Question

Two numbers are 30 % and 40 % less than a third number. The first number as a percentage of the second is:
\[\left( a \right)75\%\]
\[\left( b \right)113\dfrac{1}{3}\%\]
\[\left( c \right)116\dfrac{2}{3}\%\]
\[\left( d \right)166\dfrac{2}{3}\%\]

Answer 1

Hint: First let us assume the three numbers as x, y and z. Then we will find x and y in terms of z using the information given in the question. Now, we will assume x as m% of y and use the relation, \[m=\dfrac{x}{y}\times 100\] to determine the value of m.

Complete step-by-step solution:
Here, we have been provided with two numbers that are 30% and 40% less than a third number. We have to determine the first number as a percentage of the second number. Now, let us assume x, y, and z as the first, second, and third numbers respectively. So, let us come to the given conditions one by one.
(1) It is given that the first number is 30% less than the third number that means x is 30% less than z. Therefore, mathematically,
\[x=z-30\%\text{ of z}\]
\[\Rightarrow x=z-\dfrac{30}{100}\times z\]
\[\Rightarrow x=z-\dfrac{3z}{10}\]
\[\Rightarrow x=\dfrac{7z}{10}.....\left( i \right)\]
(2) It is given that the second number is 40% less than the third number that means y is 40% less than z. Therefore, mathematically,
\[y=z-40\%\text{ of z}\]
\[\Rightarrow y=z-\dfrac{40}{100}\times z\]
\[\Rightarrow y=z-\dfrac{4z}{10}\]
\[\Rightarrow y=\dfrac{6z}{10}.....\left( ii \right)\]
Now, we are assuming that the first number (x) is m% of the second number (y). Therefore, mathematically, we have,
\[x=m\%\text{ of y}\]
\[\Rightarrow x=\dfrac{m}{100}\times y\]
\[\Rightarrow m=\dfrac{x}{y}\times 100......\left( iii \right)\]
Now dividing the equation (i) by equation (ii), we get,
\[\Rightarrow \dfrac{x}{y}=\dfrac{\dfrac{7z}{10}}{\dfrac{6z}{10}}\]
\[\Rightarrow \dfrac{x}{y}=\dfrac{7}{6}\]
So, substituting the value of \[\dfrac{x}{y}\] in equation (iii), we get,
\[\Rightarrow m=\dfrac{7}{6}\times 100\]
\[\Rightarrow m=\dfrac{7}{3}\times 50\]
\[\Rightarrow m=\dfrac{350}{3}\]
\[\Rightarrow m=116\dfrac{2}{3}\%\]
Hence, the option (c) is the right answer.

Note: One may note that to convert \[\dfrac{350}{3}\] into the improper fraction, we have divided 350 by 3 and the quotient is 116 and remainder is 2. So it is written as \[116\dfrac{2}{3}.\] To solve the above question we must know how to calculate the percentage. We do not need to find the exact values of x, y, and z to calculate the percentage because the percentage is a ratio multiplied by 100, and when we consider a ratio the common terms get canceled. So, no exact values for x, y, and z can be determined.