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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 \(1\): \(10 \div 10 = 1\).