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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