how do I set up this problem?

Question

Mulching. Anita can mulch the college gardens in 3 fewer days than it takes Tori to mulch the same areas. When they work together, it takes them 2 days. How long would it take each of them to do the job alone?

Answer 1

Hi M, let x be the number of days it takes Tori to mulch the gardens. Then it would take Anita x - 3 days to mulch the same gardens.

 

Set up the problem by how much work each would do in a day:

 

Anita would do 1 / (x - 3) of the work in one day.

 

Tori would do 1 / x of the work in one day

 

Together, they would do 1/2 the work in one day.

 

Then the equation would be:

 

1 / (x - 3) + 1 / x = 1 / 2

 

Solve for x by finding the common denominator of (x - 3), x, and 2. The simple way to clear the fractions is to multiply the entire equation by (x - 3) * x * 2.

 

Doing this, the (x - 3) cancels in the first term leaving x * 2. The x cancels in the second term leaving (x - 3) * 2, and finally, the 2 cancels in the third term leaving (x - 3) * x.

 

The equation becomes:

 

(x * 2) + (x - 3) * 2 = (x - 3) * x

 

Clear the parenthesis giving:

 

2x + 2x - 6 = x² - 3x

 

Combine all terms on one side of the equal sign giving: 

 

x² - 7x + 6 = 0

 

(x - 6) * (x - 1) = 0

 

Use the zero product rule giving:

 

x - 6 = 0, x = 6 and x - 1 = 0, x = 1

 

We can discard the x = 1 solution since it would give Anita negative days, (x - 3) would be (1 - 3) or -2.

 

Solution:

 

Tori can mulch the gardens in 6 days, Anita can mulch the gardens in x - 3, or 3 days, and together, then can mulch the gardens in 2 days. 

 

Questions?

 

 

Answer 2

Hi M....

 

These types of problems are most easily solved when you first figure out what part of the job is done by each person in one day.

 

Let X = the number of days that Tori needs to mulch

 

Then Tori finishes 1/X of the job in one day.

 

Anita needs X - 3 days to finish the job

 

Anita finishes 1/(X-3) of the job in one day.

 

Now divide 1 by the sum of those two fractions to get how long it will take to finish the job together.  If this confuses you, let's say that each girl finishes 1/4 of the job in one day. . . Then together they would finish 1/2 of the job in one day and they would need 2 days to finish the job together. . . that's 1(1/4 + 1/4) = 1/(2/4) = 1/(1/2) = 2

 

So the question says that Anita and Tori need 2 days to finish the job together. . .

 

2 = 1 / (1/X + 1/(X-3))

 

Now the math gets tricky. . .

 

Cross multiply--> 2/X + 2/(X-3) = 1

 

Multiply each expression by X-3--> (2X-6)/X + 2 = X - 3

 

Subtract 2 from both sides--> (2X-6)/X = X - 5

 

Multiply each expression by X--> 2X - 6 = X^2 - 5X

 

Rearrange --> X^2 - 7X + 6 = 0

 

Factor with FOIL--> (X - 6) (X - 1) = 0

 

X = 6 days for Tori to mulch alone

 

Then 3 days for Anita to mulch alone

 

X can't be 1 because then Anita would finish the job in negative days.

 

Dan