
  • Averages: total ÷ number of items, when new items are added.
  • Work: how much of a job gets done per hour, alone and together.
  • Sharing costs: cost per person when the group size changes.
  • Mixtures: the fraction of a solution that is one ingredient.

Why it matters on the GED. Questions ask, "Which expression represents the time…?" or "Which expression gives the new average…?" The wrong choices are built from real slips: multiplying instead of dividing, flipping the fraction, adding times instead of rates, or forgetting to update the count. Once you know the patterns, these become quick points.

Real life. Splitting a rental among friends, figuring out how long a trip takes if you drive faster, or estimating how long a job takes with a helper are all rational-expression problems.

Words to know

  • Rate — an amount per unit, such as 55 miles per hour or $12 per ticket.
  • Rational expression — a fraction with a variable in the denominator (and maybe the numerator), such as \dfrac{240}{r}.
  • Distance–rate–time — d = rt; solving for time gives t = \dfrac{d}{r}.
  • Upstream / downstream — against / with the current. A boat with still-water speed 12 and current c goes 12 - c upstream and 12 + c downstream.
  • Average (mean) — the total divided by how many: \dfrac{\text{sum}}{\text{count}}.
  • Work rate — the fraction of a job done per hour. A job that takes 5 hours has a rate of \tfrac{1}{5} job per hour.
  • Combined rate — the sum of individual rates when people or machines work together.
  • Per person / per unit — total divided by the number of people or units.
  • Concentration — the part of a mixture that is one ingredient, as a fraction of the whole: \dfrac{\text{amount of ingredient}}{\text{total amount}}.
  • Reciprocal — a fraction flipped over: the reciprocal of \dfrac{x + 5}{5x} is \dfrac{5x}{x + 5}.

All Math words in the glossary

Rates and time

Start from the formula on the GED sheet, d = rt, and solve it for time:

t = \frac{d}{r}

Time = distance on top, speed on the bottom. If you drive 240 miles at r mph, the time is \dfrac{240}{r} hours. Sanity check: faster speed (bigger r) should mean less time, and dividing by a bigger number does give a smaller result. 240r would grow with speed — wrong. \dfrac{r}{240} is upside down — also wrong.

Changed speeds. Write the new speed first, then divide.

Graph of driving time for 240 miles. The solid blue curve t = 240/r falls as speed r rises from 20 to 80 mph: faster means less time. The dashed green curve t = 240/(r + 10) is the time 10 mph faster. At r = 50 the times are 4.8 h and 4.0 h; the red gap, 0.8 h, equals 2400/(50 times 60).02468101220304050607080speed r (mph)time for 240 mi (hours)t = 240/rt = 240/(r + 10)r = 50: saved 0.8 h= 2400/(50 · 60)
  • 10 mph faster: \dfrac{240}{r + 10}.
  • Twice as fast: \dfrac{240}{2r}.
  • Time saved by going 10 mph faster: \dfrac{240}{r} - \dfrac{240}{r + 10}.

Currents and wind. The current helps when you go with it and hurts when you go against it (see the figure below). A boat with still-water speed 12 mph on a river with current c:

  • Upstream speed: 12 - c, so 30 miles upstream takes \dfrac{30}{12 - c} hours.
  • Downstream speed: 12 + c, so 30 miles back takes \dfrac{30}{12 + c} hours.
  • Round-trip time: \dfrac{30}{12 - c} + \dfrac{30}{12 + c}.

Combining into one fraction. If a question wants a single fraction, use the LCD. For time saved:

\frac{240}{r} - \frac{240}{r + 10} = \frac{240(r + 10) - 240r}{r(r + 10)} = \frac{2400}{r(r + 10)}
A river flows left to right with a current of c miles per hour. A boat that goes 12 mph in still water travels at 12 minus c mph going upstream (against the current) and 12 plus c mph going downstream (with the current). Time equals distance divided by speed.current: c mph →Upstream (against current)speed = 12 − cDownstream (with current)speed = 12 + cboat speed in still water: 12 mphtime = distance ÷ speed

With a current of c mph, a 12-mph boat moves at 12 - c upstream and 12 + c downstream. Divide each distance by its own speed.

Averages and sharing costs

Average = total ÷ count. When the question adds new items, update both the total and the count.

Example: A student's 4 quiz scores add up to 328. She takes one more quiz and scores x.

\text{new average} = \frac{328 + x}{5}

The total grows by x and the count grows from 4 to 5.

If she takes n more quizzes and scores 95 on each, the total grows by 95n and the count by n:

\frac{328 + 95n}{4 + n}

Average of a mixed trip. Driving 2 hours at 55 mph and then t hours at 65 mph: total distance = 110 + 65t, total time = 2 + t, so the average speed is \dfrac{110 + 65t}{2 + t}. (Do not average 55 and 65.)

Sharing a cost. A $1200 cabin split equally among n friends costs \dfrac{1200}{n} each. If 3 more friends join, each pays \dfrac{1200}{n + 3}. The "+3" goes in the denominator because it changes the number of people, not the price.

Mixtures. A 40-ounce drink contains 8 ounces of juice concentrate. If x ounces of concentrate are added, the concentration is

\frac{8 + x}{40 + x}

Adding concentrate increases both the concentrate and the total.

Mixture picture, drawn for x = 10: a 40-ounce drink with 8 ounces of concentrate (red). Adding x ounces of concentrate raises both the concentrate to 8 + x and the total to 40 + x, so the concentration is (8 + x)/(40 + x).8 oz40 oz totaladd x ozconcentrate8 + x40 + x totalconcentration = (8 + x)/(40 + x)

Work problems: add rates, not times

The key idea: rate = 1 job ÷ time. If Maria paints a fence in 5 hours, she paints \tfrac{1}{5} of the fence each hour. If José takes x hours, he paints \tfrac{1}{x} per hour.

Working together, rates add:

\text{combined rate} = \frac{1}{5} + \frac{1}{x} = \frac{x + 5}{5x} \text{ fence per hour}

Time together is the reciprocal of the combined rate:

\text{time together} = \frac{5x}{x + 5} \text{ hours}
Work-rate flow: Maria's rate 1/5 plus José's rate 1/x equals the combined rate (x + 5)/(5x) fence per hour. Flip the combined rate to get the time together, 5x/(x + 5) hours. Add rates, not times.Maria's rate15+José's rate1x=combined ratex + 55xfliptime together5xx + 5Add RATES, not times.5 + x hours would be slowerthan either person alone.
StepExpression
Maria's rate\dfrac{1}{5}
José's rate\dfrac{1}{x}
Combined rate\dfrac{1}{5} + \dfrac{1}{x} = \dfrac{x + 5}{5x}
Time together\dfrac{5x}{x + 5}

Two traps to avoid.

  • Adding the times (5 + x) would mean working together takes longer than either person alone. That makes no sense.
  • Stopping at the combined rate answers "how much per hour," not "how long." Flip it for the time.

Part of a job. In h hours, a worker with rate \tfrac{1}{x} finishes \dfrac{h}{x} of the job. In 3 hours, a printer that needs 12 hours for a job finishes \tfrac{3}{12} = \tfrac{1}{4} of it.

A job bar split into 12 one-hour parts; a printer that needs 12 hours does 1/12 of the job each hour. After 3 hours, 3 parts are shaded: 3/12 = 1/4 of the job.3 hourswhole job = 12 hours (1/12 per hour)done in 3 h: 3/12 = 1/4 of the job
Formula

Formulas

d = rt \;\Rightarrow\; t = \frac{d}{r} \qquad \text{average} = \frac{\text{total}}{\text{count}} \text{rate of work} = \frac{1}{\text{time alone}} \qquad \frac{1}{a} + \frac{1}{b} = \frac{1}{t} \;\Rightarrow\; t = \frac{ab}{a + b} \text{concentration} = \frac{\text{amount of ingredient}}{\text{total amount}} \qquad \text{cost per person} = \frac{\text{total cost}}{\text{number of people}}
  • d = distance, r = rate (speed), t = time; a, b = times for each worker alone.

On the GED formula sheet: d = rt and "total cost = (number of units) × (price per unit)." Memorize the work-rate idea (add \tfrac{1}{a} + \tfrac{1}{b}), the concentration fraction, and per-person cost. They are not printed on the sheet.

Worked example

Example 1 (easy): Driving time

Leah drives 150 miles at r miles per hour. (a) Write her driving time. (b) Write her time if she drives 10 mph faster.

Example 1 time bars for 150 miles: at r = 50 mph the trip takes 150/50 = 3 hours; 10 mph faster, at 60 mph, it takes 150/60 = 2.5 hours. The faster trip has the shorter bar.150/r = 150/50 = 3 hr = 50 mph150/60 = 2.5 hr + 10 = 600 h1 h2 h3 hSame 150 miles: faster speed, shorter bar
  1. (a) t = \dfrac{d}{r} = \dfrac{150}{r} hours.
  2. (b) New speed: r + 10. New time: \dfrac{150}{r + 10} hours.

Answer: (a) \dfrac{150}{r}; (b) \dfrac{150}{r + 10}

Check: If r = 50: (a) 3 hours ✓ (50 \times 3 = 150); (b) \dfrac{150}{60} = 2.5 hours ✓ — faster means less time.

Worked example

Example 2 (real world): Round trip on a river

A boat travels 12 mph in still water. It goes 30 miles upstream against a current of c mph and then returns. Write the total time as a single fraction.

Example 2 river diagram with current c = 3 mph flowing right. Upstream against the current the 12-mph boat goes 12 − 3 = 9 mph, so 30 miles takes 3 and 1/3 hours. Downstream it goes 12 + 3 = 15 mph, so 30 miles takes 2 hours. Total 5 and 1/3 hours = 720/(144 − 9).currentc = 3 mphupstream (against): 12 − 3 = 9 mph; 30/9 = 3⅓ hdownstream (with): 12 + 3 = 15 mph; 30/15 = 2 htotal 5⅓ h = 720/(144 − 9) = 720/135
  1. Upstream speed: 12 - c; time \dfrac{30}{12 - c}.
  2. Downstream speed: 12 + c; time \dfrac{30}{12 + c}.
  3. Total: \dfrac{30}{12 - c} + \dfrac{30}{12 + c}. LCD = (12 - c)(12 + c) = 144 - c^2.
  4. Numerators: 30(12 + c) + 30(12 - c) = 360 + 30c + 360 - 30c = 720.

Answer: \dfrac{720}{144 - c^2} hours

Check: c = 3: \dfrac{30}{9} + \dfrac{30}{15} = 3.\overline{3} + 2 = 5.\overline{3}; and \dfrac{720}{135} = 5.\overline{3} ✓

Worked example

Example 3 (work backwards): The score needed

Jordan's 4 test scores average 82. (a) Write an expression for his average after a fifth test with score x. (b) What score does he need for an average of 85?

Example 3 tape model: four tests totalling 328 plus a fifth score x = 97 make 425, the same length as five equal parts of 85. New average = (328 + x)/5.4 tests: 4 × 82 = 328x = 978585858585total 425 = 5 × 85new average = (328 + x)/5
  1. Total so far: 4 \times 82 = 328.
  2. (a) New average: \dfrac{328 + x}{5}.
  3. (b) Set it equal to 85: \dfrac{328 + x}{5} = 85.
  4. Multiply by 5: 328 + x = 425, so x = 97.

Answer: (a) \dfrac{328 + x}{5}; (b) 97

Check: \dfrac{328 + 97}{5} = \dfrac{425}{5} = 85 ✓

Worked example

Example 4 (real world): Painting together

Maria can paint a fence in 5 hours. José can paint it in x hours. Write an expression for the time they need working together.

Example 4 bar model with x = 20: the fence is split into 20 equal parts. Each hour Maria paints 4 parts (1/5, blue) and José paints 1 part (1/20, green). After 4 hours all 20 parts are done, matching 5x/(x + 5) = 100/25 = 4 hours.hour 1hour 2hour 3hour 4Maria: 1/5 = 4/20 per hourJosé (x = 20): 1/20 per hour4 hours: Maria 16/20 = 4/5, José 4/20 = 1/5, total 1 fence5x/(x + 5) = 100/25 = 4 hours
  1. Rates: Maria \dfrac{1}{5}, José \dfrac{1}{x} fence per hour.
  2. Combined rate: \dfrac{1}{5} + \dfrac{1}{x} = \dfrac{x}{5x} + \dfrac{5}{5x} = \dfrac{x + 5}{5x}.
  3. Time = reciprocal of rate: \dfrac{5x}{x + 5}.

Answer: \dfrac{5x}{x + 5} hours

Check: If x = 20: \dfrac{100}{25} = 4 hours. In 4 hours Maria does \tfrac{4}{5} and José \tfrac{4}{20} = \tfrac{1}{5}; total 1 fence ✓

Worked example

Example 5 (GED-hard): How much each person saves

A group of n friends rents a cabin for $1200, split equally. Then 3 more friends join and the cost is re-split. Write a single fraction for how much each original friend saves.

Example 5 with n = 5: a 1200 dollars bar split into 5 shares of 240 dollars, and the same bar split into 8 shares of 150 dollars. The red bracket shows each original friend saves 240 dollars − 150 dollars = 90 dollars, matching 3600/(n(n + 3)) = 3600/40 = 90.$240$240$240$240$240n = 5 friends$150$150$150$150$150$150$150$150n + 3 = 8$1200 cabinsaves $901200/n − 1200/(n + 3) = 3600/(n(n + 3)) = 3600/40 = 90
  1. Original share: \dfrac{1200}{n}.
  2. New share: \dfrac{1200}{n + 3}.
  3. Savings: \dfrac{1200}{n} - \dfrac{1200}{n + 3}. LCD = n(n + 3).
  4. Numerator: 1200(n + 3) - 1200n = 1200n + 3600 - 1200n = 3600.

Answer: \dfrac{3600}{n(n + 3)} dollars

Check: n = 5: original $240, new \dfrac{1200}{8} = \$150, savings $90; formula \dfrac{3600}{5 \cdot 8} = 90 ✓

Calculator tip

Calculator tip: test the expression with real numbers

The GED's TI-30XS MultiView cannot write expressions, but it can confirm one. Choose an easy value for the variable, work out the story directly, and compare with the expression.

  1. Pick a friendly value, such as r = 50 in a driving problem or n = 5 in a sharing problem.
  2. Use the n/d key for fractions: press n/d, type the numerator, press the down arrow, type the denominator, press the right arrow.
  3. Example 5 check: n/d 3600 ▼ 5 × 8 ▶ enter → 90.
  4. Press toggle to switch between fraction and decimal results.

When not to use it: choosing the right expression is about reading the situation, so set up the expression on paper first. Avoid test values that make a denominator 0 (such as c = 12 in the river problem).

Common mistake

Common mistakes

  • Multiplying instead of dividing. Wrong: time = 240r. Right: \dfrac{240}{r}. Check: faster must mean less time.
  • Flipping the fraction. Wrong: time = \dfrac{r}{240}. Right: distance on top, speed on the bottom.
  • Adding times in a work problem. Wrong: together they take 5 + x hours. Right: add the rates, \dfrac{1}{5} + \dfrac{1}{x}, then flip: \dfrac{5x}{x + 5}.
  • Forgetting to update the count. Five scores total 410, then n more scores of 95 are added. Wrong: new average = \dfrac{410 + 95n}{5}. Right: \dfrac{410 + 95n}{5 + n}.
  • Putting a change outside the fraction. A $90 bill is split among n people, then 2 more people join. Wrong: each pays \dfrac{90}{n} - 2. Right: \dfrac{90}{n + 2}.
  • Mixing up upstream and downstream. Upstream is against the current: subtract c. Downstream is with it: add c.
Test strategy

GED test strategy

  • Ask "divided by what?" Find the quantity that does the dividing (speed, number of people, number of tests). It goes in the denominator.
  • Test with a number. Choose a value, work the story in plain arithmetic, and see which choice matches. This beats algebra on many questions.
  • Use common sense to eliminate. More helpers should mean less time; more people should mean a smaller share; a faster speed should mean a shorter trip.
  • Check what is asked: time or rate? new share or savings? average or total?
  • Units: if speeds are in mph and the answer is wanted in minutes, multiply hours by 60 at the end.
  • Time management: set up the expression in under a minute; if two choices look alike, test both with the same number.

Check yourself

Write rational expressions from contextNo calculator

A truck driver travels 240 miles at an average speed of r miles per hour. Which expression represents the driving time, in hours?

Answer choices
Hint

Start with d = rt and solve for t.

Check yourself

Write rational expressions from contextNo calculator

A group of n coworkers plans to split the cost of a $90 gift equally. Then 2 more coworkers join the group. Which expression represents the amount each person now pays, in dollars?

Answer choices
Hint

How many people are sharing the cost now?

Summary

  • Time = distance ÷ speed; change the speed first (e.g. r + 10, 12 - c), then divide.
  • Average = total ÷ count; when items are added, update both.
  • Work: rates add (\tfrac{1}{a} + \tfrac{1}{b}); time together is the reciprocal, \dfrac{ab}{a + b}.
  • Per person = total ÷ people; changes in group size go in the denominator.
  • Combine differences like "time saved" with an LCD, and always check with a test number.
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