Problem Solving

can someone please review this question and more specifically - the way i did it. i need confirmation that i solved the problem correctly. i know there are many many ways of solving rate problems; with respect - @ this point it's too late for me to learn new ways - just hoping an expert can follow the logic below and say whether i stumbled across the answer by accident or whether this is a correct way of solving this problem (note i said "a way" not "THE way"). thanks!

"Faucet 1 working alone can fill a tub in 20 minutes. Faucet 2 working alone can do the same job in 15 minutes. The hole at the bottom of the tub can empty a full tub in 30 minutes. If both faucets are working at their constant rates given above and the hole is not covered, how long does it take to fill an empty tub?"

(a) 5 minutes
(b) 6 minutes
(c) 10 minutes
(d) 12 minutes
(e) 20 minutes

Ok - here is now I did it:

R x T = W
1/3 x 3/1 = 1
1/4 x 4/1 = 1

After accounting for LCD, the combined rate is 7/12. Then we need to subtract 30 minutes (rate = 1/2) which is represented below (6/12).

Combined rate minue rate of loss:
7/12 - 6/12 = 1/12

So now, we're left with the combined rate plus the rate at which the hole is emptying water at 1/12, which means we need 12 minutes in order for the job do be done (see below).

R x T = W
1/12 x 12 = 1

So the answer is 12 minutes. The correct answer is 12 minutes - just want to make sure that I didn't get the correct answer "by accident" (which has happened to me on occasion. Rare, but it does happen.) Thanks again to whoever checks this.

(1/20)+(1/15)-(1/30)=1/12

answ is 12

jzw wrote:can someone please review this question and more specifically - the way i did it. i need confirmation that i solved the problem correctly. i know there are many many ways of solving rate problems; with respect - @ this point it's too late for me to learn new ways - just hoping an expert can follow the logic below and say whether i stumbled across the answer by accident or whether this is a correct way of solving this problem (note i said "a way" not "THE way"). thanks!

"Faucet 1 working alone can fill a tub in 20 minutes. Faucet 2 working alone can do the same job in 15 minutes. The hole at the bottom of the tub can empty a full tub in 30 minutes. If both faucets are working at their constant rates given above and the hole is not covered, how long does it take to fill an empty tub?"

(a) 5 minutes
(b) 6 minutes
(c) 10 minutes
(d) 12 minutes
(e) 20 minutes

Ok - here is now I did it:

R x T = W
1/3 x 3/1 = 1
1/4 x 4/1 = 1

After accounting for LCD, the combined rate is 7/12. Then we need to subtract 30 minutes (rate = 1/2) which is represented below (6/12).

Combined rate minue rate of loss:
7/12 - 6/12 = 1/12

So now, we're left with the combined rate plus the rate at which the hole is emptying water at 1/12, which means we need 12 minutes in order for the job do be done (see below).

R x T = W
1/12 x 12 = 1

So the answer is 12 minutes. The correct answer is 12 minutes - just want to make sure that I didn't get the correct answer "by accident" (which has happened to me on occasion. Rare, but it does happen.) Thanks again to whoever checks this.

I think you did, indeed, get lucky here unfortunately. Always remember to keep the units consistent...you chose to convert minutes to hours for some reason. All that does is confuse the issue. Here is your approach without switching to hours:

Faucet 1: 1 tub in 20 minutes -> 1/20 tub per minute
Faucet 2: 1 tub in 15 minutes -> 1/15 tub per minute
Hole: Minus 1 tub in 30 minutes -> -1/30 tub per minute

Net rate = 1/20+1/15-1/30 = 5/60 = 1/12 tubs per minute

12 minutes it is!


In your case, it looks like you took the times in minutes and converted them to hours:

1 tub in 20 minutes -> 1/3 hours/tub
1 tub in 15 minutes -> 1/4 hours/tub
-1 tub in 30 minutes -> -1/2 hours/tub

This step is fine but probably not the best use of time since they're looking for minutes in the end anyway (ALWAYS pay attention to units of measure when doing rate problems). Where you started to get off track is by skipping the next step: you never "flipped" to get the rate you WANT (in this case, tubs per hour).

Following this step, you would have gotten:

Faucet 1 -> 3 tubs per hour
Faucet 2 -> 4 tubs per hour
Hole -> -2 tubs per hour

Net rate = 3+4-2 = 5 tubs per hour

Now divide by 60 minutes/hour to get: 1/12 tubs per minute -> 12 minutes per tub

thank you for once again lifting a cloud of confusion that i hadn't realized was there. i've been converting everything to fractions of an hour - and you're right, it's just a colossal waste of time, and it causes mistakes. i was about to post a problem seperately in another post, one in which i couldn't quite figure out why i was screwing it up - and then i tried it just keeping within the same units the problem gave and i did it correctly in under a minute. i'll post the question below and the easy way to solve it (rather than the complex fractions back and forth).

"When working alone, Dominic takes twice as much time as Nick does to mow the lawn. When working together, they can mow half of the same lawn in 15 minutes. How long does it take Dominic to mow the lawn by himself?"

(a) 90 minutes
(b) 60 minutes
(c) 45 minutes
(d) 30 minutes
(e) 20 minutes

R x T = W
x 1/2x = 1
x 1/x = 1
x 1/30 = 1

1/2x + 2/2x = 1/30
x = 45
2x = 90

answer = 90 minutes

(in the original way i had 30 minutes as 1/2 and i was going back and forth to minutes)