Question:
Let \(f\) be a function defined by
\[
f(xy)=\frac{f(x)}{y}
\]
for all positive real numbers \(x\) and \(y\). If \(f(30)=20\), then \(f(40)=\)
Let \(f\) be a function defined by
\[
f(xy)=\frac{f(x)}{y}
\]
for all positive real numbers \(x\) and \(y\). If \(f(30)=20\), then \(f(40)=\)
Show Hint
In functional equations, try to rewrite the required value in terms of the given value so that the functional relation can be applied directly.
Updated On: Jun 25, 2026
- \(10\)
- \(15\)
- \(25\)
- \(17\)
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The Correct Option is B
Solution and Explanation
Step 1: Use the given functional relation.
We are given \[ f(xy)=\frac{f(x)}{y} \] This relation is true for all positive real numbers \(x\) and \(y\).
We know \[ f(30)=20 \] We need to find \[ f(40) \]
Step 2: Express \(40\) in terms of \(30\).
Write \[ 40=30\times \frac{4}{3} \] Now compare with \[ f(xy)=\frac{f(x)}{y} \] Take \[ x=30 \] and \[ y=\frac{4}{3} \] Then, \[ f(40)=f\left(30\times \frac{4}{3}\right) \] Using the functional equation, \[ f(40)=\frac{f(30)}{\frac{4}{3}} \]
Step 3: Substitute the given value.
Since \[ f(30)=20, \] we get \[ f(40)=20\times \frac{3}{4} \] \[ f(40)=15 \]
Step 4: Final conclusion.
Therefore, \[ \boxed{15} \]
We are given \[ f(xy)=\frac{f(x)}{y} \] This relation is true for all positive real numbers \(x\) and \(y\).
We know \[ f(30)=20 \] We need to find \[ f(40) \]
Step 2: Express \(40\) in terms of \(30\).
Write \[ 40=30\times \frac{4}{3} \] Now compare with \[ f(xy)=\frac{f(x)}{y} \] Take \[ x=30 \] and \[ y=\frac{4}{3} \] Then, \[ f(40)=f\left(30\times \frac{4}{3}\right) \] Using the functional equation, \[ f(40)=\frac{f(30)}{\frac{4}{3}} \]
Step 3: Substitute the given value.
Since \[ f(30)=20, \] we get \[ f(40)=20\times \frac{3}{4} \] \[ f(40)=15 \]
Step 4: Final conclusion.
Therefore, \[ \boxed{15} \]
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