Numerator/Denominator confusion

I observed that a bit of confusion prevails while simplifying fractions. The confusion is about which number will go to numerator part and which number will go to denominator part. This post is intended to clear the confusion.

Consider a fraction in the form (a/b)/(c/d).

Let us call the variables like this:

a – Numerator-numerator

b – Numerator-denominator

c – Denominator-numerator

d – Denominator-denominator

The rules:

Numerator-numerator remains in the the numerator part of the result fraction

Numerator-denominator remains in the the denominator part of the result fraction

Denominator-numerator always goes to the denominator part of the result fraction

Denominator-denominator always goes to the numerator part of the result fraction

The thumb rule here is:

If the number is either of type “Numerator-numerator” or “Denominator-denominator”, then it will go to numerator part. In other cases, it will go to denominator part.

So applying these rules, the given fraction takes the following form:

'a' remains in numerator, 'b' goes to denominator, 'c' remains in denominator and 'd' comes up to the numerator.

(a/b)/(c/d) = (a*d)/(b*c)

It's not like always there exist four parts in such fractions. Sometimes it is like this: (x/y)/z. This contains only three numbers. Here

x is Numerator-numerator,

y is Numerator-denominator,

and z is Denominator-numerator.

Applying the thumb rule, x remains in numerator while y and z settle in denominator.

=> (x/y)/z = x/(y*z)

If we take the other form p/(q/r), it changes like this: (p*r)/q

Try out the following examples:

Question set:

1.      (3/4)/(7/8)

2.      3/(4/5)

3.      (3/4)/5

4.      (2/3)/4

5.      (8/9)/(2/3)

Solution set:

1.      (3/4)/(7/8) = (3*8)/(4*7) = 6/7

2.      3/(4/5) = (3*5)/4 = 15/4

3.      (3/4)/5 = 3/(4*5) = 3/20

4.      (2/3)/4 = 2/(3*4) = 1/6

5.      (8/9)/(2/3) = (8*3)/(9*2) = 4/3