## Fractions are Hard!

### 3.6 Multiplying and dividing by numbers smaller and larger than 1

MULTIPLYING BY A POSITIVE NUMBER LARGER THAN 1

People say that multiplying a quantity by a number bigger than one gives an answer bigger than the quantity. Is this true?

For instance, is $$\dfrac{5}{4}\times N$$ larger than $$N$$?

Well, yes.

$$\dfrac{5}{4}\times N = \left(1+\dfrac{1}{4}\right)N = N + \dfrac{1}{4}N = N + more$$.

In general, any number larger that $$1$$ can be written as $$1 + \epsilon$$ , with $$\epsilon$$ positive, and

$$\left(1+\epsilon\right) \times N = N + more$$.

MULTIPLYING BY A POSITIVE NUMBER SMALLER THAN 1

In the same way, multiplying a positive quantity by a positive number smaller than $$1$$ is sure to give an answer smaller than the original quantity. For example,

$$\dfrac{4}{5}\times N = \left(1-\dfrac{1}{5}\right)N = N – something.$$

In general, any positive quantity smaller than $$1$$ can be written in the form $$1 – \epsilon$$, with $$\epsilon$$ positive, and we have

$$\left(1-\epsilon\right)N = N – something$$.

DIVIDING BY A POSITIVE NUMBER SMALLER THAN 1

Does dividing a quantity by a positive number smaller than $$1$$ give a bigger or smaller result?

Let’s try $$100 \div \dfrac{4}{5}$$ as an example.

$$\dfrac{100}{\dfrac{4}{5}}=\dfrac{100\times 5}{\dfrac{4}{5}\times 5}=\dfrac{500}{4}=125$$

which is larger than $$100$$.

In general, can we say that $$\dfrac{N}{1-\epsilon}$$ is sure to be larger than $$N$$?

To compare these quantities, write them each with a common denominator.

$$\dfrac{N}{1-\epsilon}$$

$$N=\dfrac{N}{1}= \dfrac{N\times \left(1-\epsilon\right)}{1-\epsilon}=\dfrac{N-something}{1-\epsilon}$$.

We see that $$\dfrac{N}{1-\epsilon}$$ is larger than $$N$$.

DIVIDING BY A POSITIVE NUMBER LARGER THAN 1

 Question: Is a quantity $$\dfrac{N}{1+\epsilon}$$ sure to be larger or smaller than $$N$$?

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