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### Which Barrel is Unsold?

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Date: 1 Mar 1995 20:48:35 -0500
From: 91S17200807C
Subject: Which barrel is unsold?

Hello there!

I came across this particular problem in a quiz book.  Can
anybody help me with this?

The problem is as follow:

There are six barrels with volumes of 8, 13 15, 17, 19 and
31 litres respectively. These barrels contain either petrol or
diesel oil. The price of petrol is twice the price of diesel oil
per litre.  If \$140 were received from the sales of petrol,
\$140 from the sales of diesel oil and with only one barrel of
fuel left unsold, which barrel is left unsold?

Regards,
Wai Leng
2/3/95
9.45 a.m. (Singapore time)
```

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Date: 2 Mar 1995 11:21:23 -0500
From: Anonymous
Subject: Re: Which barrel is unsold?

Hi Wai Leng,

So, that's strange.  The same amount is earned but petrol costs
twice as much.  So what does that tell us about the number of
litres of diesel sold compared to that of petrol?

Once you've figure that out we now have to find five of the
barrels that combine to give us the right proportion of litres
of petrol to litres of diesel, right?

I think that once you have figured out how many more litres
of diesel than petrol there should be, you can fairly quickly
play with combinations of five barrels to narrow down the
possibilities.  At which point you can use trial and error to
pick the correct one.

Let me know how far you get with this help and what more
you need.

-- Steve "Chief of Staff"
```

```
Date: Fri, 07 Mar 1997 07:37:19 -0700
From:
Subject: A logical solution

Hello 'Doctors'!

I must thank you for all the fun I always get solving some of
your old problems. Here is a logical solution to this one:

There are six barrels with volumes of 8, 13 15, 17, 19 and
31 litres respectively. These barrels contain either petrol or
diesel oil. The price of petrol is twice the price of diesel oil
per litre.  If \$140 is received from the sales of petrol,
\$140 from the sales of diesel oil and with only one barrel of
fuel left unsold, which barrel is left unsold?

------------------------------------------------------------------
Of course trial and error can be used here. But one can use a
logical approach that is much quicker and lot more fun!!

Since Petrol costs twice as much as Diesel but fetches the same
total amount, it must be half the volume sold. This also means
that the total volume of fuel sold must be divisible by three!

Knowing this, divide the barrels into three types :
(a) Type	3*n-1	: 8, 17
(b) Type	3*n	: 15
(c) Type	3*n+1	: 13, 19, 31

Since the total volume is 103 a number of type (c), to get a
volume divisible by three you must keep a barrel of type (c) out!
So the unsold barrel should be 13 or 19 or 31

Now look at the three cases:

Unsold        Total
barrel     volume sold       Petrol sold   Diesel sold

13       103 - 13 = 90          30            60
19       103 - 19 = 84          28            56
31       103 - 31 = 72          24            48

So now the problem boils down to:

Which of the numbers 30,28,24 can you make up by adding up the
barrels?

It is easy to see that only 28 = 15 + 13 is possible.
Hence the barrel with capacity 19 was unsold!

Note : In fact you can reduce trial and error by using logic
even in the last step. The three possible volumes of petrol can
also be classified as

(b) Type : 24, 30 (Only possible with a+c or a+b+c)
(c) Type : 28	(Only possible with b+c)

Thanks! - Ujjwal Rane

```
Associated Topics:
High School Basic Algebra

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