Question 1208723: Ruth has a beaker containing a solution of $800$ mL of acid and $200$ mL of water. She thinks the solution is a little strong, so she drains $300$ mL from the beaker, adds $300$ mL of water, and stirs the solution. Ruth thinks the solution is still too strong, so again she drains $400$ mL from the beaker, and adds $400$ mL of water. Ruth again thinks the solution is still too strong, so again she drains $100$ mL from the beaker, and adds $100$ mL of water. How many mL of water are now in the beaker?
Found 4 solutions by Edwin McCravy, MathTherapy, ikleyn, mccravyedwin:
Answer by Edwin McCravy(20054)   (Show Source):
You can put this solution on YOUR website!
First, I'll copy and paste your problem here so I can delete those "$" dollar
marks around every number. They confuse me. We didn't do that where I came
from. J
Ruth has a beaker containing a solution of 800 mL of acid and 200 mL of water.
She thinks the solution is a little strong, so she drains 300 mL from the
beaker, adds 300 mL of water, and stirs the solution. Ruth thinks the solution
is still too strong, so again she drains 400 mL from the beaker, and adds 400 mL
of water. Ruth again thinks the solution is still too strong, so again she
drains 100 mL from the beaker, and adds 100 mL of water. How many mL of water
are now in the beaker?
I'll keep track of the liquid, acid, water, % acid, and % water, because in
other problems like this you might be asked different amounts and percentages.
Ruth has a beaker containing a solution of 800 mL of acid and 200 mL of water.
liquid = 800+200 = 1000 mL
acid = 800 mL
water = 200 mL
% acid = 800/1000 = 0.8 = 80%
% water = 200/1000 = 0.2 = 20%
She thinks the solution is a little strong, so she drains 300 mL from the beaker,
liquid = 1000-300 = 700 mL
acid = (0.8)(700) = 560 mL
water = (0.2)(700) = 140 mL
% acid = 80%
% water = 20%
adds 300 mL of water, and stirs the solution.
liquid = 700+300 = 1000 mL
acid = 560 mL
water = 140+300 = 440 mL
% acid = 560/1000 = 0.56 = 56%
% water = 440/1000 = 0.44 = 44%
Ruth thinks the solution is still too strong, so again she drains 400 mL from the beaker,
liquid = 1000-400 = 600 mL
acid = (600)(0.56) = 336 mL
water = (600)(0.44) = 264 mL
% acid = 56%
% water = 44%
and adds 400 mL of water.
liquid = 600+400=1000 mL
acid = 336 mL
water = 264+400=664 mL
% acid = 336/1000=33.6%
% water = 664/1000=66.4%
Ruth again thinks the solution is still too strong, so again she drains 100 mL from the beaker,
liquid = 1000-100=900 mL
acid = (900)(0.336) = 302.4 mL
water = 664-100=564 mL
% acid = 33.6%
% water = 66.4%
and adds 100 mL of water.
liquid = 900+100=1000 mL
acid = 302.4 mL
water = 564+100=664 mL
% acid = 302.4/1000 0.3024=30.24%
% water = 664/1000 = 0.664 = 66.4%
How many mL of water are now in the beaker?
664 mL <-- ANSWER
Edwin
Answer by MathTherapy(10551)  (Show Source):
You can put this solution on YOUR website!
Ruth has a beaker containing a solution of $800$ mL of acid and $200$ mL of water. She thinks the solution is a little strong, so she drains $300$ mL from the beaker, adds $300$ mL of water, and stirs the solution. Ruth thinks the solution is still too strong, so again she drains $400$ mL from the beaker, and adds $400$ mL of water. Ruth again thinks the solution is still too strong, so again she drains $100$ mL from the beaker, and adds $100$ mL of water. How many mL of water are now in the beaker?
 .
From the above TABLE:
Start: Amount of acid: 800 mL ; amount of water: 200 mL ; Total in solution: 800 + 200 = 1,000 mL ; % acid: 
% water: 
1. DRAINING 300 mL of solution: 1,000 - 300 = 700 mL of solution left, of which 560 mL is acid and 140 mL is water
2. ADDING 300 mL of water: Solution = 700 + 300 = 1,000 mL, of which 560 mL is STILL acid and 140 + 300 = 440 mL, is water.
Acid in solution:  ; Water in solution: 
3. DRAINING 400 mL of solution: 1,000 - 400 = 600 mL of solution left, of which 56%, or 336 mL is acid and 264 mL is water
4. ADDING 400 mL of water: Solution = 600 + 400 = 1,000 mL, of which 336 mL is STILL acid and 264 + 400 = 664 mL, is water.
Acid in solution:  ; Water in solution: 
5. DRAINING 100 mL of solution: 1,000 - 100 = 900 mL of solution left, of which 33.6%, or 302.4 mL is acid, and 900 - 302.4, or 597.6 mL is water
6. Finally, ADDING 100 mL of water: Solution = 900 + 100 = 1,000 mL, of which 302.4 mL is STILL acid and 1,000 - 302.4 = 697.6 mL, is water.
I hope you're able to follow this!
Sir Edwin,
Where @Ikleyn says the error is, is NOT where it is. I'll copy part of yours to show you the errors.
Ruth again thinks the solution is still too strong, so again she drains 100 mL from the beaker,
liquid = 1000-100=900 mL
acid = (900)(0.336) = 302.4 mL <==== The error is NOT HERE, as per @Ikleyn. This is FINE!
water = 664-100=564 mL <==== The error is HERE. 100 mL was DRAINED, so 100 mL should NOT HAVE been
subtracted from water. In fact, the REMAINING 900 mL, after 100 mL was drained
includes 302.4 mL (33.6%) acid, and 597.6 (900 - 302.4) mL, or 66.4% water.
% acid = 33.6%
% water = 66.4%
and adds 100 mL of water.
liquid = 900+100=1000 mL
acid = 302.4 mL
water = 564+100=664 mL <==== Instead of 564 mL HERE, the STARTING amount of water should be 597.6 mL; and,
with 100 mL ADDED, water becomes 597.6 + 100 = 697.6 mL!
% acid = 302.4/1000 0.3024=30.24%
% water = 664/1000 = 0.664 = 66.4% <==== Instead of 664/1,000, this should be 697.6/1,000 = 69.76%, which makes
FINAL acid and water percentages, 30.24 and 69.76, for a TOTAL of 100%.
How many mL of water are now in the beaker?
664 mL <-- ANSWER
697.6 mL <-- ANSWER <==== Should be.
Edwin
Answer by ikleyn(52777)  (Show Source):
You can put this solution on YOUR website! .
Ruth has a beaker containing a solution of 800 mL of acid and 200 mL of water.
She thinks the solution is a little strong, so she drains 300 mL from the beaker, adds 300 mL of water, and stirs the solution.
Ruth thinks the solution is still too strong, so again she drains 400 mL from the beaker, and adds 400 mL of water.
Ruth again thinks the solution is still too strong, so again she drains 100 mL from the beaker, and adds 100 mL of water.
How many mL of water are now in the beaker?
~~~~~~~~~~~~~~~~~~~~~~
The answers in the posts by Edwin and by @MathThearapy are different - so, there is the need to check their solutions.
Here I make this check, but I use slightly different methodology, which better suits to the problem.
In this problem, after each step, Ruth keeps the total volume of the mixture in the beaker unchangeable, 1000 mL.
Therefore, in my solution, I calculate and watch/track for the concentration/amount of the acid, only, in the beaker.
I will calculate the amount of water in the beaker after the last step, but will not calculate/watch/track it at the intermediate steps.
It will allow me to reduce the volume of calculations and will allow me to concentrate
my attention on one component, only (which is acid). It will diminish the volume
of calculations and will diminish possible errors.
(0) Starting amount of the acid is 800 mL; the starting concentration is  = 0.8.
(1) First step is to drain 300 mL of the mixture from the beaker.
With it, 0.8*300 = 240 mL of the acid goes out.
The amount of the acid remained in the beaker is 800-240 = 560 mL.
The amount of water is added to keep the total volume of the mixture in the beaker 1000 mL.
The concentration after step 1 is  = 0.56.
(2) Second step is to drain 400 mL of the mixture from the beaker.
With it, 0.56*400 = 224 mL of the acid goes out.
The amount of the acid remained in the beaker is 560-224 = 336 mL.
The amount of water is added to keep the total volume of the mixture in the beaker 1000 mL.
The concentration after step 2 is  = 0.336.
(3) Third step is to drain 100 mL of the mixture from the beaker.
With it, 0.336*100 = 33.6 mL of the acid goes out.
The amount of the acid remained in the beaker is 336-33.6 = 302.4 mL.
The amount of water is added to keep the total volume of the mixture in the beaker 1000 mL.
Hence, the amount of water in the beaker is 1000 - 302.4 = 697.6 mL.
ANSWER. The amount of water in the beaker after step 3 is 697.6 mL.
Solved.
The answer by @MathTherapy is confirmed.
Calculations are made by the most economic/effective way.
Of course, the most important element of my post
is the METHODOLOGY, which EXCLUDES unnecessary calculations.
The answer by Edwin is incorrect.
//////////////////////////////
To help Edwin to identify his error, I copied and pasted here his solution
from his post, and pointed the precise place/line with the error.
Ruth has a beaker containing a solution of 800 mL of acid and 200 mL of water.
She thinks the solution is a little strong, so she drains 300 mL from the
beaker, adds 300 mL of water, and stirs the solution. Ruth thinks the solution
is still too strong, so again she drains 400 mL from the beaker, and adds 400 mL
of water. Ruth again thinks the solution is still too strong, so again she
drains 100 mL from the beaker, and adds 100 mL of water. How many mL of water
are now in the beaker?
I'll keep track of the liquid, acid, water, % acid, and % water, because in
other problems like this you might be asked different amounts and percentages.
Ruth has a beaker containing a solution of 800 mL of acid and 200 mL of water.
liquid = 800+200 = 1000 mL
acid = 800 mL
water = 200 mL
% acid = 800/1000 = 0.8 = 80%
% water = 200/1000 = 0.2 = 20%
She thinks the solution is a little strong, so she drains 300 mL from the beaker,
liquid = 1000-300 = 700 mL
acid = (0.8)(700) = 560 mL
water = (0.2)(700) = 140 mL
% acid = 80%
% water = 20%
adds 300 mL of water, and stirs the solution.
liquid = 700+300 = 1000 mL
acid = 560 mL
water = 140+300 = 440 mL
% acid = 560/1000 = 0.56 = 56%
% water = 440/1000 = 0.44 = 44%
Ruth thinks the solution is still too strong, so again she drains 400 mL from the beaker,
liquid = 1000-400 = 600 mL
acid = (600)(0.56) = 336 mL
water = (600)(0.44) = 264 mL
% acid = 56%
% water = 44%
and adds 400 mL of water.
liquid = 600+400=1000 mL
acid = 336 mL
water = 264+400=664 mL
% acid = 336/1000=33.6%
% water = 664/1000=66.4%
Ruth again thinks the solution is still too strong, so again she drains 100 mL from the beaker,
liquid = 1000-100=900 mL
acid = (900)(0.336) = 302.4 mL <<<---=== The error is HERE. The correct calculation should be acid = 1000*0.336 = 336 mL,
with relevant correction in all consequential calcs.
water = 664-100=564 mL
% acid = 33.6%
% water = 66.4%
and adds 100 mL of water.
liquid = 900+100=1000 mL
acid = 302.4 mL
water = 564+100=664 mL
% acid = 302.4/1000 0.3024=30.24%
% water = 664/1000 = 0.664 = 66.4%
How many mL of water are now in the beaker?
664 mL <-- ANSWER
Edwin
...................................
This shows once again how dangerous is to split your attention
by making unnecessary calculations.
Professional writers in such cases say: how good that this mistake was made naturally;
otherwise, it would have been worth to make it intentionally, and then talking, discussing
and teaching on how to organize calculations in a secure way.
@ikleyn
Answer by mccravyedwin(406)  (Show Source):
You can put this solution on YOUR website!
Ikleyn says:
The answer by Edwin is incorrect.
I can't find any mistake. I've gone through it several times and cannot find it.
Edwin
|
|
|
