Julia had 150 fewer bonus points than Kelly.Bothgirls used tthe same number of bonus points. Julia was left with 1 out of 2 of her bonus pointand Kelly was left with 3 out of 4 of her bonus points.how many bonus did Kelly have at first
Kelly's original number of bonus point --- x
Julia's original number of bonus points -- x-150
let the number used by both be y each.
now Kelly has x - y
Julia has x-150 - y
for Kelly:
(x-y)/x = 3/4
4x - 4y = 3x
x - 4y = 0 or x = 4y
for Julia:
(x-150-y)/(x-150) = 1/2
2x - 300 - 2y = x- 150
x - 2y = 150
but x = 4y
4y - 2y = 150
y = 75
then x = 4(75) = 300
so originally,
Kelly had 300 and Julia had 150
check:
so they each used up 75
leaving Kelly with 225 and Julia with 75
Does Julia have 1/2 or her original ? YES
Does Kelly have 3/4 of her original ? YES
To find out how many bonus points Kelly had at first, we need to set up an equation based on the given information.
Let's assume that Julia had x bonus points initially.
We know that Julia had 150 fewer bonus points than Kelly, so we can express Kelly's bonus points as x + 150.
Both Julia and Kelly used the same number of bonus points, and Julia was left with 1 out of 2 of her bonus points, which means she used half of her bonus points. So, Julia used x/2 bonus points.
Kelly was left with 3 out of 4 of her bonus points, which means she used only 1 - 3/4 = 1/4 of her bonus points. So, Kelly used (x + 150)/4 bonus points.
Since Julia and Kelly used the same number of bonus points, we can set up an equation: x/2 = (x + 150)/4.
To solve this equation, we can cross-multiply:
4x = 2(x + 150)
4x = 2x + 300
Subtract 2x from both sides:
2x = 300
Divide both sides by 2:
x = 150
Therefore, Julia initially had 150 bonus points.
Kelly initially had x + 150 = 150 + 150 = 300 bonus points.
So, Kelly had 300 bonus points at first.