Directions : Study the following information carefully and answer the questions given below:
There are 450 coupons which can be used in Pedicure and Hair cutting. The ratio of Males to Females who use their coupons in Hair cutting is 13: 7. The number of males who use their coupons in Pedicure is 72 more than the number of females who use their coupon in Hair cutting. Total number of males who use their coupon in Pedicure and Hair cutting together is 174 more than the total number of females who use their coupon in Pedicure and Hair cutting together.
The ratio of Males who use their coupon in Pedicure to those who use it in Spa is 4:5, while the ratio of Females who use their coupon in Hair cutting to those who use it in Spa is 6:11. Find the total number of people who use their coupons in Spa.

Options

A. 349
B. 481
C. 300
D. 440

Answer

To solve the problem step by step, we will first define the variables based on the information provided and then set up equations to find the required values.

### Step 1: Define Variables
Let:
- \( M_H \) = number of males using coupons for Hair cutting
- \( F_H \) = number of females using coupons for Hair cutting
- \( M_P \) = number of males using coupons for Pedicure
- \( F_P \) = number of females using coupons for Pedicure
- \( M_S \) = number of males using coupons for Spa
- \( F_S \) = number of females using coupons for Spa

### Step 2: Set Up Ratios and Equations
From the problem, we know:
1. The ratio of males to females using coupons for Hair cutting is \( 13:7 \):
\[ M_H = 13x \quad \text{and} \quad F_H = 7x \]

2. The number of males who use their coupons for Pedicure is 72 more than the number of females who use their coupons for Hair cutting:
\[ M_P = F_H + 72 = 7x + 72 \]

3. The total number of males using coupons for Pedicure and Hair cutting together is 174 more than the total number of females using their coupons for Pedicure and Hair cutting together:
\[ M_P + M_H = F_P + F_H + 174 \]

### Step 3: Total Coupons Equation
We know that the total number of coupons is 450:
\[ M_H + F_H + M_P + F_P + M_S + F_S = 450 \]

### Step 4: Substitute and Simplify
Substituting the values we have:
1. From the equation \( M_P + M_H = F_P + F_H + 174 \):
\[ (7x + 72) + 13x = F_P + 7x + 174 \]
Simplifying gives:
\[ 20x + 72 = F_P + 7x + 174 \]
Rearranging gives:
\[ 20x - 7x - F_P = 174 - 72 \]
\[ 13x - F_P = 102 \quad \text{(Equation 1)} \]

2. From the total coupons equation:
\[ 13x + 7x + (7x + 72) + F_P + M_S + F_S = 450 \]
Simplifying gives:
\[ 27x + 72 + F_P + M_S + F_S = 450 \]
Rearranging gives:
\[ 27x + F_P + M_S + F_S = 378 \quad \text{(Equation 2)} \]

### Step 5: Solve the Equations
Now, we can solve Equations 1 and 2 together:
1. From Equation 1:
\[ F_P = 13x - 102 \]

2. Substitute \( F_P \) into Equation 2:
\[ 27x + (13x - 102) + M_S + F_S = 378 \]
Simplifying gives:
\[ 40x - 102 + M_S + F_S = 378 \]
Rearranging gives:
\[ 40x + M_S + F_S = 480 \quad \text{(Equation 3)} \]

### Step 6: Find Values of Males and Females in Spa
We know the ratios for Spa:
- The ratio of males who use their coupon in Pedicure to those who use it in Spa is \( 4:5 \):
\[ M_P : M_S = 4 : 5 \implies M_S = \frac{5}{4}M_P \]

- The ratio of females who use their coupon in Hair cutting to those who use it in Spa is \( 6:11 \):
\[ F_H : F_S = 6 : 11 \implies F_S = \frac{11}{6}F_H \]

### Step 7: Calculate Total Users in Spa
Using the values of \( M_S \) and \( F_S \) from the ratios:
1. Calculate \( M_P \):
\[ M_P = 7x + 72 \]

2. Substitute \( M_P \) into the ratio for \( M_S \):
\[ M_S = \frac{5}{4}(7x + 72) \]

3. Calculate \( F_S \):
\[ F_S = \frac{11}{6}(7x) \]

4. Total number of people who use their coupons in Spa:
\[ Total_S = M_S + F_S \]

### Final Calculation
Now, you can substitute the value of \( x \) (which can be determined from previous equations) to find the exact numbers for \( M_S \) and \( F_S \), and then sum them up to find the total number of people using coupons in Spa.