Question

Two numbers are respectively $30\%$ and $40\%$ less than a third number. What percent is the second of the first?

Answer 1

Percentage:

A number or ratio that may be expressed as a fraction of 100 is called a percentage.

Calculating the first number:

Consider x, y, and z to be the first, second, and third numbers respectively.

The first number is $30\%$ less than a third number.

.$$\begin{gathered} \Rightarrow \mathrm{x}=\mathrm{z}-\frac{30}{100}\mathrm{z} \\ \Rightarrow \mathrm{x}=\frac{100\mathrm{z}-30\mathrm{z}}{100} \\ \Rightarrow \mathrm{x}=\frac{70\mathrm{z}}{100} \\ \Rightarrow \mathrm{x}=\frac{7\mathrm{z}}{10} \end{gathered}$$

Calculating the second number:

The second number is $40\%$ less than a third number.

.$$\begin{gathered} \Rightarrow \mathrm{y}=\mathrm{z}-\frac{40}{100}\mathrm{z} \\ \Rightarrow \mathrm{y}=\frac{100\mathrm{z}-40\mathrm{z}}{100} \\ \Rightarrow \mathrm{y}=\frac{60\mathrm{z}}{100} \\ \Rightarrow \mathrm{y}=\frac{6\mathrm{z}}{10} \end{gathered}$$

Calculating the percentage of the second of the first:

$$\begin{gathered} \mathrm{Required}\mathrm{percentage}=\frac{\left({\displaystyle \frac{6\mathrm{z}}{10}}\right)}{\left({\displaystyle \frac{7\mathrm{z}}{10}}\right)}\times 100 \\ =\left(\frac{6\mathrm{z}}{10}\right)\times \left(\frac{10}{7\mathrm{z}}\right)\times 100 \\ =\frac{600}{7} \\ =85\frac{5}{7}\% \end{gathered}$$

Therefore $\mathbf{85}\frac{\mathbf{5}}{\mathbf{7}}\mathbf{\%}$ is the second of the first.