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# Two numbers are less than the third number by $30%$ and $37%$ respectively. How much percent is the second number less than by the first?(a) 8(b) 9(c) 10(d) 11  Hint: Assume that the third number is z and the first and second numbers are x and y respectively. Use the fact that $a%$ of b is given by $\dfrac{ab}{100}$. Write x and y in terms of z. Subtract the two terms to calculate the difference between them and thus find the percentage difference between them.

We have data regarding three numbers, relating them by the percentage difference between them. We have to calculate the percentage difference between the two numbers.
Let’s assume that the third number is z and the first and second numbers are x and y respectively.
We know that x is $30%$ less than z and y is $37%$ less than z.
We know that the value of $a%$ of b is given by $\dfrac{ab}{100}$.
Substituting $a=30,b=z$ in the above expression, we have $30%$ of z equal to $\dfrac{30z}{100}=0.3z$.
Thus, the value of x $=z-0.3z=0.7z$.

Similarly, we know that y is $37%$ less than z.
Thus, the value of $37%$ of z is $\dfrac{37z}{100}=0.37z$.
So, the value of y $=z-0.37z=0.63z$.
Now, we know that $x=0.7z$ and $y=0.63z$.
We will now calculate the percentage difference between values of x and y.
Subtracting y from x, we have $x-y=0.7z-0.63z=0.07z$.
To calculate the percentage difference between the two numbers, we will multiply the ratio of $x-y$ to x by 100.
Thus, we have the percentage difference between x and y $=\dfrac{0.07z}{0.7z}\times 100=0.1\times 100=10%$.
Hence, the second number is less than the first number by $10%$, which is option (c).

Note: One must clearly know the concepts of percentage to find the percentage difference between two numbers. Percentage is a number or a ratio that represents the fraction out of 100. We can’t find the exact value of the three numbers as we have data regarding the ratio between the numbers.
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