Kyra and Sage, working together can paint a room in 5 hours. If Melora

Question

Kyra and Sage, working together can paint a room in 5 hours. If Melora helps Kyra and Sage paint the room, the three of them can paint the room in 4 hours. What amount of time (in hours) would it take Melora, working alone, to paint the room?

(A) 6
(B) 8
(C) 10
(D) 15
(E) 20

(A) 6
(B) 8
(C) 10
(D) 15
(E) 20

Bunuel · Answer

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Gladiator59 · Answer

Rate of painting room for K + S = 1/5 ( how much part of the room they can paint in one hour )

When M is working with K & S -> New rate is 1/4 ( as they take 4 hrs to paint room , they paint 1/4th room in one hour)

Let M alone finish paint job in x hours. So her rate is 1/x.

So Rate of K&S + Rate of M = Rate of K&S&M together.

We have...

1/5 + 1/x = 1/4 
 1/x = 1/4 - 1/5 
 1/x = (5-4)/20 
 x = 20 
Hence E.

Please give Kudos if you liked the explanation...

Abhishek009 · Answer

Let the total work be 20 units

So, Efficiency of Kyra and Sage is 4 units
And, Efficiency of Melora , Kyra and Sage is 5 units

So, We have ( k + s ) + m = 5 ; 4 + s = 5

So, Efficiency of m is 1 , Thus time required by Melora to paint the room is 20 Hours, Answer must be (E)

hoang221 · Answer

In 1 hour, Kyra and Sage can paint 1/5 of the room
In 1 hour, Kyra, Sage and Melora can paint 1/4 of the room

Therefore, in 1 hour, Melora can paint 1/4 - 1/5 = 1/20 of the room

Therefore Melora needs 20 hours to paint the room => Answer (E)

generis · Answer

The combined rate of two persons is given. When the third person, Melora, joins, the equation is simple.

1)  Combined rate  of Sage and Kyra =  \frac{1}{5} 
Substitute that rate in the next step

2)  Combined rate  of Sage, Kyra, and Melora: Together all three can paint the room in 4 hours:
  (\frac{1}{S}+\frac{1}{K})+\frac{1}{M}=\frac{1}{4} , that is,
 \frac{1}{5}+\frac{1}{M}=\frac{1}{4}

3) Melora's  rate , working alone*:
  \frac{1}{M}=(\frac{1}{4}-\frac{1}{5}) = (\frac{(5-4)}{20})=\frac{1}{20}

4) M's  time ?
Work is 1 (room). Rate is inversely proportional to time. Flip the rate fraction:   \frac{1room}{20hrs} -> \frac{20hrs}{1room}

Melora takes 20 hours to paint the room alone.

Answer E

*  Shortcuts 
Rates: \frac{1}{A}-\frac{1}{B} =\frac{(B-A)}{AB} 
Use LCM to see proof.
For added rates, the shortcut is  \frac{1}{A}+\frac{1}{B}=\frac{(A+B)}{AB}

Inverting (flipping) the rate fraction to get time
 W = 1  room,  r 
 R * T = W , so  T = \frac{W}{R} 
 T=\frac{1r}{\frac{1r}{20hrs}}= (1r * \frac{20hrs}{1r}) = 20 hrs

EgmatQuantExpert · Answer

Solution

Let the total amount of work done to paint the room is  LCM (5,4) = 20  units .

•  Kyra and Sage, working together can paint a room in  5  hours .
 o Thus, Kyra and Sage complete  4  units of work in  1  hour.
o  K+S=4 ---------------------- (1)

•  If Melora helps Kyra and Sage paint the room, the three of them can paint the room in  4  hours .
 o Thus, Kyra, Sage, and Melora complete  5  units of work in  1  hour.
o  K+S+M=5 ------------------- (2)

By subtracting equation  (1)  from equation  (2) , we get:
 •  M=1  unit, OR,  M  completes  1  unit of work in  1  hour.

• Thus, to complete  20 units of work Melora will take  \frac{20}{1} = 20  hours.

Answer: E

gracie · Answer

K and S paint 1/5 of the room in 1 hour
M saves them 1 hour by painting 1/5 of the room in 4 hours
4/(1/5)=20 hours for M to paint room alone
E

dhruvin917 · Answer

Remember: Rate * Time = Work
K = Kyra | S = Sage | M = Melora

(K+S) * 5 = 1 --> (K + S) working together at their own rate took 5 hours to paint 1 room.
 This means the rate at which (K + S) work is (1/5).

(M+K+S) * 4 = 1 --> (M + K + S) working together at their own rate took 4 hours to paint 1 room.
 This means the rate at which (M + K + S) work is (1/4).

Now we need to figure out at what rate Melora (M) works at: 
(M + K + S) = (1/4) --> This is the rate at which when (M + K + S) work together.
(M) + (1/5) = (1/4) --> We already know that (K + S) = 1/5 since this is the rate at which they work together.
M = (1/4) - (1/5)
M = (1/20) --> This is the rate at which Melora (M) works alone.

Now to solve for how long it will take Melora to paint 1 room: 
R * T = W --> Our standard rate formula
(1/20) * T = 1 --> Plug in the rate for which Melora works (1/20) and solve for T (time)
 T = 20

The answer is  E .

JeffTargetTestPrep · Answer

We can let K, S, and M = the time it takes Kyra, Sage, and Melora to paint the room respectively and create the equations:

1/K + 1/S = 1/5

And

1/M + 1/K + 1/S = 1/4

Subtracting the first equation from the second we have:

1/M = 1/4 - 1/5

1/M = 5/20 - 4/20 = 1/20

So it would take Melora 20 hours to paint the room.

Answer: E

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Kyra and Sage, working together can paint a room in 5 hours. If Melora

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