If 65/3 is subtracted by a rational number and the quotient of 3 and 10, the result is the rational number itself, find the number.

Steps to Solve

1. Set Up the Equation

Let's denote the unknown rational number as ( x ). According to the problem, we can set up the equation as follows:

$$ \frac{65}{3} - \left( \frac{3}{10} \right) = x $$

2. Combine and Simplify the Left Side

To simplify the left-hand side, we need a common denominator to combine the fractions. The common denominator for ( 3 ) and ( 10 ) is ( 30 ).

Convert ( \frac{65}{3} ) to have a denominator of ( 30 ):

$$ \frac{65}{3} = \frac{65 \times 10}{3 \times 10} = \frac{650}{30} $$

Convert ( \frac{3}{10} ) to have a denominator of ( 30 ):

$$ \frac{3}{10} = \frac{3 \times 3}{10 \times 3} = \frac{9}{30} $$

Now, substitute these back into the equation:

$$ \frac{650}{30} - \frac{9}{30} = x $$

3. Perform the Subtraction

Now, we can subtract the two fractions:

$$ x = \frac{650 - 9}{30} = \frac{641}{30} $$

This gives us the value for ( x ).