# Is there more than one way to multiply 25 xx 14 ?

Dec 16, 2017

The product is $350$, but there are several ways to get that answer.

#### Explanation:

Vertical Multiplication
$\left.\begin{matrix}\text{ " & color(white)(x)25 \\ xx & underline(color(white)(x)14) \\ " " & 100 \\ " " & underline(250) \\ " } & 350\end{matrix}\right.$

That is the same as

$25 \times 14 = 25 \times \left(4 + 10\right) = \left(25 \times 4\right) + \left(25 \times 10\right) = 100 + 250 = 350$

We could instead use

$25 \times 14 = \left(20 + 5\right) \times 14 = \left(20 \times 14\right) + \left(5 \times 14\right) = 280 + 70 = 350$
(Although the two multiplications will be challenging for many.)

Or we could use Factoring and re-arrange the factors

$25 \times 14 = 25 \times \left(2 \times 7\right) = \left(25 \times 2\right) \times 7 = 50 \times 7$

$= \left(5 \times 10\right) \times 7 = \left(5 \times 7\right) \times 10 = 35 \times 10 = 350$

Or

$25 \times 14 = \left(5 \times 5\right) \times \left(2 \times 7\right) = \left(5 \times 2\right) \times \left(5 \times 7\right) = 10 \times 35 = 350$

There are other ways of factoring, but I think those two are the most helpful.

Dec 16, 2017

Yes

#### Explanation:

Here are just a few...

Long multiplication

$25 \textcolor{w h i t e}{00000} \leftarrow 25 \times 10$
$\underline{100} \textcolor{w h i t e}{0000} \leftarrow 25 \times 4$
$350$

or:

$28 \textcolor{w h i t e}{00000} \leftarrow 14 \times 20$
$\underline{\textcolor{w h i t e}{0} 70} \textcolor{w h i t e}{0000} \leftarrow 14 \times 5$
$350$

Russian multiplication

$25 \textcolor{w h i t e}{000} \textcolor{red}{14}$
$\textcolor{g r e y}{12} \textcolor{w h i t e}{000} \textcolor{g r e y}{28}$
$\textcolor{w h i t e}{0} \textcolor{g r e y}{6} \textcolor{w h i t e}{000} \textcolor{g r e y}{56}$
$\textcolor{w h i t e}{0} 3 \textcolor{w h i t e}{00} \textcolor{red}{112}$
$\textcolor{w h i t e}{0} 1 \textcolor{w h i t e}{00} \underline{\textcolor{red}{224}}$
$\textcolor{w h i t e}{0000} 350$

The left hand column is formed by repeatedly halving and discarding any remainder. The right hand column is formed by repeatedly doubling. Then ignoring any row where the number in the left hand column is even, add up the numbers in the right hand column.

Squares and difference

$25 \times 14 = {\left(\frac{25 + 14}{2}\right)}^{2} - {\left(\frac{25 - 14}{2}\right)}^{2}$

$\textcolor{w h i t e}{25 \times 14} = \frac{1521}{4} - \frac{121}{4}$

$\textcolor{w h i t e}{25 \times 14} = \frac{1400}{4}$

$\textcolor{w h i t e}{25 \times 14} = 350$

(Note this method works better when both multiplicands are odd or both even)