# If matt and isabella work together, how long will it take them to rake leaves? *fill in tableIt takes 9 hours for isabella to rake leaves by herself, but her brother matthew can work 3 times as...

If matt and isabella work together, how long will it take them to rake leaves? *fill in table

It takes 9 hours for isabella to rake leaves by herself, but her brother matthew can work 3 times as fast.

Rate Time Work Done

isabella

matthew

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### 1 Answer

Note that the rate of a person refers to how fast he/she could finish one complete work alone. It is express as a ratio of amount of work done to the amount of time the person has worked.

For a completed work, assume that the amount of work done is 1.

So, the rate of Isabelle `(r_i)` is:

r= (amount of work done)/(amount of time the person work)

`r_i= 1/9`

Since Mathew can work 3 time as fast as Isabelle, then his rate is:

`r_m=3r_i=3*1/9`

`r_m=1/3`

To determine the time it takes them to rake the leaves when working together, add the amount of work done by each person.

The amount of work done by each person is the product of their rate and the time working together (t).

For Isabelle,

Amount of work done = 1/9 t

For Mathew,

Amount of work done = 1/3 t

So,

`1/9t+1/3t ` = Total amount of work done

Again assume that for a completed work, amount done is 1.

`1/9t+1/3t=1`

To simplify, multiply both sides by the LCD which is 9.

`9*(1/9t+1/3t)=1*9`

`t+3t=9`

`4t=9`

`(4t)/4=9/4`

`t=2.25`

**Hence, it takes 2.25 hours for Isabelle and Mathew to rake the leaves when working together.**

Then, multiply 2.25 hours with the rate of each person, to get the work done by each.

For Isabelle, amount of work done = `1/9*2.25=0.25` .

For Mathew, the amount of work done = `1/3*2.25=0.75` .

Thus the table would have the following details when working together.

Rate Time(hrs) Work Done

Isabelle `1/9` 2.25 0.25 (Do 25% of raking the leaves)

Mathew `1/3` 2.25 0.75 (Do 75% of raking the leaves)