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Unit Rates and Best Buys

Turn any rate into a 'per one' number so you can compare prices, speeds, and pay — and pick the best buy.

Practice this skill 22 min Unit rates Leer en español
Key idea

What you'll learn

A unit rate tells you how much of something you get for one of something else: miles per hour, dollars per pound, words per minute. Once you can find a unit rate, a huge number of everyday questions become simple division and multiplication.

In this lesson you will learn to:

  • find a unit rate from any pair of numbers, including decimals and fractions;
  • use a unit rate to predict a total ("at this pace, how far in 6 hours?");
  • work backwards from a unit rate to a missing amount;
  • read a unit rate from a table or a graph;
  • compare prices and choose the best buy.

Why it matters on the GED: Unit rates appear all over the Mathematical Reasoning test — in shopping questions ("Which package is the better deal?"), in work and pay questions, in speed and gas-mileage questions, and in graphs where the rate is hidden as the slope of a line. Many of these questions are in the calculator section, but some simple ones are in the short no-calculator part, so you should be comfortable dividing by hand too.

Why it matters in life: The shelf tag at the grocery store, the miles-per-gallon sticker on a car, and your hourly pay are all unit rates. Knowing how to find them protects your wallet.

Words to know

  • Ratio — a comparison of two quantities, like 3 cups of flour to 2 cups of sugar (3:2 or \tfrac{3}{2}).
  • Rate — a ratio that compares two quantities with different units, like 120 miles in 2 hours.
  • Unit rate — a rate where the second quantity is exactly 1: 60 miles per 1 hour, written 60\text{ mi/h} or 60 mph.
  • Per — means "for each" or "divided by." "$12 per hour" means $12 for each hour.
  • Unit price — a unit rate for money: the cost of one item, ounce, pound, etc. ($3.00 for 6 cans → $0.50 per can).
  • Best buy — the option with the lowest unit price (you pay the least for each unit).
  • Constant rate — a rate that does not change, so equal amounts of time (or items) always give equal amounts of the other quantity.
  • Proportional — two quantities are proportional when their ratio is always the same unit rate (2 h → 120 mi, 3 h → 180 mi: always 60 mph).
  • Slope — on a straight-line graph through the origin, the unit rate is the slope: rise ÷ run.

All Math words in the glossary

Finding a unit rate: divide by the 'per' quantity

Rule: To find a unit rate, divide the first quantity by the second quantity — the one that comes after the word per.

\text{unit rate} = \frac{\text{amount of A}}{\text{amount of B}} \quad\text{gives}\quad \text{A per 1 B}

The units travel with the numbers. Miles ÷ hours = miles per hour. Dollars ÷ pounds = dollars per pound.

Examples

  • 150 miles in 3 hours: 150 \div 3 = 50 → 50 miles per hour.
Tape diagram: a bar for 150 miles is split into 3 equal parts, one per hour. Each part is 50 miles, so the unit rate is 150 divided by 3 = 50 miles per hour.150 miles in 3 hours50 mihour 150 mihour 250 mihour 3total: 150 mi ÷ 3 h = 50 miles per 1 hour
  • $84 for 7 hours of work: 84 \div 7 = 12 → $12 per hour.
  • $4.50 for 6 apples: 4.50 \div 6 = 0.75 → $0.75 per apple.

Which number goes on top? Ask yourself what the question wants "per one" of. If it asks for dollars per pound, pounds go on the bottom. If it asks for pounds per dollar, dollars go on the bottom. Both are correct rates — they just answer different questions.

Question asks forDivideExample ($6 for 3 lb)
price per pounddollars ÷ pounds6 \div 3 = \$2 per lb
pounds per dollarpounds ÷ dollars3 \div 6 = 0.5 lb per dollar

Decimals and money. Money unit prices are often small decimals. Keep at least three or four decimal places while comparing, and round only at the very end. For example, $2.99 for 12 oz is 2.99 \div 12 = 0.2491\ldots dollars per ounce — about 24.9 cents per ounce.

Fractions. Rates can involve fractions: "She walks \tfrac{1}{2} mile in \tfrac{1}{4} hour." Divide the fractions just like whole numbers:

\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = 2 \text{ miles per hour}

Remember: dividing by a fraction means multiplying by its reciprocal (flip the second fraction).

Time units. Watch for minutes versus hours. 30 miles in 40 minutes is 30 \div 40 = 0.75 miles per minute. To get miles per hour, multiply by 60: 0.75 \times 60 = 45 mph. (Or change 40 minutes to \tfrac{2}{3} hour first: 30 \div \tfrac{2}{3} = 45.)

Using a unit rate: multiply forward, divide backward

Once you know the unit rate, you can answer two kinds of follow-up questions.

1. Find a total (multiply).

\text{total} = \text{unit rate} \times \text{number of units}
  • A car gets 32 miles per gallon. How far on 11 gallons? 32 \times 11 = 352 miles.
  • Grapes cost $2.40 per pound. Cost of 3.5 pounds? 2.40 \times 3.5 = \$8.40.

2. Work backwards to find the number of units (divide).

\text{number of units} = \frac{\text{total}}{\text{unit rate}}
  • A printer prints 18 pages per minute. How long for 270 pages? 270 \div 18 = 15 minutes.
  • You earn $16.50 per hour. How many hours to earn $495? 495 \div 16.50 = 30 hours.

Two-step problems. Many GED questions hide the unit rate. You first find it from one situation, then use it in another:

A machine fills 420 bottles in 6 minutes. How many bottles in 25 minutes?

Step 1: 420 \div 6 = 70 bottles per minute. Step 2: 70 \times 25 = 1{,}750 bottles.

Double number line for bottles and minutes, drawn to scale. 0 minutes = 0 bottles, 1 minute = 70 bottles, 6 minutes = 420 bottles, 25 minutes = 1,750 bottles. The unit rate 70 bottles per minute times 25 gives 1,750.420 bottles in 6 minutes → 70 per minuteminutesbottles001706420251,750

Rates in tables. If a table shows two quantities that are proportional, divide any row to get the unit rate — every row gives the same answer.

Hours worked5812
Pay ($)92.50148.00222.00

92.50 \div 5 = 18.50, 148 \div 8 = 18.50, 222 \div 12 = 18.50. The unit rate is $18.50 per hour.

Graph of pay versus hours worked from the table. A straight line from the origin passes through (5, 92.50), (8, 148) and (12, 222). Each point gives pay divided by hours = 18.50, so the slope and unit rate is 18.50 dollars per hour.Pay = $18.50 per hour02468101214$0$40$80$120$160$200$240$280Hours workedPay (dollars)(5, 92.50)(8, 148)(12, 222)

Rates on graphs. On a straight-line graph that starts at the origin (0, 0), pick any marked point (x, y). The unit rate is y \div x — this is the slope of the line. If the line passes through (3, 45) with x = hours and y = dollars, the rate is 45 \div 3 = \$15 per hour.

Best buys: compare unit prices

Stores sell the same product in different sizes. Bigger is not always cheaper per unit. To find the best buy:

  1. For every option, divide price ÷ quantity to get the unit price.
  2. Make sure every unit price uses the same unit (all per ounce, or all per can).
  3. The lowest unit price is the best buy.

Example. Laundry detergent:

SizePriceUnit price (price ÷ oz)
16 oz$3.293.29 \div 16 \approx \$0.2056
32 oz$5.995.99 \div 32 \approx \$0.1872
48 oz$8.498.49 \div 48 \approx \$0.1769

The 48-oz bottle has the lowest price per ounce, so it is the best buy — even though it costs the most at the register.

Different units. If one package is measured in pounds and another in ounces, convert first (1 pound = 16 ounces). If one deal is "3 for $5" and another is "$1.75 each," change both to price per item: 5 \div 3 \approx \$1.67 vs $1.75.

Bar chart comparing price per item: 3 for 5 dollars is 5 dollars divided by 3, about 1.67 dollars per item; the other deal is 1.75 dollars each. The 3 for 5 dollars bar is shorter, so it is the better buy.Price per item3 for $5≈ $1.67$1.75 each$1.75$5 ÷ 3 ≈ $1.67 < $1.75

Coupons and sales. Apply the coupon or discount to the price first, then divide by the quantity. A $1 coupon on a $7.00, 50-oz bottle makes it 6.00 \div 50 = \$0.12 per ounce.

The "per dollar" shortcut. You can also compare quantity per dollar (ounces ÷ price). Then the highest number is the best buy, because you get more for each dollar. Just don't mix the two methods in the same problem.

Formula

Formulas to know

\text{unit rate} = \frac{\text{quantity A}}{\text{quantity B}} \qquad\qquad \text{unit price} = \frac{\text{total price}}{\text{number of units}} \text{total cost} = (\text{number of units}) \times (\text{price per unit}) \qquad\qquad d = rt
  • unit rate — amount of A for one B (e.g. miles per hour).
  • unit price — cost of one unit (one ounce, one can, one pound).
  • d — distance, r — rate (speed), t — time. Make sure the time unit matches the rate (hours with miles per hour).

On the GED formula sheet? The sheet lists distance d = rt and total cost = number of units \times price per unit. The plain unit-rate idea ("divide by the per quantity") is not written there — you must know it. Rearranging is your job too: r = \dfrac{d}{t} and t = \dfrac{d}{r}.

Worked example

Example 1 (easy): speed

A bus travels 357 miles in 7 hours at a steady speed. What is its speed in miles per hour?

  1. The question asks for miles per hour, so hours go on the bottom.
  2. \frac{357\text{ mi}}{7\text{ h}} = 51\text{ mi/h}
Double number line for the bus: 7 hours matches 357 miles. The 7 hours are split into 7 equal steps, and 1 hour matches 357 divided by 7 = 51 miles.Bus: 357 miles in 7 hourshoursmiles001517357

Answer: 51 miles per hour.

Check: 51 \times 7 = 357 ✓

Worked example

Example 2: best buy at the grocery store

Peanut butter comes in three sizes: 16 oz for $3.29, 32 oz for $5.99, and 48 oz for $8.49. Which size is the best buy?

  1. Unit price of 16 oz: 3.29 \div 16 \approx 0.2056 dollars per ounce.
  2. Unit price of 32 oz: 5.99 \div 32 \approx 0.1872 dollars per ounce.
  3. Unit price of 48 oz: 8.49 \div 48 \approx 0.1769 dollars per ounce.
  4. Compare: 0.1769 < 0.1872 < 0.2056.
Bar chart of unit prices for peanut butter: 16 oz for 3.29 dollars is about 20.56 cents per ounce, 32 oz for 5.99 dollars is about 18.72 cents per ounce, 48 oz for 8.49 dollars is about 17.69 cents per ounce. The 48-ounce jar has the shortest bar and is the best buy.Unit price (cents per ounce)16 oz, $3.2920.56¢32 oz, $5.9918.72¢48 oz, $8.4917.69¢Shortest bar = lowest unit price = best buy (48 oz)

Answer: the 48-oz jar (about 17.7¢ per ounce).

Check: If the 48-oz jar cost the same per ounce as the 32-oz jar, it would cost 0.1872 \times 48 \approx \$8.99. It actually costs $8.49, which is less ✓

Worked example

Example 3: a rate with fractions

A painter covers \tfrac{3}{4} of a wall in \tfrac{1}{6} of an hour. At this rate, how many walls can he paint per hour?

  1. Walls per hour means walls ÷ hours.
  2. \frac{3}{4} \div \frac{1}{6} = \frac{3}{4} \times \frac{6}{1} = \frac{18}{4} = 4.5
Painter rate model drawn to scale. Top row: 6 pieces, each 1/6 hour long and each covering 3/4 of a wall. Bottom row: the same length is 4 whole walls and one half wall. So 6 times 3/4 = 4.5 walls per hour.Each 1/6 hour paints 3/4 of a wall1/6 h1/6 h1/6 h1/6 h1/6 h1/6 hwall 1wall 2wall 3wall 4½ wall6 × 3/4 = 4.5 walls in 1 hour

Answer: 4.5 walls per hour.

Check: In \tfrac{1}{6} hour he paints 4.5 \times \tfrac{1}{6} = 0.75 = \tfrac{3}{4} wall ✓

Worked example

Example 4 (two steps): pay from a rate

Darnell earned $1,035 for 45 hours of work at a constant hourly wage. How much will he earn for 32 hours?

  1. Find the unit rate: \frac{\$1{,}035}{45\text{ h}} = \$23\text{ per hour}
  2. Multiply by the new number of hours: 23 \times 32 = 736
Graph of Darnell pay at 23 dollars per hour. A straight line from the origin passes through (45 hours, 1,035 dollars). Dashed guide lines show that 32 hours gives 736 dollars.Darnell: $23 per hour01020304050$0$200$400$600$800$1,000$1,200Hours workedPay (dollars)(45, $1,035)(32, $736)

Answer: $736.

Check: 32 hours is a bit more than \tfrac{2}{3} of 45 hours, and $736 is a bit more than \tfrac{2}{3} of $1,035 (about $690) ✓

Worked example

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

Store A sells an 18-pack of water for $6.30. Store B sells a 32-pack for $11.52. Maria needs 160 bottles for an event. (a) Which store has the lower unit price? (b) How much does she save by buying all 160 bottles at the cheaper rate instead of the other? (c) With $1.60 left over, how many extra bottles could she buy at the cheaper rate?

  1. Store A: 6.30 \div 18 = 0.35 → $0.35 per bottle.
  2. Store B: 11.52 \div 32 = 0.36 → $0.36 per bottle.
  3. Store A is cheaper per bottle — even though Store B has the bigger pack.
  4. Cost of 160 bottles: A: 0.35 \times 160 = \$56.00; B: 0.36 \times 160 = \$57.60.
  5. Savings: 57.60 - 56.00 = \$1.60.
  6. Work backwards: 1.60 \div 0.35 \approx 4.57, so she can afford 4 more whole bottles.
Bar chart of price per bottle: Store A sells 18 for 6.30 dollars, which is 0.35 dollars per bottle; Store B sells 32 for 11.52 dollars, which is 0.36 dollars per bottle. For 160 bottles Store A costs 56.00 dollars and Store B costs 57.60 dollars, a saving of 1.60 dollars.Price per bottleStore A: 18 for $6.30$0.35Store B: 32 for $11.52$0.36160 bottles: A $56.00 vs B $57.60 → save $1.60

Answer: (a) Store A; (b) $1.60; (c) 4 bottles.

Check: 4 \times 0.35 = \$1.40 \le \$1.60, but 5 \times 0.35 = \$1.75 > \$1.60 ✓ (Round down — you can't buy part of a bottle.)

Calculator tip

Calculator tip (TI-30XS MultiView)

  • Plain division: type the first quantity, press ÷, type the "per" quantity, press enter. For 5.99 \div 32 you will see 0.1871875.
  • Fractions: press n/d to build a fraction such as \tfrac{3}{4}, then use the right arrow to leave the fraction before typing ÷. If the answer shows as a fraction like \tfrac{9}{2}, press toggle to see the decimal 4.5.
  • Best buy: compute each unit price on its own line. Use the up arrow to scroll back and compare the results — don't round them first.
  • When not to bother: simple rates like 150 \div 3 or 84 \div 7 are faster in your head, and they may appear in the no-calculator part of the test. Practice short division by hand.
Common mistake

Common mistakes

  • Dividing the wrong way. $6 for 3 lb: 3 \div 6 = 0.5 is pounds per dollar, not dollars per pound. ✔ Put the "per" unit on the bottom: 6 \div 3 = \$2 per lb.
  • Thinking the biggest package is always the best buy. ✔ Always compute every unit price — sometimes the middle size wins.
  • Rounding too early. $0.2749 and $0.2701 both look like "27¢" if you round to cents first. ✔ Keep 3–4 decimal places until you compare.
  • Mixing time units. 12 miles in 20 minutes is not 0.6 mph. 12 \div 20 = 0.6 miles per minute, which is 0.6 \times 60 = 36 mph. ✔ Check the units in the answer choices.
  • Stopping after step one. In a two-step problem, the unit rate ($23 per hour) is often a wrong answer choice. ✔ Reread what the question actually asks for.
  • Forgetting the coupon or sale. ✔ Subtract the coupon from the price before dividing by the quantity.
Test strategy

GED test strategy

  • Underline the word "per" (or "each," "for every"). Whatever comes after it goes on the bottom of the fraction.
  • Estimate first. 357 miles in 7 hours is about 350 \div 7 = 50. If you get 2,499 or 0.02, you multiplied or flipped.
  • Plug answers back in. If a choice says 51 mph, check: does 51 \times 7 give 357?
  • Check the units in the choices. If the choices are in dollars per ounce and your package is in pounds, convert.
  • Eliminate trap answers. The answer you get if you stop after step one, or multiply instead of divide, is almost always among the choices.
  • Best-buy tables: make a quick list of unit prices on your erasable note board before looking at the choices.
  • Time: most unit-rate questions should take about a minute. If one is taking longer, mark it and come back.

Check yourself

Unit ratesNo calculator

Keisha drove 240 miles and used 8 gallons of gas. How many miles per gallon did her car get?

Answer choices
Hint

"Miles per gallon" means miles for each one gallon. Which quantity should go on the bottom?

Check yourself

Unit ratesNo calculatorFill in

A 12-pack of sparkling water costs $5.40. What is the price of one can?

$

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

Hint

You want the cost per can. Divide the total price by the number of cans.

Summary

Key points

  • A unit rate is an amount per one unit: divide by the quantity that comes after "per."
  • Multiply the unit rate to find a total; divide by the unit rate to work backwards.
  • The best buy has the lowest price per unit — compute every unit price, in the same units, before choosing.
  • Keep extra decimal places while comparing; round only at the end.
  • On a straight-line graph through the origin, the unit rate is y \div x (the slope).
  • The GED sheet gives d = rt and total cost = units \times price per unit; the "divide by per" idea you must remember.
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