If $40\%\text{ of }P=0.5\text{ of }Q=\frac{1}{8}\text{ of }R$, find P : Q : R.

The correct option is 5 : 4 : 16.

To find the ratio $P:Q:R$, we first express each part of the given equation as a single fraction multiplied by the respective variable.

Given equation:

$40\%\text{ of }P=0.5\text{ of }Q=\frac{1}{8}\text{ of }R$

Step 1: Convert all coefficients into simple fractions.

$40\%\text{ of }P=\frac{40}{100}P=\frac{2}{5}P$

$0.5\text{ of }Q=\frac{5}{10}Q=\frac{1}{2}Q$

$\frac{1}{8}\text{ of }R=\frac{1}{8}R$

So, the given relation becomes:

$\frac{2}{5}P=\frac{1}{2}Q=\frac{1}{8}R=k$

Step 2: Express $P$, $Q$, and $R$ in terms of a constant $k$.

$P=\frac{5}{2}k$

$Q=2k$

$R=8k$

Step 3: Calculate the ratio $P:Q:R$.

$P:Q:R=\frac{5}{2}k:2k:8k$

Dividing through by $k$:

$P:Q:R=\frac{5}{2}:2:8$

Multiply each term by 2 to clear the fraction:

$P:Q:R=\left(\frac{5}{2}\times 2\right):\left(2\times 2\right):\left(8\times 2\right)$

$P:Q:R=5:4:16$

Thus, the ratio $P:Q:R$ is equal to 5 : 4 : 16.