Decimal Arithmetic

5.C.1Add decimals01/04

Add decimals (tenths and hundredths).

Step01of 04

Line up the decimal points. Stack the numbers vertically so the decimal points are in a column. Then add column by column, just like whole numbers.

4.27 + 1.85 6.12

5.C.1 — Add decimals (tenths and hundredths).

Step 01 of 04

Line up the decimal points. Stack the numbers vertically so the decimal points are in a column. Then add column by column, just like whole numbers.

4.27 + 1.85 6.12

Step 02 of 04

If decimals have different lengths, pad with trailing zeros.

5.7 + 2.34 = 5.70 + 2.34 = 8.04

Trailing zeros after a decimal don't change the value, but they help line up the columns.

Step 03 of 04

The decimal point in the answer goes STRAIGHT DOWN. Don't shift it.

Step 04 of 04

Word problem. A receipt shows $12.45 + $3.78 + $0.99. What's the total?

12.45 + 3.78 + 0.99 = 17.22

Key insight

Line up decimals. Pad with trailing zeros if lengths differ. Add column by column. Decimal point in answer goes straight down.

5.C.2 — Subtract decimals.

Step 01 of 04

Line up decimal points, pad with zeros if needed, subtract column by column. Borrow when the top digit is smaller — same as with whole numbers.

Step 02 of 04

Worked example. 7.5 − 2.83.

Pad: 7.50 − 2.83.

7.50 - 2.83 4.67

Hundredths: 0 − 3 → borrow. 10 − 3 = 7. Tenths is now 4 (was 5). Tenths: 4 − 8 → borrow. 14 − 8 = 6. Ones is now 6 (was 7). Ones: 6 − 2 = 4.

Step 03 of 04

Subtract from a whole number. Pad the whole number with a decimal and zeros.

10 - 3.45 = 10.00 - 3.45 = 6.55

Step 04 of 04

Word problem. A board is 8 ft. You cut off 3.75 ft. How much remains?

8 - 3.75 = 4.25 ft

Key insight

Same rules as whole-number subtraction, plus: line up decimals, pad with zeros. Decimal point in answer goes straight down. Always include UNITS in word-problem answers.

5.C.3 — Multiply decimals by whole numbers.

Step 01 of 04

Multiply as if both were whole numbers. Then place the decimal so the result has the SAME NUMBER of decimal places as the original decimal.

Step 02 of 04

Worked example. 4.7 × 6.

Compute 47 × 6 = 282. The decimal 4.7 has ONE decimal place. So the answer has one decimal place: 28.2.

4.7 \times 6 = 28.2

Step 03 of 04

Two decimal places. 0.35 × 4.

35 \times 4 = 140 \to 1.40 = 1.4

Two decimal places in 0.35 → two in the answer. Trailing zero can drop.

Step 04 of 04

Word problem. Each pencil costs $0.45. Cost of 8 pencils?

0.45 \times 8 = 3.60 = $3.60

Key insight

Multiply as whole numbers, count total decimal places in the original decimal, place the decimal in the answer to match.

5.C.4 — Multiply two decimals.

Step 01 of 04

Multiply as if both were whole numbers. Then place the decimal so the answer has the SUM of decimal places from the two factors.

Step 02 of 04

Worked example. 2.4 × 1.3.

Whole-number product: 24 × 13 = 312. Decimal places: 2.4 has 1 + 1.3 has 1 = 2 total. So answer has 2 decimal places: 3.12.

2.4 \times 1.3 = 3.12

Step 03 of 04

More decimal places. 0.06 × 0.4.

6 × 4 = 24. Decimal places: 2 + 1 = 3. Answer: 0.024 (pad with leading zero to make 3 decimal places).

0.06 \times 0.4 = 0.024

Step 04 of 04

Sanity-check by estimation. 2.4 × 1.3 ≈ 2 × 1 = 2 (or rounder, 3). Our 3.12 is in the ballpark.

Key insight

Multiply as whole numbers. Decimal places in answer = sum of decimal places in factors. Pad with leading zeros if needed.

5.C.5 — Divide decimals by whole numbers.

Step 01 of 04

Set up long division. Place the decimal point in the QUOTIENT directly above the decimal point in the dividend. Then divide as usual.

Step 02 of 04

Worked example. 7.5 ÷ 5.

Decimal in answer goes above the decimal in 7.5. Divide: 5 into 7 → 1, remainder 2. Bring down 5 to make 25. 5 into 25 → 5. No more digits.

7.5 \div 5 = 1.5

Step 03 of 04

Add zeros if dividend doesn't end cleanly.

6.4 \div 8 : 8 into 6 \to 0, into 64 \to 8. Answer 0.8.

Sometimes you need to extend the dividend with trailing zeros to keep dividing.

Step 04 of 04

Word problem. 4 friends share $11.20 equally.

11.20 \div 4 = $2.80 each

Key insight

Decimal in quotient goes ABOVE the decimal in the dividend. Then divide normally. Add trailing zeros if the dividend ends before the division does.

5.C.6 — Divide decimals by decimals.

Step 01 of 04

Convert the divisor to a whole number. Multiply both divisor and dividend by the same power of 10 (shift both decimals the same number of places). Then divide normally.

Step 02 of 04

Worked example. 6.4 ÷ 0.8.

Multiply both by 10: $64 \div 8$ . Easy: $= 8$ .

6.4 \div 0.8 = 8

Step 03 of 04

Two decimal places in divisor. 1.25 ÷ 0.05.

Multiply both by 100: $125 \div 5 = 25$ .

Step 04 of 04

Word problem. A ribbon 3.6 ft long is cut into 0.4-ft pieces. How many pieces?

3.6 \div 0.4 = 36 \div 4 = 9 pieces

Key insight

Shift both decimals the same number of places to make the divisor a whole number. Then divide normally. Same answer, easier mechanics.

5.C.7 — Decimal word problems.

Step 01 of 04

Same word-problem strategies, with decimal arithmetic instead of whole-number arithmetic.

"How much?"	→add or multiply
"How much remaining?"	→subtract
"How many groups?"	→divide
"Each X"	→multiplication or division

Step 02 of 04

Money example. A book costs $14.95, and tax is $1.20. You pay with $20. Change?

14.95 + 1.20 = 16.15

20.00 - 16.15 = $3.85

Step 03 of 04

Multiplication example. Each gallon of paint covers 350 sq ft. You have 2.5 gallons. Total coverage?

2.5 \times 350 = 875 sq ft

Step 04 of 04

Division example. A car uses 0.04 gallons per mile. With 8.4 gallons, how far can it go?

8.4 \div 0.04 = 840 \div 4 = 210 miles

Key insight

Identify the operation, do the decimal arithmetic carefully (line up decimals, count places), answer with units. Estimating first is a great sanity check.

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