Answer: The mixed number 5 7/9
We can write the mixed number as 5 & 7/9 to better separate the whole part (5) from the fractional part (7/9)
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Explanation:
Use this formula
[tex]a \frac{b}{c} = \frac{a*c + b}{c}[/tex]
to convert from a mixed number to an improper fraction.
Here are the steps in converting 5 & 1/3
[tex]a \frac{b}{c} = \frac{a*c+b}{c}\\\\5 \frac{1}{3} = \frac{5*3+1}{3}\\\\5 \frac{1}{3} = \frac{16}{3}\\\\[/tex]
Through similar steps you should also have
[tex]4 \frac{5}{9} = \frac{41}{9}\\\\6 \frac{5}{9} = \frac{59}{9}\\\\[/tex]
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Let's add the amounts needed for the family room and the living room.
[tex]\frac{41}{9}+\frac{59}{9}\\\\\frac{41+59}{9}\\\\\frac{100}{9}\\\\[/tex]
Subtract off the amount the painter already has
[tex]\frac{100}{9} - \frac{16}{3}\\\\\frac{100}{9} - \frac{3*16}{3*3}\\\\\frac{100}{9} - \frac{48}{9}\\\\\frac{100-48}{9}\\\\\frac{52}{9}\\\\[/tex]
This improper fraction is the number of gallons of paint that's missing, i.e. the amount the painter needs to buy.
Converting it to a mixed number gets us:
[tex]\frac{52}{9}=\frac{45+7}{9}\\\\\frac{52}{9}=\frac{45}{9}+\frac{7}{9}\\\\\frac{52}{9}=5+\frac{7}{9}\\\\\frac{52}{9}=5\frac{7}{9}\\\\[/tex]
Note that 52/9 leads to a quotient of 5 and a remainder of 7. The 5 and 7 match with 5 & 7/9
The painter needs to buy 5 full gallons, plus another 7/9 of a gallon. Or we can state it as him needing 5 & 7/9 gallons of paint.
