Exercises: multiplying decimal numbers

Multiplying decimal fractions

$2.5\cdot 10$

Since you are multiplying a decimal fraction with $10$, you can simply shift the decimal mark by one place to the right.

$=25$

Multiplying decimal fractions

$2.5\cdot 0.1$

Since you are multiplying the decimal fraction with $0.1$ you can simply shift the decimal mark by one place to the left.

$=0.25$

Multiplying decimal fractions

$5.1\cdot 2.9$

Ignore the decimal marks and multiply the two figures.

$\begin{array}{l}\underline{\phantom{0.}51\cdot 29}\\ \phantom{\mathrm{5,10}}459\\ \underline{\phantom{\mathrm{5,1}}1020}\\ \phantom{500}1479\end{array}$

Place the decimal mark in the result in a way so that the solution contains as many decimal places as both coefficients combined, in this case $1+1=2$ places.

$\to 14.79$

Multiplying decimal fractions

$2.45\cdot 0.671$

Ignore the decimal marks and multiply the two figures.

$\begin{array}{l}\underline{\phantom{,0,}245\cdot 671}\\ \phantom{\mathrm{2,45000}}245\\ \phantom{\mathrm{2,450}}17150\\ \underline{\phantom{\mathrm{2,45}}147000}\\ \phantom{\mathrm{24,5}}164395\end{array}$

Place the decimal mark in the result in a way so that the solution contains as many decimal places as both coefficients combined, in this case $2+3=5$ places.

$\to 1.64395$

Multiplying decimal fractions

$9\cdot 0.686$

Ignore the decimal marks and multiply the two figures.

$\begin{array}{l}\underline{\phantom{0,}9\cdot 686}\\ \phantom{90000}54\\ \phantom{9000}720\\ \underline{\phantom{900}5400}\\ \phantom{\mathrm{9,0}}6174\end{array}$

Place the decimal mark in the result in a way so that the solution contains as many decimal places as both coefficients combined, in this case $0+3=3$ places.

$\to 6.174$

Multiplying decimal fractions

$2.5\cdot 100$

Since you are multiplying the decimal fraction with $100$ you can simply shift the decimal mark two places to the right.

$=250$

Division of decimal fractions

$8.45:100$

Since $100$ is a multiple of ten, you can simply shift the decimal mark two places (one for each zero) to the left.

$8.45:100=0.0845$

Division of decimal fractions

$16:0.25$

Multiply both figures with $100$ to turn $0.25$ into an integer.

$\begin{array}{l}16:0.25\\ =(16\cdot 100):(0.25\cdot 100)\\ =1600:25\end{array}$

Utilize long division.

$\begin{array}{l}\phantom{25)}\underline{\phantom{.16}64}\\ 25)\phantom{-}1600\\ \phantom{25)}\underline{-150}\\ \phantom{25)}\phantom{-1}100\\ \phantom{25)}\phantom{0}\underline{-100}\\ \phantom{25)}\phantom{-000}0\end{array}$

$\Rightarrow 16:0.25=64$

Division of decimal fractions

$8.5:0.160$

Write the denominator as $0.16$ instead of $0.160$. Omitting zeros at the end of a decimal fraction does not change the figure.

$8.5:0.160=8.5:0.16$

Multiply both figures with $100$ to turn $0.16$ into an integer.

$\begin{array}{l}8.5:0.16\\ =(8.5\cdot 100):(0.16\cdot 100)\\ =850:16\end{array}$

Utilize long division.

$\begin{array}{l}\phantom{25)}\underline{\phantom{60}53.125}\\ 16)\phantom{-}850.000\\ \phantom{16)}\underline{-80}\\ \phantom{16)}\phantom{-1}50\\ \phantom{16)}\phantom{0}\underline{-48}\\ \phantom{16)}\phantom{-00}2.0\\ \phantom{16)}\phantom{00}\underline{-1.6}\\ \phantom{16)}\phantom{-00.0}40\\ \phantom{16)}\phantom{0.00}\underline{-32}\\ \phantom{16)}\phantom{-0.000}80\\ \phantom{16)}\phantom{000.0}\underline{-80}\\ \phantom{16)}\phantom{-000.00}0\end{array}$

$\Rightarrow 8.5:0.160=53.125$

Division of decimal fractions

$0.125:0.5$

Multiply both figures with $1000$ to turn $0.125$ into an integer.

$\begin{array}{l}\phantom{=.}0.125:0.5\\ =(0.125\cdot 1000):(0.5\cdot 1000)\\ =125:500\end{array}$

Utilize long division.

$\begin{array}{l}\phantom{251)}\underline{\phantom{160}0.25}\\ 500)\phantom{-}125.00\\ \phantom{500)}\underline{-\phantom{12}0}\\ \phantom{500)}\phantom{-}125.0\\ \phantom{500)}\underline{-100.0}\\ \phantom{500)}\phantom{-0}25.00\\ \phantom{500)}\phantom{0}\underline{-25.00}\\ \phantom{500)}\phantom{-000.0}0\end{array}$

$\Rightarrow 0.125:0.5=0.25$

Division of decimal fractions

$8.45:0.01$

Dividing a decimal fraction by $0.01$ shifts the decimal mark two places (one for each zero) to the right.

$8.45:0.01=845$

Introduce a variable $x$ that corresponds to the number by which $0.0123$ must be multiplied to obtain $1230$.

Then, set up the equation:

Introduce a variable $x$ that corresponds to the number by which $0.0123$ must be divided to obtain $0.123$.

Then, set up the equation:

Introduce a variable $x$ that corresponds to the number by which $0.0123$ must be divided to obtain $1000$.

Then, set up the equation:

If we multiply any number $x$ by $0.01$, it looks like this:

$\phantom{=.}x\cdot 0.01$

$=x\cdot \frac{1}{100}$

$=x:100$

You only have to move the decimal point two positions to the left inside the number $x$, as you divide by 100.

Example: $13.7\cdot 0.01$

$\phantom{Bei.:}=13.7:100$

$\phantom{Bei.:}=0.137$

If we divide any number $x$ by $0.01$, it looks like this:

$x:0.01$

$=x:\frac{1}{100}$

$=x\cdot 100$

You only have to move the decimal point two positions to the right inside the number $x$, as you multiply by 100.

Example: $13.7:0.01$

$\phantom{Bei.:}=13.7\cdot 100$

$\phantom{Bei.:}=1370$

As soon as you "pull down" the $1$ after the decimal point on the left, you also have to put a decimal point in the result on the right.

$\begin{array}{l}\phantom{-}8.1:9=0.9\\ \underline{-0}\\ \phantom{-}81\\ \underline{-81}\\ \phantom{-8}0\end{array}$

As soon as you "pull down" the $5$ after the decimal point on the left, you also have to put a decimal point in the result on the right.

$\begin{array}{l}\phantom{-}7.5:2=3.75\\ \underline{-6}\\ \phantom{-}15\\ \underline{-14}\\ \phantom{-1}10\\ \phantom{0}\underline{-10}\\ \phantom{-10}0\end{array}$

As soon as you "pull down" the $4$ after the decimal point on the left, you also have to put a decimal point in the result on the right.

$\begin{array}{l}\phantom{-}29.48:11=2.68\\ \underline{-22}\\ \phantom{-2}74\\ \phantom{0}\underline{-66}\\ \phantom{-66}88\\ \phantom{88}\underline{-88}\\ \phantom{-888}0\end{array}$

As soon as you "pull down" the $8$ after the decimal point on the left, you also have to put a decimal point in the result on the right.

$\begin{array}{l}\phantom{-}13.811:7=1.973\\ \phantom{0}\underline{-7}\\ \phantom{-0}68\\ \phantom{0}\underline{-63}\\ \phantom{-00}51\\ \phantom{00}\underline{-49}\\ \phantom{-000}21\\ \phantom{000}\underline{-21}\\ \phantom{-0000}0\end{array}$