Work and Motion Applications

Question

McKenzie and Lindy work on a landscaping crew. They can complete the landscaping job in 4 hours if they work together. McKenzie generally takes 6 hours less Lindy. How long would it take McKenzie to complete the landscaping job if Lindy calls in sick?

_______ Hours

Answer 1

Let M be the number of hours that McKenzie needs to complete the job alone.

Let L be the number of hours that Lindsey needs to complete the job alone.

The problem tells us that M = L - 6.

McKenzie works at a rate of 1/M job per hour.

This can be written as 1/(L-6) job per hour.

Lindsey works at a rate of 1/L job per hour.

They work at a combined rate of 1/4 job per hour.

So,

(1/L) + (1/M) = 1/4

Substitute L - 6 for M:

(1/L) + [1/(L-6)] = 1/4

Note here that we must exclude L = 6 as a final solution, since if this were the case, we would be dividing by zero, and that is never allowed.

Obtain a common denominator and begin adding on the left-hand side:

[ L + (L-6) ] / [ L×(L-6) ] = 1/4

Simplify by combining like terms:

[ 2L -6 ] / [ L² -6L ] = 1/4

Cross multiply:

4 × ( 2L - 6 ) = L² - 6L

Expand using the distributive law:

8L - 24 = L² - 6L

Combine all the non-zero parts on the left-hand side of the equation:

L² -14L + 24 = 0

Factor:

( L - 2 ) × ( L - 12 ) = 0

So, either L = 2 or L= 12.

If L = 2, then M = L - 6 = 2 - 6 = -4, and this does not correspond to the logic of the problem. We can't work a negative 4 hours.

If L = 12, then M = L - 6 = 12 - 6 = 6. (Since we did not determine L= 6 as a potential answer, we do not need to exclude it.)

So, it would take McKenzie 6 hours complete an entire landscaping job on her own.

Checking:

[1/(12)] + [1/((12)-6)] = (1/12) + (1/6)

= (1/12) + (2/12) = (3/12) = 1/4

And so this confirms that L = 12 and M = 6 are valid solutions.

Answer 2

L represents the time Lindsey needs working alone

L - 6 represents the time McKenzie needs working alone

[4 / L] + [4 / (L - 6)] = 1

Can you solve for L and answer?