If $a = b \times \frac{d}{c}$; $b$, $c$ and $d$ are each increased by $10\%$, then by how much does $a$ increase?

### Concept & Logic When evaluating how percentage changes in individual variables affect a combined equation, you can use multiplying factors. An increase of $10\%$ corresponds to multiplying by $1.1$. $$a_{new} = b_{new} \times \frac{d_{new}}{c_{new}}$$ ### Step-by-step Solution **Step 1: Determine the new values for each variable** Since $b$, $c$, and $d$ all increase by $10\%$, we can express their new values as: $b_{new} = 1.1b$ $c_{new} = 1.1c$ $d_{new} = 1.1d$ **Step 2: Substitute these into the original equation** $a_{new} = (1.1b) \times \frac{1.1d}{1.1c}$ **Step 3: Simplify the expression** Notice that the $1.1$ in the numerator (from $d$) and the $1.1$ in the denominator (from $c$) cancel each other out: $a_{new} = 1.1b \times \frac{d}{c}$ **Step 4: Compare with the original equation** Since original $a = b \times \frac{d}{c}$, we can substitute $a$ back in: $a_{new} = 1.1a$ A multiplier of $1.1$ means the value has increased by $10\%$. ### Exam Strategy & Shortcut Look at the structure of the equation $a = \frac{bd}{c}$. The variables $d$ and $c$ have a direct and inverse relationship, respectively. If both increase by the exact same percentage, their effects perfectly cancel out. This leaves only $b$, which increases by $10\%$. Therefore, $a$ must also simply increase by $10\%$. You can solve this in $5$ seconds without writing anything down. ### Common Pitfall A very common mistake is assuming that multiple $10\%$ increases compound across the whole equation (e.g., guessing $21\%$ or more) without noticing that the variable $c$ is in the denominator, which effectively reduces the overall value. Always set up the multipliers to see what cancels. ### Final Answer **Therefore, the correct answer is $10\%$.**