Multiplication of decimal numbers - II — lesson. Mathematics State Board, Class 7.

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ii) Multiplication of Decimal Numbers by $10$, $100$ and $1000$:

In the previous lesson, we explored the conversion of decimal into fraction.

Let us recall the conversion of decimal into fraction.

Converting $66.89$ into a fraction we get:

\begin{array}{l} 66.89 = 66 + \frac{8}{10} + \frac{9}{100} \\ 66.89 = \frac{6689}{1000} \end{array}

Similarly, let us express $265.876$ as a fraction.

\begin{array}{l} 265.876 = 265 + \frac{8}{10} + \frac{7}{100} + \frac{6}{1000} \\ 265.876 = \frac{265876}{1000} \end{array}

Inference:

From the above decimal to fraction conversions, we can notice the following:

If there is one digit after the decimal point, then it's corresponding fraction's denominator is $10$.
If there is two digits after the decimal point, then it's corresponding fraction's denominator is $100$.
If there is three digits after the decimal point, then it's corresponding fraction's denominator is $1000$.

Let us see what happens if a certain decimal number is multiplied by $10$, $100$ and $1000$.

$75.03 \times 10 = 750.3$	$465.956 \times 10 = 4659.56$
$75.03 \times 100 = 7503$	$465.956 \times 100 = 46595.6$
$75.03 \times 1000 = 75030$	$465.956 \times 1000 = 465956$

From the table, we can observe that:

If a decimal number is multiplied by $10$, the decimal point in the product is shifted by one-place to the right side.
If a decimal number is multiplied by $100$, the decimal point in the product is shifted by two-places to the right side.
If a decimal number is multiplied by $1000$, the decimal point in the product is shifted by three-places to the right side.

Therefore, we conclude that when a decimal number is divided by $10$, $100$ or $1000$, the digits in the product are same as in the decimal number but the decimal point in the product is shifted to the right side by as many places as there are zeros followed by $1$.

Similarly, when we multiply the decimal number by the following numbers:

$0.1$, the decimal point moves one-place to the left.
$0.01$, the decimal point moves two-places to the left.
$0.01$, the decimal point moves three-places to the left.

Let us see some decimal numbers multiplied by $0.1$, $0.01$, and $0.001$.

$42.21 × 0.1 = 4.221$	$899.75 × 0.1 = 89.975$
$42.21 × 0.01 = 0.4221$	$899.75 × 0.01 = 8.9975$
$42.21 × 0.001 = 0.04221$	$899.75 × 0.001 = 0.89975$

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\(42.21 × 0.1 = 4.221\)	\(899.75 × 0.1 = 89.975\)
\(42.21 × 0.01 = 0.4221\)	\(899.75 × 0.01 = 8.9975\)
\(42.21 × 0.001 = 0.04221\)	\(899.75 × 0.001 = 0.89975\)

2. Multiplication of decimal numbers - II

Theory: