Question

In a small town, there are 4 times as many left-handed males as there are left-handed females, and there are 3 times as many right-handed females as there are right-handed males. There are a total of 204 males and 348 females in the town.

Let x represent the number of left-handed females, and let y represent the number of right-handed males. Write a system of equations to represent the situation. What is the value of x, the number of left-handed females?
6
24
96
108

Answer 1

Let's define the variables based on the problem statement:

- Let \( x \) be the number of left-handed females.
- Let \( y \) be the number of right-handed males.

From the problem, we can gather the following information:

1. **Left-handed males and females**: There are 4 times as many left-handed males as left-handed females. Therefore, the number of left-handed males can be expressed as \( 4x \).

2. **Right-handed males and females**: There are 3 times as many right-handed females as right-handed males. Hence, the number of right-handed females can be given as \( 3y \).

Now, we know the total number of males and females in the town:

- Total males = 204
- Total females = 348

Using the definitions and relationships established, we can write the following equations for males and females:

From the total number of males:
\[
y + 4x = 204 \quad \text{(1)}
\]
(This states that the total number of males is the sum of right-handed males \( y \) and left-handed males \( 4x \).)

From the total number of females:
\[
x + 3y = 348 \quad \text{(2)}
\]
(This states that the total number of females is the sum of left-handed females \( x \) and right-handed females \( 3y \).)

Now, we have the system of equations:

1. \( y + 4x = 204 \)
2. \( x + 3y = 348 \)

**Now we'll solve this system of equations.**

Starting with equation (1):
\[
y = 204 - 4x \quad \text{(3)}
\]

We can substitute equation (3) into equation (2):
\[
x + 3(204 - 4x) = 348
\]
Expanding this gives:
\[
x + 612 - 12x = 348
\]
Combining like terms:
\[
-11x + 612 = 348
\]
Subtracting 612 from both sides:
\[
-11x = 348 - 612
\]
\[
-11x = -264
\]
Dividing both sides by -11:
\[
x = 24
\]

Thus, the number of left-handed females, \( x \), is 24.

To find \( y \), we substitute \( x \) back into equation (3):
\[
y = 204 - 4(24) = 204 - 96 = 108
\]

Finally, let's summarize:
- The number of left-handed females, \( x \), is **24**.
- The number of right-handed males, \( y \), is **108**.

Thus, the answer is:
\[ \boxed{24} \]