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In a class of 255 students, 125 studied English, only 80 of them studied Mathematics only. Each student studied at least one of the two subjects. a) How many students studied (i) both subjects (ii) Mathematics? b) Draw a Venn diagram to represent the information.

Mathematics

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Answers

2 parts
a(i)

50\text{50}

a(ii)

130\text{130}

Step-by-step solution

4 steps

The problem provides the following information:

  • Total number of students in the class, N(U)=255N(U) = 255.
  • Number of students who studied English only, N(E only)=125N(E \text{ only}) = 125.
  • Number of students who studied Mathematics only, N(M only)=80N(M \text{ only}) = 80.
  • Each student studied at least one of the two subjects, which means the total number of students is equal to the number of students in the union of the two subjects: N(E∪M)=255N(E \cup M) = 255.

We will use the formula for the union of two sets:

N(E∪M)=N(E only)+N(M only)+N(E∩M)N(E \cup M) = N(E \text{ only}) + N(M \text{ only}) + N(E \cap M)

Part a(i)· 2 steps

How many students studied both subjects?

  1. Substitute the given values into the union formula.

    We are looking for N(E∩M)N(E \cap M), the number of students who studied both subjects.

    255=125+80+N(E∩M)255 = 125 + 80 + N(E \cap M)

  2. Simplify the equation and solve for N(E∩M)N(E \cap M).

    255=205+N(E∩M)255 = 205 + N(E \cap M) N(E∩M)=255−205N(E \cap M) = 255 - 205 N(E∩M)=50N(E \cap M) = 50

    The number of students who studied both subjects is 50\boxed{\text{50}}.

Part a(ii)

130\text{130}

How many students studied Mathematics?

  1. Calculate the total number of students who studied Mathematics.

    The total number of students who studied Mathematics, N(M)N(M), includes those who studied Mathematics only and those who studied both English and Mathematics.

    N(M)=N(M only)+N(E∩M)N(M) = N(M \text{ only}) + N(E \cap M)

  2. Substitute the known values into the formula for N(M)N(M).

    N(M)=80+50N(M) = 80 + 50 N(M)=130N(M) = 130

    The number of students who studied Mathematics is 130\boxed{\text{130}}.

Part b

Draw a Venn diagram to represent the information.

A Venn diagram for this information would consist of two overlapping circles, one for English (E) and one for Mathematics (M), inside a rectangle representing the universal set (U).

  • The region representing students who studied English only (part of circle E not overlapping M) would contain 125.
  • The region representing students who studied Mathematics only (part of circle M not overlapping E) would contain 80.
  • The overlapping region (intersection of E and M) representing students who studied both subjects would contain 50.
  • The region outside both circles (students who studied neither subject) would contain 0, as every student studied at least one subject.

To verify, the sum of students in all regions should equal the total number of students:

125(E only)+80(M only)+50(Both)=255125 (\text{E only}) + 80 (\text{M only}) + 50 (\text{Both}) = 255

This matches the total number of students in the class.

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