Math Core

Prices, measurements and sports stats are full of decimals. You already know how to add, subtract, multiply and divide whole numbers. With decimals, the steps are the same. The only new job is keeping track of the decimal point.

Adding and subtracting

Line up the decimal points so that tenths are under tenths, hundredths under hundredths, and so on. Fill empty places with zeros so every number has the same number of decimal places. Then add or subtract like whole numbers, and bring the decimal point straight down.

Worked example: Adding and subtracting

Find $52.7 + 8.394$ and $40 - 17.286$ .

\begin{array}{r} 52.700 \\ +\;\;8.394 \\ \hline 61.094 \end{array} \qquad\qquad \begin{array}{r} 40.000 \\ -\,17.286 \\ \hline 22.714 \end{array}

The whole number 40 has a decimal point after the 0, so it becomes $40.000$ .

Check the subtraction by adding: $22.714 + 17.286 = 40.000$ . ✓

Common mistake

Don't line up the right edges. In $52.7 + 8.394$ , putting the 7 above the 4 would add tenths to thousandths. Always line up the decimal points.

Multiplying

When you multiply, you do not line up the decimal points.

Multiplying decimals

Multiply as if there were no decimal points.
Count the decimal places in both factors together.
Give the product that many decimal places.

Why does this work? $0.3 \times 0.05$ means 3 tenths times 5 hundredths. Tenths times hundredths makes thousandths, and $3 \times 5 = 15$ , so the product is 15 thousandths, or $0.015$ . One decimal place plus two decimal places gives three.

Worked example: Multiplying decimals

Find $3.46 \times 2.8$ .

Multiply the whole numbers: $346 \times 28 = 9{,}688$ .

Count decimal places: $3.46$ has 2 and $2.8$ has 1, for a total of 3. So the product is $9.688$ .

Estimate to check: $3.46 \times 2.8$ is about $3 \times 3 = 9$ . ✓

Dividing by a whole number

Divide exactly as you would with whole numbers. Put the decimal point in the quotient directly above the decimal point in the dividend.

Worked example: A decimal divided by a whole number

Find $57.12 \div 16$ .

\begin{array}{r} 3.57 \\ 16\,\overline{)\,57.12} \\ \underline{-\,48\phantom{.00}} \\ 9\phantom{.}1\phantom{0} \\ \underline{-\,8\phantom{.}0\phantom{0}} \\ 1\phantom{.}12 \\ \underline{-\,1\phantom{.}12} \\ 0 \end{array}

$57 \div 16$ is 3 with 9 left. Bring down the 1 to make 91: $16 \times 5 = 80$ , leaving 11. Bring down the 2 to make 112: $16 \times 7 = 112$ , leaving 0.

The decimal point goes straight up, so $57.12 \div 16 = 3.57$ . Check: $16 \times 3.57 = 57.12$ . ✓

If you run out of digits but still have a remainder, write a 0 at the end of the dividend and keep going. For example, $9 \div 4$ : write $9.00$ , and you get $2.25$ .

Dividing by a decimal

To divide by a decimal, turn the divisor into a whole number first. Multiply the divisor and the dividend by the same power of 10. This is like writing an equivalent fraction, so the quotient doesn't change:

8.64 \div 0.36 = \frac{8.64}{0.36} = \frac{8.64 \times 100}{0.36 \times 100} = \frac{864}{36}

Dividing by a decimal

Move the decimal point in the divisor to the right until it is a whole number. Move the decimal point in the dividend the same number of places, adding zeros if needed. Then divide as usual.

Worked example: Dividing by a decimal

Find $8.64 \div 0.36$ .
Find $7 \div 0.4$ .

Solutions.

Move both decimal points two places: $864 \div 36 = 24$ . Check: $0.36 \times 24 = 8.64$ . ✓
Move both decimal points one place: $7 \div 0.4$ becomes $70 \div 4$ . $4 \times 17 = 68$ , leaving 2. Write $70.0$ and bring down the 0: $20 \div 4 = 5$ . So $7 \div 0.4 = 17.5$ .

Tip

Dividing by a number less than 1 gives an answer bigger than the dividend. There are 17.5 groups of 0.4 in 7, which makes sense because 0.4 is small. Use this to catch a misplaced decimal point.