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
- 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:
$$ \frac{65}{3} - \left( \frac{3}{10} \right) = x $$
- 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:
$$ \frac{650}{30} - \frac{9}{30} = x $$
- 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.
