Question

Adding and subtracting with unlike denominator
6/7 + 1/3

Answer 1

`\displaystyle\\(6)/(7)+(1)/(3)=(^(3))6~)/(~~7~)+(^(7))1~)/(~~3~)=(3*6)/(3*7)+ (7*1)/(7*3)=(18)/(21)+ (7)/(21)=(18+7)/(21)=\boxed{(25)/(21)}`

Answer 2

We can indeed only sum and subtract fractions with the same denominator, so we can't perform the sum as it is presented to us.

We can, though, transform both fractions so that they still yield the same numeric value, but written in a more convenient form.

In particular, we know that multiplying by 1 does not affect the value of a number, and we can express 1 as a fraction with the same numerator and denominator:
`1=(a)/(a)`

So, let's multiply both fractions by 1, written in a convenient way, so that they have the same denominator:

`(6)/(7) = (6)/(7)\cdot(3)/(3) = (18)/(21)`

`(1)/(3) = (1)/(3)\cdot(7)/(7) = (7)/(21)`

Now the fractions have the same denominator, and their sum is simply the sum of the numerators:

`(18)/(21)+(7)/(21) = (25)/(21)`