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Calculations with decimal fractions

14.4 Calculations with decimal fractions

To add and subtract decimal fractions, it is important to know that:

  • tenths may only be added to (or subtracted from) tenths
  • hundredths may only be added to (or subtracted from) hundredths
  • thousandths may only be added to (or subtracted from) thousandths.

Worked Example 14.4: Adding and subtracting decimals

Estimate your answer first and then calculate:

\[\text{160,55} + \text{22,36}\]

Estimate the answer by rounding off each number.

  • \(\text{160,55}\) rounded off to the nearest whole number \(\approx 161\)
  • \(\text{22,36}\) rounded off to the nearest whole number \(\approx 22\)

Now we can add the whole numbers to estimate the answer:

\[161 + 22 = 183\]

Add the decimal numbers.

Align the two numbers on the comma, according to the place value of each digit in the number:

\[\begin{array}{lcccccc} & 1 & 6 & 0 & , & ^{1}5 & 5 \\ + & & 2 & 2 & , & 3 & 6 \\ \hline & 1 & 8 & 2 & , & 9 & 1 \end{array}\]

Compare the estimate to the final answer.

\(\text{182,91} \approx 183\), so the calculation is correct.

Worked Example 14.5: Adding and subtracting decimals

Estimate your answer first and then calculate:

\[\text{150,35} - \text{106,31}\]

Estimate the answer by rounding off each number.

  • \(\text{150,35}\) rounded off to the nearest whole number \(\approx 150\)
  • \(\text{106,31}\) rounded off to the nearest whole number \(\approx 106\)

Now we can subtract the whole numbers to estimate the answer:

\[150 - 106 = 44\]

Subtract the decimal numbers.

Align the two numbers on the comma, according to the place value of each digit in the number:

\[\begin{array}{lcccccc} & 1 & ^{4}5 & ^{1}0 & , & 3 & 5 \\ - & 1 & 0 & 6 & , & 3 & 1 \\ \hline & & 4 & 4 & , & 0 & 4 \end{array}\]

Compare the estimate to the final answer.

\(\text{44,04} \approx 44\), so the calculation is correct.

Exercise 14.2

Calculate.

  1. \(\text{16,52} + \text{2,35}\)
  2. \(\text{160,52} + \text{9,38}\)
  3. \(\text{216,52} + \text{9,78}\)
  4. \(\text{300,08} + \text{21,9}\)
  5. \(\text{0,042} + \text{0,103}\)
  6. \(\text{9,99} + \text{0,99}\)
  1. \[\text{16,52} + \text{2,35} = \text{18,87}\]
  2. \[\text{160,52} + \text{9,38} = \text{169,90}\]
  3. \[\text{216,52} + \text{9,78} = \text{226,30}\]
  4. \[\text{300,08} + \text{21,9} = \text{321,98}\]
  5. \[\text{0,042} + \text{0,103} = \text{0,145}\]
  6. \[\text{9,99} + \text{0,99} = \text{10,98}\]

Calculate.

  1. \(\text{450,67} - \text{23,25}\)
  2. \(\text{405,67} - \text{203,80}\)
  3. \(\text{187,6} - \text{98,45}\)
  4. \(\text{1,009} - \text{0,998}\)
  5. \(\text{0,9} - \text{0,045}\)
  6. \(\text{65,7} - \text{37,6}\)
  1. \[\text{450,67} - \text{23,25} = \text{427,42}\]
  2. \[\text{405,67} - \text{203,80} = \text{201,87}\]
  3. \[\text{187,6} - \text{98,45} = \text{89,15}\]
  4. \[\text{1,009} - \text{0,998} = \text{0,011}\]
  5. \[\text{0,9} - \text{0,045} = \text{0,855}\]
  6. \[\text{65,7} - \text{37,6} = \text{28,1}\]

Four consecutive stages in a cycling race are the following lengths:

\(\text{21,4} \text{ km}\); \(\text{14,7} \text{ km}\); \(\text{31} \text{ km}\) and \(\text{18,6} \text{ km}\)

How long is the whole race?

\[\text{21,4} \text{ km} + \text{14,7} \text{ km} + \text{31} \text{ km} + \text{18,6} \text{ km} = \text{85,7} \text{ km}\]

The following set of length measurements (in centimetres) was recorded during an experiment:

\[\text{56,8};\ \text{55,4};\ \text{78,9};\ \text{57,8};\ \text{34,2};\ \text{67,6};\ \text{45,5};\ \text{34,5};\ \text{64,5};\ \text{88}\]
  1. Find the sum of the measurements and round it off to the nearest whole number.
  2. First round off each measurement to the nearest whole number and then find the sum.
  3. Which of your answers is closest to the actual sum? Explain why.
  1. \(\text{56,8} + \text{55,4} + \text{78,9} + \text{57,8} + \text{34,2} + \text{67,6} + \text{45,5} + \text{34,5} + \text{64,5} + \text{88} = \text{583,2}\)
    \(\text{583,2} \approx \text{583}\) to the nearest whole number
  2. \[57 + 55 + 79 + 58 + 34 + 68 + 46 + 35 + 65 + 88 = 585\]
  3. \(583\) is closer to the actual answer \(\text{583,2}\). This is because we rounded the final answer instead of rounding each number before adding. The rounding error is bigger if rounding is done prior to calculations.

By how much is \(\text{0,7}\) greater than \(\text{0,07}\)?

\(\text{0,7} - \text{0,07} = \text{0,63}\), so \(\text{0,7}\) is greater than \(\text{0,07}\) by \(\text{0,63}\).

The difference between two numbers is \(\text{0,75}\). The bigger number is \(\text{18,4}\). What is the other number?

\(\text{18,4} - \text{number} = \text{0,75}\) So, the other number \(= \text{18,4} - \text{0,75} = \text{17,65}\).

Multiplying and dividing decimals

To multiply fractions written as decimals:

  • Convert the fractions to whole numbers by multiplying by powers of \(10\) (for example, \(0,3 \times 10 = 3\)).
  • Do your calculations with the whole numbers.
  • Then convert back to decimals again.

For example: \(\text{13,1} \times \text{1,01}\)

\[\begin{align} \left(\text{13,1}\ \mathbf{\times\ 10} \right) \times \left( \text{1,01}\ \mathbf{\times\ 100} \right) &= 131 \times 101 \\ &= 13\ 231 \end{align}\]

To convert back to decimals, divide the answer by all the powers of \(10\) you multiplied. In this example, divide by \(10\) and by \(100\):

\[\begin{align} 13\ 231\ \mathbf{\div\ 10 \div 100} &= 13\ 231 \div 1\ 000 \\ &= \text{13,231} \end{align}\]

So, \(\text{13,1} \times \text{1,01} = \text{13,231}\).

You can also count the total number of decimal places in both numbers before multiplying. The final answer will have that many decimal places after the comma.

In the example above:

  • \(\text{13,1}\) has one decimal place
  • \(\text{1,01}\) has two decimal places

The answer will have one + two = three decimal places.

\[131 \times 101 = 13\ 231\]

Then

\[13,\mathbf{1} \times 1,\mathbf{01} = 13,\mathbf{231}\]

Worked Example 14.6: Multiplying and dividing decimals

Multiply the following:

\[\text{1,3} \times \text{12,01}\]

Determine the number of decimal places in the answer.

  • \(\text{1,3}\) has one decimal place
  • \(\text{12,01}\) has two decimal places

The answer will have three decimal places.

Multiply the whole numbers (decimals without the comma).

Use a calculator to work out the answer:

\[\text{13} \times \text{1 201} = \text{15 613}\]

Place the comma back into the answer.

There are three decimal places, so the answer is \(15,\mathbf{613}\).

\[\text{1,30} \times \text{12,01} = \text{15,613}\]

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:

\[\begin{align} \text{21,7} \div \text{0,7} &= (\text{21,7}\ \mathbf{\times\ 10}) \div (\text{0,7}\ \mathbf{\times\ 10}) \\ &= 217 \div 7 \\ &= 31 \end{align}\]

The rule is the same when you divide a decimal number by a whole number. Any whole number can be written as a decimal with \(0\) after the comma.

For example: (\(14 = \text{14,0}\))

\[\begin{align} \text{172,2} \div 14 &= (\text{172,2} \times 10) \div (\text{14,0} \times 10) \\ &= \text{1 722} \div 140 \\ &= \text{12,3} \end{align}\]

Worked Example 14.7: Multiplying and dividing decimals

Divide the following:

\[\text{325,65} \div \text{12,5}\]

Move the decimal comma to the biggest number of decimal places.

  • \(\text{325,65}\) has two decimal places
  • \(\text{12,5}\) has one decimal place

“Move” the comma two decimal places to the right by multiplying both numbers by \(100\):

  • \(\text{325,65} \times \text{100} = \text{32 565}\)
  • \(\text{12,5} \times \text{100} = \text{12 500}\)

Divide the whole numbers (decimals without a comma).

Use a calculator to work out the answer:

\[\begin{align} \text{32 565} \div \text{12 500} &= \text{2,6052} \\ \text{325,65} \div \text{12,5} &= \text{2,6052} \end{align}\]

It is important to notice that, unlike with multiplication, you do not need to do anything else after dividing the whole numbers. This is because your first step was essentially to multiply the numbers by \(1\).

\[\text{325,65} \div \text{12,5} = (\text{325,65} \times \mathbf{100})\mathbf{\div}(\text{12,5} \times \mathbf{100})\] \[100 \div 100 = 1\]
Exercise 14.3

Calculate each of the following. You may use fraction notation if you wish.

  1. \(\text{0,12} \times \text{0,3}\)
  2. \(\text{0,12} \times \text{0,03}\)
  3. \(\text{1,2} \times \text{0,3}\)
  4. \(\text{350} \times \text{0,043}\)
  5. \(\text{0,35} \times \text{0,43}\)
  6. \(\text{0,13} \times \text{0,16}\)
  7. \(\text{1,3} \times \text{1,6}\)
  8. \(\text{0,13} \times \text{1,6}\)
  1. \[\text0,12 \times \text{0,3} = \text{0,036}\]
  2. \[\text0,12 \times \text{0,03} = \text{0,0036}\]
  3. \[\text{1,2} \times \text{0,3} = \text{0,36}\]
  4. \[\text{350} \times \text{0,043} = \text{15,05}\]
  5. \[\text{0,35} \times \text{0,43} = \text{0,1505}\]
  6. \[\text{0,13} \times \text{0,16} = \text{0,0208}\]
  7. \[\text{1,3} \times \text{1,6} = \text{2,08}\]
  8. \[\text{0,13} \times \text{1,6} = \text{0,208}\]

\(\text{305} \times \text{13} = \text{3 965}\). Use this answer to work out each of the following.

  1. \(\text{3,05} \times \text{1,3}\)
  2. \(\text{305} \times \text{1,3}\)
  3. \(\text{0,305} \times \text{0,13}\)
  4. \(\text{30,5} \times \text{1,3}\)
  5. \(\text{39,65} \div \text{30,5}\)
  6. \(\text{39,65} \div \text{0,305}\)
  7. \(\text{39,65} \div \text{0,13}\)
  1. \[\text{3,05} \times \text{1,3} = \text{3,965}\]
  2. \[\text{305} \times \text{1,3} = \text{396,5}\]
  3. \[\text{0,305} \times \text{0,13} = \text{0,03965}\]
  4. \[\text{30,5} \times \text{1,3} = \text{39,65}\]
  5. \[\text{39,65} \div \text{30,5} = \text{1,3}\]
  6. \[\text{39,65} \div \text{0,305} = \text{130}\]
  7. \[\text{39,65} \div \text{0,13} = \text{305}\]

\(\text{35} \times \text{43} = \text{1 505}\). Use this answer to work out each of the following.

  1. \[\text{3,5} \times \text{43}\]
  2. \[\text{0,35} \times \text{43}\]
  3. \[\text{3,5} \times \text{0,043}\]
  4. \[\text{0,35} \times \text{0,43}\]
  5. \[\text{15,05} \div \text{0,35}\]
  6. \[\text{15,05} \div \text{0,043}\]
  1. \[\text{3,5} \times 43 = \text{150,5}\]
  2. \[\text{0,35} \times 43 = \text{15,05}\]
  3. \[\text{3,5} \times \text{0,043} = \text{0,1505}\]
  4. \[\text{0,35} \times \text{0,43} = \text{0,1505}\]
  5. \[\text{15,05} \div \text{0,35} = 43\]
  6. \[\text{15,05} \div \text{0,043} = 350\]

Calculate each of the following. You may convert to whole numbers to make it easier.

  1. \[\text{62,5} \div \text{2,5}\]
  2. \[\text{6,25} \div 25\]
  3. \[\text{6,25} \div \text{0,25}\]
  4. \[\text{0,625} \div \text{2,5}\]
  1. \[\text{62,5} \div \text{2,5} = 25\]
  2. \[\text{6,25} \div 25 = \text{0,25}\]
  3. \[\text{6,25} \div \text{0,25} = 25\]
  4. \[\text{0,625} \div \text{2,5} = \text{0,25}\]

Calculate the following.

  1. \[\text{7,35} \times \text{1,1}\]
  2. \[\text{0,45} \times \text{2,15}\]
  3. \[\text{2,54} \times \text{3,45}\]
  4. \[\text{21,05} \times \text{30,3}\]
  1. \[\text{7,35} \times \text{1,1} = \text{8,085}\]
  2. \[\text{0,45} \times \text{2,15} = \text{0,9675}\]
  3. \[\text{2,54} \times \text{3,45} = \text{8,763}\]
  4. \[\text{21,05} \times \text{30,3} = \text{637,815}\]

Divide the following decimal numbers. First, rewrite the division of decimals as the equivalent form of whole number division.

  1. \[\text{345,15} \div \text{32,5}\]
  2. \[\text{1 689,1} \div \text{13,3}\]
  3. \[\text{378,35} \div 161\]
  4. \[\text{775,455} \div \text{15,205}\]
  1. \[\text{345,15} \div \text{32,5} = \text{34 515} \div \text{3 250} = \text{10,62}\]
  2. \[\text{1 689,1} \div \text{13,3} = \text{16 891} \div \text{133} = \text{127}\]
  3. \[\text{378,35} \div \text{161} = \text{37 835} \div \text{16 100} = \text{2,35}\]
  4. \[\text{775,455} \div \text{15,205} = \text{775 455} \div \text{15 205} = \text{51}\]

Calculate the squares, cubes, square roots and cube roots of decimal fractions

Squaring or cubing a fraction or a decimal fraction is no different from squaring or cubing an integer. With decimal numbers, you just need to keep count of the number of decimal places. For example, \({0,3}^{2} = 0,\mathbf{3} \times 0,\mathbf{3} = 0,\mathbf{09}\) (two decimal places in the answer) and \({0,3}^{3} = 0,\mathbf{3} \times 0,\mathbf{3} \times 0,\mathbf{3} = 0,\mathbf{027}\) (three decimal places in the answer).

When looking for a square root or a cube root, one method you can use to work out the answer is to rewrite the decimal number as a common fraction.

For example, the decimal number \(\text{0,49}\) is equal to the fraction \(\frac{49}{100}\), so you can rewrite the expression with the square root:

\[\begin{align} \sqrt{\text{0,49}} &= \sqrt{\frac{49}{100}} \\ &= \frac{\sqrt{49}}{\sqrt{100}} \\ &= \frac{7}{10} \end{align}\]

Similarly, for example, the decimal number \(\text{0,008} = \frac{8}{\text{1 000}}\), so the cube root is:

\[\begin{align} \sqrt[3]{\text{0,008}} &= \sqrt[3]{\frac{8}{\text{1 000}}} \\ &= \frac{\sqrt[3]{8}}{\sqrt[3]{\text{1 000}}} \\ &= \frac{2}{10} \\ &= \frac{1}{5} \end{align}\]

These problems are easy to work out on a calculator, but it is very useful to understand how the decimal numbers work, so you will benefit from learning how to solve this type of question in your head. You might not always have a calculator!

Worked Example 14.8: Powers and roots of decimal numbers

Determine the value of this expression:

\[\sqrt[3]{\text{0,064}}\]

To answer this question, you must evaluate (work out the answer for) this expression.

Rewrite the expression as a fraction.

The decimal number \(\text{0,064}\) is equal to the fraction \(\frac{64}{1\ 000}\), so you can rewrite the expression as:

\[\sqrt[3]{\frac{64}{\text{1 000}}}\]

Work out the value of the cube root.

Take the cube root of the numerator and the denominator to get the answer.

\[\begin{align} \sqrt[3]{\frac{64}{1\ 000}} &= \frac{\sqrt[3]{64}}{\sqrt[3]{1\ 000}} \\ &= \frac{4}{10} \end{align}\]

You can give the answer as either a decimal or a fraction, but if you choose to write it as a fraction, it must be simplified. In this case, the fraction can indeed be simplified, and it becomes \(\frac{4}{10} = \frac{2}{5}\). If you choose to write the answer as a decimal, then \(\frac{4}{10} = \text{0,4}\)

So, the answer is: \(\sqrt[3]{\text{0,064}} = \frac{2}{5} = \text{0,4}\).

Exercise 14.4

Calculate the following powers of decimal numbers.

  1. \[{\text{0,7}}^{3}\]
  2. \[{\text{0,25}}^{2}\]
  3. \[{\text{12,2}}^{2}\]
  4. \[- {\text{2,6}}^{3}\]
  5. \[( - \text{10,7})^{2}\]
  1. \[{\text{0,7}}^{3} = \text{0,343}\]
  2. \[{\text{0,25}}^{2} = \text{0,0625}\]
  3. \[{\text{12,2}}^{2} = \text{148,84}\]
  4. \[- {\text{2,6}}^{3} = - \text{17,576}\]
  5. \[( - \text{10,7})^{2} = \text{114,49}\]

Calculate the value of the following roots.

  1. \[\sqrt{\text{0,49}}\]
  2. \[\sqrt[3]{\text{0,125}}\]
  3. \[\sqrt[3]{\text{0,216}}\]
  4. \[\sqrt{\text{0,0625}}\]
  5. \[\sqrt[3]{\text{1,331}}\]
  1. \[\sqrt{\text{0,49}} = \text{0,7}\]
  2. \[\sqrt[3]{\text{0,125}} = \text{0,5}\]
  3. \[\sqrt[3]{\text{0,216}} = \text{0,6}\]
  4. \[\sqrt{\text{0,0625}} = \text{0,25}\]
  5. \[\sqrt[3]{\text{1,331}} = \text{1,1}\]

Equivalent forms between common fraction, decimal fraction and percentages

Decimal fractions, common fractions and percentages are simply different ways of expressing the same number. We call them different notations.

Remember, to write a common fraction as a decimal fraction or as a percentage, we must first express the common fraction with a power of ten (\(10\), \(100\), \(1\ 000\), etc.) as the denominator. For percentages, the denominator must always be \(100\)!

For example:

\[\begin{align} \frac{3}{20} &= \frac{3}{20} \times \frac{5}{5} \\ &= \frac{15}{100} \\ &= \text{0,15} \\ &= 15\% \end{align}\]

Similarly, 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:

  • \(\text{0,25} = \frac{25}{100} = 25\%\)
  • \(\text{0,25} = \frac{25}{100} = \frac{25 \div 5}{100 \div 5} = \frac{5}{20} = \frac{1}{4}\)
Exercise 14.5

Write each of the following in three ways: in decimal notation, in percentage notation and in common fraction notation in its simplest form.

  1. \(80\) hundredths
  2. \(5\) hundredths
  3. \(60\) hundredths
  4. \(35\) hundredths
  1. \(80\) hundredths \(= \frac{80}{100} = \text{0,8} = 80\% = \frac{8}{10} = \frac{4}{5}\)
  2. \(5\) hundredths \(= \frac{5}{100} = \text{0,05} = 5\% = \frac{1}{20}\)
  3. \(60\) hundredths \(= \frac{60}{100} = \text{0,6} = 60\% = \frac{6}{10} = \frac{3}{5}\)
  4. \(35\) hundredths \(= \frac{35}{100} = \text{0,35} = 35\% = \frac{7}{50}\)

Complete the following table.

Common fraction Decimal fraction Percentage
  \(\text{0,3}\)  
\(\frac{1}{4}\)    
    \(15\%\)
  \(\text{0,55}\)  
    \(1\%\)
\(\frac{1}{8}\)    
Common fraction Decimal fraction Percentage
\(\frac{3}{10}\) \(\text{0,3}\) \(30\%\)
\(\frac{1}{4}\) \(\text{0,25}\) \(25\%\)
\(\frac{15}{100} = \frac{3}{20}\) \(\text{0,15}\) \(15\%\)
\(\frac{55}{100} = \frac{11}{20}\) \(\text{0,55}\) \(55\%\)
\(\frac{1}{100}\) \(\text{0,01}\) \(1\%\)
\(\frac{1}{8}\) \(1 \div 8 = \text{0,125}\) \(\frac{125}{1\ 000} = \frac{125 \div 10}{1\ 000 \div 10} = \frac{12,5}{100} = \text{12,5}\%\)