6.3 Ratios, Proportions, and Unit Rates
Key Takeaways
- OPM names proportions explicitly as Occupational Math content.
- A ratio compares two quantities; a proportion states that two ratios are equal and is solved by cross-multiplication.
- Setting up a proportion correctly requires keeping like units in matching positions on both sides.
- A unit rate expresses how much of one quantity corresponds to a single unit of another, and makes different offers directly comparable.
- Part-to-part ratios must be converted to part-to-whole before they can be applied to a total.
6.3 Ratios, Proportions, and Unit Rates
Quick Summary: OPM lists proportions among the topics the Occupational Math Assessment covers. A ratio compares two quantities; a proportion sets two ratios equal and is solved by cross-multiplication. The skill that decides these items is setting the proportion up with matching units in matching positions, not the algebra of solving it.
Ratios
A ratio compares quantities and can be written 3:5, 3 to 5, or 3/5. Ratios are usually reduced to lowest terms, so 12:20 becomes 3:5.
Part-to-part versus part-to-whole is the distinction that decides most ratio items.
A unit's staff are assigned to intake and review in the ratio 3:5.
That is part to part. It does not say three-fifths are in intake. To reach the whole, add the parts: 3 + 5 = 8 total shares. So intake is 3/8 of staff and review is 5/8.
If the unit has 48 staff, how many work in intake?
48 ÷ 8 = 6 staff per share. Intake = 3 × 6 = 18. Review = 5 × 6 = 30. Check: 18 + 30 = 48. ✓
The trap is computing 3/5 of 48 = 28.8, treating a part-to-part ratio as part-to-whole. Whenever you see a ratio applied to a total, add the shares first.
Dividing a total in a given ratio
The method generalises to any number of parts.
Supplies costing $1,260 are shared between three offices in the ratio 2:3:4.
Total shares: 2 + 3 + 4 = 9. Per share: 1,260 ÷ 9 = 140. Amounts: 2 × 140 = $280, 3 × 140 = $420, 4 × 140 = $560. Check: 280 + 420 + 560 = 1,260. ✓
Proportions
A proportion states two ratios are equal. Solve by cross-multiplying.
If 4 staff process 260 forms in a day, how many forms would 7 staff process at the same rate?
Set up with matching units in matching positions:
4 staff / 260 forms = 7 staff / x formsCross-multiply: 4x = 260 × 7 = 1,820. So x = 1,820 ÷ 4 = 455 forms.
Check the magnitude: more staff should process more forms, and 455 > 260. ✓
Setting up correctly
The only real difficulty is arrangement. Two rules make it reliable:
- Same units in the same position. Staff over forms on the left means staff over forms on the right. Never staff over forms on one side and forms over staff on the other.
- Sanity-check the direction. Before computing, decide whether the answer should be larger or smaller than the known value.
That second check catches inverted setups. If you had written 4/260 = x/7 you would get x ≈ 0.108, which is not a plausible number of forms.
Inverse relationships
Not every relationship scales directly. If a fixed amount of work is shared, more workers means less time each.
If 3 staff complete a task in 8 hours, how long would 4 staff take at the same rate?
Total work is 3 × 8 = 24 staff-hours. With 4 staff: 24 ÷ 4 = 6 hours.
A direct proportion would wrongly give 8 × 4/3 ≈ 10.7 hours — more staff taking longer, which fails the direction check. Ask whether the quantities move together or in opposite directions before setting up.
Unit rates
A unit rate expresses a quantity per one unit of another: forms per hour, cost per page, minutes per request. Unit rates make unlike offers comparable.
Vendor A: 500 folders for $92. Vendor B: 750 folders for $135. Which is cheaper per folder?
A: 92 ÷ 500 = $0.184 per folder. B: 135 ÷ 750 = $0.18 per folder. Vendor B is cheaper, by $0.004 per folder.
The margin is small, which is the point: comparing totals ($92 versus $135) or quantities tells you nothing about value. Only the rate does.
An employee processes 91 forms in 3.5 hours. What is the rate per hour, and how long for 156 forms?
Rate: 91 ÷ 3.5 = 26 forms per hour. Time: 156 ÷ 26 = 6 hours.
Watch the direction of the rate
"Forms per hour" and "hours per form" are reciprocals and answer different questions.
- 26 forms per hour.
- 1 ÷ 26 ≈ 0.0385 hours per form, which is about 2.3 minutes per form.
When a question asks how long one item takes, you want the second form. When it asks how many are done in a period, you want the first.
Scaling recipes and mixtures
A mixture requires cleaning concentrate and water in a ratio of 1:12. How much concentrate is needed for 39 litres of finished mixture?
Total shares: 1 + 12 = 13. Per share: 39 ÷ 13 = 3. Concentrate = 1 × 3 = 3 litres; water = 12 × 3 = 36 litres. Check: 3 + 36 = 39. ✓
Again the part-to-whole conversion does the work. A candidate computing 39 ÷ 12 = 3.25 has used the wrong denominator by treating the 12 as the whole rather than as one part of 13.
A unit's 48 staff are assigned to intake and review in the ratio 3:5. How many work in intake?
If 3 staff complete a task in 8 hours, how long would 4 staff take working at the same rate?
Vendor A offers 500 folders for $92 and Vendor B offers 750 folders for $135. Which is cheaper per folder?