We have many question testing our concepts on effects of arithmetic operations on a fraction.

It may be straightforward as

\(\frac{x+1}{y+1}>\frac{x}{y}\)
OR
it may be hidden ..

which is the greatest - \(\frac{7}{9}, \frac{31}{33},\frac{21}{24}\)?

so lets check the different scenarios....

x and y are integers and both are increased/decreased with same number
I. both are >0 or both are <0
1) x and y are positive numbers and x<y or fraction \(\frac{x}{y}<1\)

a) If you ADD same positive number to x and y, the fraction\(\frac{x+a}{y+a}>\frac{x}{y}\)
b) If you subtract same positive number from x and y, the fraction\(\frac{x-a}{y-a}<\frac{x}{y}\)

2) x and y are positive numbers and x>y or fraction \(\frac{x}{y}>1\)

a) If you ADD same positive number to x and y, the fraction\(\frac{x+a}{y+a}<\frac{x}{y}\)
b) If you subtract same positive number from x and y, the fraction\(\frac{x-a}{y-a}>\frac{x}{y}\)

3) x and y are positive numbers and x<y or fraction \(\frac{x}{y}<1\)

a) If you MULTIPLY or DIVIDE x and y with same positive number, the fraction\(\frac{x*a}{y*a}=\frac{x}{y}\)
b) If you MULTIPLY or DIVIDE x and y with same negative number , the fraction\(\frac{x*-a}{y*-a}=\frac{x}{y}\)

2) x and y are positive numbers and x>y or fraction \(\frac{x}{y}>1\)

a) If you ADD same negative number to x and y, the effect will be as subtracting a positive number and the fraction\(\frac{x+a}{y+a}>\frac{x}{y}\), where a is -x and x is a positive number
b) If you SUBTRACT same negative number to x and y, the effect will be as adding a positive number and the fraction\(\frac{x-a}{y-a}<\frac{x}{y}\), where a is -x and x is a positive number

what can you answer now?
which is greater- \(\frac{31}{33}-or-\frac{31+10}{33+10}\)

why does this happens...because since x<y, any number will be a larger % of smaller number, so numerator has a BIGGER % change
example 1 is 50% of 2 but 1 is only 33.33% of 3

II. If x and y are integers and they undergo arithmetic operations with different number..
1. x and y are positive and x<y

a) If numerator is increased but denominator is decreased with positive numbers,the fraction\(\frac{x+1}{y-2}>\frac{x}{y}\).. after the arithmetic operations too, the fraction should be positive... say x/y = 1/3 add 1 to numerator and subtract 4 from denominator so \(\frac{1+1}{3-4}=-2\)
b) If numerator is decreased and denominator is increased with positive numbers,the fraction\(\frac{x-2}{y+1}<\frac{x}{y}\)
c) similar observation can be deduced of subtracting a number

Would add few more observations that would be useful in tackling such problems

questions testing these concepts will be added in the thread