Question

A rational number with a denominator of 9 is divided by (-2/3). The result is multiplied by 4/5 and then -5/6 is added. The final value is 1/10. What is the original rational?

Answer 1

`- frac(7)(9)`

Explanation:

"Rational numbers" are fractional numbers of the form `frac(x)(y)` where both the numerator and denominator are integers, i.e. `frac(x)(y);` `x, y in ZZ`.

We know that some rational number with a denominator of `9` is divided by `- frac(2)(3)`.

Let's consider this rational to be `frac(a)(9)`:

`" " " " " " " " " " " " " " " " " " " frac(a)(9) div - frac(2)(3)`

`" " " " " " " " " " " " " " " " " " " frac(a)(9) times - frac(3)(2)`

`" " " " " " " " " " " " " " " " " " " " - frac(3 a)(18)`

Now, this result is multiplied by `frac(4)(5)`, and then `- frac(5)(6)` is added to it:

`" " " " " " " " " " " " " " (- frac(3 a)(18) times frac(4)(5)) + (- frac(5)(6))`

`" " " " " " " " " " " " " " " " " " - frac(12 a)(90) - frac(5)(6)`

`" " " " " " " " " " " " " " " " " - (frac(12 a)(90) + frac(5)(6))`

`" " " " " " " " " " " " " " - (frac(6 times 12 a + 90 times 5)(90 times 6))`

`" " " " " " " " " " " " " " " " - (frac(72 a + 450)(540))`

Lastly, we know that the final value is `frac(1)(10)`:

`" " " " " " " " " " " " " " - (frac(72 a + 450)(540)) = frac(1)(10)`

`" " " " " " " " " " " " " " frac(72 a + 450)(540) = - frac(1)(10)`

`" " " " " " " " " " " " " " 72 a + 450 = - frac(540)(10)`

`" " " " " " " " " " " " " " 72 a + 450 = - 54`

`" " " " " " " " " " " " " " " 72 a = - 504`

`" " " " " " " " " " " " " " " " " a = - 7`

Let's substitute `- 7` in place of `a` in our rational number:

`" " " " " " " " " " " " " " " " frac(a)(9) = - frac(7)(9)`

Therefore, the original rational number is `- frac(7)(9)`.