Question

When we multiply a fraction's numerator and denominator by the same number, why is the fraction still the same value?

Answer 1

See below:

Explanation:

Idea 1 - What happens if I multiply anything by the number 1?

• I get the same number:

`2xx1=2=2xx1`

`16xx1=16=16xx1`

So I can multiply any number by 1 and get the same value and anything multiplied by 1 is simply that value.

• The same thing works when I divide by 1 - I get the same number.

For instance, I can say that:

`2=2/1=2`

`6=6/1=6`

So I can divide anything by 1 and get the same value and anything divided by 1 is simply that value.

~~~~~

Idea 2 - If I take any number and divide it by itself, I get 1

• Let's talk a second about the number 1:

`"any number"/"the same number"=1`

and I can write it the other way too:

`1="any number"/"the same number"`

and so I can write something like:

`1=2/2=1=6/6=1`

~~~~~~

Idea 3 - How we multiply fractions

We need one more idea and then we'll be ready to make equivalent fractions.

• How do we multiply fractions?

When I multiply `1/2xx3/5`, I take the numerators and multiply them and the denominators and multiply them, like this:

`1/2xx3/5=(1xx3)/(2xx5)=3/10`

~~~~~

Putting it all together

To make an equivalent fraction, we want to take a fraction (let's say `1/2`), multiply it by a form of 1 (let's say `4/4`) so that the value of the fraction doesn't change, but the appearance of the fraction will:

`1/2xx4/4=(1xx4)/(2xx4)=4/8`

So what have I done? Let's describe both `1/2` and `4/8` in terms of pizza.

With `1/2`, I've taken a pizza and sliced it in 2 (thus the denominator 2) and I get 1 of those slices (thus the numerator 1).

With `4/8`, I've taken a pizza and sliced it into 8 pieces and I have 4 of those pieces.

Notice I have the same amount of pizza, it's just in the first example it was one big slice and in the second example I have 4 smaller slices.

Here's an image to showcase what we've just done with math above:

Answer 2

Please see below.

Explanation:

A fraction, say `a/b`, where `a` is called as numerator and `b` is called denominator, assuming `a < b`, represents a part of a whole object, wherein the object is divided in `b` equal parts, of whom `a` are chosen.

For example, in the figure below, shows a full bar divided into `12` parts of which `9` parts (coloured blue) have been chosen and they represent `9/12` of the whole bar.

Let us consider a simple example, say `1/2`, which is say object divided in two equal parts of which one is chosen, like in the figure below and this blue portion represents `1/2` of whole.

What if we had divided the object in `4` parts and then chosen `2` parts. It would have appeared as shown below.

Although it is `2/4`, it is quite obvious that whether one gets `1/2` of a whole or gets `2/4` of the whole, there is no difference i.e. `1/2=2/4`.

What if we had divided the same in `6` parts and for getting equivalent amount, one would have to choose `3` parts (i.e. `3/6`) as is obvious from following figure.

Hence one can say that `3/6=2/4=1/2`.

Similarly dividing by `8` and choosing `4` out of it i.e. `4/8` appears as below.

It is quite obvious that `1/2=2/4=3/6=4/8`

Also observe that in this case we are just multiplying numerator and denominator same number (in above case `2`, `3` and `4`) and

`1/2=(1xx2)/(2xx2)=(1xx3)/(2xx3)=(1xx4)/(2xx4)`.

Now without actually drawing these figures consider `9/12` shown in the first figure. Observe that had we divided the figure in `4` parts, for equivalent portion, we would have selected `3/4`. Why?

Because `9/12=(9-:3)/(12-:3)=3/4`

Hence multiplying or dividing both numerator and denominator, make an equivalent fraction.