How do you add two fractions?

Question

I have a fraction I am trying to solve. I know the answer already, as Wolfram says it is $\frac{143}{300}$.
The fraction is: $$\frac{5}{12} + \frac{3}{50} = \space ?$$ Please explain why and how your method works.

Answer 1

The fractions need to have the same denominator in order to add them together. The denominator must be the least common multiple of the two denominators, or a multiple of it. The least common multiple of 12 and 50 is 300.

First multiply $5/12$ by $25/25$. Since $25/25 = 1$ the product is the same as the original fraction. You simply multiply the numerators together and the denominators together:

$$\frac{5}{12}\times\frac{25}{25} = \frac{125}{300}$$

Then multiply $3/50$ by $6/6$. Again, the product is the same as the original fraction:

$$\frac{3}{50}\times\frac{6}{6} = \frac{18}{300}$$

Now both fractions have the same denominator and you can simply add the numerators:

$$\frac{125}{300}+\frac{18}{300} = \frac{143}{300}$$

Answer 2

Null has given you a good way. Here's a way without worrying about the LCM: $${a\over b}+{c\over d}={ad+bc\over bd}$$ In the example, $${5\over12}+{3\over50}={(5)(50)+(12)(3)\over(12)(50)}={286\over600}={143\over300}$$

The price of not worrying about the LCM is that you get an answer, $286/600$, that isn't in lowest terms, so you have the extra step at the end of reducing the fraction.

Answer 3

Here is another equivalent idea, put $$x=\frac{5}{12}+\frac{3}{50}$$ Multiply by 12 $$12x=5+\frac{36}{50}=5+\frac{18}{25}$$ Multiply by 25 $$300x=125+18$$ Divide by 300 $$x=\frac{143}{300}$$

Answer 4

If $a=b$ then for any function $f(a)=f(b)$. Suppose $\displaystyle x=\frac{5}{12}+\frac{3}{50}$. Then $\displaystyle (12\cdot 50)\cdot x=(12\cdot 50)\cdot\left(\frac{5}{12}+\frac{3}{50}\right)$, so $\displaystyle 600x=\frac{12\cdot 50\cdot 5}{12}+\frac{12\cdot 50\cdot 3 }{50}=$ $=50\cdot 5+12\cdot 3=286$, why $\displaystyle x=\frac{286}{600}=\frac{143}{300}$.