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Comparing Proportional Relationships

Find the unit rate in a table, graph, equation, or description, then compare two rates to decide which is faster, cheaper, or better.

Practice this skill 25 min Compare proportional relationships Leer en español
Key idea

What you'll learn

In this lesson you will learn how to compare two proportional relationships when they are shown in different ways — one as a table and one as a graph, or one as an equation and one as a sentence.

The secret is simple: turn both relationships into a unit rate (the amount for exactly 1 unit), and then compare the two numbers.

Why it matters on the GED. The test loves questions like these:

  • "Pump A's data is in a table. Pump B's is on a graph. Which pump is faster, and by how much?"
  • "Store A's price is given by y = 1.80x. Store B's prices are in a table. Which store is cheaper per pound?"
  • "Ana's pay is shown on a graph; Ben earns $132 for 8 hours. How much more does Ben earn in a 40-hour week?"

Why it matters in real life. Every time you compare hourly pay, price per pound, miles per gallon, or data plans, you are comparing proportional relationships. Once you can find a unit rate from any format, you can make the better choice every time.

Words to know

  • Proportional relationship — two quantities that always have the same ratio; if one doubles, the other doubles. (Gas at $4 per gallon: 2 gal → $8, 4 gal → $16.)
  • Ratio — a comparison of two numbers by division, like \frac{60 \text{ dollars}}{4 \text{ hours}}.
  • Unit rate — the amount for 1 unit of the other quantity: $15 per hour, 28 miles per gallon.
  • Constant of proportionality (k) — the unit rate written as a number in the equation y = kx. In y = 15x, k = 15.
  • Origin — the point (0, 0) where the axes cross. Every proportional graph passes through it.
  • Slope — the steepness of a line, \frac{\text{rise}}{\text{run}}. For a proportional graph, slope = unit rate.
  • Input (x) and output (y) — x is what you choose (hours, gallons); y is what you get (pay, cost).
  • Representation — a way of showing a relationship: table, graph, equation, or verbal description.

All Math words in the glossary

1. What makes a relationship proportional?

A relationship between x and y is proportional when y is always the same multiple of x:

y = kx

Here k is the constant of proportionality, which is the same thing as the unit rate.

You can test each representation like this:

RepresentationIt is proportional when…Example
Table\frac{y}{x} is the same for every row (and x = 0 would give y = 0)\frac{30}{2} = \frac{75}{5} = \frac{120}{8} = 15
Graphthe points lie on a straight line through the origin (0, 0)line from (0,0) through (4, 60)
Equationit looks like y = kx — no number added or subtractedy = 18x ✓, y = 18x + 5 ✗
Descriptionit says "per," "each," or "for every," with no flat fee"$3.50 per gallon" ✓, "$20 plus $3 per hour" ✗

Quick example. A table shows x: 2, 5, 8 and y: 30, 75, 120. Check the ratios: 30 \div 2 = 15, 75 \div 5 = 15, 120 \div 8 = 15. All the same, so it is proportional with k = 15, and the equation is y = 15x.

Not proportional. A table with x: 1, 2, 3 and y: 5, 8, 11 has ratios 5, 4, 3.67 — they change, so the relationship is not proportional (it is y = 3x + 2, which has a starting amount of 2).

Two side-by-side panels. Left, labelled Proportional: a table with x = 2, 5, 8 and y = 30, 75, 120; the ratio y divided by x is 15, 15, 15. Below it, the graph of y = 15x is a straight line through the origin (0, 0) passing through (2, 30), (5, 75) and (8, 120). Right, labelled Not proportional: a table with x = 1, 2, 3 and y = 5, 8, 11; the ratios are 5, 4 and 3.67. Below it, the graph of y = 3x + 2 is a straight line through (1, 5), (2, 8) and (3, 11) that starts at (0, 2), not at the origin.ProportionalNot proportionalx258y3075120y ÷ x151515x123y5811y ÷ x543.670123456780306090120xy(2, 30)(5, 75)(8, 120)through (0, 0)y = 15x0123402468101214xy(1, 5)(2, 8)(3, 11)starts at (0, 2)y = 3x + 2

2. Finding the unit rate from each representation

To compare, you need the unit rate from each side. Here is how to get it from all four formats.

From an equation y = kx: read k straight off. y = 18x → 18 per unit. d = 0.2t → 0.2 per unit. A fraction works too: b = \tfrac{2}{3}w → \tfrac{2}{3} per unit.

From a table: divide any output by its input: k = \frac{y}{x}. Use a row that is not (0, 0). Example: 6 hours → $99 gives 99 \div 6 = 16.50 per hour.

From a graph: find a point on the line with easy-to-read coordinates (x, y), then divide: k = \frac{y}{x}. If the line passes through (1, k), the unit rate is simply the y-value at x = 1. You can also use slope: \frac{\text{rise}}{\text{run}} between any two points.

Two lines on the same axes. Horizontal axis: Hours, 0 to 8. Vertical axis: Pay in dollars, 0 to 120. Job A (blue line) passes through (0, 0) and (8, 120), a rate of 15 dollars per hour. Job B (orange line) passes through (0, 0) and (8, 80), a rate of 10 dollars per hour. Job A's line is steeper.01234567820406080100120HoursPay (dollars)(8, 120)(8, 80)Job AJob B

In the graph above, Job A passes through (8, 120), so its rate is 120 \div 8 = \text{\textdollar}15 per hour. Job B passes through (8, 80), so its rate is 80 \div 8 = \text{\textdollar}10 per hour. On the same axes, the steeper line has the greater unit rate.

From a description: divide the two amounts in the sentence. "$148.75 for 8.5 hours" → 148.75 \div 8.5 = 17.50 dollars per hour. "351 miles on 13 gallons" → 351 \div 13 = 27 miles per gallon.

Always divide in the same order for both relationships. If the question asks "dollars per hour," compute \frac{\text{dollars}}{\text{hours}} for both. Mixing \frac{\text{dollars}}{\text{hours}} with \frac{\text{hours}}{\text{dollars}} is the number-one trap.

3. Comparing — and answering what is really asked

Once both unit rates are in the same units, comparing is easy. Follow these four steps:

  1. Find unit rate A (from its representation).
  2. Find unit rate B (from its representation).
  3. Make the units match. Convert minutes to hours, ounces to pounds, etc. if needed.
  4. Answer the exact question. Is it "which is greater," "how much greater per unit," or "how much more in 40 hours"?

Difference per unit vs. difference in total. If A pays $16.50/hour and B pays $15/hour, the difference is $1.50 per hour. In a 40-hour week the difference is 1.50 \times 40 = \text{\textdollar}60. Read the question to see which one it wants.

"Better" depends on the situation.

RateBetter when it is…
pay per hour, miles per gallon, pages per minutehigher
price per pound, cost per gallon, minutes per milelower

Units that are flipped. "9 minutes per mile" and "7.5 miles per hour" both describe speed, but in opposite directions. To compare, convert one: 60 \div 7.5 = 8 minutes per mile. Now 8 min/mile beats 9 min/mile — fewer minutes per mile means faster.

Mixed units. "$7.20 for 12 ounces" vs. "$9 per pound": convert ounces to pounds first (16 oz = 1 lb). 7.20 \div 12 = \text{\textdollar}0.60 per ounce, and 0.60 \times 16 = \text{\textdollar}9.60 per pound.

Formula

Formulas to remember

y = kx \qquad\qquad k = \frac{y}{x} \qquad\qquad m = \frac{y_2 - y_1}{x_2 - x_1}
  • y — the output (total cost, total pay, distance…)
  • x — the input (pounds, hours, gallons, minutes…)
  • k — the constant of proportionality = the unit rate = the slope of the graph
  • m — slope between two points (x_1, y_1) and (x_2, y_2)

Formula sheet: the slope formula m = \frac{y_2 - y_1}{x_2 - x_1} and the slope-intercept form y = mx + b are on the GED formula sheet. A proportional relationship is just y = mx + b with b = 0. The idea "unit rate = \frac{y}{x}" is not written on the sheet — memorize it.

Worked example

Example 1 (easy): table vs. equation

Printer A's output is shown in the table. Printer B prints according to y = 18x, where x is minutes and y is pages. Which printer is faster, and by how many pages per minute?

Minutes (x)258
Pages (y)3075120
  1. Printer A's unit rate: \frac{30}{2} = 15, \frac{75}{5} = 15, \frac{120}{8} = 15 → 15 pages per minute.
  2. Printer B's unit rate: read k from y = 18x → 18 pages per minute.
  3. Compare: 18 - 15 = 3.
Printer A's table (2 minutes, 30 pages; 5 minutes, 75 pages; 8 minutes, 120 pages) beside a graph of pages against minutes. Printer A's points lie on a solid blue line through the origin with rate 15 pages per minute. Printer B, y = 18x, is a dashed red line through the origin that is steeper, reaching 144 pages at 8 minutes, so B is faster by 3 pages per minute.Printer AMinutesPages2305758120Printer B:y = 18x0123456780306090120150Minutes (x)Pages (y)(2, 30)(5, 75)(8, 120)B: 18 pages/minA: 15 pages/min

Answer: Printer B is faster, by 3 pages per minute.

Check: In 2 minutes, B prints 18 \times 2 = 36 pages, A prints 30 pages; 36 - 30 = 6 = 3 \times 2 ✓

Worked example

Example 2: graph vs. description (word problem)

Ana's pay is shown on the graph. Ben earns $132 for 8 hours of work. Both are paid by the hour. How much more does the higher-paid person earn in a 40-hour week?

Line graph titled Ana's pay. Horizontal axis: Hours worked, 0 to 8. Vertical axis: Pay in dollars, 0 to 120 in steps of 15. A straight line starts at the origin (0, 0) and passes through the marked points (2, 30), (4, 60) and (8, 120).012345678306090120Hours workedPay (dollars)(2, 30)(4, 60)(8, 120)
  1. Ana's rate: the line passes through (4, 60), so k = \frac{60}{4} = \text{\textdollar}15 per hour.
  2. Ben's rate: \frac{132}{8} = \text{\textdollar}16.50 per hour.
  3. Ben earns more per hour: 16.50 - 15 = \text{\textdollar}1.50 per hour.
  4. In 40 hours: 1.50 \times 40 = \text{\textdollar}60.

Answer: Ben earns $60 more per 40-hour week.

Check: Ben: 16.50 \times 40 = \text{\textdollar}660. Ana: 15 \times 40 = \text{\textdollar}600. 660 - 600 = 60 ✓

Worked example

Example 3: description vs. equation, then a cost question

Car A travels 336 miles on 12 gallons of gas. Car B's mileage is given by m = 32g, where g is gallons and m is miles. Gas costs $3.50 per gallon. For a 448-mile trip, how much less does the gas cost in the more efficient car?

  1. Car A's rate: \frac{336}{12} = 28 miles per gallon.
  2. Car B's rate: k = 32 miles per gallon. Car B is more efficient.
  3. Gallons for 448 miles: Car A needs 448 \div 28 = 16 gal; Car B needs 448 \div 32 = 14 gal.
  4. Gallons saved: 16 - 14 = 2. Money saved: 2 \times 3.50 = \text{\textdollar}7.00.
Graph of miles against gallons. Car A gets 28 miles per gallon (solid blue line through the origin and the point (12, 336)). Car B, m = 32g, is a steeper green line. A dashed red line at 448 miles meets Car B at 14 gallons and Car A at 16 gallons, so Car B uses 2 fewer gallons for the 448-mile trip.024681012141618080160240320400480560Gallons (g)Miles (m)448-mile trip14 gal16 galCar B: m = 32g (32 mi/gal)Car A: (12, 336) → 28 mi/gal

Answer: $7.00 less with Car B.

Check: Car A: 16 \times 3.50 = \text{\textdollar}56; Car B: 14 \times 3.50 = \text{\textdollar}49; 56 - 49 = 7 ✓

Worked example

Example 4: work backwards — fill in a table, then compare

Store B sells rice at a constant price per pound. Some of its table is missing. Store A's price is y = 1.80x (dollars for x pounds). Find the missing values and decide which store is cheaper per pound.

Pounds26?
Cost ($)3.90?15.60
  1. Store B's unit rate: 3.90 \div 2 = \text{\textdollar}1.95 per pound.
  2. Missing cost for 6 lb: 1.95 \times 6 = \text{\textdollar}11.70.
  3. Missing pounds for $15.60: 15.60 \div 1.95 = 8 lb.
  4. Compare: Store A charges $1.80, Store B $1.95. 1.95 - 1.80 = 0.15.

The figure shows the finished table and both stores' graphs:

Store B's completed table: 2 pounds cost 3.90 dollars, 6 pounds cost 11.70 dollars and 15.60 dollars buys 8 pounds (the two found values in green). Beside it, a graph of cost against pounds: Store B's points lie on a solid blue line through the origin with rate 1.95 dollars per pound; Store A, y = 1.80x, is a dashed red line that is less steep, so Store A is cheaper by 0.15 dollars per pound.Store B (completed)PoundsCost ($)23.90611.70815.60green = foundStore A: y = 1.80x012345678910048121620Pounds (x)Cost in dollars (y)(2, 3.90)(6, 11.70)(8, 15.60)B: $1.95 per lbA: $1.80 per lb

Answer: missing values $11.70 and 8 lb; Store A is cheaper by $0.15 per pound.

Check: \frac{11.70}{6} = 1.95 and \frac{15.60}{8} = 1.95 — the ratios match, so the table is still proportional ✓

Worked example

Example 5 (GED-hard): flipped units

Kim's running times are in the table. Lee runs at a steady 7.5 miles per hour. Who is faster?

Miles25
Minutes1845
  1. Kim's rate from the table: \frac{18}{2} = 9 and \frac{45}{5} = 9 → 9 minutes per mile (minutes on top!).
  2. Lee's rate is in miles per hour. Flip it to minutes per mile: 1 hour = 60 minutes, so 60 \div 7.5 = 8 minutes per mile.
  3. Lee takes 8 minutes per mile, Kim takes 9. Fewer minutes per mile = faster.
Kim's table (2 miles in 18 minutes, 5 miles in 45 minutes) beside a graph of minutes against miles. Kim's points lie on a solid blue line through the origin, 9 minutes per mile. Lee, 7.5 miles in 60 minutes, is a dashed red line through (7.5, 60), 8 minutes per mile. Lee's line is lower: fewer minutes per mile means faster.Kim's tableMilesMinutes218545Lee: 7.5 mi per60 minutes0123456780122436486072MilesMinutes(2, 18)(5, 45)(7.5, 60)Kim: 9 min per mileLee: 8 min per mileLower line = fewer minutes per mile = faster

Answer: Lee is faster (by 1 minute per mile).

Check: Convert Kim to miles per hour instead: 60 \div 9 \approx 6.67 mph, which is less than 7.5 mph ✓ Both methods agree.

Calculator tip

TI-30XS MultiView tips

  • Unit rate: type the output, press ÷, type the input, press enter. Example: 148.75 ÷ 8.5 enter → 17.5.
  • Fractions: if a rate is a fraction like \frac{5}{6}, use the n/d key and then the toggle key to switch to the decimal 0.8333… so it is easy to compare with \frac{2}{3} = 0.6666….
  • Check that a table is proportional by dividing each y by its x; all the answers should be identical.
  • When NOT to use it: reading a graph and deciding which number goes on top. The calculator cannot tell you whether you need "dollars per pound" or "pounds per dollar" — decide that first, on paper. Simple ratios like 60 \div 4 are faster in your head.
Common mistake

Common mistakes

  • Flipping one rate. Computing $ per pound for one store and pounds per $ for the other. ✗ Wrong: Store B "4 lb for $6.50" → 4 \div 6.5 = 0.62. ✓ Right: 6.50 \div 4 = \text{\textdollar}1.625 per pound — same order (dollars ÷ pounds) for both.
  • Using only the first row of a table. (1, 2.5), (2, 3.5): the first ratio is 2.5, but the second is 1.75, so this table is not proportional. Always check at least two rows.
  • Treating "plus a fee" as proportional. y = x + 2.5 is not y = 2.5x. If a number is added, the graph does not pass through the origin.
  • Reading the wrong axis. On a graph of distance (miles) against time (minutes), the point (30, 6) means 6 miles in 30 minutes — that is 0.2 miles per minute, not 6 miles per hour.
  • Answering the per-unit difference when the question asks for a total. $1.50 per hour is not "how much more in 40 hours" — multiply: 1.50 \times 40 = \text{\textdollar}60.
  • Forgetting to match units. $7.20 for 12 ounces cannot be compared directly with $9 per pound. Convert first.
Test strategy

GED test strategy

  • Read the question twice and circle the unit it wants ("per hour," "per pound," "in 2 hours").
  • Write both unit rates side by side on your erasable note board before looking at the answer choices: "A = 15/h, B = 16.50/h."
  • Eliminate choices that just repeat one of the rates when the question asks for a difference — that is a favorite distractor.
  • Estimate: $148.75 for 8.5 hours is "about $150 for 8.5 hours," a bit under $18/h. That alone can rule out choices.
  • Plug back in: if you think the rate is 12 gallons per minute, check one table row or graph point: 12 \times 4 = 48 ✓
  • Time: these questions should take about 1–2 minutes. If a graph is hard to read, use a point where the line crosses grid lines exactly.

Check yourself

Compare proportional relationshipsNo calculator

The table shows a proportional relationship between x and y.

x1234
y6121824

Which equation represents this relationship?

Answer choices
Hint

Find \frac{y}{x} for more than one column. Is it always the same number?

Check yourself

Compare proportional relationshipsFill in

The graph shows the cost of buying gasoline at a gas station.

Line graph of the cost of gasoline. Horizontal axis: Gallons, 0 to 8. Vertical axis: Cost in dollars, 0 to 32 in steps of 4. A straight line starts at (0, 0) and passes through the marked points (3, 12) and (5, 20).01234567848121620242832GallonsCost (dollars)(3, 12)(5, 20)

What is the price of gasoline, in dollars per gallon?

$ per gallon

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

Hint

Pick a point on the line where you can read both numbers exactly. Divide cost by gallons.

Summary

  • A proportional relationship is y = kx: straight line through the origin, constant ratio \frac{y}{x}, no added fee.
  • The unit rate k is the key number: read it from an equation, divide \frac{y}{x} in a table or graph, or divide the two amounts in a description.
  • Divide in the same order for both relationships and convert units so they match.
  • On the same axes, the steeper line has the greater unit rate.
  • Answer exactly what is asked: which is greater, the difference per unit, or the difference over many units.
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