To find the exact value of [tex]\(3 \frac{7}{15}\)[/tex], we need to follow these steps:
### Step 1: Convert the Mixed Number to an Improper Fraction
A mixed number consists of a whole number and a fraction. Here, the mixed number is [tex]\(3 \frac{7}{15}\)[/tex].
1. Multiply the whole number part by the denominator of the fractional part:
[tex]\[3 \times 15 = 45\][/tex]
2. Add this result to the numerator of the fractional part:
[tex]\[45 + 7 = 52\][/tex]
So, [tex]\(3 \frac{7}{15}\)[/tex] can be written as the improper fraction:
[tex]\[\frac{52}{15}\][/tex]
### Step 2: Simplify the Fraction if Possible
The fraction [tex]\(\frac{52}{15}\)[/tex] is already in its simplest form. The greatest common divisor of 52 and 15 is 1, so the fraction does not need further simplification.
### Step 3: Convert the Improper Fraction to a Decimal
To convert [tex]\(\frac{52}{15}\)[/tex] to a decimal, we perform the division:
[tex]\[\frac{52}{15} \approx 3.466666666666667\][/tex]
The decimal value here is [tex]\(3.\overline{466666666666667}\)[/tex] (or [tex]\(3.4666\overline{6}\)[/tex] which means 3 followed by an infinite sequence of recurring sixes).
### Conclusion
The exact value of [tex]\(3 \frac{7}{15}\)[/tex] is [tex]\(3.466666666666667\)[/tex], which can be written as [tex]\(3.46\overline{6}\)[/tex]. Thus, the closest matching answer is:
[tex]\[3.466\overline{6}\][/tex]
