Respuesta :
[tex]\Large \frac { x }{ y } [/tex] [tex]\begin{matrix} \rightarrow numerator \\ \rightarrow denominator \end{matrix}[/tex]
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:
[tex]\frac { 1 }{ 2 } [/tex] and [tex]\frac { 4 }{ 2 } [/tex] has the same number (2) as their denominator. So we will add their numerators, without changing the denominator.
[tex]\frac { 1 }{ 2 } +\frac { 4 }{ 2 } \quad =\quad \frac { 1+4 }{ 2 } \quad =\quad \frac { 5 }{ 2 } [/tex]
So we added the numerators (1 + 4 = 5) and kept the denominator same (2) and we got ([tex] \frac{5}{2} [/tex]
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 :
[tex]\frac { 1 }{ 2 } [/tex] and [tex]\frac { 3 }{ 4 } [/tex] do no have the same number as their denominator. To be able to add them, we have to equalize their denominators.
[tex]\frac { 1 }{ 2 } [/tex] 's denominator is 2 and [tex]\frac { 3 }{ 4 } [/tex] 's is 4
What can we do to equalize them ? Well, 4 is two times 2 ( [tex]4 = 2\cdot 2[/tex] ) So we cane multiply the denominator of [tex]\frac { 1 }{ 2 } [/tex] (which is 2) with 2 , to equal it to [tex]\frac { 3 }{ 4 } [/tex] '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 [tex]\frac { 4 }{ 6 } [/tex]
(1) if we multiply only its numerator with a number (let it be 3)
[tex]\frac { 3\cdot 4 }{ 6 } \quad =\quad \frac { 12 }{ 6 } \quad =\quad 2[/tex]
You got a new fraction with a different ratio than [tex] \frac{4}{6} [/tex]. And it is also equal to 2, but [tex] \frac{4}{6} [/tex] isn't equal to 2.
(2) if we multiply only its denominator with a number (let it be 3 again)
[tex]\frac { 4 }{ 3\cdot 6 } \quad =\quad \frac { 4 }{ 18 } [/tex]
You got a new fraction again, with a different ration than [tex] \frac{4}{6} [/tex] .
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.
[tex]\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 } [/tex]
So the simplified form of [tex] \frac{4}{6} [/tex] in the fraction form is [tex] \frac{2}{3} [/tex]
[tex]\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 } [/tex]
And the simplest form of [tex] \frac{4}{18} [/tex] as a fraction is [tex] \frac{2}{9} [/tex] , which is not equal to [tex] \frac{2}{3} [/tex]
[tex]\frac { 2 }{ 3 } \quad \neq \quad \frac { 2 }{ 9 } [/tex]
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 [tex]\frac { 1 }{ 2 } [/tex] and [tex]\frac { 3 }{ 4 } [/tex]
We were going to multiply [tex] \frac{1}{2} [/tex] 's denominator with 2. Now that we know, the ratio must not change, we'll also multiply the numerator with 2.
[tex]\frac { 2\cdot 1 }{ 2\cdot 2 } \quad =\quad \frac { 2 }{ 4 } [/tex]
Now we've got a number which has the same denominator as [tex]\frac { 3 }{ 4 } [/tex]
We can add them now,
[tex]\frac { 2 }{ 4 } \quad +\quad \frac { 3 }{ 4 } \quad =\quad \frac { 2+3 }{ 4 } \quad =\quad \frac { 5 }{ 4 } [/tex]
[tex]\boxed { \frac { 1 }{ 2 } \quad +\quad \frac { 3 }{ 4 } \quad =\quad \frac { 5 }{ 4 } } [/tex]
I hope this was clear, if not please ask and I'll try to explain.
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:
[tex]\frac { 1 }{ 2 } [/tex] and [tex]\frac { 4 }{ 2 } [/tex] has the same number (2) as their denominator. So we will add their numerators, without changing the denominator.
[tex]\frac { 1 }{ 2 } +\frac { 4 }{ 2 } \quad =\quad \frac { 1+4 }{ 2 } \quad =\quad \frac { 5 }{ 2 } [/tex]
So we added the numerators (1 + 4 = 5) and kept the denominator same (2) and we got ([tex] \frac{5}{2} [/tex]
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 :
[tex]\frac { 1 }{ 2 } [/tex] and [tex]\frac { 3 }{ 4 } [/tex] do no have the same number as their denominator. To be able to add them, we have to equalize their denominators.
[tex]\frac { 1 }{ 2 } [/tex] 's denominator is 2 and [tex]\frac { 3 }{ 4 } [/tex] 's is 4
What can we do to equalize them ? Well, 4 is two times 2 ( [tex]4 = 2\cdot 2[/tex] ) So we cane multiply the denominator of [tex]\frac { 1 }{ 2 } [/tex] (which is 2) with 2 , to equal it to [tex]\frac { 3 }{ 4 } [/tex] '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 [tex]\frac { 4 }{ 6 } [/tex]
(1) if we multiply only its numerator with a number (let it be 3)
[tex]\frac { 3\cdot 4 }{ 6 } \quad =\quad \frac { 12 }{ 6 } \quad =\quad 2[/tex]
You got a new fraction with a different ratio than [tex] \frac{4}{6} [/tex]. And it is also equal to 2, but [tex] \frac{4}{6} [/tex] isn't equal to 2.
(2) if we multiply only its denominator with a number (let it be 3 again)
[tex]\frac { 4 }{ 3\cdot 6 } \quad =\quad \frac { 4 }{ 18 } [/tex]
You got a new fraction again, with a different ration than [tex] \frac{4}{6} [/tex] .
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.
[tex]\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 } [/tex]
So the simplified form of [tex] \frac{4}{6} [/tex] in the fraction form is [tex] \frac{2}{3} [/tex]
[tex]\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 } [/tex]
And the simplest form of [tex] \frac{4}{18} [/tex] as a fraction is [tex] \frac{2}{9} [/tex] , which is not equal to [tex] \frac{2}{3} [/tex]
[tex]\frac { 2 }{ 3 } \quad \neq \quad \frac { 2 }{ 9 } [/tex]
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 [tex]\frac { 1 }{ 2 } [/tex] and [tex]\frac { 3 }{ 4 } [/tex]
We were going to multiply [tex] \frac{1}{2} [/tex] 's denominator with 2. Now that we know, the ratio must not change, we'll also multiply the numerator with 2.
[tex]\frac { 2\cdot 1 }{ 2\cdot 2 } \quad =\quad \frac { 2 }{ 4 } [/tex]
Now we've got a number which has the same denominator as [tex]\frac { 3 }{ 4 } [/tex]
We can add them now,
[tex]\frac { 2 }{ 4 } \quad +\quad \frac { 3 }{ 4 } \quad =\quad \frac { 2+3 }{ 4 } \quad =\quad \frac { 5 }{ 4 } [/tex]
[tex]\boxed { \frac { 1 }{ 2 } \quad +\quad \frac { 3 }{ 4 } \quad =\quad \frac { 5 }{ 4 } } [/tex]
I hope this was clear, if not please ask and I'll try to explain.
