Word problem: Mixture'd investments

[foahchon]

Hi, I have a word problem that I've been banging my head against for the past several hours, and while I've made a little progress, I'm not sure whether I'm on the right track or not. It's just driving me out of my mind. Here goes:

Tara paid one-half of her game-show winnings to the government for taxes. She invested one-third of her winnings in Jeff's copy shop at 14% interest and one-sixth of her winnings in Kaiser's German Bakery at 12% interest. If she earned a total of $4000 on the investments in one year, then how much did she win on the game show?

So far, all I've got is this:

x = original amount (won on game show)
\(\displaystyle \frac{1}{2}x\) = amount after government taxes
\(\displaystyle \frac{1}{3} * \frac{1}{2}x\) = amount invested at 14%
\(\displaystyle \frac{1}{6} * \frac{1}{2}x\) = amount invested at 12%

(Little symbols are meant to be multiplication signs)

I devised a table too, but it's a mess, I'm afraid I can't get it to show the way I want it to:

I P r t
first 1/6(x) 0.14 1
second 1/12(x) 0.12 1
total 4000 1

Not sure what to do with the blanks. I tried applying I = prt, and filling the interest blanks with \(\displaystyle \frac{1}{6} * 0.14\) for example, but the result was incorrect.

Any help would be appreciated putting it together. Thanks.

 

[foahchon]

I solved it. The problem was badly-worded. I was working backward from the solution, thinking that the solution ($60,000) was what she had originally won on the game show, when in fact $60,000 is what she had after taxes. The problem says to solve for what she originally won.

 

[tkhunny]

Your setup looks a bit confused.

Add up 1/2 , 1/2 * 1/3, and 1/2 * 1/6. I get 3/4. What is to be done with the other 1/4?

I agree that the problem should have stated the three values constituted ALL her winnings, but really that strikes me as a reasonable assumption. Other than that, the question is fine. It says "of her winnings". You translated it as "of what she had left after taxes".

x = winnings
x/2 = taxes
x/3 = cash at 14%
x/6 = cash at 12%

x/2 + x/3 + x/6 = x -- It's all accounted for.

Then, (x/3)*0.14 + (x/6)*0.12 = 4000, giving x = 60,000