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## 14.6 Chapter summary

• Decimal fractions, common fractions and percentages are simply different ways of expressing the same number. We call them different notations.
• A decimal fraction is a fraction in which the digits of the numerator are written after a decimal point.
• To write a common fraction as a decimal fraction, we must first express the common fraction with a power of ten ($$10$$, $$100$$, $$1\ 000$$, etc.) as the denominator. For example: $$\frac{9}{20} = \frac{9}{20} \times \frac{5}{5} = \frac{45}{100} = \text{0,45}$$.
• To write a decimal fraction as a common fraction, we must first express it as a common fraction with a power of ten as the denominator and then simplify if necessary. For example: $$\text{0,65} = \frac{65}{100} = \frac{65 \div 5}{100 \div 5} = \frac{13}{20}$$.
• Each digit in the number has its own place value. For example, in the number $$\text{1,23}$$ the digit “$$3$$” is in the second decimal place. The second place is called the “hundredths place”.
• Place value is the value of a digit according to its position in a number. For decimal fractions, the place values we work with are $$\frac{1}{10}$$, $$\frac{1}{100}$$, $$\frac{1}{1\ 000}$$ and $$\frac{1}{10\ 000}$$. They are called tenths, hundredths, thousandths and ten thousandths. These place values are represented from biggest to smallest, moving to the right after the decimal comma:

units (U) , tenths (t) hundredths (h) thousandths (th) ten thousandths (tth)
$$1$$ , $$\frac{1}{10}$$ $$\frac{1}{100}$$ $$\frac{1}{1\ 000}$$ $$\frac{1}{10\ 000}$$
• When rounding off decimals, the first thing is to work out which of the digits in the number to focus on: we always look at the digit one place further along than we are rounding the number to.
• If the digit that follows the rounding off place is $$5$$ or bigger, then we round up to the next number. For example: $$\text{13,5}$$ rounded to the nearest whole number is $$14$$; $$13,526$$ rounded to two decimal places (after the comma) is $$\text{13,53}$$.
• If the digit that follows the rounding off place is $$4$$ or less, then we round down to the previous number. For example: $$\text{13,4}$$ rounded to the nearest whole number is $$13$$.
• To add and subtract decimal fractions:
• tenths may be added to (or subtracted from) tenths
• hundredths may be added to (or subtracted from) hundredths
• thousandths may be added to (or subtracted from) thousandths.
• To multiply fractions written as decimals, convert the fractions to whole numbers by multiplying by powers of $$10$$ (for example, $$\text{0,2} \times 10 = 2$$), do your calculations with the whole numbers, and then convert back to decimals again.
• When you do division, you can first multiply the number and the divisor by the same power of $$10$$ to make the working easier. Choose the power of 10 in such a way that both decimal numbers will be free of a decimal comma. For example:

\begin{align} \text{20,6} \div \text{0,2} &= (\text{20,6}\ \mathbf{\times\ 10}) \div (\text{0,2}\ \mathbf{\times\ 10}) \\ &= 206 \div 2 \\ &= 103 \end{align}
• The rule is the same when you divide a decimal number by the whole number. Any whole number can be written as a decimal with 0 after the comma. For example: $$4 = \text{4,0}$$, so:

\begin{align} \text{16,8} \div \text{4,0} &= (\text{16,8} \times 10) \div (\text{4,0} \times 10) \\ &= 168 \div 4 \\ &= 42 \end{align}
• You don’t need to change the place of the decimal comma, because you are multiplying the whole expression by $$1$$: $$10 \div 10 = 1$$.