1. ## Basic Arithmetic Conundrum.

Hi all ,

I stumbled upon the following :

A bottle filled with water ( density 1 kilo per Litre (1000 cm3) weighs 950 grammes, filled with alcohol ( density 800 grammes ) this bottle weighs 800 grammes.

You are asked to give the volume of this bottle.

Could someone explain :

1 ) why is the solution : (950-800) / 1-0.8 ?

2) If there is a way to find the correct result by resorting to ratios ? As we know the ratio of the 2 densities per equal volume ( 1000 to 800 grammes for an equal volume of 1 decimeter^3 ) we could find values to multiply or divide the numbers of the altered ratio ( 950 to 800 ) with and thereby find the respective wheights of both the contained materials and the container.
The respective weights are 750 grammes for the water and 600 grammes for the alcohol ( the bottle weighs 200 gr).
I know we could resort to an equation but I'd like a purely arithmetical way to work that out , Euclid style.

Thanks a ton for your insight.

2. ## Re: Basic Arithmetic Conundrum.

HI werecat,
The bottle has a capacity of 950 ml (.95 liter) and holds the same volume of alcohol

bjh

3. ## Re: Basic Arithmetic Conundrum.

Maybe you can use:
$\rho = \frac{m}{v}$

(With $\rho$=density)

4. ## Re: Basic Arithmetic Conundrum.

@ BJHOPPER

The weights include the weights of both the contained materials (water and/or alcohol ) and the container itself ( the weight of the bottle ), it's not that simple !

5. ## Re: Basic Arithmetic Conundrum.

Hi werecat,
You are correct.
950 = B + V *1
800 = B + V* .8
Subtract 2 from 1 and solve for V and B

bjh

6. ## Re: Basic Arithmetic Conundrum.

Originally Posted by bjhopper
Hi werecat,
You are correct.
950 = B + V *1
800 = B + V* .8
Subtract 2 from 1 and solve for V and B

bjh
That's a perfect answer but what I want to find is a purely arithmetical reasoning, I don't want a use an equation. In other words, by which values ( ratios ) would you need to multiply or divide 950 and 800 so as to get the 1 : 0.8 ratio ( or 1.25 ( 1/0.8) between the two actual weights of the liquids and thereby sifting out the weight of the bottle itself ?

By the way, do you get the 1) in my initial post ? I'm struggling to picture it

7. ## Re: Basic Arithmetic Conundrum.

The weight of the bottle does not have to be known to solve for volume as shown by the two equations. It cancels out but for any gross weights it obviously affects these.You said in your post that the density of water is 1 gram per ml ( 1000 gram = 1000 cubic cm ( 1000 ml ).

8. ## Re: Basic Arithmetic Conundrum.

I found my "purely arithmetical way" finally ! You know that the weights are made of quantities ( grammes ) bound by a 1.25 ratio.You notice that from 800 grms to 950 grms there is 150 grms. You just divide 150 grms by 0.8 ( density of alcohol ), that gives you 187.5 cm 3. The density of water being 1 ,or the ratio being 1.25, you divide 187.5 by 1 or multiply 150 by 1.25, that gives you 187.5 grms of water to be added on top of the 950. That way you get an according part ( since 800 and 950 > 150 and 187.5 ) of the 1.25 ratio that links the weights, then you climb down the "weight scale" by subtracting 150 and 187.5 from the respective weights ( 950 and 800 or these + 150 and 187.5 ) until you find a common measure of both weights, this common measure is the weight of the bottle, the volume of the bottle is the number of times you've had to subtract 150 and 187.5 multiplied by one of these numbers and divided by either 0.8 or 1.

I still don't get why they found the volume by dividing 150 by 200 though...I still need help on that one.

9. ## Re: Basic Arithmetic Conundrum.

Hi Werecat,
I think you are confused. Here is another equation

150 = weight of water - weight alcohol for equal volumes of each
V*1 -V*0.8 =150
V ( 1-0.8 ) =150
V = 150/0.2 = 750 ml

bjh

10. ## Re: Basic Arithmetic Conundrum.

Originally Posted by bjhopper
Hi Werecat,
I think you are confused. Here is another equation

150 = weight of water - weight alcohol for equal volumes of each
V*1 -V*0.8 =150
V ( 1-0.8 ) =150
V = 150/0.2 = 750 ml

bjh

That's great. Thank you BJH.