I am doing this problem a+3 / a - 5 x 2a - 10 / 3a + 9 the answer given is 2/3 but as there are 3 a's in the numerator line and 4 a's in the denominator line how do these cancell out?
ok, you need to be more clear as to what you mean. if you can't use latex, type fractions like this:
(numerator)/(denominator)
example, type (2x + 5)/(x + 9) to mean $\displaystyle \frac {2x + 5}{x + 9}$
i think you mean $\displaystyle \frac {a + 3}{a - 5} \times \frac {2a - 10}{3a + 9}$ but i am not sure. that is certainly not what you typed, but it is probably what you meant
Firstly, please use parantheses when you write a problem.
I assume you are asking for this:
$\displaystyle \frac{a+3}{a-5}\cdot \frac{2a - 10}{3a + 9}$
Factorize 2a-10 and 3a+9.
$\displaystyle \frac{\not a+\not 3}{\not a-\not 5}\cdot \frac{2 (\not a- \not 5)}{3 ( \not a+\not 3)}$
$\displaystyle \frac{2}{3}$
"Cancel"? What is that? Never do that.
$\displaystyle \frac{a+3}{a-5}*\frac{2a-10}{3a+9} =$
$\displaystyle \frac{a+3}{a-5}*\frac{2(a-5)}{3(a+3)} =$ -- That's right, the Distributive Property of Multiplication over Addition
$\displaystyle \frac{a+3}{a-5}*\frac{2}{3}*\frac{a-5}{a+3} =$ -- What's that? Multiplication?
$\displaystyle \frac{2}{3}*\frac{a+3}{a-5}*\frac{a-5}{a+3} =$ -- Now we see the Commutative Property of Multiplication.
$\displaystyle \frac{2}{3}*\frac{a+3}{a+3}*\frac{a-5}{a-5} =$ -- Same thing, but only in the denominators.
$\displaystyle \frac{2}{3}*1*1 = \frac{2}{3}$ -- At last, an Identity Definition and Identity Property
There is no "cancelling" going on in there. "Cancelling" is bad. Properties of mathematics will solve math problems. There is a reason why you studied those properties. I dare you to show me a text book that describes the "Property of Cancelling".
Special Note: This is good only for $\displaystyle a \neq 5$ and $\displaystyle a \neq -3$. You tell me why.