The ratio of the number of boys to that of girls was `1:2` but when 2 boys and 2 girls left, the ratio became `1:3`. How many people were at the party originally ?

Options

A. 12
B. 15
C. 10
D. 18

Explanation

To solve the problem step by step, let's denote the number of boys as \( x \) and the number of girls as \( y \).

### Step 1: Set up the initial ratio
We know from the problem that the ratio of boys to girls is \( 1:2 \). This can be expressed mathematically as:
\[ \frac{x}{y} = \frac{1}{2} \]
From this, we can express \( y \) in terms of \( x \):
\[ y = 2x \]

**Hint:** Use the ratio to express one variable in terms of the other.

### Step 2: Set up the equation after boys and girls leave
When 2 boys and 2 girls leave, the new number of boys and girls becomes \( x - 2 \) and \( y - 2 \) respectively. The new ratio is given as \( 1:3 \):
\[ \frac{x - 2}{y - 2} = \frac{1}{3} \]
Cross-multiplying gives us:
\[ 3(x - 2) = 1(y - 2) \]
Expanding this equation:
\[ 3x - 6 = y - 2 \]

**Hint:** Use cross-multiplication to eliminate the fraction when dealing with ratios.

### Step 3: Substitute \( y \) in the equation
Now, substitute \( y \) from Step 1 into the equation:
\[ 3x - 6 = 2x - 2 \]

**Hint:** Substitute the expression you found for \( y \) into the new ratio equation.

### Step 4: Solve for \( x \)
Rearranging the equation gives:
\[ 3x - 2x = -2 + 6 \]
\[ x = 4 \]

**Hint:** Isolate the variable to find its value.

### Step 5: Find \( y \)
Now that we have \( x \), we can find \( y \) using the equation \( y = 2x \):
\[ y = 2 \times 4 = 8 \]

**Hint:** Use the relationship you established earlier to find the second variable.

### Step 6: Calculate the total number of people
The total number of people at the party originally is the sum of boys and girls:
\[ \text{Total} = x + y = 4 + 8 = 12 \]

**Hint:** Add the two quantities together to find the total.

### Final Answer
The total number of people at the party originally was **12**.