Set math help, please explain step by step.

Question

Q. In a survey conducted for three perfumes, it was found that 40% of the respondents used Fa, 25 % used Havoc, and 60 % used Byredo. 16% used none of the three perfumes, whereas 15% of the respondents used exactly two of the three perfumes.

1. What percentage of the respondents used all three perfumes?
2. What percentage of the respondents used at least two perfumes?
3. It is found that 22% of the respondents used both Havoc and Byredo. What percentage of the respondents used only Fa?

Answer 1

Answer 2

Label perfumes A, H, B

Let AcHcBc = S1, AHcBc = S2, AcHBc = S3, AcHcB = S4, AHBc = S5, AHcB = S6, AcHB = S7, and AHB = S8

1)S1 + S2 + S3 + S4 + S5 + S6 + S7 + S8 = 1

2)S1 = .16

3)S2 + S5 + S6 + S8 = .4

4)S3 + S5 + S7 + S8 = .25

5)S4 + S6 + S7 + S8 = .6

6)S5 + S6 + S7 = .15

1 From the first two equations, 7) S2+S3+S4+S5+S6+S7+S8 = .84

If we add equations, 3, 4, and 5, and subtract 6 and 7, we get

2 S8 = .4 + .25 + .6 - .15 - .84

2S8 = .26

S8 = .13. Let's call this 8)

2 At least 2 equals exactly 2 or 3 = .15 + .13 = .28

3 If 22% used Havoc and Breedo, then 9) S7 + S8 = .22

Then, as 2) S2 + S5 + S6 + S8 = .4

6) S5 + S6 + S7 = .15

8) S8 = .13

9) S7 + S8 = .22

Equation 2 - Equation 6 + Equation 9 - 2 * Equation 7 gives you

S2 = .4 - .15 + .22 - 2 * .13

S2 = .21