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.

Understand the Problem

The question is asking to find a rational number that, when subtracted from 65/3 and the quotient of 3 and 10, results in the rational number itself. This can be framed as an equation to solve for the unknown rational number.

Answer

The rational number is \( \frac{641}{30} \).
Answer for screen readers

The rational number is ( \frac{641}{30} ).

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:

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

  1. 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:

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

  1. 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 ).

The rational number is ( \frac{641}{30} ).

More Information

The rational number ( \frac{641}{30} ) is the solution to the equation where subtracting ( \frac{3}{10} ) from ( \frac{65}{3} ) results in the same number. This means that the calculation and the balance in the equation are verified.

Tips

  • Forgetting to find a common denominator: This is crucial when subtracting fractions. Make sure to align the denominators before performing operations.
  • Mismanaging subtraction: Carefully track signs and values to avoid errors during fraction subtraction.
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