how to set this up

[allegansveritatem]

Here is the problem:


Here is my attempt:


How can we solve this problem without knowing at how many incorrect answers the student had? I have spent a long time trying to set this up. I even wrote a small essay trying to explain to myself how to work it. In vain.

 

[Dr.Peterson]

Here is the problem:
View attachment 10541

Here is my attempt:
View attachment 10542

How can we solve this problem without knowing at how many incorrect answers the student had? I have spent a long time trying to set this up. I even wrote a small essay trying to explain to myself how to work it. In vain.

You've done fine. You just don't need the inequalities.

We do know the number of incorrect answers: it's 10, as you found. And if 75% of the answers are right, 25% are wrong, the total number of questions is 40. That is, 0.25x = 10, so x = 40.

You should always check your answer. The student got 20 correct and 10 incorrect, then got all of the remaining 10 questions right; so the grade is 30/40 = 75%.

 

[Denis]

Just aNUTter look at it:

20 : 30 (after 1st 30 q's)
20+x : 30+x (after extra q's)

sooooo:
(20+x) / (30+x) = 3/4

 

[allegansveritatem]

You've done fine. You just don't need the inequalities.

We do know the number of incorrect answers: it's 10, as you found. And if 75% of the answers are right, 25% are wrong, the total number of questions is 40. That is, 0.25x = 10, so x = 40.

You should always check your answer. The student got 20 correct and 10 incorrect, then got all of the remaining 10 questions right; so the grade is 30/40 = 75%.

HalHa...it was staring me in the face...but somehow it couldn't quite come into focus. I'm not quite sure what I was thinking about here--maybe it has something to do with context, I mean, this comes at the end of doing a lot of problems dealing with fractions. I was expecting, I suppose, some kind of complicated equation built of fractions full of variables to be added or subtracted.

Many thanks

 

[allegansveritatem]

Just aNUTter look at it:

20 : 30 (after 1st 30 q's)
20+x : 30+x (after extra q's)

sooooo:
(20+x) / (30+x) = 3/4

Thanks for reply. I have to look at this awhile and explain it to my slow brain.

 

[JeffM]

I find it imposes the least mental burden to assign a unique symbol to each number not yet known.

\(\displaystyle q = \text { number of question.}\)

\(\displaystyle c = \text { number of correct answers.}\)

\(\displaystyle w = \text { number of wrong answers.}\)

To solve for three unknowns calls for three equations.

\(\displaystyle c + w = q.\) Obvious.

\(\displaystyle \dfrac{c}{q} = 0.75.\) You are told this.

\(\displaystyle w = (30 - 20) + 0 = 10.\) You are told this.

Now it is just mechanics.

\(\displaystyle \therefore c + w = q \implies c + 10 = q.\)

\(\displaystyle \therefore \dfrac{c}{q} = 0.75 \implies \dfrac{c}{c + 10} = 0.75 \implies c = 0.75c + 7.5 \implies\)

\(\displaystyle 0.25c = 7.5 \implies 4 * 0.25c = 4 * 7.5 \implies c = 30 \implies q = 30 + 10 = 40.\)

Very straightforward.

 

[allegansveritatem]

I find it imposes the least mental burden to assign a unique symbol to each number not yet known.

\(\displaystyle q = \text { number of question.}\)

\(\displaystyle c = \text { number of correct answers.}\)

\(\displaystyle w = \text { number of wrong answers.}\)

To solve for three unknowns calls for three equations.

\(\displaystyle c + w = q.\) Obvious.

\(\displaystyle \dfrac{c}{q} = 0.75.\) You are told this.

\(\displaystyle w = (30 - 20) + 0 = 10.\) You are told this.

Now it is just mechanics.

\(\displaystyle \therefore c + w = q \implies c + 10 = q.\)

\(\displaystyle \therefore \dfrac{c}{q} = 0.75 \implies \dfrac{c}{c + 10} = 0.75 \implies c = 0.75c + 7.5 \implies\)

\(\displaystyle 0.25c = 7.5 \implies 4 * 0.25c = 4 * 7.5 \implies c = 30 \implies q = 30 + 10 = 40.\)

Very straightforward.

I like your idea of assigning symbols for each unknown...but I was messing with this problem again today to make sure I understood it and it became clear to me what has been bothering me, namely this: The text tells us that the first 20 questions were correctly answered and after 30 all the remaining questions got correct answers too, but nowhere does it say how many of the 10 questions between 20 and 30 were actually missed. I mean, suppose the student only missed nine of the ten...that would mean (9)/(40)=.225 or 78 percent, but (9)/(35) would be .257 or about 75 percent and would thus satisfy the data. Am I failing to see something here still?

 

[Dr.Peterson]

Here is the problem:
View attachment 10541

Here is my attempt:
View attachment 10542

How can we solve this problem without knowing at how many incorrect answers the student had? I have spent a long time trying to set this up. I even wrote a small essay trying to explain to myself how to work it. In vain.

I was messing with this problem again today to make sure I understood it and it became clear to me what has been bothering me, namely this: The text tells us that the first 20 questions were correctly answered and after 30 all the remaining questions got correct answers too, but nowhere does it say how many of the 10 questions between 20 and 30 were actually missed. I mean, suppose the student only missed nine of the ten...that would mean (9)/(40)=.225 or 78 percent, but (9)/(35) would be .257 or about 75 percent and would thus satisfy the data. Am I failing to see something here still?

You're misreading the problem. It doesn't say "the first 20 questions were correctly answered". What it says is, "A student answered 20 of the first 30 problems correctly". That means that, of the first 30 questions,exactly 20 were correct, and the rest were not.

 

[JeffM]

I'd like to supplement Dr. P's very cogent thread preceding this. Not all problems are well written. Part of the useful challenge of word problems is determining what is really intended behind the ambiguities of natural languages. (Of course some problems are so incoherent that no such determination is possible.) Figuring out what was really intended is a skill useful outside mathematics.

 

[allegansveritatem]

You're misreading the problem. It doesn't say "the first 20 questions were correctly answered". What it says is, "A student answered 20 of the first 30 problems correctly". That means that, of the first 30 questions,exactly 20 were correct, and the rest were not.

Yes, I see it now. I thought the author was saying this: the student answered the first twenty correctly and after the 30th question he got all correct so that all the mistakes were made in a range of 10 questions. But upon a closer reading just now I see that he was saying he got 20 of the first 30 and all the rest after 30.

Good. I will put this one to rest. Thanks again.

 

[allegansveritatem]

I'd like to supplement Dr. P's very cogent thread preceding this. Not all problems are well written. Part of the useful challenge of word problems is determining what is really intended behind the ambiguities of natural languages. (Of course some problems are so incoherent that no such determination is possible.) Figuring out what was really intended is a skill useful outside mathematics.

Yes, I'm beginning to see that the texts of these problems have to be pulled apart the way critics do the texts of ancient poems.