To solve the problem, let's first convert the mixed numbers into improper fractions.
Step 1: Convert the mixed numbers to improper fractions.
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Convert \(3 \frac{1}{4}\): \[ 3 \frac{1}{4} = 3 \times 4 + 1 = 12 + 1 = 13 \quad \text{so,}\quad 3 \frac{1}{4} = \frac{13}{4}. \]
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Convert \(2 \frac{3}{8}\): \[ 2 \frac{3}{8} = 2 \times 8 + 3 = 16 + 3 = 19 \quad \text{so,} \quad 2 \frac{3}{8} = \frac{19}{8}. \]
Step 2: Set up the division of fractions. Now we need to divide: \[ \frac{13}{4} \div \frac{19}{8}. \] This is equivalent to multiplying by the reciprocal: \[ \frac{13}{4} \times \frac{8}{19} = \frac{13 \times 8}{4 \times 19}. \]
Step 3: Simplify the fractions. Calculating: \[ \frac{13 \times 8}{4 \times 19} = \frac{104}{76}. \]
Step 4: Simplify \(\frac{104}{76}\). To simplify, we find the greatest common divisor (GCD) of 104 and 76.
The GCD of 104 and 76 is 4. Now we can divide both the numerator and the denominator by 4: \[ \frac{104 \div 4}{76 \div 4} = \frac{26}{19}. \]
Step 5: Express \(\frac{26}{19}\) in the required format. This fraction is already in simplest form, and we can write it as: \[ a = 1, \quad b = 7, \quad c = 19. \]
Now, let's compare this with the provided options:
- a = 38, b = 14, c = 1
- a = 1, b = 14, c = 38
- a = 1, b = 7, c = 19
- a = 19, b = 7, c = 1
The correct values of \( a, b, c \) that represent the answer in simplest form are: \[ \boxed{1, 7, 19}. \]
1 answer