In a group of 78 persons, the ratio of the number of males and females is 7 : 6 . The average height of all the persons is 160.2 cm . If the average height of males is `5%` more than that of females , then what is the average height (in cm) of females in the group ?

Options

A. `157.6`
B. 158
C. `156.4`
D. 156

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript:

### Step 1: Determine the number of males and females
Given the ratio of males to females is 7:6, we can express the number of males (M) and females (F) in terms of a common variable \( k \):
- Males = \( 7k \)
- Females = \( 6k \)

The total number of persons is given as 78:
\[ 7k + 6k = 78 \]
\[ 13k = 78 \]
\[ k = \frac{78}{13} = 6 \]

Now we can find the number of males and females:
- Males = \( 7k = 7 \times 6 = 42 \)
- Females = \( 6k = 6 \times 6 = 36 \)

### Step 2: Set up the average heights
Let the average height of females be \( x \) cm. Since the average height of males is 5% more than that of females, we can express the average height of males as:
\[ \text{Average height of males} = x + 0.05x = 1.05x \]

### Step 3: Calculate the total height of all persons
The average height of all persons is given as 160.2 cm. Therefore, the total height of all persons can be calculated as:
\[ \text{Total height} = \text{Average height} \times \text{Total number of persons} = 160.2 \times 78 \]
Calculating this gives:
\[ \text{Total height} = 12495.6 \text{ cm} \]

### Step 4: Set up the equation for total height
The total height can also be expressed as the sum of the total heights of males and females:
\[ \text{Total height} = (\text{Number of males} \times \text{Average height of males}) + (\text{Number of females} \times \text{Average height of females}) \]
Substituting the known values:
\[ 12495.6 = (42 \times 1.05x) + (36 \times x) \]

### Step 5: Simplify the equation
Now, we simplify the equation:
\[ 12495.6 = 44.1x + 36x \]
\[ 12495.6 = 80.1x \]

### Step 6: Solve for \( x \)
To find \( x \), divide both sides by 80.1:
\[ x = \frac{12495.6}{80.1} \]
Calculating this gives:
\[ x = 156 \text{ cm} \]

### Conclusion
The average height of females in the group is **156 cm**.

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