To solve the problem of multiplying the whole number [tex]\(3\)[/tex] by the fraction [tex]\(\frac{5}{9}\)[/tex] and then simplifying the answer into a mixed number, follow these detailed steps:
1. Multiply the whole number by the fraction:
Multiply the whole number [tex]\(3\)[/tex] by the fraction [tex]\(\frac{5}{9}\)[/tex]:
[tex]\[
3 \cdot \frac{5}{9} = \frac{3 \cdot 5}{9} = \frac{15}{9}
\][/tex]
2. Simplify the product (if possible):
Next, simplify the fraction [tex]\(\frac{15}{9}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is [tex]\(3\)[/tex]:
[tex]\[
\frac{15 \div 3}{9 \div 3} = \frac{5}{3}
\][/tex]
This gives us [tex]\(\frac{5}{3}\)[/tex].
3. Convert the improper fraction into a mixed number:
To convert [tex]\(\frac{5}{3}\)[/tex] into a mixed number, divide the numerator by the denominator:
[tex]\[
5 \div 3 = 1 \text{ remainder } 2
\][/tex]
This means that [tex]\(\frac{5}{3}\)[/tex] can be expressed as:
[tex]\[
1 \frac{2}{3}
\][/tex]
4. Verify the equivalent decimal representation and mixed fraction:
Multiplication result in decimal form:
[tex]\[
3 \cdot \frac{5}{9} \approx 1.6666666666666667
\][/tex]
Corresponding mixed number:
[tex]\[
1 \frac{6}{9}
\][/tex]
Simplifying [tex]\( \frac{6}{9} \)[/tex]:
[tex]\[
1 \frac{2}{3}
\][/tex]
Thus, the answer in mixed number form is indeed [tex]\(1 \frac{2}{3}\)[/tex].
So, the correct answer to the question [tex]\(3 \cdot \frac{5}{9}\)[/tex] in simplified mixed number form is:
[tex]\[ 1 \frac{2}{3} \][/tex]
