Multiplying decimals by whole numbers

In a nutshell

Multiplying decimals by whole numbers is very much like multiplying any other pair of numbers together. The only added step is to make sure that the decimal point is added to the  correct column.

Multiplying by a one-digit number

When multiplying a decimal by a one-digit number, short multiplication can be used, just like when multiplying two whole numbers.

procedure

Example 1​

What is $$1.56\times3$$?

Set out the calculation.

$$\begin{array}{cccc}& \text{T} & \text{O} &.&\text{Tth} & \text{Hth} \\& & 1 & . &5&6 \\\times & & 3&&& \\ \hline&&&& \\ \hline&&&&\end{array}$$​​

Multiply each digit of the decimal by three, carrying as necessary.

$$\begin{array}{cccc}& \text{T} & \text{O} &.&\text{Tth} & \text{Hth} \\& & 1 & . &5&6 \\\times & &3 &&& \\ \hline&&4&&6&8 \\ \hline&&^1&&^1&\end{array}$$

​

Add back in the decimal point.

$$\begin{array}{cccc}& \text{T} & \text{O} &.&\text{Tth} & \text{Hth} \\& & 1 & . &5&6 \\\times & & 3&&& \\ \hline&&4&.&6&8 \\ \hline&&^1&&^1&\end{array}$$

​​

​$$1.56\times3=\underline{4.68}$$

Multiply by a two-or-more-digit number

The method of multiply a two-or-more-digit number is the same. However, this time, long multiplication is needed.

Procedure

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Example 2

What is $$6.32\times32$$?

Set out the calculation.

$$\begin{array}{cccc}& \text{T} & \text{O} &.&\text{Tth} & \text{Hth} \\& & 6 & . &3&2 \\\times & 3& 2&&& \\ \hline&&&& \\ \hline&&&&\end{array}$$​​

Multiply each digit of the decimal by two, carrying where necessary.

$$\begin{array}{cccc}&\text{H}& \text{T} & \text{O} &.&\text{Tth} & \text{Hth} \\&& & 6 & . &3&2 \\\times& & 3& 2&&& \\ \hline&&^11&2&&6&4 \\ \\ \hline\end{array}$$​​

Add a zero to the hundredths column (to account for the thirty) and multiply each digit of the decimal by three.

​$$\begin{array}{cccc}&\text{H}& \text{T} & \text{O} &.&\text{Tth} & \text{Hth} \\&& & 6 & . &3&2 \\\times& & 3& 2&&& \\ \hline&&^11&2&&6&4 \\ &^11&8&9&&6&0 \\ \hline\end{array}$$​​

Add together each result and add back in the decimal.

​$$\begin{array}{cccc}&\text{H}& \text{T} & \text{O} &.&\text{Tth} & \text{Hth} \\&& & 6 & . &3&2 \\\times& & 3& 2&&& \\ \hline&&^11&2&.&6&4 \\ &^11&8&9&.&6&0 \\ \hline&^12&^10&^12&.&2&4\end{array}$$

​​$$6.32\times32=\underline{202.24}$$​​