Assignment #2 --- Problem Solving

Question:
Consider the following problem. Jaskom carries out a survey of Computer Science Students and the result shows the following:
a- For question "Do you have computers?", 221 answered "Yes".,
b- For question "Do you have Java programming books?", 159 answered "Yes", and
c- For question "Do you have computers and Java programming books?", 87 answered "Yes".

Do an analysis of these data and solve the following problems. Among the students of Computer Science who have at least computers and Java programming books, find the percentage (%) of students who :

a) **have at least one computers and programming books
b) have only computers
c) have only programming books
d) have both computers and programming books

Solutions:
This is a "find" type problem. So we try to identify unknowns, data and conditions.

The unknowns are the percentage of students who only have computers, the percentage of students who only have Java programming books, and the percentage of students who have both computers and Java programming books.

The data are the three numbers: 221, 159 and 87, representing the number of students who have computers, Java programming books, and both computers and Java programming books, respectively. Note that 221 includes students who have both computers and Java programming books as well as people who have only computers. Similarly for 159.

The conditions are not explicitly given in the problem statement. But one can see that the percentages must add up to 100, and they must be nonnegative.

First let us consider the unknowns in more detail. To calculate the percentage of the students who have only computers, for example, we need the number of students who have at least one of computers and Java programming books, and the number of students who have only computers. Thus actually two unknowns are involved in each of the required percentages, and the real unknowns are the number of students in each of the categories, and the number of students who have at least one of computers and Java programming books.

Next let us look at the data. First the number 221 is the number of students who have computers. But that is not necessarily that of the students who have only computers. It includes that and the number of students who have both computers and Java programming books. Similarly for the second number 159.

Let us use symbols to represent each of the unknowns: Let C represent the number of students who have only computers, J that of the students who have only Java programming books, and T that of the students who have at least one of those things.

Then we have the following relationships among the unknowns:

C + 87 = 221
B + 87 = 159
C + B + 87 = T

From these equations we can easily obtain C = 134, B = 72, and T = 293.
Thus the required percentages are 45.7%, 24.6%, and 29.7%, respectively.

a) The percentage (%) of students who have at least one computers and programming book
= (134 / 293 * 100%) + (72 /293 * 100%)
= 45.7% + 24.6%
=70.3%

b) The percentage (%) of students who have only computers
= 134 / 293 * 100%
= 45.7%

c) The percentage (%) of students who have only Java programming books
= 72 /293 * 100%
= 24.6%

d) The percentage (%) of students who have both computers and Java programming books
= 87 / 293 *100%
= 29.7%