jrosesweet 3 years ago PLEASE HELP, EASY Jeff and Lucy have been asked to wash their mom’s minivan. It takes Jeff 2 hours to wash the van by himself, and it takes Lucy 1.5 hours to wash the van by herself. How long will it take Jeff and Lucy to wash the van if they work together?

1. Calcmathlete

Ok. First find how much work is done individually in one hour.

2. Calcmathlete

So, if Jeff does the work in 2 hours, how much will he get done in 1 hour?

3. jrosesweet

the whole car?

4. Calcmathlete

Well. In 2 hours, he gets all of the work done. In 1 hour, he gets half the work done because it's half the time it takes. Does that make sense? In other words, $$\frac12$$.

5. Calcmathlete

Now, in one hour, Lucy does $$\LARGE \frac{1}{1.5}$$ amount of the work. If you combine it, you get this. $\huge\frac 12 + \frac{1}{1.5} = \frac1x$Can you solve for x here?

6. jrosesweet

so What equation is used to solve this problem? What does each variable represent?

7. jrosesweet

oh ok so thats the equation we use?

8. Calcmathlete

I just gave you the equation and the variable represents how long it takes for them to do it when they combine.

9. jrosesweet

oh thanks! so whats the solution?

10. jrosesweet

can i cross multiply to get the solution?

11. Calcmathlete

I suppose you can, but you have to in the very least get a common denominator before that. $\frac 12 \times \frac33+ \frac{1}{1.5} \times\frac44= \frac 1x \implies \frac 36 + \frac46 = \frac1x \implies \frac{7}{6} = \frac1x$Now you can cross multiply. 7x = 6 x = ?

12. jrosesweet

so it would be the fraction of 6/7 ??

13. Calcmathlete

yup :)

14. jrosesweet

:) so thats the solution?

15. Calcmathlete

I believe so :)

16. jrosesweet

and last question, so how long does it take for BOTH of them to wash the car together?

17. Calcmathlete

18. jrosesweet

i plug in the solution and thats the time it takes for both of them...right

19. Calcmathlete

The solutions 6/7 is how long it takes to wash the car.

20. jrosesweet

but that doesnt make sense lol

21. Calcmathlete

@apoorvk Can you double check my work here?

22. Calcmathlete

I'm pretty sure that it would take 6/7 of an hour to wash the car together though...

23. jrosesweet

\OHH 6/7 of an hour. ok now it makes sense. thanks for all your help :)

24. Calcmathlete

np :)

25. apoorvk

@Calcmathlete yeah you are right - well almost. :P Actually 1/2 + 1/(1.5) gives you the work that can be completed by both kids together *in an hour* - so they basically complete 6/7 of the work in an hour. Hence they will need 1/(6/7), i.e. 7/6 hours to complete the work from scratch! :]

26. Calcmathlete

Oh ok. I was always under the impressions that since $$\LARGE \frac 1x$$ represents work done in an hour, the reciprocal which is x would give you the combined rate?

27. apoorvk

Work done in an hour is itself the rate - isn't it? ;)

28. Calcmathlete

lol...I have always been confused at this part...I'll take your word for it XD

29. apoorvk

No, don't take 'my' word - understand it and take your brain's word then ;)

30. Calcmathlete

Ok. $\frac 1x = \frac 76 = \text{Work rate}$

31. Calcmathlete

Than last question. What exactly would 6/7 be then?

32. apoorvk

Oh wait - it-is-my bad! :O I completely misread what you'd written! ಠ_ಠ You had got 1/2 + 1/1.5 = 7/6, and so ofcourse the complete the work in 6/7 hours together!! I am really sorry, am stupid. -__-

33. Calcmathlete

lol. no worries ;)

34. apoorvk

I thought *1/2 + 1/1.5* was 6/7 - so I made that mix-up. Apologies once again!