In a school, the ratio of the number of boys to that of girls was 6:5. In the next year, the number of boys increased by 20%, and the number of girls increased by 10%, resulting in an increase of 119 students in the school. How many students were there in the school initially?

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Accepted Answer

#### Step 1: Represent Boys and Girls
Let the initial number of boys be $$ 6x $$ and the initial number of girls be $$ 5x $$, based on the given ratio of **6:5**.

The total number of students initially is:  
$$
	ext{Total Students} = 6x + 5x = 11x.
$$

#### Step 2: Calculate Increases
- Boys increased by **20%**, so the increase in boys is:  
  $$ 	ext{Increase in Boys} = 20\% 	imes 6x = \frac{20}{100} 	imes 6x = 1.2x. $$

- Girls increased by **10%**, so the increase in girls is:  
  $$ 	ext{Increase in Girls} = 10\% 	imes 5x = \frac{10}{100} 	imes 5x = 0.5x. $$

The total increase in students is:  
$$
	ext{Total Increase} = 	ext{Increase in Boys} + 	ext{Increase in Girls} = 1.2x + 0.5x = 1.7x.
$$

#### Step 3: Use Total Increase to Find $$ x $$
We are given that the total increase is **119 students**:  
$$
1.7x = 119.
$$

Solve for $$ x $$:  
$$
x = \frac{119}{1.7} = 70.
$$

#### Step 4: Calculate Initial Total Students
The initial total number of students is:  
$$
	ext{Total Students} = 11x = 11 	imes 70 = 770.
$$

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