6.1 Percentages: Finding the Part, the Whole, and the Rate

6.1 Percentages: Finding the Part, the Whole, and the Rate

Quick Summary: OPM names percentages twice — in the Occupational Math description and in the competency definition, which speaks of solving practical problems by choosing appropriately from a variety of mathematical techniques such as formulas and percentages. Every basic percentage item reduces to one relationship: part = rate × whole. Which of the three quantities is missing tells you which arrangement to use.

One relationship, three cases

Before anything else, identify which quantity you are missing. Most percentage errors are not arithmetic errors; they are cases of using the wrong arrangement.

A reliable clue is the word of. The number attached to "of" is the whole. In "18% of 450", the whole is 450. In "36 is 15% of what number", the whole is the unknown.

Case 1 — Finding the part

Convert the percent to a decimal and multiply.

A unit received 450 applications and 18% required additional documentation. How many required additional documentation?

0.18 × 450 = 81.

The conversion is the danger point. 18% is 0.18. A percentage such as 6.5% becomes 0.065, not 0.65 — the tenfold error flagged in Section 5.3.

Case 2 — Finding the whole

Divide the part by the rate as a decimal.

After a 15% deposit of $36 was paid, what was the full amount?

36 ÷ 0.15 = $240.

Check by reversing: 0.15 × 240 = 36. ✓

The characteristic error here is multiplying instead of dividing: 36 × 0.15 = 5.40, which is far too small to be a whole containing a $36 part. The magnitude check catches it instantly — the whole must be larger than the part whenever the rate is under 100%.

Case 3 — Finding the rate

Divide the part by the whole, then multiply by 100.

Of 240 requests, 84 were submitted by mail. What percent were submitted by mail?

84 ÷ 240 = 0.35 → 35%.

The error here is inverting the division: 240 ÷ 84 ≈ 2.857, which as a percentage would be 285.7% — impossible for a part of a whole. A part cannot exceed 100% of its whole, so any rate above 100% in this kind of problem signals an inverted division.

Percent of a percent

Problems that chain percentages require applying them in sequence.

Of 600 requests, 40% came from the online portal. Of those online requests, 25% required follow-up. How many required follow-up?

Online: 0.40 × 600 = 240. Follow-up: 0.25 × 240 = 60.

Equivalently, 25% of 40% is 10% of the total: 0.10 × 600 = 60.

The trap is applying the second rate to the original total: 0.25 × 600 = 150. As with fractions, the phrase of those tells you the second rate applies to the computed part.

Successive percentages do not add

This is the most consistently mishandled idea in the topic.

A budget line of $50,000 is reduced by 10%, and the reduced figure is then reduced by a further 10%. What remains?

First reduction: 50,000 × 0.90 = 45,000. Second reduction: 45,000 × 0.90 = $40,500.

A 20% single reduction would give $40,000. The two are not the same, because the second 10% applies to a smaller base. Two successive 10% reductions leave 0.90 × 0.90 = 0.81, or 81% of the original.

The same holds for increases. Two successive 10% increases give 1.10 × 1.10 = 1.21, a 21% overall increase.

The third and fourth rows deserve attention: an increase followed by an equal-sized decrease does not return you to the starting value. It always lands slightly below.

The multiplier shortcut

Rather than computing an amount and then adding or subtracting it, multiply once.

A caseload of 320 increases by 8%. New caseload = 320 × 1.08 = 345.6, so 346 cases if the answer must be a whole number.

One multiplication is faster than two operations and removes the risk of adding when you meant to subtract.

Percentage points versus percent

If a completion rate rises from 40% to 45%, that is a rise of 5 percentage points, but a 12.5% increase in the rate itself, because 5 ÷ 40 = 0.125. Both descriptions are correct and they answer different questions. When options include both 5 and 12.5, the question's wording decides.