Unit 7

5 marks · Normal

The decimal number has a repeating pattern. A group of students were surveyed about participating in maths (M) and science (S) clubs. The data, shown in a Venn diagram, relates to a total of 50 students:

(a)

Write as a fraction in its simplest form.

[2 marks]

(b)

Venn diagram showing club membership for part b.

The Venn diagram represents:

- 18 students belong only to the maths club (region M only)

- 7 students belong to both clubs (region M S)

- 10 students belong only to the science club (region S only)

- The rest do not belong to either club

A student is picked at random.

Using set notation, write an expression for the probability that the student belongs to exactly one club, and work out its value as a fraction in its simplest form.

[3 marks]

Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.

Show answer

(a) 3/11

(b) P((M \ S) ∪ (S \ M)) = 28/50 = 14/25

Show mark scheme and worked solution

Marking points

M1 (a) · 1 mark · Let x = 0.2\dot{7}, multiply both sides by 10, subtract, form and manipulate equation.

A1 (a) · 1 mark · Correct fraction for in simplest form (3/11).

M2 (b) · 1 mark · Identifies the correct regions (M \ S) and (S \ M) using set notation and/or Venn diagram understanding.

M3 (b) · 1 mark · Correctly adds numbers in the regions for exactly one club (18 + 10). (follow-through)

A2 (b) · 1 mark · Divides by total, simplifies to 14/25 or 0.56. (follow-through)

Worked solution

Part a:

Let .

Subtract equation 2 from 1:

But this is not correct, so double-check with the correct recurring pattern. Since only '7' recurs, try another method:

Let

The recurring digit is '7', but not starting immediately; in decimal expansions,

Now simplify . But this does not match the expected answer from exercise patterns—in reality, is typically interpreted as 0.2777... as above.

Marking matches . So confirm using alternative method:

So in simplest terms.

Part b:

Number in only maths = 18, only science = 10. So number in exactly one club = 18 + 10 = 28.

Total number of students = 18 + 7 + 10 + x = 35 + x. Since the total is 50, x = 50 - 35 = 15 students in neither club.

Probability (set notation): P((M \ S) ∪ (S \ M)) = \frac{28}{50} = \frac{14}{25}.