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.
Find the ratio between the total number of persons who use their coupons in Pedicure to the total number of persons who use their coupons in Hair cutting?

Options

A. (52:23)
B. none of these
C. (8:9)
D. (7:8)

Answer

To solve the problem step by step, we will define the variables and use the information given in the question.

### 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
From the problem, we know that the ratio of males to females using coupons for Hair cutting is 13:7. Therefore, we can express this as:
\[ M_H = \frac{13}{20} \times H \quad \text{and} \quad F_H = \frac{7}{20} \times H \]
where \( H \) is the total number of people using coupons for Hair cutting.

### Step 3: Total Coupons
The total number of coupons is 450, so:
\[ M_H + F_H + M_P + F_P = 450 \]

### Step 4: Express Males in Pedicure
According to the problem, 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 who use their coupons in Pedicure and Hair cutting together is 174 more than the total number of females who use their coupons in Pedicure and Hair cutting together:
\[ M_H + M_P = F_H + F_P + 174 \]

### Step 6: Substitute Known Values
Substituting \( M_P \) from Step 4 into the equation from Step 5:
\[ M_H + (F_H + 72) = F_H + F_P + 174 \]
This simplifies to:
\[ M_H + 72 = F_P + 174 \]
Thus:
\[ M_H - F_P = 102 \quad \text{(Equation 1)} \]

### Step 7: Total Coupons Equation
Now substituting \( M_P \) and \( F_H \) into the total coupons equation:
\[ M_H + F_H + (F_H + 72) + F_P = 450 \]
This simplifies to:
\[ M_H + 2F_H + 72 + F_P = 450 \]
Thus:
\[ M_H + 2F_H + F_P = 378 \quad \text{(Equation 2)} \]

### Step 8: Solve the Equations
Now we have two equations:
1. \( M_H - F_P = 102 \)
2. \( M_H + 2F_H + F_P = 378 \)

From Equation 1, we can express \( F_P \):
\[ F_P = M_H - 102 \]

Substituting \( F_P \) into Equation 2:
\[ M_H + 2F_H + (M_H - 102) = 378 \]
This simplifies to:
\[ 2M_H + 2F_H - 102 = 378 \]
Thus:
\[ 2M_H + 2F_H = 480 \]
Dividing by 2:
\[ M_H + F_H = 240 \quad \text{(Equation 3)} \]

### Step 9: Substitute Back
Now we can find \( M_H \) and \( F_H \) using the ratio:
Let \( M_H = 13x \) and \( F_H = 7x \):
\[ 13x + 7x = 240 \implies 20x = 240 \implies x = 12 \]
Thus:
\[ M_H = 156 \quad \text{and} \quad F_H = 84 \]

### Step 10: Find \( M_P \) and \( F_P \)
Using \( F_H \) to find \( M_P \):
\[ M_P = F_H + 72 = 84 + 72 = 156 \]
Using \( M_H \) to find \( F_P \):
\[ F_P = M_H - 102 = 156 - 102 = 54 \]

### Step 11: Calculate Totals
Total persons using coupons for Pedicure:
\[ M_P + F_P = 156 + 54 = 210 \]
Total persons using coupons for Hair cutting:
\[ M_H + F_H = 156 + 84 = 240 \]

### Step 12: Find the Ratio
The ratio of total persons using coupons for Pedicure to Hair cutting is:
\[ \frac{210}{240} = \frac{7}{8} \]

### Final Answer
The ratio between the total number of persons who use their coupons in Pedicure to the total number of persons who use their coupons in Hair cutting is \( 7:8 \).