If 1,600 is increased by `20%`, and then reduced by `y%`, yielding 1,536, what is y?

Answer

To solve the problem step by step, we will follow the process of increasing the initial amount by 20%, then reducing it by \(y\%\), and finally equate it to the final amount of 1536.

### Step-by-Step Solution

1. **Calculate the Increase of 20% on 1600:**
\[ \text{Increased Amount} = 1600 + \left(1600 \times \frac{20}{100}\right) \]
\[ = 1600 + (1600 \times 0.2) \]
\[ = 1600 + 320 = 1920 \]

**Hint:** To find 20% of a number, multiply the number by 0.2.

2. **Set Up the Equation for Reduction:**
After increasing, the amount is reduced by \(y\%\):
\[ \text{Final Amount} = 1920 - \left(1920 \times \frac{y}{100}\right) \]
According to the problem, this final amount equals 1536:
\[ 1920 - \left(1920 \times \frac{y}{100}\right) = 1536 \]

**Hint:** The expression \(1920 \times \frac{y}{100}\) represents the amount reduced from 1920.

3. **Rearranging the Equation:**
Move the reduction term to the right side:
\[ 1920 - 1536 = 1920 \times \frac{y}{100} \]
\[ 384 = 1920 \times \frac{y}{100} \]

**Hint:** Subtracting the final amount from the increased amount gives the total reduction.

4. **Isolate \(y\):**
To isolate \(y\), multiply both sides by 100:
\[ 384 \times 100 = 1920y \]
\[ 38400 = 1920y \]

**Hint:** Multiplying both sides by 100 helps eliminate the fraction.

5. **Solve for \(y\):**
Divide both sides by 1920:
\[ y = \frac{38400}{1920} \]
\[ y = 20 \]

**Hint:** Dividing gives the percentage reduction directly.

### Conclusion
The value of \(y\) is \(20\%\).