Question

A guidance counselor is planning schedules for 34 students. All students must take at least one of either Math, English, or History. 16 students say they want to take Math, 14 want to take English, and 15 want to take History. 5 say they want to take both English and History, and of these, 2 wanted to take Math as well. 8 want only History, and 10 want only math. How many students want only English

Answer 1

Answer: 7 students are ready to offer only English

Explanation:

We were told that 5 students were interested in taking history and English. Among these 5 students, 2 wanted to offer maths as well. This means that 2 students will offer the three subjects and thereby pegging the number of students who want to offer both English and history at 3 (5 - 2).

Before, we can find the number of students who want only English, we must first find the number that wants maths and history.

Let's denote the students who want to offer maths and history by "k".

Total History students - (History and English students + students who want to offer all three + k) = 8 (those that want to offer only history)

That is: 15 - (3+2+k) = 8

15 - (5+k) = 8

15 - 5 - k = 8

10 - k = 8

k = 2

Therefore, 2 students want to offer maths and history.

We will then find the number of students that want to offer maths and English but first; let's denote the students that want to offer maths and English by "v".

Total maths students - (maths and history students + students that want to offer all three + v) = 10 (those that want to offer only maths).

That is 16 - (2+2+v) = 10

16 - (4 + v) = 10

16 - 4 - v = 10

12 - v =10

v = 2

Therefore, the number of students that want to offer maths and English is 2.

Now, we can calculate the number of students that want to offer only English.

Those that want to offer only English =

Total English students - (English and history students + maths and English students + students that want to offer all three)

= 14 - (3 + 2 + 2)

= 14 - 7

= 7 students.

Therefore, the number of students want to offer only English is 7.

Answer 2

7 Students want only English

Explanation:

Total Number of Students=34

n(students say they want to take Math)=16

n(students say they want to take English)=14

n(students say they want to take History)=15

n(both English and History)=5

Since 2 out of those who want to take English and History also wants Maths, The number of those who want the three subjects=2

n(who want only History)=8

n(who want only Maths)=10

This available information is placed on the Venn Diagram below.

n(H)=2+3+8+n(M∩H)

15=2+3+8+n(M∩H)

n(M∩H)=15-13=2

n(M)=10+2+2+n(M∩E)

16=10+2+2+n(M∩E)

n(M∩E)=16-14=2

n(E)=2+3+n(M∩E)+n(E only)

14=2+3+2+n(E only)

n(E only)=14-7=7