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Solve the following expression:

[tex]\[ 3 \frac{2}{3} + 2 \overline{3} \][/tex]

A. [tex]\(5\)[/tex]
B. [tex]\(5 \frac{2}{3}\)[/tex]
C. [tex]\(6\)[/tex]
D. [tex]\(6 \frac{1}{3}\)[/tex]


Sagot :

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].