Points Per Answer Word Problem

[geekily]

My math final's tomorrow, and I'm just reviewing some problems that my teacher indicated might be helpful. This one's got me stumped:

On Hideki's history exams, he gets 4 points for each problem answered correctly, he loses 2 points for each incorrect answer, and he gets 0 points for each question left blank. On a 25-question test, Hideki received a score of 70.

Well, the first thing that confuses me is by "a score of 70", I assume they mean 70%. However, 70% of 25 is 17.5, which is not an option because everything is by 2's and 4's.

I tried setting the problem up as 4c - 2i + 0b = 17.5, and then immediately getting rid of 0b. Once I solved for either c or i, though, I didn't know what to do next - I couldn't come up with another formula to plug it in to. Any suggestions?

Thanks so much!

 

[pka]

Using your notation, it would be:
c+i+b=25 & 4c-2i=70.

That is all the information you gave us.
It seems that you need more to solve it.
However one solution is c=20 & i=5.

 

[geekily]

Oops, I'm so sorry! The questions that accompanied it were "what is the largest number of questions that he could have answered correctly?", "what is the fewest number of questions that he could have answered correctly?", and "what is the largest number of questions that he could have left blank?"

I've tried playing around with the formulas you gave me, but I still can't seem to figure it out.

Thanks so much for your help with this!

 

[soroban]

Hello, geekily!

If your teacher suggested this problem,
. . then he/she should have taught you certain techniques.
It is not a "normal" problem.

On Hideki's history exams, he gets 4 points for each problem answered correctly,
loses 2 points for each incorrect answer, and gets 0 points for each question left blank.
On a 25-question test, Hideki received a score of 70.

Well, the first thing that confuses me is by "a score of 70".
I assume they mean 70%.. . No, it means he got 70 points.

Let \(\displaystyle c\) = number of correct answers, \(\displaystyle i\) = number of incorrect answers,
. . and \(\displaystyle b\) = number of questions left blank.

Since there were 25 questions: \(\displaystyle \:c\,+\,i\,+\,b\:=\:25\;\) [1]

From the scoring, we have: \(\displaystyle \:4c\,-\,2i \:=\:70\;\;\Rightarrow\;\;i\:=\:2c\,-\,35\;\) [2]
. . Since \(\displaystyle i\,\geq\,0\), we have: \(\displaystyle \:2c\,-\,35\:\geq\:0\;\;\Rightarrow\;\;c\:\geq\:18\)

Substitute [2] into [1]: \(\displaystyle \:c\,+\,(2c\,-\,35)\,+\,b\:=\:25\;\;\Rightarrow\;\;b\:=\:60\,-\,3c\;\) [3]
. . Since \(\displaystyle b\,\geq\,0\), we have: \(\displaystyle \:60\,-\,3c\,\geq\,0\;\;\Rightarrow\;\;c\,\leq\,20\)


There are only three values for \(\displaystyle c\) . . .

. . \(\displaystyle c\,=\,18\), then \(\displaystyle b\,=\,6,\:i\,=\,1\)

. . \(\displaystyle c\,=\,19\), then \(\displaystyle b\,=\,3,\:i\,=\,3\)

. . \(\displaystyle c\,=\,20\), then \(\displaystyle b\,=\,0,\:i\,=\,5\)

 

[geekily]

soroban, thank you so much for your explanation! It was really clear and it helped a lot. My teacher just gave us a long list of problems in the book to go over for the final, so we never dealt with a problem like this in class before. Thanks so much for your help, I really appreciate it!