The number of boys, girls and adults at a marathon was in the ratio 5 : 8 : 2. When some girls left the marathon, the new ratio became 10 : 9 : 4. If there were 150 participants at first, how many girls left the marathon?

Step 1: Understand the problem

The problem is about ratios and proportions. We are given the initial ratio of boys, girls, and adults at a marathon, which is 5:8:2. We are also given the new ratio after some girls left, which is 10:9:4. We are asked to find out how many girls left the marathon, given that there were 150 participants at first.

Step 2: Set up the equations

Let's denote the number of boys, girls, and adults at the marathon as B, G, and A respectively. We know that B + G + A = 150.

From the initial ratio, we can write the following equations:

B = 5x G = 8x A = 2x

where x is a common factor.

From the new ratio, we can write the following equations:

B = 10y G - g = 9y A = 4y

where y is a new common factor and g is the number of girls who left.

Step 3: Solve the equations

First, we need to find the value of x. We can do this by substituting the equations for B, G, and A into the equation B + G + A = 150:

5x + 8x + 2x = 150 15x = 150 x = 10

So, initially there were 50 boys, 80 girls, and 20 adults.

Next, we need to find the value of y. We can do this by substitifying the equations for B and A into the equation B + G - g + A = 150:

10y + 9y + 4y = 150 23y = 150 y = 150 / 23

So, after some girls left, there were 50 boys, (9 * 150 / 23) girls, and 40 adults.

Step 4: Find the number of girls who left

The number of girls who left is the difference between the initial number of girls and the new number of girls:

g = G - (9 * 150 / 23) g = 80 - (9 * 150 / 23) g = 80 - (1350 / 23) g = 80 - 58.7 g = 21.3

So, approximately 21 girls left the marathon.