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To find the most reduced form of a fraction, you need to find the common factors of the numerator (top) and denominator (bottom).
In this case we have a numerator of 45 and a denominator of 80.
The best way to find common factors is to find the prime factorization of each number by dividing out numbers until we have only prime numbers left (numbers whose only integer factors are themselves and 1). To be honest, there's no really good way of doing this besides "guess and check" or "trial and error."
We'll start by finding the prime factors of 45. I know 45 ends in 5, so that means I can divide 5 out:
`45-:5 = 9`
That means 5 is one of my factors. Because 5 is prime, I'm going to leave it alone. 9, though, is not prime and is divisible by 3:
`9-:3 = 3`
This means that 3 is another one of my factors. The result of factoring out 3 has given me another prime number (3), so now I know I'm done finding the prime factorization, given below:
`45 = 5*3*3`
Now, I need to factor the denominator, 80, to see whether I have any prime factors that match.
I know 80 ends in "0," so that must mean it is divisible by 10:
`80-:10 = 8`
Now, neither 10 nor 8 are prime, so I'll split them up into their factors, knowing that both can be divided by 2:
`10-:2 = 5`
`8-:2 = 4`
5 is a prime number, so I can stop factoring the 10. 4 is not prime, though, so I continue to factor it, knowing it is divisible by 2:
`4-:2 = 2`
Now, we have another prime number, 2, that results from dividing 4 by 2. Because all of our factors are now prime, we can stop factoring, because we found the prime factorization of 80, shown below:
`80 = 5*2*2*2*2`
That may have been difficult to follow, I know. Basically all I did was put in any prime number that I divided by (2's for 80) and any prime number that came after dividing by something else. It works better as a tree, and I linked to a copy of one at the end of this, so you can see better how the factors break down. Granted, the picture factors in a slightly different way than I did, but the idea remains the same.
Now, we have the prime factorization of both 45 and 80:
`45 = 5*3*3`
`80 = 5*2*2*2*2`
You can see that the only prime factors the two numbers share are 5. This means when we divide the two numbers, we can divide out 5 from both, giving us a new set of factors:
Top (numerator): `3*3`
Bottom (denominator): `2*2*2*2`
Now, they don't share any factors, so we just simplify and put back together our fraction:
`(3*3)/(2*2*2*2) = 9/16`
And that's how you get 9/16.
Don't feel bad about finding this difficult. Mathematicians still haven't found a good, efficient way of finding the prime factors of numbers in the thousands of years since math was founded as a concept!
On the other hand, our inability to factor based on something besides trial and error keeps our banking information/social security numbers/lots of other data on us safe! See the third link below to see what I mean.
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