Decimals

What each digit means#

In the number 3.478 the whole part is 3, and then come the place values: 4 is tenths, 7 is hundredths, 8 is thousandths. You can read the whole thing as one mixed number: $3.478 = 3\frac{478}{1000}$. The denominator is a one followed by as many zeros as there are digits after the point.

An important consequence: adding zeros on the right does not change a decimal. $0.5 = 0.50 = 0.500$, because $\frac{5}{10} = \frac{50}{100}$. You will use this trick all the time — when comparing and when adding.

Comparing decimals#

Step 1. Compare the whole parts. Whichever is larger wins, and the digits after the point do not matter.

Step 2. If the whole parts are equal, pad with zeros so both numbers have the same number of digits after the point.

Step 3. Compare the resulting numbers place by place, from left to right.

Which is larger, $0.45$ or $0.5$? Pad: $0.50$ against $0.45$. The first number has five tenths, the second has four, so $0.5 > 0.45$. The answer "0.45 is larger because it has more digits" is the most common mistake in the whole topic: the number of digits after the point says nothing about the size of the number.

Adding and subtracting#

There is one rule: point under point. Place values have to line up in columns, otherwise you end up adding tenths to hundredths.

Work out $3.4 + 12.75$.

Step 1. Even out the digits after the point: $3.4 = 3.40$.

Step 2. Write the numbers in a column so the points line up: 3.40 over 12.75.

Step 3. Add as if they were whole numbers: $340 + 1275 = 1615$.

Step 4. Put the point in the answer in the same column as in the numbers you added: $16.15$.

If you right-align the numbers the way you do with whole numbers, you get $13.09$ — the hundredths get added to the tenths. That is the second most common slip in the topic. Subtraction works the same way: $12.75 - 3.4 = 12.75 - 3.40 = 9.35$.

Multiplying decimals#

Here you do not line up the points at all — in fact, you ignore them at first.

Work out $2.5 \times 0.4$.

Step 1. Drop the points and multiply the whole numbers: $25 \times 4 = 100$.

Step 2. Count the digits after the point in both factors: one and one, two in total.

Step 3. Count off two digits from the right of the product and place the point: $1.00$.

Step 4. Drop the trailing zeros: the answer is 1.

Another example: $0.6 \times 0.7$. Without points $6 \times 7 = 42$, two decimal places in total, so $0.42$. Notice that the product is smaller than either factor — that is exactly what should happen when you multiply by a number less than one.

Multiplying and dividing by 10, 100 and 1000 is just moving the point: $4.7 \times 10 = 47$, and $4.7 \div 100 = 0.047$. Right when you multiply, left when you divide, by as many places as there are zeros.

Dividing by a decimal#

Dividing by a number with a decimal point is awkward, so you get rid of the point in the divisor.

Work out $7.2 \div 0.8$.

Step 1. The divisor has one digit after the point. Multiply both the dividend and the divisor by 10: $72 \div 8$.

Step 2. The quotient does not change when you do this — it is a property of division: you scaled both numbers by the same factor.

Step 3. Divide the whole numbers: $72 \div 8 = 9$.

The answer 9 is larger than the dividend, and that is fine: you divided by a number less than one. If you had got $0.9$, that should have made you suspicious — a quick estimate works as well as a full check here.

Mistakes and checks#

Comparing by the length of the number rather than place by place. Cure: pad with zeros.

Losing the point in a product. Count the decimal places out loud: "one and one — two".

Moving the point only in the divisor and forgetting the dividend. The answer then comes out ten times off.

To check, convert the decimals to ordinary fractions and calculate a second time: $2.5 \times 0.4 = \frac{25}{10} \times \frac{4}{10} = \frac{100}{100} = 1$. The reverse conversion helps when one problem mixes both kinds of fraction — bringing them to a common form is covered in adding fractions with unlike denominators.

You will need decimals again in percentage word problems: a percentage is simply a decimal written in hundredths.

Step-by-step plan

1. Step 1 — place valuesRead tenths, hundredths and thousandths out loud and write each decimal as a fraction: 0.07 = 7/100.
2. Step 2 — comparingCompare twenty pairs of numbers, padding with zeros so both have the same number of decimal places.
3. Step 3 — columnsAdd and subtract in columns with point under point, ten examples of each.
4. Step 4 — multiplyingMultiply without the points, then count off the decimal places in the answer; check with an estimate using the whole parts.
5. Step 5 — dividingRemove the point from the divisor by multiplying both numbers by 10 or 100, then use long division.

Check yourself

Sources

• Khan Academy — ArithmeticPractice on adding, multiplying and dividing decimals with hints
  free
• OpenStax Prealgebra 2eFree textbook; chapter 5 covers decimals with worked examples and answers
  free

Was this helpful?

Yes No