It takes Frank 2 hours longer than Jane to carpet a certain type of room. Together they can carpet that type of room in 1(7/8) hours. How long would it take for Frank to do the job alone?
Let's assume Jane takes x hours to carpet the room.
Since Frank takes 2 hours longer than Jane, his time will be x + 2 hours.
Together, they can carpet the room in 1(7/8) hours, but we need to convert this mixed number to an improper fraction.
1(7/8) = (8 * 1 + 7) / 8 = 15/8 hours.
So, their combined rate is 1 job per 15/8 hours.
Now, let's set up the equation:
1/x + 1/(x + 2) = 1/(15/8)
To simplify the equation, we can multiply every term by the least common denominator, which is 8(x)(x + 2):
8(x + 2) + 8x = x(x + 2)
Expanding the equation:
8x + 16 + 8x = x^2 + 2x
Simplifying:
16x + 16 = x^2 + 2x
Rearranging the equation:
0 = x^2 - 14x - 16
To solve this quadratic equation, we can factor it or use the quadratic formula. Factoring is not ideal in this case, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = -14, c = -16:
x = (-(-14) ± √((-14)^2 - 4(1)(-16))) / (2(1))
= (14 ± √(196 + 64)) / 2
= (14 ± √260) / 2
Simplifying further:
x = (14 ± 2√65) / 2
= 7 ± √65
Since x represents the time taken, we can ignore the negative root. Therefore, x = 7 + √65.
Recall that x represents the time it takes for Jane to carpet the room. So, it would take Frank x + 2 = 9 + √65 hours to carpet the room alone.
To find out how long it would take for Frank to do the job alone, we need to first determine Jane's time to complete the job.
Let's assign variables to represent the time it takes for Jane and Frank to carpet the room alone. Let's say it takes Jane x hours to carpet the room. Since it takes Frank 2 hours longer than Jane, it would take him (x + 2) hours.
Now, let's consider their combined work rate. When Jane and Frank work together, they can carpet the room in 1(7/8) hours, which is equivalent to (15/8) hours.
The work rate can be defined as the amount of work completed per hour. Therefore, Jane's work rate is 1/x, and Frank's work rate is 1/(x + 2).
When working together, their work rates are added up, so we have the equation:
1/x + 1/(x + 2) = 8/15
To solve this equation, we need to find a common denominator, which is (x)(x + 2). Multiplying every term by this denominator gives us:
(x + 2) + x = 8/15 * (x)(x + 2)
Simplifying this equation:
2x + 2 = 8/15 * (x^2 + 2x)
Now, let's cross multiply and simplify further:
30x + 30 = 8x^2 + 16x
Rearranging this equation to be in quadratic form:
8x^2 + 16x - 30x - 30 = 0
Simplifying:
8x^2 - 14x - 30 = 0
Dividing every term by 2:
4x^2 - 7x - 15 = 0
Now, we can solve this quadratic equation. The solutions to this equation represent the possible values of x, which is the time it takes for Jane to carpet the room alone.
Factoring the quadratic equation:
(4x + 5)(x - 3) = 0
Using the zero product property, we have two possible solutions:
4x + 5 = 0, which gives x = -5/4 (extraneous solution)
x - 3 = 0, which gives x = 3
Since x represents the time it takes for Jane to carpet the room alone, Jane would take 3 hours to do the job alone.
To find out how long Frank would take to complete the job alone, we can substitute the value of x back into the equation (x + 2):
3 + 2 = 5
Therefore, it would take Frank 5 hours to carpet the room alone.
they each do a fraction , together the fractions add to one (the whole)
(15/8) / F + (15/8) / J = 1
(15/8) (J + F) = J * F
substituting ... (15/8) (2 F - 2) = F^2 - 2 F
0 = F^2 - 23/4 F + 15/4
0 = 4 F^2 - 23 F + 15