2.3 Decimals: Place Value, Operations & Conversions

Decimal Place Value

Decimals are another way to write fractions whose denominator is a power of 10 (10, 100, 1000, ...). To the right of the decimal point: tenths (0.1), hundredths (0.01), thousandths (0.001), ten-thousandths (0.0001), each one-tenth of the place to its left.

Worked example. In 7.304:

• 7 is in the ones place (7)
• 3 is in the tenths place (0.3)
• 0 is in the hundredths place (0.00)
• 4 is in the thousandths place (0.004)

So 7.304 = 7 + 0.3 + 0.004.

Trailing zeros after the decimal do not change value: 5.20 = 5.200 = 5.2. But leading zeros inside the decimal do matter: 0.05 ≠ 0.5.

Comparing and Ordering Decimals

Line up the decimal points and compare from left to right. Padding with trailing zeros makes this easier.

Worked example. Order 0.6, 0.608, and 0.68 from least to greatest.

1. Pad: 0.600, 0.608, 0.680.
2. Compare tenths: all are 6. Compare hundredths: 0, 0, 8. So 0.600 and 0.608 come before 0.680.
3. Compare thousandths for the first two: 0 < 8, so 0.600 < 0.608.
4. Least to greatest: 0.6, 0.608, 0.68.

Rounding Decimals

To round to a given place, look at the digit immediately to its right.

• If it is 5 or greater, round up (add 1 to the rounding place).
• If it is less than 5, leave the rounding place unchanged.
• Drop all digits to the right of the rounding place.

Worked example. Round 3.476 to the nearest hundredth. The hundredths place is 7; the next digit is 6 (≥5), so round up: 3.48.

Worked example. Round 12.0349 to the nearest tenth. The tenths place is 0; the next digit is 3 (<5), so leave it: 12.0.

Nursing example. A weight of 62.348 kg rounded to the nearest tenth is 62.3 kg (next digit 4 < 5).

Adding and Subtracting Decimals

Line up the decimal points. Add or subtract as with whole numbers. Place the decimal point in the answer directly under the decimal points of the numbers.

Worked example. 4.25 + 3.8.

• Line up: 4.25 + 3.80 = 8.05.

Worked example. 12.4 - 5.73.

• Line up: 12.40 - 5.73 = 6.67.

Common trap: not lining up decimals. Writing 4.25 + 3.8 as if 5 and 8 align gives a wrong answer. Always line up the points, not the right edges.

Multiplying Decimals

Multiply as with whole numbers, then count the total decimal places in the factors. The answer has that many decimal places.

Worked example. 2.4 × 0.3.

• 24 × 3 = 72. One decimal place in 2.4 plus one in 0.3 = two total. Answer: 0.72.

Worked example. 0.05 × 4.

• 5 × 4 = 20. Two places in 0.05 plus zero in 4 = two total. Answer: 0.20 = 0.2.

Worked example. 1.25 × 0.4.

• 125 × 4 = 500. Two places plus one place = three total. Answer: 0.500 = 0.5.

Dividing Decimals

If the divisor is a whole number, divide and place the decimal point in the quotient directly above the decimal point in the dividend.

If the divisor is a decimal, move the decimal point in both numbers the same number of places to the right until the divisor is a whole number. Then divide.

Worked example. 6.4 ÷ 0.8.

• Move one place right in both: 64 ÷ 8 = 8.

Worked example. 0.75 ÷ 0.25.

• Move two places: 75 ÷ 25 = 3.

Worked example. 4.2 ÷ 0.07.

• Move two places: 420 ÷ 7 = 60.

If the division does not come out even, add zeros to the dividend and keep dividing.

Terminating vs. Repeating Decimals

A terminating decimal ends (e.g., 1/4 = 0.25). A repeating decimal has a digit or block that repeats forever: 1/3 = 0.333... = 0.3̄, 2/3 = 0.666... = 0.6̄, 1/6 = 0.1666... = 0.16̄.

Use a bar over the repeating block or an ellipsis: 0.8181... = 0.81̄.

Converting Fractions to Decimals

Divide the numerator by the denominator.

• 3/4 = 3 ÷ 4 = 0.75.
• 1/5 = 1 ÷ 5 = 0.2.
• 2/3 = 2 ÷ 3 = 0.6̄.
• 5/8 = 5 ÷ 8 = 0.625.

To check: the decimal times the denominator should give back the numerator. 0.75 × 4 = 3. ✓

Converting Decimals to Fractions

Say the decimal in place-value words, write it as a fraction over that power of 10, then simplify.

• 0.6 → "six tenths" → 6/10 → 3/5.
• 0.45 → "forty-five hundredths" → 45/100 → 9/20.
• 0.125 → "one hundred twenty-five thousandths" → 125/1000 → 1/8.

Converting Percents and Decimals

A percent is a ratio out of 100.

• Percent → decimal: divide by 100 (move the decimal two places left). 45% = 0.45; 8% = 0.08.
• Decimal → percent: multiply by 100 (move the decimal two places right). 0.3 = 30%; 0.075 = 7.5%.
• Percent → fraction: write the percent over 100 and simplify. 60% = 60/100 = 3/5.

The full percents chapter covers percent change, discounts, and dosage percent problems in depth.

Common Traps

• Forgetting to count decimal places when multiplying.
• Not shifting the divisor's decimal in division.
• Writing 0.5 and 0.05 as if they are equal (they differ by a factor of 10).
• Leaving the answer unrounded when the problem specifies a place.

Nursing example. A dose is 0.25 mg and the supply concentration is 0.5 mg/mL. Volume needed = 0.25 ÷ 0.5 = 0.5 mL. Decimals are everywhere in dosage math, so precision matters.