Question

What is the value of the expression
Picture above
Thanks

Answer 1

First, we want the fractions in the numerator to have the same denominator.

Find the LCM (least common multiple) of 5 and 10:

Multiples of 5: 5, 10, 15, 20

Multiples of 10: 10, 20

10 < 20

The LCM is 10. Divide the LCM by the smaller denominator for 2/5:

`10 / 5 = 2`

Multiply the numerator and denominator of the fraction by this number:

`(2)/(5) \cdot (2)/(2) = (4)/(10)`

We can now solve for the overall numerator:

`(4)/(10) + (3)/(10) = (7)/(10)`

The expression should now look like this:

`((7)/(10))/(-(7)/(9))`

We can multiply the numerator by the fraction's denominator within the expression's denominator:

`((7)/(10))/(-(7)/(9)) = ((7)/(10) \cdot 9)/(-7) = ((63)/(10))/(-7)`

We can eliminate the denominator of the expression's numerator fraction by multiplying it by the expression's denominator:

`((63)/(10))/(-7) = (63)/(-7 \cdot 10) = (63)/(-70)`

Find the GCF (greatest common factor) between 63 and 70:

Factors of 63: 1, 3, 7, 9, 21, 63

Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70

7 > 1

GCF : 7

Divide the numerator and denominator by the GCF:

`(63)/(-70) / (7)/(7) = (9)/(-10)`

There is a negative sign in the denominator. Move this negative sign outside of the fraction:

`(9)/(-10) = \boxed{-(9)/(10)}`

The answer in simplest form will be -9/10.