Question

A group of college students are volunteering for Help the Homeless during their spring break. They are putting the finishing touches on a house they built. Working alone, Irina can paint a certain room in 9 hours. Paulo can paint the same room in 8 hours. Write an equation that can be used to find how long it will take them working together to paint the room. How many hours will it take them to paint the room? If necessary, round your answer to the nearest hundredth

Answer 1

Since Irina can paint 1 room in 9 hours, that means she paints 1/9 of the room in 1 hour.  Her portion of the equation would be 1/9x, with x being the number of hours she works and 1/9 of a room per hour being her speed.
Since Paulo can paint 1 room in 8 hours, that means he paints 1/8 of the room in 1 hour.  His portion of the equation would be 1/8x, with x being the number of hours he works and 1/8 of a room per hour being his speed.
The equation would then be 1/9x + 1/8x = 1 (Irina's portion of the room, plus Paulo's portion of the room, equal to one whole room).
Find a common denominator.  72 is the first number that both 9 and 8 divide evenly into.  Since 9*8 = 72, we multiply the top of 1/9 by 8 to convert the fraction and get 8/72x.  Since 8*9 = 72, we multiply the top of 1/8 by 9 to convert the fractio and get 9/72x.  We now have 8/72x+9/72x=1
17/72x=1
Divide both sides by 17/72:
17/72x ÷ 17/72 = 1÷17/72
x=1/1 ÷ 17/72
x=1/1 * 72/17
x=72/17=4.24
Together it should take them 4.24 hours.

Answer 2

The combined work rate can be found by using the equation for combined rate.

The correct responses;

• The time it will take them working together is; $\displaystyle \underline{\frac{1}{t_b} = \frac{1}{t_1} + \frac{1}{t_2} }$

• The number of hours it will take them working together to paint the room is approximately 4.24 hours.  

The method by which the above responses are obtained:

The time it takes Irina to paint the room alone, t₁ = 9 hours

Time it takes Paulo to paint the room, t₂ = 8 hours

Required:

To write the equation for the combined rate of Irina and Paulo

Solution:

The equation that gives the time, $\mathbf{t_b}$, it will take both Irina and Paulo to paint the room together is given by the combined rate formula, as follows;

• $\displaystyle \underline{ \frac{1}{t_b} = \frac{1}{t_1} + \frac{1}{t_2}}$

Required:

The duration it will take them to paint the room when working together.

Solution:

The time it will take them to paint the room together is found as follows;

$\displaystyle \frac{1}{t_b} = \mathbf{\frac{1}{9} + \frac{1}{8}}$

Which gives;

$\displaystyle \frac{1}{t_b} = \frac{8 + 9}{9 \times 8} = \mathbf{\frac{17}{72}}$

Therefore;

$\displaystyle t_b = \frac{72}{17} \approx \mathbf{4.24}$

• The time it takes Irina and Paulo to paint together, $t_b$ ≈ 4.24 hrs.

Learn more about combined rate of work here:

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