Question

1. $3\frac{2}{5} \div \frac{2}{5}$
2. $2\frac{1}{2} \div \frac{3}{4}$
3. $2\frac{2}{9} \div \frac{5}{6}$

Explanation:

Problem 7: \( 3\frac{2}{5} \div \frac{2}{5} \)

Step 1: Convert mixed number to improper fraction

A mixed number \( a\frac{b}{c} \) can be converted to an improper fraction using the formula \( \frac{a\times c + b}{c} \). For \( 3\frac{2}{5} \), we have \( a = 3 \), \( b = 2 \), and \( c = 5 \). So, \( 3\frac{2}{5}=\frac{3\times5 + 2}{5}=\frac{15 + 2}{5}=\frac{17}{5} \).

Step 2: Divide by a fraction (multiply by reciprocal)

Dividing by a fraction \( \frac{d}{e} \) is the same as multiplying by its reciprocal \( \frac{e}{d} \). So, \( \frac{17}{5}\div\frac{2}{5}=\frac{17}{5}\times\frac{5}{2} \).

Step 3: Multiply the fractions

When multiplying fractions \( \frac{m}{n}\times\frac{p}{q}=\frac{m\times p}{n\times q} \). Here, \( m = 17 \), \( n = 5 \), \( p = 5 \), and \( q = 2 \). So, \( \frac{17}{5}\times\frac{5}{2}=\frac{17\times5}{5\times2} \). The 5 in the numerator and denominator cancels out, leaving \( \frac{17}{2} \), which is also equal to \( 8\frac{1}{2} \).

Step 1: Convert mixed number to improper fraction

For \( 2\frac{1}{2} \), using the formula \( \frac{a\times c + b}{c} \) with \( a = 2 \), \( b = 1 \), and \( c = 2 \), we get \( 2\frac{1}{2}=\frac{2\times2+1}{2}=\frac{4 + 1}{2}=\frac{5}{2} \).

Step 2: Divide by a fraction (multiply by reciprocal)

Dividing by \( \frac{3}{4} \) is multiplying by \( \frac{4}{3} \). So, \( \frac{5}{2}\div\frac{3}{4}=\frac{5}{2}\times\frac{4}{3} \).

Step 3: Multiply the fractions

\( \frac{5}{2}\times\frac{4}{3}=\frac{5\times4}{2\times3}=\frac{20}{6} \). Simplify this fraction by dividing numerator and denominator by their greatest common divisor, which is 2. So, \( \frac{20\div2}{6\div2}=\frac{10}{3} \), or \( 3\frac{1}{3} \).

Step 1: Convert mixed number to improper fraction

For \( 2\frac{2}{9} \), using \( a = 2 \), \( b = 2 \), \( c = 9 \), we have \( 2\frac{2}{9}=\frac{2\times9+2}{9}=\frac{18 + 2}{9}=\frac{20}{9} \).

Step 2: Divide by a fraction (multiply by reciprocal)

Dividing by \( \frac{5}{6} \) is multiplying by \( \frac{6}{5} \). So, \( \frac{20}{9}\div\frac{5}{6}=\frac{20}{9}\times\frac{6}{5} \).

Step 3: Multiply the fractions and simplify

\( \frac{20}{9}\times\frac{6}{5}=\frac{20\times6}{9\times5} \). We can simplify before multiplying: 20 and 5 have a common factor of 5, and 6 and 9 have a common factor of 3. So, \( \frac{20\div5}{9\div3}\times\frac{6\div3}{5\div5}=\frac{4}{3}\times\frac{2}{1}=\frac{8}{3} \), or \( 2\frac{2}{3} \).

Answer:

\( \frac{17}{2} \) (or \( 8\frac{1}{2} \))

Problem 8: \( 2\frac{1}{2} \div \frac{3}{4} \)