5.3 Decimals, Money, and Rounding

5.3 Decimals, Money, and Rounding

Quick Summary: OPM names decimals as Occupational Math content. In practice decimals appear most often as money, and money problems bring a specific hazard: because currency is conventionally shown to two places, candidates round early and drift into a distractor. The rule is to carry full precision and round once, at the end, to the precision the question requires.

Place value

Cents are hundredths. A rate quoted to three or four places — for instance a cost of 0.0725 per unit — is common in practical problems, and truncating it to 0.07 before multiplying by a large quantity produces a materially wrong total.

The four operations

Adding and subtracting: align the point

Write the numbers so decimal points line up, filling with zeros as needed.

• 12.4 + 3.75 + 0.6: align as 12.40, 3.75, 0.60 → 16.75.
• 100 − 37.85: align as 100.00 − 37.85 → 62.15.

With a calculator this is automatic; the reason to know it is checking a result by hand when a total looks wrong.

Multiplying: count the decimal places

Multiply as whole numbers, then place the point so the product has as many decimal places as the factors combined.

• 3.5 × 0.4: 35 × 4 = 140. One decimal place plus one gives two, so 1.40 = 1.4.
• 12.25 × 3: 1225 × 3 = 3675. Two decimal places, so 36.75.

This rule is a magnitude check. If a calculator entry slips and you get 14 instead of 1.4, the place-value count catches it.

Dividing: shift both numbers

Move the decimal point in the divisor until it is a whole number, and move it the same number of places in the dividend.

• 4.68 ÷ 0.4 → 46.8 ÷ 4 = 11.7.

Again, note that dividing by a number less than 1 increases the value.

Converting between forms

The percent-to-decimal conversion is the one worth over-learning, because Chapter 6 uses it constantly. 6.5% is 0.065, not 0.65. A tenfold error here is the most common single mistake in percentage problems.

Rounding correctly

Standard rounding looks at the digit immediately to the right of the place you are rounding to. If it is 5 or more, round up; if less than 5, round down.

The 8.995 row is worth pausing on. Rounding to two decimals gives 9.00 because the third decimal is 5, which cascades through the 9s. Cascading rounds catch people out.

Round once, at the end

A unit orders 1,450 units of a supply item at $0.0725 each, and pays a $12.50 handling charge. What is the total cost to the nearest cent?

Correct: 1,450 × 0.0725 = 105.125. Add 12.50 → 117.625. Round to the cent → $117.63.

Wrong (rounded early): 1,450 × 0.07 = 101.50, plus 12.50 → $114.00.

The gap is $3.63 on a small order, and it grows with volume. A distractor built on the early round is a standard construction.

A second discipline: round to the precision the question asks for. If a question asks for a whole number of forms, an answer of 46.7 forms means 47 forms only if the context is a count achieved; if it is "how many complete batches", the answer is 46, because a partial batch is not complete. Read the context.

Money-specific habits

• Keep dollars and cents in one unit. Mixing $4 and 75¢ in the same expression invites a factor-of-100 error. Work in dollars: 4.75.
• Watch "per" carefully. "$0.23 per page" times 340 pages is 78.20, but "$0.23 per page for the first 100 pages and $0.18 thereafter" is 100 × 0.23 + 240 × 0.18 = 23.00 + 43.20 = $66.20. Tiered rates are common and are computed piecewise.
• Sanity-check the magnitude. Before submitting, ask whether the answer is roughly the size you would expect. An order of 1,450 items at about seven cents each should land near $100, so an answer of $1,051 or $10.51 signals a misplaced decimal point.