Length of a rectangular solid is increased by $10\%$ and breadth is decreased by $10\%$. Then the volume of the solid

### Concept & Formula When two multiplying variables are changed by $+x\%$ and $-x\%$, the net effect on their product is always a decrease of $\frac{x^2}{100}\%$. Since height is unmentioned, it is assumed constant, meaning the volume scales directly with the base area. $$ \text{Net Change} = \left( \frac{x^2}{100} \right) \% \text{ Decrease} $$ ### Step-by-Step Solution 1. **Set up the initial variables:** Let the original dimensions be $L, B,$ and $H$. Original Volume $V_1 = L \times B \times H$. 2. **Determine the new dimensions:** New Length $L_2 = L + 0.10L = 1.1L$ New Breadth $B_2 = B - 0.10B = 0.9B$ New Height $H_2 = H$ (remains unchanged) 3. **Calculate the new volume:** $V_2 = (1.1L) \times (0.9B) \times (H) = 0.99 LBH$ 4. **Determine the percentage change:** Change $= 0.99 V_1 - V_1 = -0.01 V_1$ A negative $0.01$ multiplier means a $1\%$ decrease. ### Exam Strategy & Shortcut Use the successive percentage formula: $A + B + \frac{AB}{100}$. Here, $A = +10$ and $B = -10$. $\text{Net effect} = 10 - 10 + \frac{10 \times -10}{100} = 0 - \frac{100}{100} = -1\%$. The negative sign indicates a decrease. ### Common Pitfall Assuming that an equal percentage increase and decrease cancel each other out, leading to the incorrect "remains unchanged" option. ### Final Answer Therefore, the correct answer is **decreases by 1%**.