The total number of people visiting a museum over three days (Monday, Tuesday, and Sunday) is 135. On Sunday, the number of males visiting is 40\% of the females visiting on Monday. The females visiting on Sunday are 5 fewer than those on Monday. On Tuesday, 20 males and 40 females visit the museum. The ratio of total visitors on Sunday to Tuesday is 1:2.

What is the total number of females visiting the museum over all three days?

Question

The total number of people visiting a museum over three days (Monday, Tuesday, and Sunday) is 135. On Sunday, the number of males visiting is 40\% of the females visiting on Monday. The females visiting on Sunday are 5 fewer than those on Monday. On Tuesday, 20 males and 40 females visit the museum. The ratio of total visitors on Sunday to Tuesday is 1:2.

What is the total number of females visiting the museum over all three days?

Accepted Answer

**Solution:**

1. Let the number of females visiting on Monday be $ F_M $.

2. Then, females visiting on Sunday = $ F_S = F_M - 5 $.

3. Males visiting on Sunday = 40\% of females on Monday = $ M_S = 0.4 F_M $.

4. On Tuesday, males $ M_T = 20 $ and females $ F_T = 40 $.

5. Total visitors on Tuesday = $ M_T + F_T = 20 + 40 = 60 $.

6. The ratio of visitors on Sunday to Tuesday is 1:2, so total visitors on Sunday = $ $\frac{1}{2} \times 60 = 30$ $.

7. Total visitors on Sunday = males + females = $ M_S + F_S = 0.4 F_M + (F_M - 5) = 1.4 F_M - 5 $.

8. Equate this to 30:
   $$ 1.4 F_M - 5 = 30 $$
   $$ 1.4 F_M = 35 $$
   $$ F_M = $\frac{35}{1.4} = 25$ $$

9. Females on Monday $ F_M = 25 $.

10. Females on Sunday $ F_S = 25 - 5 = 20 $.

11. Males on Sunday $$ M_S = 0.4 \times 25 = 10 $$.

12. Total visitors on Monday = males + females. Let males on Monday be $ M_M $.

13. Total visitors over three days = 135:
    $$ (M_M + 25) + (20 + 40) + (10 + 20) = 135 $$
    $$ M_M + 25 + 60 + 30 = 135 $$
    $$ M_M + 115 = 135 $$
    $$ M_M = 20 $$

14. Total females over all days = $ F_M + F_T + F_S = 25 + 40 + 20 = 85 $.

**Answer:** 85

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