To solve the multiplication of the mixed numbers [tex]\(6 \frac{2}{5} \times 2 \frac{5}{8}\)[/tex], we need to follow several steps. Here is the step-by-step solution:
### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert [tex]\(6 \frac{2}{5}\)[/tex] to an improper fraction:
- First, multiply the whole number part (6) by the denominator (5):
[tex]\(6 \times 5 = 30\)[/tex].
- Then, add the numerator (2) to this product:
[tex]\(30 + 2 = 32\)[/tex].
- So, [tex]\(6 \frac{2}{5}\)[/tex] is equivalent to [tex]\(\frac{32}{5}\)[/tex].
2. Convert [tex]\(2 \frac{5}{8}\)[/tex] to an improper fraction:
- First, multiply the whole number part (2) by the denominator (8):
[tex]\(2 \times 8 = 16\)[/tex].
- Then, add the numerator (5) to this product:
[tex]\(16 + 5 = 21\)[/tex].
- So, [tex]\(2 \frac{5}{8}\)[/tex] is equivalent to [tex]\(\frac{21}{8}\)[/tex].
### Step 2: Multiply the Fractions
To multiply the fractions [tex]\(\frac{32}{5}\)[/tex] and [tex]\(\frac{21}{8}\)[/tex]:
[tex]\[ \frac{32}{5} \times \frac{21}{8} \][/tex]
Multiply the numerators:
[tex]\[ 32 \times 21 = 672 \][/tex]
Multiply the denominators:
[tex]\[ 5 \times 8 = 40 \][/tex]
Thus, the resulting fraction is:
[tex]\[ \frac{672}{40} \][/tex]
### Step 3: Simplify the Fraction
To simplify [tex]\(\frac{672}{40}\)[/tex], we need to find the greatest common divisor (GCD) of 672 and 40:
- The GCD of 672 and 40 is 8.
Divide both the numerator and the denominator by the GCD to simplify the fraction:
[tex]\[ \frac{672 \div 8}{40 \div 8} = \frac{84}{5} \][/tex]
### Final Answer:
Thus, [tex]\(6 \frac{2}{5} \times 2 \frac{5}{8}\)[/tex] simplifies to [tex]\(\frac{84}{5}\)[/tex].
