at a playground, the ratio of boys to girls was 5:6. if 2 over 5 of the boys and 1 over 2 of the girls left, there were 54 children in the playground. how many boys were at the playground at first?
let the number of boys be 5x
let the number of girls be 6x
(notice 5x :6x = 5:6 )
after departures:
number of boys = (3/5)(5x) or 3x
numer of girls = (1/2)(6x) or 3x
then 3x + 3x = 54
x = 9
So at first there were 45 boys and 54 girls
check:
(2/5) of 45 or 18 boys left, leaving 27 boys
(1/2) of 54 girls left, leaving 27 girls
and 27+27 = 54
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Let's start by setting up the equations based on the given information.
Let's say the initial number of boys is 5x and the initial number of girls is 6x, where x represents the common ratio.
After some children left, there were 2/5 * (5x) boys remaining and 1/2 * (6x) girls remaining.
The total number of children remaining is (2/5 * 5x) + (1/2 * 6x) = 54.
Now let's solve this equation to find the value of x.
(2/5 * 5x) + (1/2 * 6x) = 54
(2/5 * 5x) + (3/5 * 5x) = 54
(10x + 15x) / 5 = 54
25x / 5 = 54
25x = 54 * 5
25x = 270
x = 270 / 25
x = 10.8
Since we can't have a partial number of children, we round x down to the nearest whole number, which is 10.
So, the initial number of boys is 5x = 5 * 10 = 50.
Therefore, there were 50 boys at the playground at first.
To solve this problem, we will set up an algebraic equation based on the given information.
Let's assume that the number of boys at the playground initially is represented by "5x" and the number of girls initially is represented by "6x."
After some boys and girls leave, the new ratio of boys to girls at the playground becomes (5/5 - 2/5):(6/6 - 1/2). Simplifying this ratio, we get 3/5:7/12.
Since the number of boys after some leave is (3/5) * 5x = 3x and the number of girls after some leave is (7/12) * 6x = 7x/2, we can set up the following equation:
3x + 7x/2 = 54
To solve for x, we can multiply both sides of the equation by 2 to get rid of the fraction:
6x + 7x = 108
13x = 108
x ≈ 108/13 ≈ 8.31
Since we cannot have a fraction of a child, we will round x down to the nearest whole number, giving us x = 8.
Therefore, initially, the number of boys at the playground was 5x = 5 * 8 = 40.