Division with decimal in the divisor

Question

I understand HOW to do division with a decimal in the divisor, but my question is MUST we remove the decimal in the divisor and if so, why?

Thanks.

Answer 1

Remember that the word decimal is a shortened form of decimal fraction, meaning a fraction whose denominator is a power of $10$. For example, $$ 3.14 = \frac{314}{100}, \qquad 0.0625 = \frac{625}{1000}, \qquad 5280.1 = \frac{52801}{10}. $$

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As an example, consider the division problem $$ \frac{9.6}{3.84} = \frac{\frac{96}{10}}{\frac{384}{100}} = \frac{96}{10} \cdot \frac{100}{384} = \frac{96}{384} \cdot \frac{100}{10} = \frac{96}{384} \cdot 10. $$ This last form shows you that the original division calculation is equivalent the quotient of the whole numbers$96$ and $384$, as long as you adjust by a factor of $10$ afterwards. (This is usually described as moving the decimal point.)

In order to actually do this calculation, you can multiply the numerator by a sufficiently large power of $10$ to make it a multiple of the denominator, making sure to divide by that same power of $10$ afterwards to compensate. Note: it won't always be the case that you get precisely a multiple, no matter how large a power of $10$ you multiply by. In this case, $$ \frac{96}{384} \cdot 10 = \frac{96}{384} \cdot \frac{100}{100} \cdot 10 = \frac{9600}{384} \cdot \frac{10}{100} = 25 \cdot \frac{1}{10} = \frac{25}{10} = 2.5. $$

The upshot: the integer fact that $384 \cdot 25 = 9600$ is responsible for all of the following: $$ \cdots = \frac{0.0096}{0.00384} = \frac{0.096}{0.0384} = \frac{0.96}{0.384} = \frac{9.6}{3.84} = \frac{96}{38.4} = \frac{960}{384} = \frac{9600}{3840} = \cdots = 2.5 $$