Algebraic Proof for the Units and Tens Method

The Units and Tens Method is also known as the two finger method and uses .
First we will look at a three digit number multiplied by a single digit number.
We will refer to our three digit number as $abc$ and the single digit multiplier as $n$ .

$abc = \left(a\cdot 100 + b\cdot 10 + c\cdot 1\right)$

So multiplying this number by $n$ we get:

$n\cdot abc = n\cdot\left(a\cdot 100 + b\cdot 10 + c\cdot 1\right)$

Expanding this out we get:

$n\cdot abc = n\cdot a\cdot 100 + n\cdot b\cdot 10 + n\cdot c\cdot 1$

The $n\cdot a$ , $n\cdot b$ and $n\cdot c$ are all pairs of single digit numbers. When multiplying two single digit numbers we normally get a two digit result, the exceptions are for the very low value digits where the result is a single digit number. However, these single digit results can be treated as a two digit result by adding a leading zero.
For example.

$\begin{equation*} \begin{split} 9\times 9 &= 81\\ 4\times 3 &= 12\\ 2\times 4 &= 08\\ 1\times 5 &= 05 \end{split} \end{equation*}$

Two digit numbers can be represented like this:

$xy = \left(x\cdot 10 + y\cdot 1\right)$

Where $x$ is the tens digit, the digit multiplied by ten, and $y$ is the units digit, the digit multiplied by one.

We will use the letter $T$ to represent the tens digit and $U$ to represent the units digit of our two digit numbers.

As we have seen with our three digit number multiplied by a single digit number we get three pairs of numbers multiplied together so we need to include a subscript with our $T$ and $U$ so we can keep track of which pair they belong to.
We will use the following:

$\begin{equation*} \begin{split} n\cdota &= T_a\cdot 10 + U_a\\ n\cdotb &= T_b\cdot 10 + U_b\\ n\cdotc &= T_c\cdot 10 + U_c \end{split} \end{equation*}$

Where $T_a$ represents the tens digit of the result when $a$ is multiplied by $n$ and $U$ represents the unit digit of that result.

Putting this into our equation and expanding the parentheses we get:

$\begin{equation*} \begin{split} n\cdot abc &= n\cdot a\cdot 100 + n\cdot b\cdot 10 + n\cdot c\cdot 1\\ &= \left(T_a\cdot 10+U_a\right)\cdot 100 + \left(T_b\cdot 10+U_b\right)\cdot 10 + \left(T_c\cdot 10+U_c\right)\cdot 1\\ &= T_a\cdot 1000+U_a\cdot 100 + T_b\cdot 100+U_b\cdot 10 + T_c\cdot 10+U_c\cdot 1\\ &= T_a\cdot 1000+\left(U_a + T_b\right)\cdot 100+\left(U_b + T_c\right)\cdot 10+U_c \end{split} \end{equation*}$

This describes the method for the units and tens multiplication.
Looking at the term $\left(U_b + T_c\right)\cdot 10$ :

$U_b =$ units of $b$ times the multiplier $n$
$T_c =$ tens of $c$ times the multiplier $n$

The normal way we write out our equation is :

$\underline{a\quad b\quad c}\quad \times\quad n$

We will place the $U_b$ and $T_c$ above the digits they refer to

$\begin{equation*} \begin{split} &\text{\quad \enspace $U_b$\enspace $T_c$}\\ &\underline{a\quad b\quad c}\quad \times\quad n \end{split} \end{equation*}$

Once in place the subscripts are not needed so we can simply put:

$\begin{equation*} \begin{split} &\text{\quad \enspace $U$\enspace $T$}\\ &\underline{a\quad b\quad c}\quad \times\quad n\\ &\text{\quad \enspace *} \end{split} \end{equation*}$

The * indicates which figure in the answer the $U$ and $T$ gives us.
The rest of the answer comes from the other terms in the equation in exactly the same way.
Thus we have proved the method of units and tens multiplication when multiplying by a single digit.

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