Name the reference quantity
A ratio compares two quantities in a specific order. A unit rate puts the reference quantity on a one-unit basis. If a price of $8.40 pays for six items, the rate in dollars per item is 8.40 ÷ 6. Write the unit beside the result to check which way you divided. In a different question, items per dollar would reverse the ratio. When comparing two offers, convert both to the same direction and the same unit before deciding which is smaller.
Worked example with a changing count
A community workshop fills 96 seats one month and 108 the next. The change is 12 seats, and the starting count is 96. Dividing 12 by 96 gives 1/8, or 12.5%. You can check the result by multiplying 96 by 1.125, which returns 108. The ratio 108/96 is the new count as a share of the old count, or 112.5%. Subtract 100 percentage points from that ratio expressed as a percent to state the percent increase.
Contrast two rates on a common basis
Suppose seven blank notebooks cost $9.10 and five notebooks of the same type cost $6.25. The rates are $1.30 and $1.25 per notebook. The price difference between the two packs compares different quantities, so it cannot answer the per-notebook question directly. The five-notebook pack has the lower unit rate. If a shopper needs eight notebooks and the store sells only full packs, the lowest unit rate does not by itself determine the cheapest purchase that meets the need.
Test a proportional model before scaling
A proportional model keeps the same ratio between output and input. Charging $2 per copy with no initial fee gives cost equal to 2 times the number of copies. An $8 setup fee changes the relationship to an initial 8 dollars plus 2 dollars per copy. The price still increases by $2 when one copy is added. The average price per copy changes with the order size. Check whether the context includes a fixed charge before you double a total.
Transfer check with the direction reversed
A workshop roster changes from 42 names to 35. What percent decrease is that, and what percent increase takes 35 back to 42? Pause after writing a denominator for each direction. The decrease uses 7/42, about 16.7%. The return uses 7/35, exactly 20%. The headcount change has the same size in each direction, but the starting groups differ. Label the reference count when you explain your answer.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original OpenExam explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
College Board ACCUPLACER Math test contentOpenStax Prealgebra 2eOpenStax Elementary Algebra 2e