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.
Females who use their coupon in Hair cutting are how much per cent more than Females who use their coupon in Pedicure?

Options

A. 15
B. 45
C. 30
D. none of these

Answer

To solve the problem step by step, we can break it down as follows:

### 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

### Step 2: Set Up the Ratios
According to the problem, the ratio of males to females using coupons for Hair cutting is 13:7. Therefore, we can express the number of males and females in Hair cutting as:
- \( M_H = 13x \)
- \( F_H = 7x \)

### Step 3: Total Coupons
The total number of coupons used for both Pedicure and Hair cutting is given as 450:
\[ M_H + F_H + M_P + F_P = 450 \]

### Step 4: Express Males in Pedicure
The problem states that the number of males who use their coupons in Pedicure is 72 more than the number of females who use their coupons in Hair cutting:
\[ M_P = F_H + 72 \]

### Step 5: Total Males and Females
The total number of males using coupons for Pedicure and Hair cutting is 174 more than the total number of females:
\[ (M_H + M_P) = (F_H + F_P) + 174 \]

### Step 6: Substitute Known Values
Substituting the expressions we have into the equations:
1. From the total coupons equation:
\[ 13x + 7x + (F_H + 72) + F_P = 450 \]
Simplifying gives:
\[ 20x + F_H + F_P + 72 = 450 \]
\[ F_H + F_P = 450 - 20x - 72 \]
\[ F_H + F_P = 378 - 20x \quad \text{(Equation 1)} \]

2. From the total males and females equation:
\[ (13x + (F_H + 72)) = (7x + F_P) + 174 \]
Simplifying gives:
\[ 13x + F_H + 72 = 7x + F_P + 174 \]
\[ 6x + F_H - F_P = 102 \quad \text{(Equation 2)} \]

### Step 7: Solve the Equations
Now we have two equations:
1. \( F_H + F_P = 378 - 20x \)
2. \( 6x + F_H - F_P = 102 \)

From Equation 1, we can express \( F_P \) in terms of \( F_H \):
\[ F_P = 378 - 20x - F_H \]

Substituting this into Equation 2:
\[ 6x + F_H - (378 - 20x - F_H) = 102 \]
\[ 6x + F_H - 378 + 20x + F_H = 102 \]
\[ 26x + 2F_H = 480 \]
\[ 13x + F_H = 240 \quad \text{(Equation 3)} \]

### Step 8: Find Values of x and F_H
Substituting \( F_H \) from Equation 3 into Equation 1:
\[ F_H + (378 - 20x) = 378 - 20x \]
This allows us to solve for \( x \):
\[ F_H = 240 - 13x \]

### Step 9: Calculate Females in Pedicure and Hair Cutting
Substituting \( x = 12 \) (from solving the equations):
- \( F_H = 7x = 84 \)
- \( F_P = 378 - 20(12) = 378 - 240 = 138 \)

### Step 10: Find Percentage Increase
Now we need to find how much percentage more females who use their coupon in Hair cutting are compared to those who use it in Pedicure:
\[ \text{Difference} = F_H - F_P = 84 - 54 = 30 \]
\[ \text{Percentage Increase} = \left( \frac{30}{54} \right) \times 100 = 55.56\% \]

### Final Answer
The percentage of females who use their coupon in Hair cutting is approximately **55.56%** more than those who use it in Pedicure.