Income of Als 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. `11%`
D. `10%`

Answer

To solve the problem step by step, let's denote the incomes of A and B.

1. **Let B's income be denoted as \( B \).**
Since A's income is 10% more than B's income, we can express A's income as:
\[ A = B + 0.10B = 1.10B \]

**Hint:** Understand the relationship between A's income and B's income based on the percentage increase.

2. **Now, we need to express B's income as a percentage less than A's income.**
According to the problem, B's income is \( x\% \) less than A's income. This can be expressed as:
\[ B = A - \frac{x}{100}A = A \left(1 - \frac{x}{100}\right) \]

**Hint:** Remember that if something is \( x\% \) less than another quantity, you subtract that percentage from the whole.

3. **Substituting A's income into the equation for B's income:**
We already have \( A = 1.10B \). Now we substitute this into the equation for B:
\[ B = (1.10B) \left(1 - \frac{x}{100}\right) \]

**Hint:** Use substitution to relate the two equations involving A and B.

4. **Now, simplify the equation:**
Distributing \( (1 - \frac{x}{100}) \):
\[ B = 1.10B - \frac{1.10Bx}{100} \]

**Hint:** Keep track of the terms involving B on both sides of the equation.

5. **Rearranging the equation:**
Move all terms involving B to one side:
\[ B + \frac{1.10Bx}{100} = 1.10B \]
Factor out B:
\[ B \left(1 + \frac{1.10x}{100}\right) = 1.10B \]

**Hint:** Factor out common terms to simplify the equation.

6. **Dividing both sides by B (assuming B is not zero):**
\[ 1 + \frac{1.10x}{100} = 1.10 \]

**Hint:** Ensure that you are not dividing by zero, which is not allowed.

7. **Now, isolate \( x \):**
Subtract 1 from both sides:
\[ \frac{1.10x}{100} = 0.10 \]
Multiply both sides by 100:
\[ 1.10x = 10 \]
Finally, divide by 1.10:
\[ x = \frac{10}{1.10} = \frac{1000}{110} = \frac{100}{11} \]

**Hint:** Use basic algebraic operations to isolate the variable.

8. **Convert \( \frac{100}{11} \) into a mixed fraction:**
Dividing gives:
\[ 100 \div 11 = 9 \quad \text{remainder } 1 \]
Thus, \( \frac{100}{11} = 9 \frac{1}{11} \).

**Hint:** Converting improper fractions into mixed numbers can help in understanding the result better.

So, the value of \( x \) is:
\[ x = 9 \frac{1}{11} \% \]

**Final Answer:** \( x = 9 \frac{1}{11} \% \)