Question

mai used a total of $3\frac{5}{6}$ gallons of gas while driving her car. each hour she was driving, she used $\frac{7}{9}$ gallons of gas. what was the total number of hours she was driving?
write your answer as a mixed number in simplest form.
\boxed{} hours

Explanation:

Step1: Convert mixed number to improper fraction

First, convert \( 3\frac{5}{6} \) 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 \( 3\frac{5}{6} \), we have \( a = 3 \), \( b = 5 \), \( c = 6 \). Then \( 3\times6 + 5 = 18 + 5 = 23 \), so \( 3\frac{5}{6}=\frac{23}{6} \).

Step2: Divide total gas by gas per hour

We need to find the number of hours, which is total gas used divided by gas used per hour. So we divide \( \frac{23}{6} \) by \( \frac{7}{9} \). Dividing by a fraction is the same as multiplying by its reciprocal, so \( \frac{23}{6}\div\frac{7}{9}=\frac{23}{6}\times\frac{9}{7} \).

Step3: Multiply the fractions

Multiply the numerators and denominators: \( \frac{23\times9}{6\times7}=\frac{207}{42} \).

Step4: Simplify the fraction

Simplify \( \frac{207}{42} \) by dividing numerator and denominator by their greatest common divisor, which is 3. \( \frac{207\div3}{42\div3}=\frac{69}{14} \).

Step5: Convert back to mixed number

Convert \( \frac{69}{14} \) to a mixed number. Divide 69 by 14: \( 14\times4 = 56 \), \( 69 - 56 = 13 \), so \( \frac{69}{14}=4\frac{13}{14} \).

Answer:

\( 4\frac{13}{14} \)