A number is increased by 10%. To obtain the old number, by how much percent should it be decreased?

Options

A. 9.09
B. 11.11
C. 10
D. `12.5`

Answer

To solve the problem step by step, let's denote the old number as \( x \).

### Step 1: Define the Old Number
Let the old number be \( x \).

**Hint:** Start by assigning a variable to the old number to make calculations easier.

### Step 2: Calculate the New Number After Increase
When the old number is increased by 10%, the new number becomes:
\[ \text{New Number} = x + 0.1x = 1.1x \]

**Hint:** Remember that increasing a number by a percentage involves adding that percentage of the number to itself.

### Step 3: Determine the Decrease Needed to Get Back to the Old Number
To find out how much we need to decrease the new number \( 1.1x \) to return to the old number \( x \), we can set up the equation:
\[ \text{Decrease} = 1.1x - x = 0.1x \]

**Hint:** The decrease is simply the difference between the new number and the old number.

### Step 4: Calculate the Percentage Decrease
Now, we need to express this decrease as a percentage of the new number \( 1.1x \):
\[ \text{Percentage Decrease} = \left(\frac{\text{Decrease}}{\text{New Number}}\right) \times 100 = \left(\frac{0.1x}{1.1x}\right) \times 100 \]

**Hint:** To find the percentage, divide the decrease by the new number and multiply by 100.

### Step 5: Simplify the Expression
Now, simplify the expression:
\[ \text{Percentage Decrease} = \left(\frac{0.1}{1.1}\right) \times 100 \]
Calculating this gives:
\[ \text{Percentage Decrease} = \frac{10}{11} \times 100 \approx 9.09\% \]

**Hint:** When simplifying fractions, remember to multiply by 100 to convert it to a percentage.

### Conclusion
Thus, to obtain the old number after a 10% increase, the new number should be decreased by approximately \( 9.09\% \).

**Final Answer:** The number should be decreased by \( 9.09\% \).

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