We have many question testing our concepts on effects of arithmetic operations on a fraction.
It may be straightforward as
x+1y+1>xyOR
it may be hidden ..
which is the greatest -
79,3133,2124?
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 xy<1a) If you ADD same positive number to x and y, the fractionx+ay+a>xy
b) If you subtract same positive number from x and y, the fractionx−ay−a<xy
2)
x and y are positive numbers and x>y or fraction xy>1a) If you ADD same positive number to x and y, the fractionx+ay+a<xy
b) If you subtract same positive number from x and y, the fractionx−ay−a>xy
3)
x and y are positive numbers and x<y or fraction xy<1a) If you MULTIPLY or DIVIDE x and y with same positive number, the fractionx∗ay∗a=xy
b) If you MULTIPLY or DIVIDE x and y with same negative number , the fractionx∗−ay∗−a=xy
2)
x and y are positive numbers and x>y or fraction xy>1a) If you ADD same negative number to x and y, the effect will be as subtracting a positive number and the fractionx+ay+a>xy, 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 fractionx−ay−a<xy, where a is -x and x is a positive number
what can you answer now?which is greater-
3133−or−31+1033+10why does this happens...
because since x<y, any number will be a larger % of smaller number, so numerator has a BIGGER % changeexample 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<ya) If numerator is increased but denominator is decreased with positive numbers,the fractionx+1y−2>xy.. 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 1+13−4=−2
b) If numerator is decreased and denominator is increased with positive numbers,the fractionx−2y+1<xy
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