Question

operations with mixed numbers
divide: $6\frac{2}{3} \div 3$
options: $2\frac{5}{9}$, $\frac{7}{9}$, $2\frac{2}{9}$, $6\frac{1}{3}$

Explanation:

Step1: Convert mixed number to improper fraction

To divide a mixed number by a whole number, first convert the mixed number \( 6\frac{2}{3} \) 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 \( 6\frac{2}{3} \), we have \( a = 6 \), \( b = 2 \), \( c = 3 \). Then \( 6\frac{2}{3}=\frac{6\times3 + 2}{3}=\frac{18 + 2}{3}=\frac{20}{3} \).

Step2: Divide by the whole number

Now we need to divide \( \frac{20}{3} \) by 3. Dividing by a number is the same as multiplying by its reciprocal. The reciprocal of 3 is \( \frac{1}{3} \), so \( \frac{20}{3}\div3=\frac{20}{3}\times\frac{1}{3}=\frac{20}{9} \).

Step3: Convert improper fraction to mixed number

Next, convert \( \frac{20}{9} \) back to a mixed number. To do this, divide the numerator by the denominator. \( 20\div9 = 2 \) with a remainder of \( 20-9\times2 = 20 - 18 = 2 \)? Wait, no, wait. Wait, \( 9\times2 = 18 \), \( 20-18 = 2 \)? Wait, no, that's not right. Wait, \( 9\times2 = 18 \), \( 20 - 18 = 2 \)? Wait, no, I made a mistake. Wait, \( \frac{20}{9} \): 9 times 2 is 18, 20 minus 18 is 2? Wait, no, that would be \( 2\frac{2}{9} \)? Wait, no, wait, let's check again. Wait, \( 6\frac{2}{3}\div3 \): Wait, maybe I made a mistake in step 1. Wait, \( 6\frac{2}{3} \) is \( 6 + \frac{2}{3} \). Dividing by 3 is the same as dividing each part by 3. So \( 6\div3+\frac{2}{3}\div3 = 2+\frac{2}{9}=2\frac{2}{9} \). Wait, but earlier when I converted to improper fraction, I got \( \frac{20}{3}\div3=\frac{20}{9}=2\frac{2}{9} \). Wait, but let's check the options. The options are \( 2\frac{5}{9} \), \( \frac{7}{9} \), \( 2\frac{2}{9} \), \( 6\frac{1}{3} \). So the correct answer should be \( 2\frac{2}{9} \)? Wait, no, wait, let's recalculate step 1. \( 6\frac{2}{3} \): 6 times 3 is 18, plus 2 is 20, so \( \frac{20}{3} \). Then \( \frac{20}{3}\div3=\frac{20}{3}\times\frac{1}{3}=\frac{20}{9} \). Now, \( 20\div9 = 2 \) with a remainder of \( 20 - 9\times2 = 20 - 18 = 2 \), so \( \frac{20}{9}=2\frac{2}{9} \). So that's the correct mixed number.

Wait, but let's check the options. The first option is \( 2\frac{5}{9} \), second is \( \frac{7}{9} \), third is \( 2\frac{2}{9} \), fourth is \( 6\frac{1}{3} \). So the correct answer is \( 2\frac{2}{9} \), which is the third option. Wait, but let's do the division another way. \( 6\frac{2}{3}\div3 \): split the mixed number into whole number and fraction: \( 6\div3+\frac{2}{3}\div3 = 2+\frac{2}{9}=2\frac{2}{9} \). Yes, that's correct. So the correct answer is \( 2\frac{2}{9} \), which is the option \( 2\frac{2}{9} \).

Answer:

\( 2\frac{2}{9} \) (the option with \( 2\frac{2}{9} \))