What is the simplest form of the fraction (8a^2+10ab)/(12a+15b)?

These kinds of questions often come up in GCSE papers and involve algebraic manipulation. The key to answering these questions is finding a common factor in the numerator and denominator that can be cancelled out.

Let's start with factorising the numerator, 8a2+10ab. Immediately we can see that both terms contain the variable a and the coefficients are divisible by 2. Taking 2a as a common factor between the terms, we can factorise this expression to get 2a(4a+5b).

Let's look at the denominator now, 12a+15b. Here, unlike the numerator, we have no common variable but both terms are divisible by 3. Taking 3 as a common factor, we get the expresion 3(4a+5b).

Let's rewrite the original fraction using the factorised numerator and denominator. Doing so gives us (2a(4a+5b))/(3(4a+5b)). Immediately we can see that the numerator and denominator have a common factor of (4a+5b) that can be cancelled. Cancelling out the factor gives us 2a/3 which is the simplest form. 

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