A step up from short multiplication is long multiplication.
The long multiplication approach is very similar to short multiplication, but a bit more involved with a few more steps.
Long multiplication is used when we encounter multiplication sums that involve larger numbers that are of size two digits or greater.
It assists us in learning how to multiply 3 digits in certain sums in Math.
Long Multiplication Steps:
Before observing some further examples of how to multiply 3 digits in sums.We can learn the concept by considering a sum such as, 12 × 24.
This sum can be set up in relevant columns, the same way as with short multiplication.
Generally the larger number in the sum is placed above.
\begin{array}{r} &2\space4\\ \times &1\space2\\ \hline \end{array}
The first step to take is to multiply the entire top number by only the 2 in the lower units column, as if it was a short multiplication sum.
\begin{array}{r} &2\space4\\ \times &\space\space\space2\\ \hline &4\space8 \end{array}
The next step is to multiply the top number again, this time by the lower 1 in the lower tens column.
Placing the result of this multiplication beneath the first multiplication result obtained.
However to account for this 2nd multiplication being with the lower tens column, a 0 is placed in the answer section under the units before multiplying.
\begin{array}{r} &2\space4\\ \times &1\space2\\ \hline &4\space8\\ &\space\space\space0 \end{array} => \begin{array}{r} &\space2\space4\\ \times &\space1\space2\\ \hline &\space4\space8\\ &1\space2\space0\space\space \end{array}
Lastly, we add the two separate multiplication results together, and this gives the complete answer to the whole multiplication sum.
\begin{array}{r} &\space2\space4\\ \times &\space1\space2\\ \hline &\space4\space8\\ + &2\space4\space0\space\space\\ \hline &2\space8\space8\space\space \end{array} 12 × 24 = 288
How to Multiply 3 Digits, Long Multiplication
Examples
(1.1)
12 × 18
Solution
This isn’t a particularly large multiplication sum example, but the long multiplication method can be used to solve.
We set the sum up with columns as usual.
\begin{array}{r} &1\space8\\ \times &1\space2\\ \hline \end{array} Firstly multiplication is the 1 and the 8 on the top row, by the 2 below.
1)
\begin{array}{r} &{\tiny{1}}\space\space\space\\ &1\space8\\ \times &\space\space\space2\\ \hline &3\space6\\ \\ \\ \end{array} => \begin{array}{r} &\space1\space8\\ \times &\space1\space2\\ \hline &\space3\space6\\ &1\space8\space0\space\space \end{array}
2)
Now, for the final step in the long multiplication, the two separate multiplication results are added together.
\begin{array}{r} &\space1\space8\\ \times &\space1\space2\\ \hline &\space3\space6\\ + &1\space8\space0\space\space\\ \hline &2\space1\space6\space\space\\ &{\tiny{1}}\space\space\space\space\space\space\space\\ \end{array} Thus, 12 × 18 = 216.
(1.2)
245 × 34
Solution
1)
\begin{array}{r} &{\tiny{1}}\space\space{\tiny{2}}\space\space\\ &\space2\space4\space5\\ \times &\space\space\space\space\space\space\space4\\ \hline &\space9\space8\space0\\ &\space \end{array} => \begin{array}{r} &{\tiny{1}}\space\space{\tiny{1}}\space\space\\ &\space2\space4\space5\\ \times &\space3\\ \hline &\space9\space8\space0\\ &7\space3\space5\space0\space\space \end{array}
2)
Now adding the two results together:
\begin{array}{r} &\space2\space4\space5\\ \times &\space\space\space\space3\space4\\ \hline &\space9\space8\space0\\ + &7\space3\space5\space0\space\space\\ \hline &1\space2\space7\space4\space4\space\space\space\space\space\\ \end{array} Thus, 354 × 36 = 12744
(1.3)
517 × 343
Solution
1)
\begin{array}{r} &\space\space\space{\tiny{2}}\\ &\space\space\space5\space1\space7\\ \times &\space\space\space\space\space\space\space\space\space3\\ \hline &1\space5\space5\space1 \end{array}
2)
\begin{array}{r} &{\tiny{2}}\\ &5\space1\space7\\ \times &4\\ \hline &\space\space\space1\space5\space5\space1\space\space\space\space\space\\ &2\space0\space6\space8\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\space1\space7\\ \times &4\space\space\space\space\space\space\\ \hline &\space\space\space1\space5\space5\space1\space\space\space\space\space\\ &2\space0\space6\space8\space0\space\space\space\space\space\\ &1\space5\space5\space1\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\space1\space5\space5\space1\space\space\space\space\\ &2\space0\space6\space8\space0\space\space\space\space\\ + &1\space5\space5\space1\space0\space0\space\space\space\space\space\space\space\\ \hline &1\space7\space7\space3\space3\space1\space\space\space\space\space\space\space\space\\ &{\tiny{1}}\space\space{\tiny{1}}\space\space\space\space\space\space\space\space\\ \end{array}
517 × 343 = 177331
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