There are a total of 450 coupons used for Pedicure and Haircutting combined. The ratio of males to females using coupons for Haircutting is 13:7. The number of males using coupons for Pedicure exceeds the number of females using coupons for Haircutting by 72. Additionally, the total number of males using coupons for both Pedicure and Haircutting is 174 more than the total number of females using coupons for both services.

If 25\% of the males who use coupons for Haircutting belong to city A, find the number of males who use coupons for Haircutting but do not belong to city A.

Question

There are a total of 450 coupons used for Pedicure and Haircutting combined. The ratio of males to females using coupons for Haircutting is 13:7. The number of males using coupons for Pedicure exceeds the number of females using coupons for Haircutting by 72. Additionally, the total number of males using coupons for both Pedicure and Haircutting is 174 more than the total number of females using coupons for both services.

If 25\% of the males who use coupons for Haircutting belong to city A, find the number of males who use coupons for Haircutting but do not belong to city A.

Accepted Answer

**Solution:**

1. Let the number of males using coupons for Haircutting be $M_h$ and females using coupons for Haircutting be $F_h$.

2. Given the ratio $M_h : F_h = 13 : 7$. So, $M_h = 13x$ and $F_h = 7x$.

3. Number of males using coupons for Pedicure is 72 more than number of females using coupons for Haircutting:
   $$M_p = F_h + 72 = 7x + 72$$

4. Total males using coupons for both services:
   $$M = M_h + M_p = 13x + (7x + 72) = 20x + 72$$

5. Total females using coupons for both services:
   $$F = F_h + F_p$$

6. Given total males exceed total females by 174:
   $$M - F = 174 \ (20x + 72) - (7x + F_p) = 174 \ 13x + 72 - F_p = 174 \ F_p = 13x + 72 - 174 = 13x - 102$$

7. Total coupons:
   $$M + F = 450 \ (20x + 72) + (7x + F_p) = 450 \ 27x + 72 + F_p = 450$$

Substitute $F_p = 13x - 102$:
   $$27x + 72 + 13x - 102 = 450 \ 40x - 30 = 450 \ 40x = 480 \ x = 12$$

8. Calculate $M_h$:
   $$M_h = 13x = 13 	imes 12 = 156$$

9. Males using coupons for Haircutting who do not belong to city A (75\% of $M_h$):
   $$156 	imes \frac{75}{100} = 117$$

**Answer:** $117$

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