Question

Megan is planting a garden with two beds with her mother and father. Megan can plant 1 garden bed in 8 hours. Her mother can plant 2 garden beds in that time, and her father can plant 1 1 3 garden beds in that time.
The three begin working together, but after 3 hours, Megan's father leaves for work. Her mother works half an hour longer and then leaves for work. How long will it take Megan to finish planting the two garden beds by herself?

Answer 1

Answer:

$1.5\text{ hours}$

Step-by-step explanation:

We know that Megan can plant 1 garden bed in 8 hours. Let M represent Megan's rate. So:

$M=\frac{1\text{ b}}{8\text{ hr}}$

We know that her mother can plant 2 garden beds in that time (8 hours). Let A represent the mother's rate. So:

$A=\frac{2\text{ b}}{8\text{ hr}}=\frac{1\text{ b}}{\text{ 4hr}}$

We can reduce this to 1 flower bed every 4 hours.

We also know that Megan's father can plant 1 1/3 or 4/3 garden beds in 8 hours. Let D represent the father's rate. So:

$D=\frac{4/3 \text{ hr}}{8 \text{ hr}}=\frac{1\text{ b}}{6\text{ hr}}$

We can reduce (4/3)/8 to 1/6.

We know that the three began working together. They worked together for 3 hours. So, after 3 hours, the amount of beds they planted all together is 3 hours times their respective rates. So, we can write the following expression:

$3(\frac{1}{8}+\frac{1}{4}+\frac{1}{6})$

We know that at this point, Megan's father left, leaving only Megan and her mother. We know that they worked together for another 30 minutes, or 1/2 of an hour. So, after this, they will have planted:

$3(\frac{1}{8}+\frac{1}{4}+\frac{1}{6})+\frac{1}{2}(\frac{1}{8}+\frac{1}{4})$

Garden beds.

Now, Megan's mother leaves, leaving only Megan. Let's let x represent the number of hours. So, we can write the last part of our expression:

$3(\frac{1}{8}+\frac{1}{4}+\frac{1}{6})+\frac{1}{2}(\frac{1}{8}+\frac{1}{4})+x(\frac{1}{8})$

We know that in the end, they planted 2 flower beds. So, our entire expression equals 2:

$3(\frac{1}{8}+\frac{1}{4}+\frac{1}{6})+\frac{1}{2}(\frac{1}{8}+\frac{1}{4})+x(\frac{1}{8})=2$

To find out how long it took Megan, we will solve for x.

Let's do each term individually:

First Term:

We have:

$3(\frac{1}{8}+\frac{1}{4}+\frac{1}{6})$

Make the fractions with common denominators. Our common denominator here is 24. So:

$3(\frac{3}{24}+\frac{6}{24}+\frac{4}{24})$

Add:

$=3(\frac{13}{24})$

Multiply. So, our first term is:

$=\frac{39}{24}$

Second Term:

We have:

$\frac{1}{2}(\frac{1}{8}+\frac{1}{4})$

Again, let's turn the fractions into fractions with common denominators so we can add them. The common denominator here is 8. So:

$\frac{1}{2}(\frac{1}{8}+\frac{2}{8})$

Add:

$=\frac{1}{2}(\frac{3}{8})$

Multiply:

$=\frac{3}{16}$

So, our equation is now:

$\frac{39}{24}+\frac{3}{16}+\frac{1}{8}x=2$

Add on the left. Use the common denominator of 48. So:

$\frac{78}{48}+\frac{9}{48}+\frac{1}{8}x=2$

Add:

$\frac{87}{48}+\frac{1}{8}x=2$

Subtract 87/48 from both sides:

$\frac{1}{8}x=2-\frac{87}{48}$

Let turn into a fraction with a denominator of 48. So:

$\frac{1}{8}x=\frac{96}{48}-\frac{87}{48}$

Subtract:

$\frac{1}{8}x=\frac{9}{48}$

Reduce the right using 3:

$\frac{1}{8}x=\frac{3}{16}$

Multiply both sides by 8:

$x=\frac{24}{16}$

Reduce using 8. So, the time it will take Megan to finish planting the garden beds by herself is:

$x=3/2=1.5\text{ hours}$

And we're done!