To solve the equation \( w - 5 \frac{2}{5} = 3 \frac{1}{4} \), we first convert the mixed numbers to improper fractions.
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Convert \( 5 \frac{2}{5} \) to an improper fraction: \[ 5 \frac{2}{5} = 5 + \frac{2}{5} = \frac{5 \times 5 + 2}{5} = \frac{25 + 2}{5} = \frac{27}{5} \]
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Convert \( 3 \frac{1}{4} \) to an improper fraction: \[ 3 \frac{1}{4} = 3 + \frac{1}{4} = \frac{3 \times 4 + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4} \]
Now, rewrite the equation with the improper fractions: \[ w - \frac{27}{5} = \frac{13}{4} \]
Next, we will isolate \( w \) by adding \( \frac{27}{5} \) to both sides: \[ w = \frac{13}{4} + \frac{27}{5} \]
To perform this addition, we need a common denominator. The least common multiple of 4 and 5 is 20.
Convert both fractions: \[ \frac{13}{4} = \frac{13 \times 5}{4 \times 5} = \frac{65}{20} \] \[ \frac{27}{5} = \frac{27 \times 4}{5 \times 4} = \frac{108}{20} \]
Now, add the two fractions: \[ w = \frac{65}{20} + \frac{108}{20} = \frac{65 + 108}{20} = \frac{173}{20} \]
Thus, the solution is: \[ w = \frac{173}{20} \]
If you want to convert it back to a mixed number: \[ \frac{173}{20} = 8 \frac{13}{20} \]
So, the solution to the equation is: \[ w = 8 \frac{13}{20} \quad \text{or} \quad w = \frac{173}{20} \]
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