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Math Expert V
Joined: 02 Aug 2009
Posts: 8638
Effects of arithmetic operations on Fractions  [#permalink]

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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

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

_________________
Math Expert V
Joined: 02 Aug 2009
Posts: 8638
Re: Effects of arithmetic operations on Fractions  [#permalink]

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2
x and y are integers and both are increased/decreased with same number
I. both are
<0
1) x and y are negative numbers and x>y then 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 negative numbers and x<y then 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 negative numbers and x>y then 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 negative numbers and x<y then 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 = -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 = -x and x is a positive number

_________________
##### General Discussion
Math Expert V
Joined: 02 Aug 2009
Posts: 8638
Re: Effects of arithmetic operations on Fractions  [#permalink]

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2
1
1) https://gmatclub.com/forum/if-a-0-and-b-0-is-a-2p-b-2p-a-b-267075.html

If $$a<0$$ and $$b<0$$ , is $$\frac{(a-2p)}{(b-2p)}$$ $$>$$ $$\frac{a}{b}$$ ?

(1) $$p>0$$

(2) $$a<b$$

2) https://gmatclub.com/forum/if-m-0-and-n-0-is-m-x-n-x-m-n-93967.html

If m > 0 and n > 0, is (m+x)/(n+x) > m/n?

(1) m < n.

(2) x > 0.

3) https://gmatclub.com/forum/is-x-1-y-1-x-y-1-0-x-y-2-xy-268847.html
Is $$\frac{x + 1}{y + 1} > \frac{x}{y}$$?

(1) $$0 < x < y$$

(2) $$xy > 0$$

NEW question from GMAT® Official Guide 2019

(DS09315)
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Non-Human User Joined: 09 Sep 2013
Posts: 15110
Re: Effects of arithmetic operations on Fractions  [#permalink]

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_________________ Re: Effects of arithmetic operations on Fractions   [#permalink] 09 Jul 2019, 07:12

# Effects of arithmetic operations on Fractions  