It takes 7 hours for Isabella to rake leaves by herself.Her brother can work 3 times as fast.if they work together,how long will its take them to rake leaves?

My answer: Isabella can do one job in 7 hours (1/7in one hour).Matthew does 3/4 of the job in one hour. Together they do 1/7 + 3/7 =4/7 of the job in one hour. They do the entire job in 7/4 or 1.75 hours.

My question: how do I show just the Rate,Time,and Work Done by each person?

t1 = 7 hrs.

t2 = 1/3 * 7 = 7/3 hrs.

T = t1*t2/(t1+t2) = (7*7/3)/(7+7/3).
T = (49/3)/(28/3)
T = 49/3 * 3/28 = 49/28 = 7/4 = 1.75 hrs.

Another Method:

t1 = 7 hrs.
t2 = 1/3 * 7 = 7/3 hrs.

1/T = 1/t1 + 1/t2.
1/T = 1/7 + 3/7 = 4/7,
T = 7/4 = 1.75 hrs.

To show the rate, time, and work done by each person, you can follow these steps:

1. Let's assume Isabella's rate of work is R tasks per hour, and her brother's rate of work is 3R tasks per hour.
2. Since Isabella takes 7 hours to complete the job alone, her total work done is 1 job.
3. Isabella completes 1 job in 7 hours, so her rate of work can be calculated as R = 1/7 job per hour.
4. Isabella's brother works 3 times faster, so his rate of work is 3R = 3/7 job per hour.
5. Now, let's denote the time taken to complete the job when they work together as T hours.
6. When they work together, their combined rate of work is (R + 3R) = 4R = 4/7 job per hour.
7. The work done by Isabella in time T is R * T, and the work done by her brother in time T is 3R * T.
8. Together, they complete the entire job, which is 1 job, so their total work done is 1.
9. Therefore, the equation becomes R * T + 3R * T = 1.
10. Simplifying, we get 4R * T = 1.
11. Substituting the value of R (1/7) from earlier, we have (4/7) * T = 1.
12. Solving for T, we find T = 7/4 or 1.75 hours.

So, when they work together, it will take Isabella and her brother 1.75 hours to rake the leaves.

To show the rate, time, and work done by each person, you can use the formula:

Work = Rate * Time

Let's denote:
- Isabella's rate as R1 (work done by Isabella in 1 hour)
- Isabella's time as T1 (time taken by Isabella to complete the job)
- Matthew's rate as R2 (work done by Matthew in 1 hour)
- Matthew's time as T2 (time taken by Matthew to complete the job)

We know the following information:
1) It takes Isabella 7 hours to complete the job, so T1 = 7.
2) Matthew works 3 times as fast as Isabella, so his rate is 3 times Isabella's rate, R2 = 3*R1.

Now, let's calculate the work done by each person in terms of hours:

Work done by Isabella = R1 * T1 = (1/7) * 7 = 1 job
Work done by Matthew = R2 * T2

Since we want to find out how long it will take them to complete the job together, we need to find a common unit for work or time.

Since Isabella completes the job in 1 hour, we can use this as a common unit. So, let's express Matthew's time in terms of Isabella's time:

R1 * T1 = R2 * T2
(1/7) * 1 = (3*1/7) * T2
1/7 = 3/7 * T2
T2 = 1/3 hour

Now we have the following information:
- Isabella's rate (R1) = 1/7 job per hour
- Isabella's time (T1) = 7 hours
- Matthew's rate (R2) = 3/7 job per hour
- Matthew's time (T2) = 1/3 hour

To calculate the time it will take for them to complete the job together, we can add their individual times:

Total Time = T1 + T2
Total Time = 7 + 1/3
Total Time = 21/3 + 1/3
Total Time = 22/3

Therefore, it will take them a total of 22/3 hours (or approximately 7.333 hours) to complete the job together.