Question

operations with mixed numbers
multiply: $4 \cdot 2\frac{2}{3}$
options: $8\frac{2}{3}$, $10\frac{1}{3}$, $10\frac{2}{3}$, $\frac{3}{2}$

Explanation:

Step1: Convert mixed number to improper fraction

The mixed number \(2\frac{2}{3}\) can be converted to an improper fraction. The formula for converting a mixed number \(a\frac{b}{c}\) to an improper fraction is \(\frac{a\times c + b}{c}\). So for \(2\frac{2}{3}\), we have \(a = 2\), \(b=2\), \(c = 3\). Then \(\frac{2\times3+ 2}{3}=\frac{6 + 2}{3}=\frac{8}{3}\).

Step2: Multiply the integer with the improper fraction

We need to multiply \(4\) (which can be written as \(\frac{4}{1}\)) with \(\frac{8}{3}\). When multiplying fractions \(\frac{m}{n}\times\frac{p}{q}=\frac{m\times p}{n\times q}\), so \(\frac{4}{1}\times\frac{8}{3}=\frac{4\times8}{1\times3}=\frac{32}{3}\).

Step3: Convert the improper fraction back to a mixed number

To convert \(\frac{32}{3}\) to a mixed number, we divide \(32\) by \(3\). \(3\times10 = 30\), and \(32-30=2\). So \(\frac{32}{3}=10\frac{2}{3}\).

Answer:

\(10\frac{2}{3}\) (corresponding to the option with \(10\frac{2}{3}\))