Unit rates

1. The question asks for miles per gallon, so gallons go on the bottom of the fraction.
2. \frac{240\text{ miles}}{8\text{ gallons}} = 240 \div 8
3. 240 \div 8 = 30

Her car got 30 miles per gallon.

Check: At 30 miles for each gallon, 8 gallons take her 30 \times 8 = 240 miles ✓

Tip: The word per tells you what to divide by. "Miles per gallon" → miles ÷ gallons. If your answer is huge (like 1,920) you multiplied; if it is tiny (like 0.03) you divided the wrong way.

1. Find the unit rate (pay per hour): \frac{\$138}{8\text{ h}} = \$17.25\text{ per hour}
2. Multiply by the new number of hours: 17.25 \times 30 = 517.50

He will earn $517.50.

Check: 30 hours is 30 \div 8 = 3.75 times as long as 8 hours, and 138 \times 3.75 = 517.50 ✓

Tip: Two-step rate problems follow the same pattern: divide to get the unit rate, then multiply by the new amount. The unit rate itself ($17.25) is usually a trap answer — make sure you finish the second step.

1. The question asks for pages per minute, so minutes go on the bottom.
2. \frac{168\text{ pages}}{7\text{ minutes}} = 168 \div 7
3. Divide in parts: 140 \div 7 = 20 and 28 \div 7 = 4, so 168 \div 7 = 20 + 4 = 24.

The printer prints 24 pages per minute.

Check: 24 \times 7 = 168 pages in 7 minutes ✓

Tip: The unit after per is the divisor. Splitting 168 into 140 + 28 (both easy multiples of 7) makes mental division fast on the no-calculator part.

1. Find the unit rate: \frac{54\text{ cupcakes}}{6\text{ min}} = 9\text{ cupcakes per minute}
2. Multiply the unit rate by the new time: 9 \times 25 = 225

The machine frosts 225 cupcakes in 25 minutes.

Check: Set up a proportion: \frac{54}{6} = \frac{225}{25}, and both sides equal 9 ✓

Tip: Rates grow by multiplying, not by adding. Going from 6 minutes to 25 minutes does not add 19 cupcakes; each extra minute adds 9 cupcakes. Find the unit rate first, then scale it.

1. Divide price by ounces for each box:
   • 12 oz: 3.48 \div 12 = 0.290 dollars per ounce
   • 18 oz: 4.86 \div 18 = 0.270 dollars per ounce
   • 24 oz: 6.60 \div 24 = 0.275 dollars per ounce
   • 32 oz: 9.28 \div 32 = 0.290 dollars per ounce
2. Compare: 0.270 < 0.275 < 0.290.

The 18-oz box is the best buy, at $0.27 per ounce.

Check: If the 24-oz box were priced at the 18-oz rate it would cost 0.27 \times 24 = \$6.48. It actually costs $6.60, which is more, so the 18-oz box really is the better deal ✓

Tip: Bigger is not always cheaper per unit, and the lowest sticker price is not the best buy. Compute every unit price, and don't round 0.275 down to 0.27 — that half a cent per ounce decides the question.

1. The line starts at (0, 0), so distance and time are proportional.
2. Use the point (2, 130): in 2 hours the car traveled 130 miles. \text{speed} = \frac{130\text{ mi}}{2\text{ h}} = 65\text{ mi/h}
3. Confirm with the other point (4, 260): \frac{260}{4} = 65

The car's speed is 65 miles per hour.

Check: The slope between the two points is \dfrac{260 - 130}{4 - 2} = \dfrac{130}{2} = 65 ✓ — the same as the unit rate.

Tip: On a graph through the origin, the unit rate is the slope: y \div x for any point on the line. Reading a y-value alone (130 or 260) gives a distance, not a rate.

1. Convert the time to minutes: 8\text{ h} \times 60\ \tfrac{\text{min}}{\text{h}} = 480\text{ min}
2. Divide milliliters by minutes: \frac{1{,}000\text{ mL}}{480\text{ min}} = 2.0833\ldots\ \text{mL/min}
3. Round to the nearest hundredth: 2.08.

The flow rate is about 2.08 mL per minute.

Check: 2.08 \times 480 = 998.4 \approx 1{,}000 mL ✓ (the small gap is from rounding).

Tip: Before dividing, make the units match what the question asks for. Here the per-hour rate (1{,}000 \div 8 = 125 mL/h) is a tempting wrong choice; dividing it by 60 also works: 125 \div 60 \approx 2.08.

1. Her pace says 1 mile takes 20 minutes. Speed in miles per hour means miles ÷ hours.
2. Change 20 minutes to hours: 20\text{ min} = \frac{20}{60}\text{ h} = \frac{1}{3}\text{ h}
3. Divide: \frac{1\text{ mile}}{\tfrac{1}{3}\text{ h}} = 1 \times 3 = 3\text{ miles per hour}

Rosa hikes at 3 miles per hour.

Check: In one hour (60 minutes) she completes 60 \div 20 = 3 twenty-minute miles ✓

Tip: "Minutes per mile" (a pace) is upside down compared with "miles per hour" (a speed). To switch, divide 60 by the pace: 60 \div 20 = 3. A bigger pace number means a slower hiker.

Convert each result to words per minute.

1. Andre: 3{,}000 \div 60 = 50\text{ words per minute}
2. Bianca: 270 \div 5 = 54\text{ words per minute}
3. Chen: 0.85 \times 60 = 51\text{ words per minute}
4. Dana: 104 \div 2 = 52\text{ words per minute}
5. Compare: 54 > 52 > 51 > 50.

Bianca typed fastest, at 54 words per minute.

Check: In 10 minutes Bianca would type 540 words, Dana 520, Chen 510 and Andre 500 — Bianca is still ahead ✓

Tip: Don't compare the raw numbers (3,000 looks huge, but it covers a whole hour). Convert every rate to the same units first, then compare.

1. Job A passes through (4,\ 60): \frac{\$60}{4\text{ h}} = \$15\text{ per hour}
2. Job B passes through (5,\ 90): \frac{\$90}{5\text{ h}} = \$18\text{ per hour}
3. Difference: \$18 - \$15 = \$3

Job B pays $3 more per hour.

Check: After 10 hours Job A pays 10 \times 15 = \$150 and Job B pays 10 \times 18 = \$180; the gap is $30, which is $3 for each of the 10 hours ✓

Tip: To compare rates on a graph, find each unit rate (y \div x) from a point you can read exactly. The steeper line has the greater rate. Comparing two points with different x-values compares totals, not rates.

1. Use the first row: \frac{\$55.50}{3\text{ h}} = \$18.50\text{ per hour}
2. Check the second row: \$92.50 \div 5 = \$18.50 ✓
3. Check the third row: \$148.00 \div 8 = \$18.50 ✓

She earns $18.50 per hour.

Check: From 3 hours to 5 hours, earnings rise by 92.50 - 55.50 = \$37.00 for 2 more hours, and 37.00 \div 2 = \$18.50 ✓

Tip: In a proportional table, earnings ÷ hours is the same in every row — that constant is the unit rate. If you use the change between two rows, remember to divide by the change in hours too.

1. Center A: in C = 12.5h, the number multiplying h is the unit rate: $12.50 per hour.
2. Center B: \frac{\$27}{2\text{ h}} = \$13.50\text{ per hour} and \$54 \div 4 = \$13.50 ✓
3. Compare: 13.50 - 12.50 = 1.00, so Center A costs $1.00 less per hour.

Center A is cheaper by $1.00 per hour.

Check: For 4 hours, Center A costs 12.5 \times 4 = \$50 and Center B costs $54. The $4 gap is $1 for each of the 4 hours ✓

Tip: In a proportional equation y = kx, the number multiplying x is the unit rate. To compare it with a table or graph, turn that representation into a unit rate too, then compare like with like.

1. Brand X after the coupon: 7.68 - 1.00 = \$6.68
2. Brand X unit price: \frac{\$6.68}{64\text{ oz}} = \$0.104375 \approx 10.4\text{ cents per oz}
3. Brand Y unit price: \frac{\$5.28}{48\text{ oz}} = \$0.11 = 11.0\text{ cents per oz}
4. Compare: 10.4 < 11.0 cents, so Brand X with the coupon is the better buy.

The better buy costs about 10.4¢ per ounce (Brand X with the coupon).

Check: At 10.4375¢ per ounce, 64 oz costs 64 \times 0.104375 = \$6.68, and adding back the coupon gives $7.68 ✓

Tip: Without the coupon, Brand X is 12.0¢ per ounce and Brand Y wins. Coupons and sales can flip the answer — always find the price you will actually pay, then divide.

1. Unit rate: \frac{45\text{ gal}}{20\text{ min}} = 2.25\text{ gal per minute}
2. Change the time to minutes: 1.5\text{ h} \times 60 = 90\text{ min}
3. Multiply: 2.25 \times 90 = 202.5\text{ gal}

The sprinkler uses 202.5 gallons in 1.5 hours.

Check: 90 minutes is 90 \div 20 = 4.5 twenty-minute blocks, and 4.5 \times 45 = 202.5 gallons ✓

Tip: Before you multiply a rate by a time, make sure the time units match. The rate is per minute, so 1.5 hours must become 90 minutes first.

1. Write the bill as fee + rate × hours: 85 + 62h = 333
2. Subtract the fee: 62h = 333 - 85 = 248
3. Divide by the hourly rate: h = 248 \div 62 = 4

The plumber worked 4 hours.

Check: 85 + 62 \times 4 = 85 + 248 = 333 ✓

Tip: When a total includes a one-time fee, only the part after the fee depends on the rate. Undo the steps in reverse order: subtract the fee, then divide by the rate.

1. Unit rate from a complete column: \frac{26\text{ pages}}{4\text{ min}} = 6.5\text{ pages per minute} Check: 65 \div 10 = 6.5 ✓
2. Work backwards: minutes = pages ÷ rate: 117 \div 6.5 = 18\text{ minutes}

It takes 18 minutes to print 117 pages.

Check: 6.5 \times 18 = 117 ✓, so the ratio \tfrac{117}{18} = 6.5 matches the other columns.

Tip: Don't continue a pattern in one row of the table (4, 10, 16, …). In a proportional table, the ratio pages ÷ minutes stays the same, so use the unit rate to fill in a missing value.

1. Write the time in hours. 30 minutes is \tfrac{30}{60} = 0.5 hour, so 2\text{ h }30\text{ min} = 2.5\text{ h}
2. Divide gallons by hours: \frac{1{,}250\text{ gal}}{2.5\text{ h}} = 500\text{ gal per hour}

The tank fills at 500 gallons per hour.

Check: 500 \times 2.5 = 1{,}250 gallons ✓ (500 in the first hour, 500 in the second, and 250 in the last half hour)

Tip: "2 hours 30 minutes" is 2.5 hours, not 2.3 hours. Minutes are sixtieths of an hour, so divide the minutes by 60 before writing a decimal.

1. Change 2 pounds to ounces: 2 \times 16 = 32\text{ oz}
2. 12-ounce bag: \frac{\$8.88}{12\text{ oz}} = \$0.74\text{ per ounce}
3. 2-pound bag: \frac{\$22.40}{32\text{ oz}} = \$0.70\text{ per ounce}
4. Difference: 0.74 - 0.70 = 0.04

The 2-pound bag costs $0.04 less per ounce, so it is the better buy.

Check: 32 ounces bought as 12-ounce-bag coffee would cost 32 \times 0.74 = \$23.68, which is more than $22.40 ✓

Tip: Unit prices can be compared only in the same unit. Convert pounds to ounces (×16) first, then divide each price by its number of ounces.

1. Unit rate (miles per day): 2\tfrac{1}{4} \div \tfrac{3}{4} = \tfrac{9}{4} \times \tfrac{4}{3} = \tfrac{36}{12} = 3\text{ miles per day}
2. Work backwards: days = miles ÷ (miles per day): 7\tfrac{1}{2} \div 3 = \tfrac{15}{2} \times \tfrac{1}{3} = \tfrac{15}{6} = 2\tfrac{1}{2}\text{ days}

The crew needs 2\tfrac{1}{2} days.

Check: 3 \times 2\tfrac{1}{2} = 7\tfrac{1}{2} miles ✓

Tip: Change mixed numbers to improper fractions before dividing (2\tfrac{1}{4} = \tfrac{9}{4}, 7\tfrac{1}{2} = \tfrac{15}{2}). Find the unit rate first; then decide whether to multiply (to find miles) or divide (to find days) by looking at the units.

1. Quarts per second: 3 \div 40 = 0.075\ \tfrac{\text{qt}}{\text{s}}
2. Quarts per hour (3,600 seconds in an hour): 0.075 \times 3{,}600 = 270\ \tfrac{\text{qt}}{\text{h}}
3. Gallons per hour (divide by 4, because 4 quarts make a gallon): 270 \div 4 = 67.5\ \tfrac{\text{gal}}{\text{h}}

An equivalent rate is 67.5 gallons per hour.

Check: In 40 seconds the machine fills 3 quarts = 0.75 gallon. There are 3{,}600 \div 40 = 90 forty-second blocks in an hour, and 90 \times 0.75 = 67.5 gallons ✓

Tip: Change one unit at a time and ask whether each change should make the number bigger or smaller. A gallon is bigger than a quart, so the number of gallons must be smaller (divide by 4).