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:
Correct Answer: 68%
Let the total number of candidates be x
Number of capable candidates who got admitted to college :
= 80% of 40% of x
$$\eqalign{ & = \left( {\frac{{80}}{{100}} \times \frac{{40}}{{100}} \times x} \right) \cr & = \frac{{8x}}{{25}} \cr} $$
Number of incapable college students :
= 25% of (100 - 40)% of x
$$\eqalign{ & = \left( {\frac{{25}}{{100}} \times \frac{{60}}{{100}} \times x} \right) \cr & = \frac{{3x}}{{20}} \cr} $$
Total number of candidates who got admitted to college :
$$=$$ $$\frac{{8x}}{{25}}$$ + $$\frac{{3x}}{{20}}$$
$$=$$ $$\frac{{47x}}{{100}}$$
∴ Required percentage :
$$\eqalign{ & = \left( {\frac{{8x}}{{25}} \times \frac{{100}}{{47x}} \times 100} \right)\% \cr & = \left( {\frac{{3200}}{{47}}} \right)\% \cr & = 68.09\% \approx 68\% \cr} $$
Number of capable candidates who got admitted to college :
= 80% of 40% of x
$$\eqalign{ & = \left( {\frac{{80}}{{100}} \times \frac{{40}}{{100}} \times x} \right) \cr & = \frac{{8x}}{{25}} \cr} $$
Number of incapable college students :
= 25% of (100 - 40)% of x
$$\eqalign{ & = \left( {\frac{{25}}{{100}} \times \frac{{60}}{{100}} \times x} \right) \cr & = \frac{{3x}}{{20}} \cr} $$
Total number of candidates who got admitted to college :
$$=$$ $$\frac{{8x}}{{25}}$$ + $$\frac{{3x}}{{20}}$$
$$=$$ $$\frac{{47x}}{{100}}$$
∴ Required percentage :
$$\eqalign{ & = \left( {\frac{{8x}}{{25}} \times \frac{{100}}{{47x}} \times 100} \right)\% \cr & = \left( {\frac{{3200}}{{47}}} \right)\% \cr & = 68.09\% \approx 68\% \cr} $$