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April 19th, 2018, 05:42 PM   #1
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Exclamation Employee time word problem (urgent, please help soon)

Hi, I am having trouble with this question and am wondering whether anyone can show me what equation I should use. I know how to solve algebra decently well; it's just I am confused how to find what equation to originally use.

Three employees work at Starbucks. Ryan can fill an order two minutes faster than Uzair, but Hassan fills an order one minute slower than Uzair. When Uzair and Ryan work together, they can fill an order in one minute and twenty seconds. When Ryan and Hassan work together, they take one minute and thirty seconds to fill an order. How long would it take all three of them working together to fill an order?

Normally you would use something along the line of:
R = U-2
H = U+1
R+U = 4/3
R+H = 3/2
R+H+U = ?
but R+H+U needs to be *less*, not more, so I don't know what to do.

Last edited by skipjack; April 20th, 2018 at 12:27 AM.
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April 19th, 2018, 06:28 PM   #2
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rate = (1 order)/(time in minutes)

$U = \dfrac{1}{t}$

$R = \dfrac{1}{t-2}$

$H = \dfrac{1}{t+1}$

... where $t$ is time in minutes


(combined rates)(time in minutes) = 1 order filled

$(U + R) \cdot \dfrac{4}{3} = 1$ order filled

$(R + H) \cdot \dfrac{3}{2} = 1$ order filled

I worked this out using the first equation ... determined U fills an order in 4 minutes, R fills one in 2 minutes, and H fills one in 5 minutes

the second equation doesn't work out to be a time of 1.5 minutes ... it's close, but actually 1 and 3/7 minutes (approx 1 min 25.7 sec)
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April 20th, 2018, 11:56 AM   #3
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That kinda problem is usually hard to "grasp" at first; was for me anyway...

Rewording your problem slightly:
Three employees, Art, Ben and Cam, work at Tim Horton's: OK?!
Art can fill an order two minutes faster than Ben.
Cam fills an order one minute slower than Ben.
When Art and Ben work together, they fill an order in one minute and twenty seconds.
When Art and Cam work together, they take one minute and thirty seconds.
How long would it take all three of them working together to fill an order?

Let "filling an order" be "travelling 1 mile" (same thing).
Keep ALL times in seconds.
Let speeds be (mps = miles per second):
Art @ a mps, Ben @ b mps, Cam @ c mps

Art(@ a)................1.................>1/a sec.
Ben(@ b)...............1.................>1/b = 1/a + 120 sec.
Cam(@ c)...............1................>1/c = 1/a + 180 sec.
Given:
(@ a+b).................1.................> 1/(a+b) = 80 sec.
(@ a+c).................1.................> 1/(a+c) = 90 sec.
Solve:
(@ a+b+c)..............1................> ? sec

If that confuses you more, throw it in the waste paper basket!!
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April 22nd, 2018, 12:01 AM   #4
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Quote:
Originally Posted by TheRealJosh View Post
Hi, I am having trouble with this question and am wondering whether anyone can show me what equation I should use. I know how to solve algebra decently well; it's just I am confused how to find what equation to originally use.

Three employees work at Starbucks. Ryan can fill an order two minutes faster than Uzair, but Hassan fills an order one minute slower than Uzair. When Uzair and Ryan work together, they can fill an order in one minute and twenty seconds. When Ryan and Hassan work together, they take one minute and thirty seconds to fill an order. How long would it take all three of them working together to fill an order?

Normally you would use something along the line of:
R = U-2
H = U+1
R+U = 4/3
R+H = 3/2
R+H+U = ?
but R+H+U needs to be *less*, not more, so I don't know what to do.
That makes no sense because you have not said what "R", "U", and "H" mean! If you meant to say "Let R be the length of time, in minutes, it takes Ryan to fill an order, let "U" be the time it takes Uzair fill an order, and let "H" be the time it take Hassan to fill the order", then your error is in writing "R+U = 4/3" and "R+H = 3/2" their times do not add! As you said, that would mean two people working together would take longer than one alone which doesn't make sense.

" Ryan can fill an order two minutes faster than Uzair":
U= R+ 2.

"but Hassan fills an order one minute slower than Uzair."
H= U+ 1.

"When Uzair and Ryan work together, they can fill an order in one minute and twenty seconds."
When two people work together (or two pipes fill a tank, etc.) their rates add. If Uzair can fil an order in U minutes, the he works at rate 1/U "orders per minute". If Ryan can fill an order in R minutes then he works at rate 1/R orders per minute. Their rate when they work together is 1/U+ 1/R= (R+ U)/RU. The time it takes to fill one order at that rate is $\displaystyle \frac{1}{\frac{R+U}{RU}}= \frac{RU}{R+ U}= 1+ 28/60= 1+ 7/15= \frac{22}{15}$ minutes.

Replace U with R+ 2, from the first condition to get an equation in R only: $\displaystyle \frac{R(R+ 2)}{R+ R+ 2}= \frac{R^2+ 2R}{2R+ 2}a= \frac{22}{15}$]. $\displaystyle 15(R^2+ 2R)= 22(2R+ 2)$. Solve that for R. Then U= R+ 2 and H= U+ 1.

When Ryan and Hassan work together, they take one minute and thirty seconds to fill an order.
As above, $\displaystyle \frac{RH}{R+ H}= \frac{3}{2}$.

This is a fourth equation, but you only need three equations to solve for three unknown values. You can check to see if your values for H and R, from above, satisfy this equation- that is, check if the information given is consistent.
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Last edited by Country Boy; April 22nd, 2018 at 12:08 AM.
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April 22nd, 2018, 12:04 AM   #5
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Here's a similar problem worked and explained:
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April 22nd, 2018, 08:46 AM   #6
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I get $\frac{20}{19}$ minutes ($\approx 63$ seconds) for all three together.

One of Country Boy's solutions is discarded because it leads to a negative rate of work for one of the Baristas.
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April 22nd, 2018, 09:28 AM   #7
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Hmmm...I get 720/11 seconds (65.454545....)

Well, I'm close to yours !
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April 22nd, 2018, 12:46 PM   #8
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Quote:
Originally Posted by v8archie View Post
I get $\frac{20}{19}$ minutes ($\approx 63$ seconds) for all three together.

One of Country Boy's solutions is discarded because it leads to a negative rate of work for one of the Baristas.
Well, that's probably the coffee shop I go to!
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April 22nd, 2018, 01:35 PM   #9
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Quote:
Originally Posted by Denis View Post
Hmmm...I get 720/11 seconds (65.454545....)

Well, I'm close to yours !
Mine gave 2, 4 and 5 minutes for the Baristas solo.
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April 22nd, 2018, 02:07 PM   #10
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The word barista is an Italian word, and in Italy, a barista is a male
or female "bartender", who typically works behind a counter, serving hot drinks
(such as espresso), cold alcoholic and non-alcoholic beverages, and snacks.

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