The average height of a certain number of persons in a group is 155.5 cm. Later on 4 persons of height 154.6 cm, 158.4 cm, 152.2 cm and 153.8 cm leave the group. Asa result the average height of the remaining persons increases by 0.15 cm. What was the number of persons initially in the group?

Options

A. 18
B. 20
C. 22
D. 24

Answer

To solve the problem step by step, let's break it down:

### Step 1: Define the Variables
Let the total number of persons initially in the group be \( x \).

### Step 2: Calculate the Initial Total Height
The average height of the group is given as 155.5 cm. Therefore, the total height of all persons in the group can be calculated as:
\[ \text{Total Height} = \text{Average Height} \times \text{Number of Persons} = 155.5 \times x \]

### Step 3: Calculate the Total Height of the Persons Leaving
The heights of the 4 persons leaving the group are:
- 154.6 cm
- 158.4 cm
- 152.2 cm
- 153.8 cm

Calculating the sum of these heights:
\[ \text{Total Height of Leaving Persons} = 154.6 + 158.4 + 152.2 + 153.8 = 619 \text{ cm} \]

### Step 4: Calculate the New Total Height After 4 Persons Leave
After these 4 persons leave, the total height of the remaining persons becomes:
\[ \text{New Total Height} = 155.5x - 619 \]

### Step 5: Calculate the New Average Height
The number of remaining persons after 4 leave is \( x - 4 \). The new average height is given as:
\[ \text{New Average Height} = 155.5 + 0.15 = 155.65 \text{ cm} \]

### Step 6: Set Up the Equation for the New Average
Using the average formula, we can set up the equation:
\[ \text{New Average} = \frac{\text{New Total Height}}{\text{Number of Remaining Persons}} \]
Substituting the values we have:
\[ 155.65 = \frac{155.5x - 619}{x - 4} \]

### Step 7: Cross-Multiply to Eliminate the Fraction
Cross-multiplying gives:
\[ 155.65(x - 4) = 155.5x - 619 \]

### Step 8: Expand and Rearrange the Equation
Expanding the left side:
\[ 155.65x - 622.6 = 155.5x - 619 \]
Now, rearranging the equation:
\[ 155.65x - 155.5x = 622.6 - 619 \]
This simplifies to:
\[ 0.15x = 3.6 \]

### Step 9: Solve for \( x \)
Now, divide both sides by 0.15:
\[ x = \frac{3.6}{0.15} = 24 \]

### Conclusion
Thus, the initial number of persons in the group was \( \boxed{24} \).

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