Work rate problem. Gaz, Bert and Bob: "When Florencia went out into the garden, ..."

sgrisham

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Work rate problem. Gaz, Bert and Bob: "When Florencia went out into the garden, ..."

Hi everyone.

Here is my problem:

When Florencia went out into the garden, she was surprised to find the men had finished the job. "That was quick work," she said. "54 minutes for the lot!" "Well, we just kept going and it was the same for all three of us," old Gaz replied. "Bob is two-thirds as fast as Bert, and young Bert's half as fast as I am. But I've had more experience." How long would Gaz have taken to do the whole job alone?

I worked out the answer to be 108 minutes. But apparently the correct answer is 99 minutes.

Can someone please tell me how to work this out correctly?

Thanks,

Steve.
 
Hi everyone.

Here is my problem:

When Florencia went out into the garden, she was surprised to find the men had finished the job. "That was quick work," she said. "54 minutes for the lot!" "Well, we just kept going and it was the same for all three of us," old Gaz replied. "Bob is two-thirds as fast as Bert, and young Bert's half as fast as I am. But I've had more experience." How long would Gaz have taken to do the whole job alone?

I worked out the answer to be 108 minutes. But apparently the correct answer is 99 minutes.

Can someone please tell me how to work this out correctly?

Thanks,

Steve.
Please show us:

How did you calculate Gaz's time of completion to be 108 minutes?
 
Hi Subhotosh and Denis. Thank you for helping me.

This is how I (incorrectly) worked it out.

If Bob is two-thirds as fast as Bert, and Bert is half as fast as Gaz.

Then Gaz is twice as fast as Bert and three times as fast as Bob.

So Gaz would have done 50% of the work.

Bert would have done 25% of the work.

And Bob would have done 16.66% of the work.

The time taken for all three people working together is 54 minutes.

So, if Bob did 50% of the work, then to do all the work himself it would take 54 x 2 = 108 minutes.

Note: I have just realised something. Those percentages I worked out do not add up to 100%. This would definitely make my answer wrong, but I still don't know how to make it correct.
 
Hi Subhotosh and Denis. Thank you for helping me.

This is how I (incorrectly) worked it out.

If Bob is two-thirds as fast as Bert, and Bert is half as fast as Gaz.

Then Gaz is twice as fast as Bert and three times as fast as Bob.

So Gaz would have done 50% of the work.

Bert would have done 25% of the work.

And Bob would have done 16.66% of the work.

The time taken for all three people working together is 54 minutes.

So, if Bob did 50% of the work, then to do all the work himself it would take 54 x 2 = 108 minutes.

Note: I have just realised something. Those percentages I worked out do not add up to 100%. This would definitely make my answer wrong, but I still don't know how to make it correct.
You posted this problem in Arithmetic section.

Do you know algebra? Rate problems are lot easier to do with algebra!

Please let us know whether algebra is an acceptable method of solution for this problem.
 
I do know some very basic algebra.

I think I might be able to understand an algebraic solution.

I will definitely do my best to try to understand it.
 
The following approach uses algebra.

Many of these "work"-type problems can be solved by considering the fractional part of the job that each worker completes per unit of time.

You're trying to find the number of minutes that it takes Gaz (working alone) to complete the job.

Let x = minutes for Gaz to do the whole job

Therefore, Gaz completes 1/x of the job per minute.

We're told that Bert works 1/2 as fast as Gaz, so Bert needs twice as much time. In other words, it takes Bert 2x minutes to do the whole job.

Therefore, Bert completes 1/(2x) of the job per minute.

We're told that Bob works 2/3 as fast as Bert (who works 1/2 as fast as Gaz).

2/3 * 1/2 = 1/3 (as fast as Gaz)

In other words, it takes Bob 3x minutes to do the whole job.

Therefore, Bob completes 1/(3x) of the job per minute.

Working together, it takes all three of them 54 minutes to do the job.

Therefore, together they complete 1/54 of the job per minute.

1/x + 1/(2x) + 1/(3x) = 1/54

Solve for x.

If this set-up seems like fuzzy math, then google the search phrase algebra work-type problems, to find lessons and worked examples. :cool:
 
Look at the 2-3-6 hint I gave you earlier...
Means together = 2+3+6 = 11 per minute.

11 what per minute? :wink:

That was for 54 minutes: 54 * 11 = 594 = total job.
SO...Gaz's = 6/11 = ?
Instead of 11/6ths of 54, it seems like you're saying 6/11ths of 594.
 
Thanks guys. I understand it now.

Many of these "work"-type problems can be solved by considering the fractional part of the job that each worker completes per unit of time.

The fractional bit quoted above helped me to work it out.

I was trying to use this equation: x + 2x + 3x = 54

and I was getting nowhere.

But when I used this equation: 1/x + 1/(2x) + 1/(3x) = 1/54

I was able to work it out.

Here we go:

1/x + 1/(2x) + 1/(3x) = 1/54

simplifies to

11/6x = 1/54

So

6x = 11 x 54 = 594

So

594/6 = x = 99

Hooray!!
 
Last edited:
Denis, I understand your method now as well as mmm4444bot's method.

Big thanks to both of you. :D
 
NO.
I'm trying to illustrate a different method,
using the less confusing speed = distance / time.

The combined speed is 2+3+6 = 11 per minute.
At that speed, time is 54 minutes.
So distance is 54*11 = 594

So alone, Gaz = 594/6 = 99 minutes.

I agree that the method you're using
is the usual method for this style of problem.
I was simply trying to illustrate a different method.

And Denis HATES fractions ....
 
I'm trying to illustrate a different method,
using the less confusing speed = distance / time.
Sir, when you switch a given scenario to an analogy (for assisting the thought process), please state that up front. :cool:

I understood that you were thinking of something else, but I could not read your mind.


11 units of distance per minute
This is a valid rate.
 
Denis, I understand your method now …
Me, too! :)

Good work, on solving your equation, btw. Here's a quick note about notation.

11/(6x) = 1/54
When we type ratios as text, we need to put grouping symbols around numerators and denominators that contain multiple terms or factors, in order to show the intended Order of Operations.

11/6x means 11/6 * x.

Cheers
 
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