Alex came home after school one day and found that his mother had left a plate of cookies. Alex ate 1/4 of the cookies. When his sister Bernice came home, she ate 1/3
of the remaining cookies, and when Carla came home, she ate 1/2
of the cookies on the plate. When their mother came home, there were 3 cookies on the plate. How many did each child eat?
Let's break down the information given step-by-step:
Step 1: Alex ate 1/4 of the cookies.
Step 2: After Alex ate his share, Bernice came home and ate 1/3 of the remaining cookies.
Step 3: After Bernice ate her share, Carla came home and ate 1/2 of the remaining cookies.
Step 4: When their mother came home, there were 3 cookies left on the plate.
Step 1: Alex ate 1/4 of the cookies.
Let's assume there were X cookies on the plate initially.
So, Alex ate (1/4)X cookies.
Step 2: After Alex ate his share, Bernice came home and ate 1/3 of the remaining cookies.
After Alex ate (1/4)X cookies, there were (3/4)X cookies left on the plate.
So, Bernice ate (1/3) * (3/4)X = (1/4)X cookies.
Step 3: After Bernice ate her share, Carla came home and ate 1/2 of the remaining cookies.
After Bernice ate (1/4)X cookies, there were (3/4)X - (1/4)X = (1/2)X cookies left on the plate.
So, Carla ate (1/2) * (1/2)X = (1/4)X cookies.
Step 4: When their mother came home, there were 3 cookies left on the plate.
After Carla ate (1/4)X cookies, there were (1/4)X - 3 cookies left on the plate.
Given that (1/4)X - 3 = 3, we can solve for X:
(1/4)X - 3 = 3
(1/4)X = 6
X = 6 * 4
X = 24
So, initially, there were 24 cookies on the plate.
Now, let's find out how many cookies each child ate:
Alex ate (1/4) * 24 = 6 cookies.
Bernice ate (1/4) * 6 = 1.5 cookies. Since we cannot have half cookies, we'll assume she ate only 1 cookie.
Carla ate (1/4) * 1 = 0.25 cookies, which is less than 1 cookie. So, we'll assume she didn't eat any cookies.
Therefore, Alex ate 6 cookies, Bernice ate 1 cookie, and Carla didn't eat any cookies.
To find out how many cookies each child ate, we need to work through the problem step by step.
1. Alex ate 1/4 of the cookies. Let's represent the number of cookies on the plate initially as "x." From the information given, we know that Alex ate 1/4 of x, which can be written as (1/4)x. After Alex ate, there were 3/4 of the cookies remaining, or (3/4)x.
2. Bernice then ate 1/3 of the remaining cookies. So, we take the (3/4)x from the previous step and calculate 1/3 of that amount. This can be written as (1/3)((3/4)x), which simplifies to (1/4)x.
3. After Bernice ate, there were 1/2 of the cookies remaining, or (1/2)((3/4)x).
Now, we have reached the final step where Carla eats, and there were 3 cookies left on the plate. So, we can set up an equation:
(1/2)((3/4)x) = 3
To solve for x, we multiply both sides of the equation by 2/3:
((3/4)x) = 3 * (2/3)
This simplifies to:
((3/4)x) = 2
To find the value of "x," we divide both sides of the equation by 3/4:
x = (2) / (3/4)
This can be simplified as:
x = (2) * (4/3)
x = 8/3
Finally, we have found that the initial number of cookies, represented by "x," is equal to 8/3 (or 2 and 2/3). Plugging this value into the previous steps, we can calculate the number of cookies each child ate.
1. Alex ate (1/4)(8/3) = 2/3 of a cookie.
2. Bernice ate (1/3)((3/4)(8/3)) = 2/3 of a cookie.
3. Carla ate (1/2)((1/2)(8/3)) = 1 1/3 of a cookie.
So, Alex ate 2/3 of a cookie, Bernice ate 2/3 of a cookie, and Carla ate 1 1/3 of a cookie.
Let x = # of cookies originally on the plate.
x - 1/4x - 1/3(1/4x) - 1/2(1/3)(1/4x) = 3
Solve for x, then solve for each one of the three subtracted terms.