Question

It takes Bob four hours longer to repair a car than it takes Ken. Working together they cab complete the job in 1.5 hours. How long would each of them take working alone?

Answer 1

Actually, this one will take a bit more thought than the previous one, but the same formula still applies.

Answer 2

B = hours Bob can repair a car alone
K = hours Ken can repair a car alone
x = What Bob and Ken can repair while working together

The formula is:
\[\frac{B \times K}{B + K} = x\]

Answer 3

Now, it takes Bob four hours longer to repair so

Answer 4

B = K + 4

Answer 5

We replace B with K + 4 in the formula
Also we know x = 1.5

Answer 6

\[\frac{(K + 4) \times K}{K + 4 + K} = 1.5\]

Answer 7

Ok so would I subtract 4 from 1.5? Or make it 4K * K?

Answer 8

Do you know how to expand K(K + 4) ?

Answer 9

You have to expand K(K + 4) for the numerator

Answer 10

SO K^2 + 4K?

Answer 11

Very good

Answer 12

For the denominator you have to add

K + K + 4

Answer 13

4K^2?

Answer 14

K + K + 4 = 2K + 4

Answer 15

So what you have is

\[\frac{K^2 + 4K}{2K + 4} = 1.5\]

Answer 16

The afterwards, you have to cross multiply to get

\[K^2 + 4K = 1.5(2K + 4)\]

Answer 17

From there, you solve for K

Answer 18

K^2+4K=3K+6

Answer 19

Very good. Now subtract 3K from both sides. Then subtract 6 from both sides. After doing that, you should get a quadratic equation of the form

ax^2 + bx + c = 0

Answer 20

When you get that, solve for K by factoring or quadratic formula.

Answer 21

K^2+K+6=0 is this right?

Answer 22

Almost. You have to SUBTRACT 6 from both sides.

Answer 23

Ok so would the factoring be. (x+3) (-2)

Answer 24

After subtracting 6 from both sides you should have gotten
K^2 + K - 6 = 0

Answer 25

Then by factoring, you would get
(K + 3)(K - 2) = 0

Answer 26

Ok I got that. Ok I meant to put K. (Used to using X).

Answer 27

When you solve for K, you must remember that time only makes sense when positive. So keep the positive result and discard the negative result.

Answer 28

Knowing this, what is the correct value of K?

Answer 29

3

Answer 30

Not exactly. You did not solve for K

Answer 31

To Solve for K, you have to use zero product property for (K + 3)(K - 2) = 0

Answer 32

You solve these two equations to find possible values of K

K + 3 = 0
K - 2 = 0

Answer 33

-3 and 2 ?

Answer 34

K = -3 or K = 2

But remember, K represents the amount of hours Ken can do the job alone. -3 hours makes no sense.

Answer 35

So only K = 2 makes sense.

Answer 36

Right. So Ken can make the repairs in 2 hours.

Answer 37

Knowing this, how fast can Bob repair the car?

Answer 38

6 hours.

Answer 39

Congratulations. That is correct.

Answer 40

Thanks so much! :)

Answer 41

We can check this

\[\frac{2 \times 6}{2 + 6} = 1.5\]

Answer 42

We should get 1.5 for the left side after calculation