the ratio of boys to girls in the gym is 7:5. when 6 more girls enter the gym, the ratio of boys to girls becomes even. how many boys and girls now in the gym?
Don't know what you mean by an "even" ratio.
If the two terms are even, they we divide by at least 2, until one of them or both are odd.
A ratio with both terms even is not in lowest terms,
just like a fraction can be reduced if both numerator or denominator are even.
original:
boys -- 7x
girls --- 5x
6 more girls ---
boys = 7x
girls = 5x+6
new ratio = 7x : (5x+6)
let x=2, ratio is 14:16 which is "even", but becomes 7 : 8
let x=3, ratio is 21 : 21 = 1 : 1
let x=4, ratio is 28 : 26 = 14 : 13
Perhaps you consider the last one an "even" ratio
so there were 28 boys and 20 girls
and now we have 28 boys and 26 girls
21 boys, 21 girls.
7:5 - 6 girsl is 21 boys to 15 girls [think -6]
We are given that the ratio of boys to girls in the gym is 7:5.
Let's assume the initial number of boys in the gym is 7x and the initial number of girls is 5x.
After 6 more girls enter the gym, the new number of girls becomes 5x + 6.
According to the given information, the new ratio of boys to girls becomes even. This means the new number of boys must be the same as the new number of girls.
So, we can set up the equation:
7x = 5x + 6
Now, let's solve this equation to find the value of x:
7x - 5x = 6
2x = 6
x = 6/2
x = 3
Now we can calculate the number of boys and girls in the gym:
The initial number of boys = 7x = 7 * 3 = 21
The initial number of girls = 5x = 5 * 3 = 15
After 6 more girls enter the gym, the new number of girls = 15 + 6 = 21, and the new number of boys = 21 as well.
Therefore, currently, there are 21 boys and 21 girls in the gym.
To solve this problem, let's assume the current number of boys and girls in the gym is represented by the ratios 7x and 5x, respectively.
Given that the ratio of boys to girls is 7:5, we can write the equation: (boys/girls) = 7/5. This can also be written as 7x/5x = 7/5.
Now let's consider the second part of the problem. When 6 more girls enter the gym, the new ratio of boys to girls becomes even. This means that the number of boys and girls must be the same.
To express this in terms of the new number of girls, we adjust the ratio: (boys / (girls + 6)) = 1/1. This can also be written as 7x/(5x + 6) = 1.
Now we have two equations:
1) 7x/5x = 7/5
2) 7x/(5x + 6) = 1
Let's solve these equations simultaneously to find the values of x, which will give us the number of boys and girls in the gym.
From equation (1), we get:
7x/5x = 7/5
Cross-multiplying, we have:
7x * 5 = 7 * 5x
35x = 35x
Therefore, equation (1) simplifies to 35x = 35x.
Now let's solve equation (2):
7x/(5x + 6) = 1
Cross-multiplying, we have:
7x * 1 = 1 * (5x + 6)
7x = 5x + 6
Subtracting 5x from both sides, we get:
7x - 5x = 5x + 6 - 5x
2x = 6
Dividing both sides by 2, we get:
x = 3
Now let's find the number of boys and girls in the gym:
Number of boys = 7x = 7 * 3 = 21
Number of girls = 5x = 5 * 3 = 15
Therefore, there are now 21 boys and 15 girls in the gym.