Mathematics, 15.09.2020 02:01 ConfusedJuliana

How to combine fraction

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$\Large \frac { x }{ y }$ $\begin{matrix} \rightarrow numerator \\ \rightarrow denominator \end{matrix}$

If two (or more) fractions have the same number as their denominator, you can just go ahead and add their numerators, without changing the denominator. For example:

$\frac { 1 }{ 2 }$ and $\frac { 4 }{ 2 }$ has the same number (2) as their denominator. So we will add their numerators, without changing the denominator.

$\frac { 1 }{ 2 } +\frac { 4 }{ 2 } \quad =\quad \frac { 1+4 }{ 2 } \quad =\quad \frac { 5 }{ 2 }$

So we added the numerators (1 + 4 = 5) and kept the denominator same (2) and we got ( $\frac{5}{2}$

This was the case where the fractions' denominator was same. What if it's not ?

If their denominator isn't equal, we're gonna have to equalize them ourself. How to do that ?

Let's show it with an example :

$\frac { 1 }{ 2 }$ and $\frac { 3 }{ 4 }$ do no have the same number as their denominator. To be able to add them, we have to equalize their denominators.

$\frac { 1 }{ 2 }$ 's denominator is 2 and $\frac { 3 }{ 4 }$ 's is 4

What can we do to equalize them ? Well, 4 is two times 2 ( $4 = 2\cdot 2$ ) So we cane multiply the denominator of $\frac { 1 }{ 2 }$ (which is 2) with 2 , to equal it to $\frac { 3 }{ 4 }$ 's denominator (which is 4).

But, there is a catch here. When multiplying a fraction's denominator before adding it to another, you should make sure that you're preserving its ratio. What does that mean ?

Let's take the number $\frac { 4 }{ 6 }$

(1) if we multiply only its numerator with a number (let it be 3)

$\frac { 3\cdot 4 }{ 6 } \quad =\quad \frac { 12 }{ 6 } \quad =\quad 2$

You got a new fraction with a different ratio than $\frac{4}{6}$ . And it is also equal to 2, but $\frac{4}{6}$ isn't equal to 2.

(2) if we multiply only its denominator with a number (let it be 3 again)

$\frac { 4 }{ 3\cdot 6 } \quad =\quad \frac { 4 }{ 18 }$

You got a new fraction again, with a different ration than $\frac{4}{6}$ .

How can we know that ? Well, if you simplify these two numbers to the simplest number, you'll get a different fraction or integer. Let's do so.

$\frac { 4 }{ 6 } \quad =\quad \frac { 2\cdot 2 }{ 2\cdot 3 } \quad =\quad \frac { 2 }{ 2 } \cdot \frac { 2 }{ 3 } \quad =\quad 1\cdot \frac { 2 }{ 3 } \quad =\quad \frac { 2 }{ 3 }$

So the simplified form of $\frac{4}{6}$ in the fraction form is $\frac{2}{3}$

$\frac { 4 }{ 18 } \quad =\quad \frac { 2\cdot 2 }{ 2\cdot 9 } \quad =\quad \frac { 2 }{ 2 } \cdot \frac { 2 }{ 9 } \quad =\quad 1\cdot \frac { 2 }{ 9 } \quad =\quad \frac { 2 }{ 9 }$

And the simplest form of $\frac{4}{18}$ as a fraction is $\frac{2}{9}$ , which is not equal to $\frac{2}{3}$

$\frac { 2 }{ 3 } \quad \neq \quad \frac { 2 }{ 9 }$

So what to do, to preserve the ratio ? Simle. We'll multiply also the numerator with the same number we're going to multiply the denominator with.

Let's get back to our example.

Adding $\frac { 1 }{ 2 }$ and $\frac { 3 }{ 4 }$

We were going to multiply $\frac{1}{2}$ 's denominator with 2. Now that we know, the ratio must not change, we'll also multiply the numerator with 2.

$\frac { 2\cdot 1 }{ 2\cdot 2 } \quad =\quad \frac { 2 }{ 4 }$

Now we've got a number which has the same denominator as $\frac { 3 }{ 4 }$

We can add them now,

$\frac { 2 }{ 4 } \quad +\quad \frac { 3 }{ 4 } \quad =\quad \frac { 2+3 }{ 4 } \quad =\quad \frac { 5 }{ 4 }$

$\boxed { \frac { 1 }{ 2 } \quad +\quad \frac { 3 }{ 4 } \quad =\quad \frac { 5 }{ 4 } }$

I hope this was clear, if not please ask and I'll try to explain.

Answer from: Quest

the answer is d

step-by-step explanation:

Answer from: Quest

idk u tell me

step-by-step explanation:

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