To solve [tex]\( 3 \frac{2}{3} + 2 \overline{3} \)[/tex] in a detailed, step-by-step manner, let's break down the components and perform the calculations systematically.
1. Convert the mixed number to an improper fraction:
[tex]\( 3 \frac{2}{3} \)[/tex] can be converted to an improper fraction by multiplying the whole number part by the denominator and then adding the numerator. So:
[tex]\[
3 \frac{2}{3} = \frac{3 \times 3 + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3}
\][/tex]
2. Convert the repeating decimal to a fraction:
The repeating decimal [tex]\( 2.\overline{3} \)[/tex] (which means [tex]\( 2.3333... \)[/tex]) can be represented as a fraction. A repeating decimal of the form [tex]\( a.bc\overline{d} \)[/tex] can be represented where [tex]\( d \)[/tex] repeats. For [tex]\( 2.\overline{3} \)[/tex]:
[tex]\[
2.\overline{3} = 2 + 0.\overline{3}
\][/tex]
The decimal [tex]\( 0.\overline{3} \)[/tex] is equivalent to [tex]\( \frac{1}{3} \)[/tex]. Therefore:
[tex]\[
2.\overline{3} = 2 + \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3}
\][/tex]
3. Sum the improper fractions:
Now, we need to add the two improper fractions [tex]\( \frac{11}{3} \)[/tex] and [tex]\( \frac{7}{3} \)[/tex]:
[tex]\[
\frac{11}{3} + \frac{7}{3} = \frac{11 + 7}{3} = \frac{18}{3}
\][/tex]
4. Simplify the resulting fraction:
Simplify [tex]\( \frac{18}{3} \)[/tex] by dividing the numerator by the denominator:
[tex]\[
\frac{18}{3} = 6
\][/tex]
Therefore, the answer is 6.
So [tex]\( 3 \frac{2}{3} + 2 \overline{3} = 6 \)[/tex].
