Operations with Decimals

1. Step 1: Line up the decimal points. Write the numbers one under the other so their decimal points are in a vertical line.
2. Step 2: Fill in the gaps. Add zeros to the end of the numbers so they all have the same length after the decimal point.
3. Step 3: Add or subtract. Work from right to left, column by column, as you would with whole numbers.
4. Step 4: Place the decimal point. Bring the decimal point straight down into your answer.

Lining up the decimal points lines up the place values: ones under ones, tenths under tenths, hundredths under hundredths. The addition then combines like units with like units, exactly as with whole numbers.

Ones	.	Tenths	Hundredths
3	.	8	3
2	.	7	0

Adding a zero at the end (\(2.7 \rightarrow 2.70\)) does not change the number, because \(2.7 = 2.70\): seven tenths are seventy hundredths. The zero only fills the empty hundredths column.

1. Step 1: Ignore the decimals. Write the calculation as if the numbers were whole numbers. You do not need to line up the decimal points.
2. Step 2: Multiply. Perform the multiplication as you would with whole numbers.
3. Step 3: Count the decimal places. Count the digits after the decimal point in each factor and add these counts (a whole-number factor adds none).
4. Step 4: Place the decimal point. In your answer, place the decimal point so it has the same number of decimal places you counted in Step 3.

A decimal number is a decimal fraction: \(3.48 = \dfrac{348}{100}\) and \(2.9 = \dfrac{29}{10}\), because dividing by \(100\) or by \(10\) moves the decimal point to the left (see the sections on multiplying and dividing by \(10\), \(100\) and \(1\,000\)). Therefore $$3.48 \times 2.9 = \frac{348}{100} \times \frac{29}{10} = \frac{348 \times 29}{1000} = \frac{10\,092}{1000} = 10.092 .$$ The \(2\) decimal places of \(3.48\) divide the product by \(100\); the \(1\) decimal place of \(2.9\) divides it by \(10\). In total, the whole-number product \(348 \times 29\) is divided by \(1000\), so the result has \(2 + 1 = 3\) decimal places. Counting the decimal places of both numbers is counting the zeros of this division by a power of \(10\).

1. Step 1: Set up the division. Write the problem in the long division format.
2. Step 2: Place the decimal point. Write the decimal point in the answer space, directly above the decimal point in the number being divided.
3. Step 3: Divide from left to right. Divide as you would with whole numbers, ignoring the decimal point now.

When the digits of the dividend are used up, the Euclidean division stops there: a whole-number quotient and a remainder. For a decimal quotient, the division is continued, because appending zeros after the decimal point does not change the value of a number: \(10 = 10.0 = 10.00\).

1. Step 1: Use up the digits. Divide as usual. If the remainder is not zero after the last digit of the dividend, continue.
2. Step 2: Write the decimal point and append a zero. Write the decimal point in the quotient (if it is not there already). Append a zero to the dividend (\(10\) becomes \(10.0\)) and bring this zero down.
3. Step 3: Keep dividing. Continue the division. Append another zero each time the digits run out, until the remainder is zero or the quotient reaches the requested number of decimal places.

If the remainder is never zero, the quotient never ends: stop at the requested place; the quotient obtained is then an approximate value.

In long division, first convert the divisor to a whole number.

1. Step 1: Make the divisor whole. Move the decimal point in the divisor to the right until the divisor is a whole number. Count how many places you moved it.
2. Step 2: Move the other decimal. Move the decimal point in the dividend the same number of places to the right, adding zeros if there are no digits left (\(6 \div 0.75\) becomes \(600 \div 75\)).
3. Step 3: Divide. Now the problem is a "division by a whole number" problem. Follow the steps of that division.

A quotient does not change when the dividend and the divisor are both multiplied by the same non-zero number: there are as many times \(1.2\) cm in \(4.56\) cm as \(12\) mm in \(45.6\) mm — changing the unit multiplies both numbers by \(10\) without changing the answer. $$4.56 \div 1.2 = (4.56 \times 10) \div (1.2 \times 10) = 45.6 \div 12$$ “Moving both decimal points one place to the right” is exactly “multiplying both numbers by 10”. The method replaces the division by a decimal with an equal division whose divisor is a whole number — a division that can already be set up.

To find what must be added to reach the next whole number, complete place by place: first the hundredths to reach a round tenth, then the tenths to reach the whole. $$5.62 + \square = 6 \qquad\rightarrow\qquad 5.62 + \textcolor{colorprop}{0.38} = 6$$ because \(62\) hundredths \(+ \textcolor{colorprop}{38}\) hundredths \(= 100\) hundredths \(= 1\).

The order of the terms in a sum, and of the factors in a product, can be changed to make round numbers appear:

• Group what completes: \(2.3 + 18.6 + 0.7 = (2.3 + 0.7) + 18.6 = 3 + 18.6 = 21.6\).
• Pair the factors that make 100: \(4.7 \times 25 \times 4 = 4.7 \times 100 = 470\).
• Go through the round number and adjust: \(26 + 399 = 26 + 400 - 1 = 425\), and \(35 \times 19 = 35 \times 20 - 35 = 665\).

Doubling and halving work on decimals exactly as on whole numbers, place by place:

• Double: the double of \(3.6\) is \(7.2\) (double \(3\), double \(6\) tenths \(= 12\) tenths).
• Half: the half of \(8.4\) is \(4.2\) — and the half of an odd number is a decimal: the half of \(9\) is \(4.5\) (\(8 \div 2 = 4\) and \(1 \div 2 = 0.5\)).
• Divide by 4 or by 8 by halving: \(\div 4\) is the half of the half; \(\div 8\) is one half more. \(60 \div 4 = 30 \div 2 = 15\).

The most frequent error in decimal calculations is not a wrong digit but a misplaced decimal point — a result ten or a hundred times too large or too small. A quick estimate catches it.

1. Step 1: Round. Round each number to the nearest whole number (or to a simple nearby number).
2. Step 2: Estimate. Compute mentally with the rounded numbers.
3. Step 3: Compare. Compare the estimate with the exact result: they must be of the same order of magnitude. If they do not, look for the error — most often a misplaced decimal point.