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Question

Painter A and Painter B each paint rooms at constant rates. Painter A takes 3 days longer than Painter B to paint h rooms alone. At these rates, if the two painters together paint 2 h rooms in 4 days, how many days will it take Painter A alone to paint 3 h rooms?

Answer Choices

A. $12$
B. $15$
C. $18$
D. $20$
E. $24$

Steps

Explanation	Calculations	Help
$We first translate "Painter A takes 3 days longer than Painter B to paint h rooms alone" by defining the time for each painter to paint h rooms.$	$d = time for Painter B to paint h rooms d + 3 = time for Painter A to paint h rooms$
$We express each painter's painting rate in rooms per day.$	$r_{B} = \frac{h}{d} r_{A} = \frac{h}{d + 3}$
$We write that together in 4 days they paint 2 h rooms.$	$4 (r_{A} + r_{B}) = 2 h$
$We replace the rates with their expressions.$	$4 (\frac{h}{d + 3} + \frac{h}{d}) = 2 h$
$We divide both sides by h to simplify.$	$4 (\frac{1}{d + 3} + \frac{1}{d}) = 2$
$We add the fractions on the left.$	$\frac{1}{d + 3} + \frac{1}{d} = \frac{2 d + 3}{d ( d + 3 )}$	Theory & Tactics Method Card FDPR0-B Click to view full details
$We remove the fractions by multiplying both sides by .$	$4 (2 d + 3) = 2 d (d + 3)$	Theory & Tactics Method Card EQUA4-A Click to view full details
$We expand and move all terms to one side to form a quadratic equation.$	$4 (2 d + 3) = 8 d + 12 2 d (d + 3) = 2 d^{2} + 6 d 8 d + 12 = 2 d^{2} + 6 d 2 d^{2} + 6 d - 8 d - 12 = 0 d^{2} - d - 6 = 0$	Theory & Tactics Method Card EQUA2-A Click to view full details
$We factor the quadratic expression.$	$d^{2} - d - 6 = (d - 3) (d + 2) = 0$	Theory & Tactics Method Card EQUA2-B Click to view full details
$We solve each factor for d .$	$d - 3 = 0 gives d = 3 d + 2 = 0 gives d = - 2$	Theory & Tactics Method Card EQUA1-A Click to view full details
$We discard the negative solution since time cannot be negative, so we accept d = 3 .$	$d = 3$
$We find Painter A's time to paint h rooms by adding 3 days to d .$	$d + 3 = 6$
$We find Painter A's rate and then find the time to paint 3 h rooms.$	$\frac{3 h}{\frac{h}{6}} = 3 h \times \frac{6}{h} = \frac{3 \times h \times 6}{h} = 3 \times 6 = 18$	Theory & Tactics Method Card FDPR0-D Click to view full details

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Final Answer

C

Question

Painter A and Painter B each paint rooms at constant rates. Painter A takes 3 days longer than Painter B to paint h rooms alone. At these rates, if the two painters together paint 2 h rooms in 4 days, how many days will it take Painter A alone to paint 3 h rooms?

Answer Choices

A. $12$
B. $15$
C. $18$
D. $20$
E. $24$

Steps

Explanation	Calculations	Help
$We first translate "Painter A takes 3 days longer than Painter B to paint h rooms alone" by defining the time for each painter to paint h rooms.$	$d = time for Painter B to paint h rooms d + 3 = time for Painter A to paint h rooms$
$We express each painter's painting rate in rooms per day.$	$r_{B} = \frac{h}{d} r_{A} = \frac{h}{d + 3}$
$We write that together in 4 days they paint 2 h rooms.$	$4 (r_{A} + r_{B}) = 2 h$
$We replace the rates with their expressions.$	$4 (\frac{h}{d + 3} + \frac{h}{d}) = 2 h$
$We divide both sides by h to simplify.$	$4 (\frac{1}{d + 3} + \frac{1}{d}) = 2$
$We add the fractions on the left.$	$\frac{1}{d + 3} + \frac{1}{d} = \frac{2 d + 3}{d ( d + 3 )}$	Theory & Tactics Method Card FDPR0-B Click to view full details
$We remove the fractions by multiplying both sides by .$	$4 (2 d + 3) = 2 d (d + 3)$	Theory & Tactics Method Card EQUA4-A Click to view full details
$We expand and move all terms to one side to form a quadratic equation.$	$4 (2 d + 3) = 8 d + 12 2 d (d + 3) = 2 d^{2} + 6 d 8 d + 12 = 2 d^{2} + 6 d 2 d^{2} + 6 d - 8 d - 12 = 0 d^{2} - d - 6 = 0$	Theory & Tactics Method Card EQUA2-A Click to view full details
$We factor the quadratic expression.$	$d^{2} - d - 6 = (d - 3) (d + 2) = 0$	Theory & Tactics Method Card EQUA2-B Click to view full details
$We solve each factor for d .$	$d - 3 = 0 gives d = 3 d + 2 = 0 gives d = - 2$	Theory & Tactics Method Card EQUA1-A Click to view full details
$We discard the negative solution since time cannot be negative, so we accept d = 3 .$	$d = 3$
$We find Painter A's time to paint h rooms by adding 3 days to d .$	$d + 3 = 6$
$We find Painter A's rate and then find the time to paint 3 h rooms.$	$\frac{3 h}{\frac{h}{6}} = 3 h \times \frac{6}{h} = \frac{3 \times h \times 6}{h} = 3 \times 6 = 18$	Theory & Tactics Method Card FDPR0-D Click to view full details

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Final Answer

C