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Unit Rate as Slope

See unit rates as the slope of a proportional graph, and compare rates given as graphs, tables and equations.

Practice this skill 20 min Unit rate as slope Leer en español
Key idea

What you'll learn

A unit rate tells you how much of one thing you get for one of another thing: $15 per hour, 29 miles per gallon, $0.67 per ounce. In this lesson you will see that a unit rate and the slope of a line are the same idea.

You will learn to:

  • recognize a proportional relationship in a table, a graph and an equation y = kx;
  • find the unit rate (the constant of proportionality, k) and explain what it means with units;
  • compare two rates given in different forms (graph vs equation vs table);
  • use the unit rate to predict values and decide which points belong on a line.

Why it matters on the GED. The test loves real-world rate questions: comparing two phone plans, two printers or two cars; "what does the slope represent?"; "which is the better buy?". You will often see one rate on a graph and the other as an equation or table, and you must compare them.

Real life. Unit prices on store shelves, hourly pay, miles per gallon, speed, and dosage per pound of body weight are all unit rates.

Words to know

  • rate — a comparison of two quantities with different units (120 miles in 4 gallons).
  • unit rate — a rate for 1 unit of the second quantity (30 miles per gallon).
  • per — "for each"; it means divide (miles per gallon = miles ÷ gallons).
  • proportional relationship — two quantities whose ratio \frac{y}{x} is always the same (2 lb cost $3, 4 lb cost $6).
  • constant of proportionality (k) — that constant ratio; k = \frac{y}{x} (in y = 3x, k = 3).
  • origin — the point (0, 0); every proportional graph passes through it (0 hours → $0).
  • slope — rise over run; for a proportional graph, slope = k = unit rate.
  • unit price — a unit rate in dollars per item, ounce, pound, etc. ($0.67 per ounce).

All Math words in the glossary

Unit rate = slope

Graph of distance in miles versus time in hours for a walker. A straight line starts at the origin and passes through (1, 3) and (4, 12). A dashed step shows that for each 1 hour to the right the line rises 3 miles, so the unit rate and slope are 3 miles per hour.01234560369121518Time (hours)Distance (miles)1 hour3 miles(1, 3)(4, 12)

A relationship is proportional when doubling one quantity doubles the other, tripling triples it, and zero gives zero. Its graph is a straight line through the origin, and its equation is

y = kx

where k is the unit rate.

Why the unit rate is the slope. Take the walker in the graph: she walks 3 miles every hour. From any point, move 1 hour to the right and the line rises 3 miles. Rise over run = \frac{3}{1} = 3. So the slope is exactly the unit rate, 3 miles per hour.

Three quick ways to read the unit rate from a proportional graph:

  1. Find the y-value when x = 1 — that is the unit rate. (At 1 hour, 3 miles.)
  2. Divide the coordinates of any marked point: \frac{y}{x}. (\frac{12}{4} = 3.)
  3. Use the slope formula with two points: \frac{12 - 3}{4 - 1} = \frac{9}{3} = 3.
Distance-time graph of a walker, y = 3x, a straight line through the origin. At x = 1 hour the height is 3 miles, which is the unit rate. The point (4, 12) gives 12/4 = 3. A dashed triangle from (1, 3) to (4, 12) shows a run of 3 hours and a rise of 9 miles: 9/3 = 3 miles per hour.1234536912150Time (hours)Distance (miles)run 3 hrise 9 mi(1, 3)(4, 12)at x = 1: 3 mi12/4 = 9/3 = 3 mph

Units of the slope: always "y-units per x-unit." Distance (miles) on the y-axis and time (hours) on the x-axis → miles per hour.

Finding the unit rate from a table, an equation or words

From a table. Divide y by x in each column. If every ratio is the same, the relationship is proportional and that ratio is the unit rate.

Gallons (x)358
Miles (y)87145232

\frac{87}{3} = 29, \frac{145}{5} = 29, \frac{232}{8} = 29 → 29 miles per gallon.

Graph of miles against gallons from the table: (3, 87), (5, 145) and (8, 232) lie on one straight line through the origin, y = 29x. A small dashed triangle shows that 1 more gallon gives 29 more miles, so the unit rate is 29 miles per gallon.12345678940801201602002400GallonsMiles1 gal29 mi(3, 87)(5, 145)(8, 232)87/3 = 145/5 = 232/8 = 29

From an equation. In y = kx, read k directly. y = 1.85x (cost of x pounds) → $1.85 per pound. If the equation looks like 3y = 51x, divide both sides by 3 first: y = 17x, so k = 17.

From words. "A 12-oz bag costs $8.04" → \frac{8.04}{12} = \$0.67 per ounce. Put the unit that follows "per" on the bottom.

Fractions. "\frac{2}{3} mile every \frac{1}{4} hour" → \frac{2}{3} \div \frac{1}{4} = \frac{2}{3} \times 4 = \frac{8}{3} mph. A fraction over a fraction is still just top ÷ bottom.

Proportional or not? A straight line that does not pass through (0, 0) — for example y = 4x + 1 — has a constant slope but is not proportional, because \frac{y}{x} changes (\frac{5}{1} = 5, \frac{9}{2} = 4.5). In real life that "+1" is a starting fee or starting amount.

Two lines with the same slope 4. The line y = 4x passes through the origin, so it is proportional: (1, 4) and (2, 8) both give y/x = 4. The line y = 4x + 1 starts at (0, 1), so it is not proportional: (1, 5) gives 5/1 = 5 but (2, 9) gives 9/2 = 4.5.12323456789101112130xy(1, 4)(2, 8)(0, 1)(1, 5)(2, 9)y = 4x: through (0, 0)proportionaly = 4x + 1: starts at 1not proportional

Comparing rates in different forms

GED questions often give one rate as a graph and another as an equation or table. The plan is always the same:

  1. Turn each one into a unit rate with the same units.
  2. Compare the numbers (bigger rate = faster, more per unit, etc.).
  3. Answer the exact question asked — it may ask "by how much per minute," or "how many more in 4 hours."

Example. Printer A's graph passes through (5, 90) → \frac{90}{5} = 18 pages/min. Printer B: y = 20x → 20 pages/min. B is faster by 20 - 18 = 2 pages per minute. In 10 minutes, B prints 2 \times 10 = 20 more pages.

Pages printed against minutes for two printers. Printer A's line passes through (5, 90): 90/5 = 18 pages per minute. Printer B follows y = 20x and passes through (5, 100). B's line is steeper. At 10 minutes A has printed 180 pages and B 200, so B prints 20 more.1234567891040801201602000MinutesPages printedA (5, 90)B (5, 100)180200B: y = 20x → 20 pages/minA: 90/5 = 18 pages/min10 min: B prints200 − 180 = 20 more

On a graph with two proportional lines, the steeper line has the larger unit rate. For "miles vs gallons," steeper = more miles per gallon = more efficient.

Better buy. For prices, the lower unit price is the better buy: 16 oz for $4.00 ($0.25/oz) beats 12 oz for $3.36 ($0.28/oz).

Cost against ounces for two sizes. The 16-oz size costs 4.00 dollars, a line through (16, 4.00) with slope 0.25 dollars per ounce. The 12-oz size costs 3.36 dollars, a line through (12, 3.36) with slope 0.28 dollars per ounce. The steeper line has the higher unit price, so the 16-oz size is the better buy.481216201234560OuncesCost ($)16 oz, $4.0012 oz, $3.36$3.36 ÷ 12 = $0.28/oz (steeper)$4.00 ÷ 16 = $0.25/oz better buy
Formula

Formulas

\text{unit rate} = k = \frac{y}{x} \qquad\qquad y = kx
  • k = unit rate (constant of proportionality), in y-units per x-unit
  • x = the input (hours, gallons, pounds…), y = the output (dollars, miles…)
m = \frac{y_2 - y_1}{x_2 - x_1}
  • For a proportional relationship, the slope m equals k.

On the GED formula sheet: the slope formula and y = mx + b (a proportional line is just b = 0), and distance = rate × time, d = rt. Not on the sheet: y = kx and k = \frac{y}{x} — memorize them (they're easy: "per" means divide).

Worked example

Example 1 — Unit rate from a graph

Problem. Use the walking graph. How far does the walker go in 2.5 hours?

Graph of distance in miles versus time in hours for a walker. A straight line starts at the origin and passes through (1, 3) and (4, 12). A dashed step shows that for each 1 hour to the right the line rises 3 miles, so the unit rate and slope are 3 miles per hour.01234560369121518Time (hours)Distance (miles)1 hour3 miles(1, 3)(4, 12)
  1. Unit rate from (4, 12): \frac{12}{4} = 3 miles per hour.
  2. Equation: y = 3x.
  3. At x = 2.5: y = 3(2.5) = 7.5.
Graph of the walker's distance y = 3x. Dashed guides mark (2, 6) and (3, 9), and the point (2.5, 7.5) halfway between them: in 2.5 hours the walker goes 7.5 miles.1234369120Time (hours)Distance (miles)(2, 6)(3, 9)(2.5, 7.5)2.57.5

Answer: 7.5 miles.

Check: 7.5 is halfway between the values at 2 hours (6 mi) and 3 hours (9 mi) ✓

Worked example

Example 2 — Better buy (real world)

Problem. Laundry detergent comes in a 64-oz bottle for $9.60 or a 100-oz bottle for $13.00. Which is the better buy?

  1. Small: \frac{\$9.60}{64 \text{ oz}} = \$0.15 per oz.
  2. Large: \frac{\$13.00}{100 \text{ oz}} = \$0.13 per oz.
  3. 0.13 < 0.15.
Cost against ounces for two detergent bottles. The 64-oz bottle at 9.60 dollars lies on y = 0.15x (0.15 dollars per ounce). The 100-oz bottle at 13.00 dollars lies on y = 0.13x (0.13 dollars per ounce). The 100-oz line is less steep, so it costs less per ounce and is the better buy.204060801002468101214160OuncesCost ($)64 oz, $9.60100 oz, $13.009.60 ÷ 64 = $0.15 per oz13.00 ÷ 100 = $0.13 per oz (better)

Answer: the 100-oz bottle is the better buy (2 cents less per ounce).

Check: 0.15 \times 64 = 9.60 ✓ and 0.13 \times 100 = 13.00 ✓

Worked example

Example 3 — Interpret the slope

Problem. A nurse uses y = 0.6x, where x is a patient's weight in kilograms and y is a dose in milligrams. What does 0.6 mean, and what dose does a 70-kg patient get?

  1. The slope 0.6 has units mg per kg: 0.6 milligram of medicine for each kilogram of body weight.
  2. Dose: y = 0.6 \times 70 = 42.
Graph of dose y = 0.6x, milligrams against body weight in kilograms, a line through the origin. A dashed triangle shows that 10 more kg adds 6 more mg (6/10 = 0.6 mg per kg). The point (70, 42) shows a 70-kg patient gets 42 mg.102030405060708010203040500Weight (kg)Dose (mg)10 kg6 mg(70, 42)slope = 6/10 = 0.6 mg per kg

Answer: 42 mg.

Check: \frac{42}{70} = 0.6 ✓

Worked example

Example 4 — Compare a table and an equation

Problem. Job A pays according to the table. Job B pays y = 17.25x dollars for x hours. For a 40-hour week, which job pays more, and by how much?

Hours51020
Pay ($)88176352
  1. Job A rate: \frac{88}{5} = 17.60 dollars per hour (check: \frac{352}{20} = 17.60 ✓).
  2. Job B rate: 17.25 dollars per hour.
  3. Difference per hour: 17.60 - 17.25 = 0.35.
  4. Per 40-hour week: 0.35 \times 40 = 14.
Job A pays 88/5 = 17.60 dollars per hour (check 352/20 = 17.60); Job B pays 17.25 dollars per hour. The graph shows the extra pay from Job A, y = 0.35x dollars after x hours, a line through the origin. At 40 hours it reaches (40, 14): Job A pays 14 dollars more per week.102030402468101214160Hours workedExtra pay from Job A ($)Job A: 88 ÷ 5 = $17.60/h; Job B: $17.25/hDifference: 17.60 − 17.25 = $0.35 per hour(40, 14)(20, 7)0.35 × 40 = $14

Answer: Job A pays $14 more per week.

Check: 17.60 \times 40 = 704 and 17.25 \times 40 = 690; 704 - 690 = 14 ✓

Worked example

Example 5 — GED-hard: work backwards from the rate

Problem. A car's distance is proportional to its gasoline use. The graph of miles vs gallons passes through (6, 171). How many gallons are needed to drive 247 miles?

  1. Unit rate: k = \frac{171}{6} = 28.5 miles per gallon. So y = 28.5x.
  2. Set y = 247: 247 = 28.5x.
  3. Divide: x = \frac{247}{28.5} = 8.\overline{6} \approx 8.67.
Graph of miles against gallons, y = 28.5x, through the origin and (6, 171): 171/6 = 28.5 miles per gallon. Dashed guides show that 247 miles on the line corresponds to 8 2/3 (about 8.67) gallons.12345678910501001502003000GallonsMiles(6, 171)(8 2/3, 247)171247≈ 8.67k = 171/6 = 28.5 mpg

Answer: about 8.67 gallons (exactly 8\frac{2}{3}).

Check: 28.5 \times 8\frac{2}{3} = 28.5 \times \frac{26}{3} = \frac{741}{3} = 247 ✓

Calculator tip

Calculator tip (TI-30XS MultiView)

  • Unit rates are divisions: 8.04 ÷ 12 enter → 0.67.
  • For fraction rates, use the n/d key for each fraction: n/d 2 ▼ 3 ▶ ÷ n/d 1 ▼ 4 enter gives \frac{8}{3}. Press toggle to see 2.666\ldots.
  • For "how many more in 12 gallons," you can do it in one line: ( 120 ÷ 4 − 110 ÷ 5 ) × 12 enter → 96.
  • When NOT to use it: reading the y-value at x = 1 off a graph, or reading k from y = kx. Also, with friendly numbers like \frac{60}{4}, mental math is faster.
Common mistake

Common mistakes

  • Dividing upside down. "Miles per gallon" is miles ÷ gallons, not gallons ÷ miles. ✗ \frac{4}{60} ✓ \frac{60}{4}. The unit after "per" goes on the bottom.
  • Using a point's y-value as the rate. The point (4, 60) does not mean $60 per hour. ✗ It means $60 for 4 hours, so \frac{60}{4} = \$15 per hour. ✓
  • Thinking every straight line is proportional. y = 4x + 1 has slope 4 but does not go through (0, 0). ✗ Proportional needs a constant \frac{y}{x} and the origin. ✓
  • Adding instead of multiplying. From (8, 6), adding 2 to both gives (10, 8) — not on the line. ✗ Multiply both by the same factor: (16, 12), (4, 3). ✓
  • Stopping at the rate. "How many more miles in 4 hours" needs (rate difference) × 4, not just the rate difference. ✓
  • Comparing totals at different times. Compare unit rates, or compare totals at the same x-value. ✓
Test strategy

GED test strategy

  • Label units on everything. Writing "mi/gal" or "$/hr" next to each number shows you instantly whether you divided the right way.
  • Use the x = 1 shortcut on proportional graphs: the height of the line at x = 1 is the unit rate.
  • Estimate first. If a car goes 120 miles on 4 gallons, the answer should be "about 30 mpg" — cross out choices like 0.03 or 480.
  • Plug points back in. To decide if a point is on y = kx, multiply k by the x-value and compare with y.
  • Answer the question asked. Rate questions often have a second step (a difference, a total for several hours, a "better buy"). Re-read the last sentence before choosing.
  • Interpretation questions: turn the slope into a sentence with units ("4.5 gallons per minute") and pick the choice that says the same thing.

Check yourself

Unit rate as slope

The graph shows how much Dana earns for the hours she works.

Graph of earnings in dollars (vertical axis, 0 to 120 in steps of 20) versus hours worked (horizontal axis, 0 to 8). A straight line starts at the origin (0, 0) and passes through the marked point (4, 60) and (8, 120).012345678020406080100120Hours workedEarnings (dollars)(4, 60)

What is Dana's hourly pay rate?

Answer choices
Hint

The unit rate is the amount earned for 1 hour. Use the marked point and divide.

Check yourself

Unit rate as slopeNo calculator

At a farm stand, the cost y (in dollars) of buying x pounds of apples is given by y = 1.85x.

What does the number 1.85 represent?

Answer choices
Hint

In y = kx, k is how much y changes when x goes up by 1.

Summary

  • A unit rate is an amount per one unit; "per" means divide, with the "per" unit on the bottom.
  • A proportional relationship has y = kx; its graph is a straight line through the origin.
  • The unit rate k is the slope of the graph: k = \frac{y}{x} for any point, or the y-value at x = 1.
  • The slope's units are "y-units per x-unit" — use them to interpret the slope in words.
  • To compare rates given as graphs, tables and equations, convert all to unit rates first; the steeper line has the greater rate.
  • A line with a starting amount (y = kx + b, b \ne 0) is linear but not proportional.
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