Question

Mark again the correct answer: What percent of $\dfrac{2}{7}$ is $\dfrac{1}{35}$? \[\]
A. $20\%$\[\]
B. $25\%$\[\]
C. $15\%$\[\]
D. $10\%$\[\]

Answer 1

Hint: We recall the definition of percentage where if we say $y$ is $p\%$ of a means $y=\dfrac{p}{a}\times 100$. We assume $\dfrac{1}{35}$ is $x\%$ of $\dfrac{2}{7}$ and use the working rule for $y=\dfrac{1}{35}, p=x, a=\dfrac{2}{7}$. We solve for $x$ to get the required percentage. \[\]

Complete step-by-step solution:
We know that percentage is derived from the word per centum in Latin which means per hundred. The percentage in mathematics is a number or ratio expressed as a fraction of 100. If we have $a$ number of elements and there are total $b$ number of elements then we can express $a$ as a percentage $p$ of $b$ using the working rule,
\[p=\dfrac{a}{b}\times 100\]
We denote the percentage of $p$ as $p\%$ where ‘$\%$’ is symbol of percentage. If we say $p\%$ of $a$ that means if we divide $a$ into hundreds we can allocate $p$ in each of the hundred, for example 45% of 200 means we can allocate 45 for each hundred in 200. We can calculate the allocation $y$ using the rule,
\[y=\dfrac{p}{100}\times a\]
We are asked in the question What percent of $\dfrac{2}{7}$ is $\dfrac{1}{35}$. Let us assume $\dfrac{1}{35}$ is $x\%$ of $\dfrac{2}{7}$. So we use the percentage rule of allocation to have;
\[\begin{align}
  & \dfrac{1}{35}=x\%\text{ of }\dfrac{2}{7} \\
 & \Rightarrow \dfrac{1}{35}=\dfrac{x}{100}\times \dfrac{2}{7} \\
\end{align}\]
Let us multiply both side of the above step by $100\times 7$ and get;
\[\begin{align}
& \Rightarrow \dfrac{1}{35}\times 100\times 7=\dfrac{2x}{100\times 7}\times 100\times 7 \\
 & \Rightarrow \dfrac{100}{5}=2x \\
 & \Rightarrow 20=2x \\
\end{align}\]
We divide both side of the above step by 2 to have;
\[\begin{align}
  & \Rightarrow 10=x \\
 & \Rightarrow x=10 \\
\end{align}\]
So $\dfrac{1}{35}$ is $10\%$ of $\dfrac{2}{7}$ and hence the correct option is D.

Note: We note that the equation we solved $\dfrac{1}{35}=\dfrac{x}{100}\times \dfrac{2}{7}$ is an equation in the variable which we should always solve by collecting variable term and the constant term at different sides of the equation. We can alternatively find the percentage by dividing $\dfrac{1}{35}$ by $\dfrac{2}{7}$ and then multiplying the result by 100.