Income of A is 10% more than Income of B. Let B's income be x% less than A's Income. Find x.

Options

A. 9 `1/11` %
B. 10 `1/11` %
C. 0.11
D. 0.1

Answer

To solve the problem step by step, we will start by defining the incomes of A and B based on the information given in the question.

### Step 1: Define A's income in terms of B's income.
Since A's income is 10% more than B's income, we can express A's income (A) as:
\[ A = B + 0.10B = 1.10B \]

### Step 2: Express B's income in terms of A's income.
We know that B's income is x% less than A's income. This can be expressed as:
\[ B = A - \frac{x}{100}A \]
This simplifies to:
\[ B = A \left(1 - \frac{x}{100}\right) \]

### Step 3: Substitute A's income into the equation for B's income.
From Step 1, we know that \( A = 1.10B \). We can substitute this into the equation for B:
\[ B = (1.10B) \left(1 - \frac{x}{100}\right) \]

### Step 4: Simplify the equation.
Now, we can simplify the equation:
\[ B = 1.10B - \frac{1.10Bx}{100} \]
To isolate B, we can rearrange the equation:
\[ B + \frac{1.10Bx}{100} = 1.10B \]
Factoring B out:
\[ B \left(1 + \frac{1.10x}{100}\right) = 1.10B \]

### Step 5: Divide both sides by B (assuming B ≠ 0).
\[ 1 + \frac{1.10x}{100} = 1.10 \]

### Step 6: Solve for x.
Now, we can solve for x:
\[ \frac{1.10x}{100} = 1.10 - 1 \]
\[ \frac{1.10x}{100} = 0.10 \]
Multiplying both sides by 100:
\[ 1.10x = 10 \]
Now, divide by 1.10:
\[ x = \frac{10}{1.10} \]
\[ x = \frac{1000}{110} \]
\[ x = \frac{100}{11} \approx 9.09 \]

### Conclusion:
Thus, B's income is approximately 9.09% less than A's income.