Percentage

[Saumyojit]

At the college entrance examination, each candidate is admitted or rejected according to whether he has passed or failed the tests. Of the candidates who are really capable, 80% pass the tests and of the incapable, 25% pass the test. Given that 40% of the candidates are really capable, the proportion of capable college students is about:

I just want to know the proportion of capable college students is to what?
To the no of candidates appeared for the exam or the no of candidates selected ?
why the latter? How to understand this type of question what they are wanting in the denom?
@Dr.Peterson @Subhotosh Khan

 

[HallsofIvy]

Yes, that's a weirdly worded question! You are told that "40% of the candidates are really capable" then asked "what proportion of the students are capable"! Is "capable" different from "really capable"? If not the question is non-sense. But if they are we are not given enough information to answer the question.

Asking "what proportion of the candidates are admitted" would be a reasonable and very common question. Are you giving us the exact wording of the problem?

 

[Saumyojit]

as 68 percent is answer i conclude that they are asking what percentage will be the ratio of really capable candidates that got selected to the total no of candidates got selected in college(capable + uncapable)

Another thing does proportion fit into this question i knew that if we compare a part to whole that is a ratio and proportion is equality of 2 ratios but i discovered that in wiki " comparison of a specific quantity to "the whole" is called a proportion." . i thought that is a ratio

 

[Saumyojit]

@Dr.Peterson if we compare a part to whole that is a ratio or proportion or both? and proportion is equality of 2 ratios but i discovered that in wiki " comparison of a specific quantity to "the whole" is called a proportion." . i thought that was known as ratio

 

[Dr.Peterson]

as 68 percent is answer i conclude that they are asking what percentage will be the ratio of really capable candidates that got selected to the total no of candidates got selected in college(capable + uncapable)

I'd say you can ignore the question entirely. Whatever they intended to ask, they did not ask it! You may be right about what was intended, but you are not responsible to figure that out for them.

@Dr.Peterson if we compare a part to whole that is a ratio or proportion or both? and proportion is equality of 2 ratios but i discovered that in wiki " comparison of a specific quantity to "the whole" is called a proportion." . i thought that was known as ratio

The English language, like all languages, is inconsistent. When we "write a proportion", we mean something like a/b=c/d. When we talk about "the proportion of students who pass", we mean a ratio (or, actually, a fraction) like 2/3 of the students. Dictionaries commonly give multiple definitions for words, which we have to distinguish by context.

In this case, I find that most dictionaries don't even mention the "mathematical" definition (a/b=c/d), but only several everyday uses. Here, for example, you can find it only if you read far enough to get to a specialized dictionary: https://www.dictionary.com/browse/proportion .

It happens that the dictionary I use most often agrees with me (definitions 3 vs. 5):

Definition of PROPORTION

harmonious relation of parts to each other or to the whole : balance, symmetry; proper or equal share; quota, percentage… See the full definition

www.merriam-webster.com

​

Definition of proportion

(Entry 1 of 2)​

​

1 : harmonious relation of parts to each other or to the whole : balance, symmetry​

2a : proper or equal share​

each did her proportion of the work​

b : quota, percentage​

3 : the relation of one part to another or to the whole with respect to magnitude, quantity, or degree : ratio​

4 : size entry 1, dimension​

5 : a statement of equality between two ratios in which the first of the four terms divided by the second equals the third divided by the fourth (as in 4/2=10/5) — compare extreme sense 1b, mean sense 1c​

But among mathematics sources, the two definitions I mentioned are still both found, though never in the same source:

• http://www.math.com/school/subject1/lessons/S1U2L2DP.html
• https://stattrek.com/statistics/dictionary.aspx?definition=proportion .

Each of these has a distinct context. So you learn to interpret accordingly.

 

[JeffM]

In English, many words have two or more meanings that are similar but slightly different. (By the way, Wikipedia, wonderful as it is, is not a dictionary.)

"Proportion" may mean "ratio" in general. It may mean more narrowly the "ratio of a part to the whole." It has other meanings that are not strictly mathematical. Which meaning is meant depends on context.

 

[Saumyojit]

can a part to part ratio be a proportion?

 

[Dr.Peterson]

can a part to part ratio be a proportion?

Again, it depends. The statistical usage seems to be part to whole; but did you read the definition I quoted in full (#3)?

 

[Saumyojit]

harmonious relation of parts to each other
this is given

 

[joshuaa]

let [MATH]T[/MATH] be the total candidates who passed and failed

[MATH]0.4T[/MATH] are really capable

[MATH]0.6T [/MATH] are incapable

[MATH]0.8 \cdot 0.4T = 0.32T[/MATH] really capable who passed

[MATH]0.25 \cdot 0.6T = 0.15T[/MATH] incapable who passed

the admitted candidates are now college students, and their total is

[MATH]0.32T + 0.15T = 0.47T[/MATH]
the proportion of capable college students is

[MATH]\frac{0.32T}{0.47T} = \frac{32}{47}[/MATH]

 

[Dr.Peterson]

...
the proportion of capable college students is

[MATH]\frac{0.32T}{0.47T} = \frac{32}{47}[/MATH]

Which is just what Saumyojit said in post #3:

as 68 percent is answer i conclude that they are asking what percentage will be the ratio of really capable candidates that got selected to the total no of candidates got selected in college(capable + uncapable)

What you are adding here is an explanation of the wording:

At the college entrance examination, each candidate is admitted or rejected according to whether he has passed or failed the tests. Of the candidates who are really capable, 80% pass the tests and of the incapable, 25% pass the test. Given that 40% of the candidates are really capable, the proportion of capable college students is about:

A better way for them to have worded this is, "the proportion of (admitted) college students who are capable is ...". Yes, I think you've both found the right interpretation; they are asking for the ratio of admitted capable students to all admitted students. The problem is still not worded very clearly.