Boys and girls in a class are writing letters. There are twice as many girls as boys in the class, and each girl writes 3 more letters than each boy. If boys write 24 of the 90 total letters written by the class, how many letters does each boy write?

Options

A. 3
B. 4
C. 5
D. 8

Answer

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

### Step 1: Define Variables
Let:
- \( b \) = number of boys in the class
- \( g \) = number of girls in the class
- \( x \) = number of letters written by each boy

### Step 2: Establish Relationships
According to the problem:
- There are twice as many girls as boys:
\[ g = 2b \]
- Each girl writes 3 more letters than each boy:
\[ \text{Letters written by each girl} = x + 3 \]

### Step 3: Total Letters Written
We know the total number of letters written by boys and girls:
- Boys write 24 letters, so:
\[ bx = 24 \]
- The total number of letters written by girls is:
\[ \text{Total letters by girls} = g \times (\text{letters per girl}) = g \times (x + 3) \]
- The total number of letters written by the class is 90:
\[ \text{Total letters} = \text{Letters by boys} + \text{Letters by girls} = 24 + g(x + 3) = 90 \]

### Step 4: Substitute for Girls
Substituting \( g = 2b \) into the total letters equation:
\[ 24 + 2b(x + 3) = 90 \]
This simplifies to:
\[ 2b(x + 3) = 90 - 24 \]
\[ 2b(x + 3) = 66 \]

### Step 5: Solve for \( bx \)
From \( bx = 24 \), we can express \( b \) in terms of \( x \):
\[ b = \frac{24}{x} \]

### Step 6: Substitute \( b \) into the Equation
Now substitute \( b \) into the equation \( 2b(x + 3) = 66 \):
\[ 2\left(\frac{24}{x}\right)(x + 3) = 66 \]
This simplifies to:
\[ \frac{48(x + 3)}{x} = 66 \]

### Step 7: Clear the Fraction
Multiply both sides by \( x \):
\[ 48(x + 3) = 66x \]
Expanding gives:
\[ 48x + 144 = 66x \]

### Step 8: Rearrange the Equation
Rearranging the equation:
\[ 144 = 66x - 48x \]
\[ 144 = 18x \]

### Step 9: Solve for \( x \)
Dividing both sides by 18:
\[ x = \frac{144}{18} = 8 \]

### Conclusion
Thus, each boy writes **8 letters**.

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