In a class of 88 students, the number of girls is 20% more than the number of boys. The average height of the boys is 164 cm and the average height of the girls is 4 cm less than the average height of all the boys and girls in the class. What is the average height (in cm) of all the students in the class ?

Options

A. 159.2
B. 155.2
C. 150
D. 163.2

Answer

To solve the problem step by step, we will first define the variables and then use the information given to find the average height of all students in the class.

### Step 1: Define Variables
Let the number of boys in the class be \( x \). Since the number of girls is 20% more than the number of boys, we can express the number of girls as:
\[ \text{Number of girls} = x + 0.2x = 1.2x \]

### Step 2: Set Up the Equation for Total Students
The total number of students in the class is 88, so we can set up the equation:
\[ x + 1.2x = 88 \]
Combining the terms gives:
\[ 2.2x = 88 \]

### Step 3: Solve for the Number of Boys
Now, we can solve for \( x \):
\[ x = \frac{88}{2.2} = 40 \]
Thus, the number of boys is 40.

### Step 4: Calculate the Number of Girls
Using the value of \( x \):
\[ \text{Number of girls} = 1.2 \times 40 = 48 \]

### Step 5: Calculate Average Heights
We know the average height of the boys is 164 cm. The average height of the girls is 4 cm less than the average height of all students.

Let the average height of all students be \( y \). Therefore, the average height of the girls can be expressed as:
\[ \text{Average height of girls} = y - 4 \]

### Step 6: Calculate Total Heights
Now, we can calculate the total height of boys and girls:
- Total height of boys:
\[ \text{Total height of boys} = \text{Number of boys} \times \text{Average height of boys} = 40 \times 164 = 6560 \text{ cm} \]
- Total height of girls:
\[ \text{Total height of girls} = \text{Number of girls} \times \text{Average height of girls} = 48 \times (y - 4) \]

### Step 7: Set Up the Equation for Total Height
The total height of all students is the sum of the total heights of boys and girls:
\[ \text{Total height of all students} = 6560 + 48(y - 4) \]
Since the total number of students is 88, we can express the total height in terms of the average height:
\[ \text{Total height of all students} = 88y \]

### Step 8: Set the Equations Equal
Now we can set the two expressions for total height equal to each other:
\[ 6560 + 48(y - 4) = 88y \]

### Step 9: Simplify and Solve for \( y \)
Expanding the left side:
\[ 6560 + 48y - 192 = 88y \]
Combining like terms:
\[ 6368 + 48y = 88y \]
Rearranging gives:
\[ 6368 = 88y - 48y \]
\[ 6368 = 40y \]
Now, solving for \( y \):
\[ y = \frac{6368}{40} = 159.2 \text{ cm} \]

### Conclusion
The average height of all the students in the class is \( \boxed{159.2} \) cm.