there were 490 altogether in 2 groups. group A consited of only boys and group b constied of only girls. ther were 2 and a half times as many girls as boys. some girls joined group b and foe every 4 boys in group a 32 more boys jioned the group. the total number og girls was then one third the total number of boys. express the number of girls who jioned group b as a fraction of boys who jioned group a

Steve, you incorrect, it should be b=5/2a,

because there are more girls than boys in the group.

sorry about the mistyped capitalization. The lines in error should read

a = 1/8 A

and

b = 1/2 * 1/8 a = a/16

and there are 108 boys at the end, not 1-8

To solve this problem, let's break it down into steps:

Step 1: Determine the number of boys and girls in each group.
Let's assume the number of boys in group A is 'x'. Therefore, the number of girls in group B would be 2.5 times 'x'. So, the total number of boys in group A is 'x', and the total number of girls in group B is 2.5 times 'x'.

Step 2: Calculate the initial total number of boys and girls.
According to the problem, there were 490 students in total. Therefore, the equation would be x + 2.5x = 490.

Step 3: Solve the equation and find the value of 'x'.
Combine similar terms in the equation: 3.5x = 490.
Divide both sides of the equation by 3.5: x = 140.

Step 4: Calculate the number of boys in group A and girls in group B.
Since we now know the value of 'x', we can substitute it back into the equations to find the actual numbers.
The number of boys in group A is 140, and the number of girls in group B is 2.5 * 140 = 350.

Step 5: Calculate the new total number of boys and girls after some changes.
The problem states that some girls joined group B, and for every 4 boys in group A, 32 more boys joined. Also, after these changes, the total number of girls became one-third the total number of boys.
Let's assume 'y' boys joined group B. Therefore, the total number of girls after the change would be 350 + y, and the total number of boys would be 140 + 32(y/4).

Step 6: Set up the equation and solve for 'y'.
The problem states that after the changes, the total number of girls is one-third the total number of boys. Therefore, we can set up the equation (350 + y) = (1/3)(140 + 32(y/4)).
Simplify the equation: 350 + y = (140 + 8(y/4))/3.
Multiply both sides of the equation by 3 to eliminate the fraction: 1050 + 3y = 140 + 8(y/4).
Simplify further: 1050 + 3y = 140 + 2y.
Subtract 2y from both sides: y = -910.

Step 7: Interpret the result.
Based on the calculation, it seems that the number of boys who joined Group B is negative (-910). However, the negative value indicates that there were 910 boys who left Group A to join Group B. In essence, the fraction of boys who joined Group A compared to those who joined Group B is 910/32, which can be further simplified if needed.

to begin, A = 5/2 B

a boys joined A
32/4 = 8, so 8A boys joined the group.so, A = 1/8 A

b girls joined B
So, now there are

B+b = 1/3(9A) = 3A
5/2 A + b = 3A
b = 1/2 A
b = 1/2 * 1/8 A = a/16

example:
A=12
B=30

A/4 = 3, so 3*32 = 96 boys joined A, making 1-8 boys in all

B+b = 108/3 = 36
so, 6 girls joined, or 96/16