Following on from short multiplication is long multiplication.
The long multiplication method is similar to the short multiplication method, but a little bit more involved.
Long multiplication is the approach to take when we encounter multiplication involving larger numbers that are two digits or more.
It’s handy for helping us learn how to multiply 3 digits in Math.
Long Multiplication Method Steps:
Before looking at examples of how to multiply 3 digits.We can introduce the concept by considering the sum, 12 × 23.
This multiplication can be set up in columns as we would do in short multiplication.
Usually the larger number goes above.
\begin{array}{r} &2\space3\\ \times &1\space2\\ \hline \end{array}
The first step of the multiplication is to multiply the whole top number only the 2 in the units column below, like if it was a short multiplication sum.
\begin{array}{r} &2\space3\\ \times &\space\space\space2\\ \hline &4\space6 \end{array}
Now the next step is to multiply the top number by the lower 1 in the tens column.
Writing the result of this multiplication below the first result.
But to account for this being the tens column, a 0 is place in the answer section under the units first.
\begin{array}{r} &2\space3\\ \times &1\space2\\ \hline &4\space6\\ &\space\space\space0 \end{array} => \begin{array}{r} &\space2\space3\\ \times &\space1\space2\\ \hline &\space4\space6\\ &2\space3\space0\space\space \end{array}
Lastly we add the two separate multiplication results together, which give the full answer to the whole multiplication sum.
\begin{array}{r} &\space2\space3\\ \times &\space1\space2\\ \hline &\space4\space6\\ &2\space3\space0\space\space\\ \hline &2\space7\space6\space\space \end{array} 12 × 23 = 276
How to Multiply 3 Digits, Long Multiplication
Examples
1.1
14 × 17
Solution
This is a smaller multiplication sum example, but the same long multiplication method approach can be used for larger sums too.
We set the sum up with columns as normal.
\begin{array}{r} &1\space4\\ \times &1\space7\\ \hline \end{array} Firstly multiply the 1 and the 4 on the top row, by the 7 below.
1)
\begin{array}{r} &{\tiny{2}}\space\space\space\\ &1\space4\\ \times &\space\space\space7\\ \hline &9\space8\\ \\ \\ \end{array} => \begin{array}{r} &\space1\space4\\ \times &\space1\space7\\ \hline &\space9\space8\\ &1\space4\space0\space\space \end{array}
2)
Now, for the last step in long multiplication, the two individual multiplication results are added together.
\begin{array}{r} &\space1\space4\\ \times &\space1\space7\\ \hline &\space9\space8\\ &1\space4\space0\space\space\\ \hline &2\space3\space8\space\space\\ &{\tiny{1}}\space\space\space\space\space\space\space\\ \end{array} Thus, 14 × 17 = 238.
2.2
354 × 36
Solution
1)
\begin{array}{r} &{\tiny{3}}\space\space{\tiny{2}}\space\space\\ &\space3\space5\space4\\ \times &\space\space\space\space\space\space\space6\\ \hline &2\space1\space2\space4\space\space \end{array} => \begin{array}{r} &{\tiny{1}}\space\space{\tiny{1}}\space\space\\ &\space3\space5\space4\\ \times &\space3\\ \hline &2\space1\space2\space4\space\space\\ &1\space0\space6\space2\space0\space\space\space\space\space \end{array}
2)
Now adding the two results together:
\begin{array}{r} &\space3\space5\space4\\ \times &\space\space\space\space3\space6\\ \hline &2\space1\space2\space4\space\space\\ &1\space0\space6\space2\space0\space\space\space\space\space\\ \hline &1\space2\space7\space4\space4\space\space\space\space\space\\ \end{array} Thus, 354 × 36 = 12744
2.3
526 × 424
Solution
1)
\begin{array}{r} &{\tiny{1}}\space\space{\tiny{2}}\\ &\space\space\space5\space2\space6\\ \times &\space\space\space\space\space\space\space\space\space4\\ \hline &2\space1\space0\space4 \end{array}
2)
\begin{array}{r} &{\tiny{1}}\\ &5\space2\space6\\ \times &2\\ \hline &\space\space\space2\space1\space0\space4\space\space\space\space\space\\ &1\space0\space5\space2\space0\space\space\space\space\space \end{array}
Now with the 5, we are multiplying by “HUNDREDS”, so two 0‘s are placed in the answer section initially.
\begin{array}{r} &{\tiny{1}}\\ &5\space2\space6\\ \times &2\\ \hline &\space\space\space2\space1\space0\space4\space\space\space\space\space\\ &1\space0\space5\space2\space0\space\space\space\space\space\\ &2\space1\space0\space4\space0\space0\space\space\space\space\space\space\space\space \end{array}
4)
Now we add the three results to obtain the answer.
\begin{array}{r} &\space\space\space\space\space\space2\space1\space0\space4\\ &\space\space\space1\space0\space5\space2\space0\\ + &2\space1\space0\space4\space0\space0\\ \hline &2\space2\space3\space0\space2\space4\\ &{\tiny{1}}\space\space\space\\ \end{array}
526 × 424 = 223024
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