Question

It takes a painter and his friend 12 hours to paint a room. If the painter was working alone, it would take him 18 hours less than if his friend was working alone. How long does it take the painter to paint the room by himself?

Answer 1

Answer:

18 hours

Step-by-step explanation:

The equation for time working is

1/a + 1/b = 1/c

where a and b are the times working alone and c is the time working together

1/b-18 + 1/b = 1/12

Multiply by 12b(b-18) to clear the fractions

12b(b-18) *(1/b-18 + 1/b) = 1/12* 12b(b-18)

12b + 12(b-18) = b(b-18)

Distribute

12b +12b - 216= b^2-18b

Combine like terms

24b-216=b^2-18b

Subtract 24b from each side

24b-24b -216 = b^2 -18b-24b

-216 = b^2 -42b

Add 216 to each side

0 = b^2 -42b +216

Factor

0 = (b-36) (b-6)

b= 36  

b=6 is not possible since this is less time then when they worked together

Using the zero product property

The painter takes b-18 = 36-18 = 18 hours

The friend takes 36 hours

b=36   b=6

Answer 2

The time it will take for painting the room by himself is going to be 18 hours.

How are number of people to time needed to complete a task related?

• More people to do a task means less time it will take.
• Less people to do a task means more time it will take.

Thus, they are inversely related.

If x men take y time for a work,

Let we define:

• Manpower needed for a work = y + y + y.. + y  = Time per man × count of men
• Manpower needed for a work = $xy$

For the given case, the work is painting the considered room.

Since this time, their capacities are different, so we will add them separately, painter's contribution was 12 times his power, and his friend's contribution was 12 times his friend's power.

It we assume

• power of painter for this room painting= x,
• power of his friend for this room painting = y ,

then: manpower of painting the room = $12(x+y)$

Suppose that it would take p hours for painter to paint the room alone, and q hours for his friend to paint the room alone, then:

Manpower for painting the room = $p \times x = q \times y$

or

$12(x+y) = px = qy$

We get two equations as:

$12(x+y) = px\\\\x = \dfrac{12y}{p-12}$

and

$12(x+y) = qy\\\\x = \dfrac{(q-12)y}{12}$

Thus, from both equations for x, we get:

$\dfrac{12y}{p-12} = \dfrac{(q-12)y}{12}\\\\144 = 144 + pq -12(p+q)\\\\pq = 12p + 12q$

Since it is given that: If the painter was working alone, it would take him 18 hours less than if his friend was working alone, thus:

$p = q - 18$

or q = p + 18.

Putting this value in previous equation, we get:

$pq = 12(p + q)\\p(p+18) = 12(p+p+18)\\p^2 + 18p = 24p + 216\\p^2 - 6p - 216 = 0\\p^2 - 18p + 12p - 216 = 0\\p(p-18) +12(p-18) = 0\\(p+12)(p-18) = 0\\p=-12, p = 18$

p cannot be negative as it is number of hours taken by painter to paint the room. Thus, p = 18

Thus, the time it will take for painting the room by himself is going to be 18 hours.

Learn more about time and work here:

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