Need help with story prob.
J.Y. Miller
In article <01bbfc0b$b6215c20$4056c7cd@default>, "David Brady"
<dbr...@sirus.com> wrote:
> Can someone please help me solve this story problem?
>
> The purity of gold is measured in karats, with pure gold being 24 karats.
> Other purities of gold are expressed as proportional parts of pure gold.
> Thus, 18 karat gold is 18/24, or 75% pure gold; 12 karat gold is 12/24 or
> 50% pure gold, and so on. How much 12 karat gold should be mixed with pure
> gold to obtain 60 grams of 16 karat gold?
>
> Thanks!
You will add X grams of pure gold. Since the total weight will be 60, you
will be adding it to (60 - X) grams of 12 karat gold.
Since 12 karat gold is only half gold (50%), the amount of gold in that
(60-X) will be 12/24 * (60 - X).
The X grams of pure gold is ALL gold.... so it has X grams of gold in it!
When you add them you get 60 grams of 16 karat. 16 karat is 16/24 pure.
So this will contain 16/24 * 60 grams of gold.
The equation then is: 12/24 * (60 -X) + X = 16/24 * 60
I would multiply each term by 24 (I hate fractions... :) )
12 * (60 - X) + 24X = 16*60
720 - 12X + 24X = 960
720 + 12X = 960
12X = 960 - 720 = 240
X = 240/12 = 20
You need to add 20 grams of 12 karat gold.
J.Y.
--
James Y. Miller
Caruthersville, MO
David Brady
Gary Scott Simon
Only because others have demonstrated the rigorous method from which
my technique is adapted, I thought I'd share how I did this in my head:
16 carat gold is 16/24 gold. 12 carat gold is 12/24 gold. The
difference between 16/24 and 12/24 is 4/24
24 carat gold is 24/24 gold. The difference between 24/24 and
16/24 is 8/24.
Since the ratio of the difference 16-24 - 12/24 to the difference
24/24 - 16/24 is one-to-two, the ratio of the element of the mixture
represented by 12/24 (i.e. the 12 carat gold) to the element of the
mixture represented by 24/24 (i.e. the pure gold) must be two-to-one.
With 60 grams total, that means 40 grams of 12 carat gold, and 20
grams of pure gold.
Checking our answer, the 40 grams of "half" (i.e. 12 carat) gold has
20 grams of gold in it. Added to the 20 grams of "pure" gold, theres a
total of 40 grams of gold in the 60 gram mixture, and, sure enough 40/60
= 16/24.
This approach is based on the the math illustrated in other responses
to this thread. Don't use this shortcut unless and until you've mastered
the underlying math.
Jeremy Soria
"David Brady" <dbr...@sirus.com> wrote:
>Can someone please help me solve this story problem?
>The purity of gold is measured in karats, with pure gold being 24 karats.
>Other purities of gold are expressed as proportional parts of pure gold.
>Thus, 18 karat gold is 18/24, or 75% pure gold; 12 karat gold is 12/24 or
>50% pure gold, and so on. How much 12 karat gold should be mixed with pure
>gold to obtain 60 grams of 16 karat gold?
First, set up a chart. Let x = # g of 12k gold & y = # g of pure gold:
(remember that 12k Gold is 12/24 pure gold, or 1/2, and 16k gold is 16/24 =
2/3 gold.)
What Number of Grams of
part grams Pure Gold
12k Gold 1/2 x (1/2)x
24K Gold 1 y y
16K Gold 2/3 60 40
Add the second and third columns to get a system:
x + y = 60
(1/2)x + y = 40
And solve!
x + y = 60
-x -2y = -80
----------------
-y = -20
y = 20
x = 40
Therefore you need to mix 40g 12k gold with 20g pure gold to get 60g 16k gold.
Hope this helps!
- Jeremy "It's nice to know Bret [Hart] is such a homophobe."
-- John Petrie, on Bret's Shawn Michaels comments
--
jes...@infi.net | Pro Wrestling. TPiR. Pinball. And Rosie. | Call Cox and
and a webpage too. | It just doesn't get any better than this. | Ask for GSN!
E-mail me for URL. |---------------------------------------------------------
------------------- If this is the last one, it'll be... TAPED TO THE DESK!
Bruiser, Buddy, Andre, Kerry, Art, Eddie, Big John, Dick Murdoch ... Sigh ...
NLN
> 16 carat gold is 16/24 gold. 12 carat gold is 12/24 gold. The
> difference between 16/24 and 12/24 is 4/24
>
> 24 carat gold is 24/24 gold. The difference between 24/24 and
> 16/24 is 8/24.
>
> Since the ratio of the difference 16-24 - 12/24 to the difference
> 24/24 - 16/24 is one-to-two, the ratio of the element of the mixture
> represented by 12/24 (i.e. the 12 carat gold) to the element of the
> mixture represented by 24/24 (i.e. the pure gold) must be two-to-one.
>
>
> to this thread. Don't use this shortcut unless and until you've mastered
> the underlying math.
Gary,
Before you (or anyone else with short fuse) get disgruntled by this
question, keep in mind that I am a math-lover, Hofstadter-reader, and
Literature major.
I understand your solution because it makes practical sense to me. That
is how I almost literally "see" the problem. I understand that someone
learning algebra for algebra's sake would not want to go about it that
way without learning the underlying math. However, isn't it important to
actually comprehend mathematical situations and solutions rather than
apply a model or textbook prepared algebraic formula? This was always my
beef with math class. . . what's your take on it?
Thanks,
Nic
David
NLN <nne...@english.umass.edu> wrote:
[snip]
>However, isn't it important to
>actually comprehend mathematical situations and solutions rather than
>apply a model or textbook prepared algebraic formula? This was always my
>beef with math class. . . what's your take on it?
>Thanks,
>Nic
Yes. However, "text-book" methodology is usually a generalisation of
a problem. That means that all of the necessary thought processes
have been accomplished and condensed. The idea is "Why re-invent the
wheel" every time one is faced with the same, or a similar problem.
Thus the methods discovered and retained are applicable to a diverse
selection of problems. However, one should always retain awareness
ofthe source of underlying principles. [Use a formula, but if you
forget it, be prepared to re-develop it at any time, so to speak.]
I hope that this is on track with what you are thinking.
Gary Scott Simon
>> 16 carat gold is 16/24 gold. 12 carat gold is 12/24 gold. The
>> difference between 16/24 and 12/24 is 4/24
>>
>> 24 carat gold is 24/24 gold. The difference between 24/24 and
>> 16/24 is 8/24.
>>
>> Since the ratio of the difference 16-24 - 12/24 to the difference
>> 24/24 - 16/24 is one-to-two, the ratio of the element of the mixture
>> represented by 12/24 (i.e. the 12 carat gold) to the element of the
>> mixture represented by 24/24 (i.e. the pure gold) must be two-to-one.
>>
>>
>> This approach is based on the the math illustrated in other responses
>> to this thread. Don't use this shortcut unless and until you've mastered
>> the underlying math.
>Gary,
>Before you (or anyone else with short fuse) get disgruntled by this
>question, keep in mind that I am a math-lover, Hofstadter-reader, and
>Literature major.
> I understand your solution because it makes practical sense to me. That
>is how I almost literally "see" the problem. I understand that someone
>learning algebra for algebra's sake would not want to go about it that
>way without learning the underlying math. However, isn't it important to
>actually comprehend mathematical situations and solutions rather than
>apply a model or textbook prepared algebraic formula? This was always my
>beef with math class. . . what's your take on it?
I intended my closing comment to indicate my strong agreement with
the sentiments expressed by Nic. As I recall, I'd presented a fairly
detailed derivation of my shortcut no more than a day or two earlier, and
(perhaps carelessly) I chose to refer to it rather than re-post it.
Perhaps I could/should have been clearer, but I absolutely agree that
mindless application of formulae and shortcuts is, an a word, bad.