How to solve a unit-rate word problem

A rate answers “how much for each one?” Miles per hour, dollars per item, and words per minute all compare two different units. A unit rate rewrites that comparison so the second quantity is one. The units tell you which number goes on top and whether the answer makes sense.

Find the unit rate

Divide the amount being measured by the number of units. If a car travels 180 miles in 3 hours, then 180 miles ÷ 3 hours = 60 miles per hour. If 4 notebooks cost $7.20, then $7.20 ÷ 4 notebooks = $1.80 per notebook. Write the units in every line; “60” alone is incomplete because it does not say 60 what per what.

1. Identify the two quantities and their units.
2. Choose the quantity you want “per 1” and divide by its count.
3. Reduce or convert units if the problem uses different scales, such as minutes and hours.
4. Check the answer against the story: miles per hour should be distance divided by time, while hours per mile reverses that rate.

Compare two rates fairly

Before deciding which option is faster, cheaper, or more productive, put both rates in the same units. If one printer produces 42 pages in 6 minutes and another produces 55 pages in 10 minutes, the rates are 7 pages per minute and 5.5 pages per minute. The first printer has the higher output rate. Comparing 42 directly with 55 would ignore the different time periods.

Scale a rate to solve for a missing quantity

Once you know a unit rate, multiply it by the number of units in the question. At 60 miles per hour, 2.5 hours corresponds to 60 × 2.5 = 150 miles if the rate remains constant. For a unit price of $1.80 per notebook, 7 notebooks cost 1.80 × 7 = $12.60. Match units before multiplying so the unwanted unit cancels.

Common errors

• Reversing the division and answering in the opposite units.
• Comparing totals when the time, distance, number of items, or other base differs.
• Mixing minutes with hours or cents with dollars without converting.
• Dropping the units from the answer and losing the meaning of the rate.
• Assuming the rate stays constant when the question describes changing speed, price, or output.

On the FTCE, translate the rate into words before calculating: “miles for each hour,” “dollars for each item,” or “pages for each minute.” Then write the matching division. The unit labels catch most setup errors before they reach the arithmetic.

Reduce a rate to one unit

A unit rate compares quantities with a denominator of 1. If 240 miles are traveled in 4 hours, the speed is 240 ÷ 4 = 60 miles per hour. If 6 notebooks cost $9, the unit price is $9 ÷ 6 = $1.50 per notebook. Write the units with the calculation so you divide the correct quantities and interpret the result correctly.

A fraction such as 3/5 of a mile per minute may already be a rate, but it is not yet expressed per one minute as a whole-number or decimal unit rate. Divide 3 by 5 to get 0.6 mile per minute. If the question asks for feet per minute, convert miles to feet as well. A numerical answer without its unit can hide a conversion error.

Compare rates using a common basis

To compare two packages, convert both prices to the same unit. A 12-ounce package for $3 costs $0.25 per ounce; a 20-ounce package for $4.60 costs $0.23 per ounce. The second package has the lower unit price, assuming quantity and quality are comparable. Do not compare total prices alone when package sizes differ.

Rates should use compatible units before comparison. A cyclist traveling 18 miles in 1.5 hours averages 12 miles per hour. Another traveling 5 kilometers in 20 minutes averages 15 kilometers per hour, but the figures cannot be compared directly until distance or time units are converted. Use 1 mile ≈ 1.609 kilometers or convert both distances to one unit.

Use unit rates in proportional reasoning

When a situation has a constant rate, multiply the unit rate by the number of units to find a total. At 60 miles per hour, 2.5 hours corresponds to 150 miles. Reverse the relationship by dividing total distance by speed to find time. These operations work only if the rate remains constant over the interval described.

A unit rate can also reveal whether a table is proportional: if the ratio of output to input remains constant, the relationship is direct proportionality. A fixed fee changes the structure; a taxi fare with a base charge plus a per-mile charge is not modeled by one constant cost-per-mile from zero, even though the additional distance has a unit rate. Separate fixed and variable costs when the problem includes both.

• Divide the quantity in the numerator by the denominator quantity.
• Keep the units and state the result per one denominator unit.
• Convert rates to the same units before comparing.
• Multiply or divide by a constant rate only when the context supports constancy.
• Account for fixed charges separately from variable unit costs.

Use a unit rate to scale up or down

Once a unit rate is known, multiply by the number of units for a total or divide a total by the rate to find how many units fit. If a machine fills 18 containers in 6 minutes, its rate is 3 containers per minute. At a steady pace it fills 45 containers in 15 minutes. If the rate changes with setup time or a fixed fee, the direct scaling may no longer hold.

For a better-buy problem, compare unit prices in a common unit and check package quality, usable quantity, and any fees. A lower price per ounce does not help if much of the product is unusable or the package includes a one-time charge. The calculation provides the numerical comparison; context determines whether the options are equivalent.