Reciprocal Income?

Let their incomes be denoted by $A$ and $B$ respectively. Then, we have,

$A=B+\dfrac{10}{100}B=\dfrac{110B}{100} \\ \implies \dfrac{B}{A}=\dfrac{10}{11}$

Required percentage $=\dfrac{A-B}{A}\times 100 = 100(1-\dfrac{B}{A})= 100(1-\dfrac{10}{11})=100\times \dfrac{1}{11} = \dfrac{100}{11}\%$

An easy way is to take a fixed value. Example - B's income is 100. So A will earn 110 (10% more than B). Now B's income is 10 less than A's (110). In percentage form that is

$\frac{10}{110} \times 100$

that is equal to

$\frac{100}{11}$

let B's income is 100... then A's income is 110. (100+10% of 100)... so B's income is 10 less than 110. .. B's income is (10/110)*100=100/11 % less.... so simple....

Let $x = Ans$ , $A = 100$ and $B = A - x$

From the information given in the question we can conclude that

$A = B + \frac {B}{10}$

This can also be written as

$A = \frac {11B}{10}$

Using this new equation and the fact that $A = 100$ we can use simultaneous equations to work out $B$

$\frac {11B}{10} = 100$

Re-arranging this gives us

$11B = 100*10 \Rightarrow B = \frac {1000}{11}$

And since $B = A - x$ we can substitute it into the equation for $B$

$A - x = \frac {1000}{11}$

We can substitute $A = 100$ back in to get

$100 - x = \frac {1000}{11}$

Now we re-arrange to get an equation for $-x$

$-x = \frac {1000}{11} - 100 \Rightarrow -x = \frac {1000}{11} - \frac {1100}{11} \Rightarrow -x = \frac {-100}{11}$

And finally multiply by $-1$ to get $x$

$x = \frac {100}{11}$

So the answer is $\frac {100}{11}$

Let B's income be X

Then

A's income = X + (10/100) X = (11/10) X

Now

Let B's income lesser than A's by a %

Then

B's income = X = (11/10) X - (a/100)(11/10) X

Then

a = 100/11