When 6-year-old Melanie arrived home from school, she was the first to eat cookies from a freshly baked batch. When 8-year-old Felipe arrived home, he ate twice as many cookies as Melanie had eaten. When
9-year-old Hillary arrived home, she ate 3 fewer cook-jes than Felipe. When 12-year-old Nicholas arrived, he ate 3 times as many cookies as Hillary. Nicholas left 2 cookies, one for each of his parents. If Nicholas had eaten only 5 cookies, there would have been 3 cookies for each of his parents. How many cookies were in the original batch?
Let x be the number of cookies Melanie ate.
Felipe ate 2 * x cookies.
Hillary ate 2 * x - 3 cookies.
Nicholas ate 3 * (2 * x - 3) + 5 cookies.
Nicholas left 2 cookies, so he ate x - 2 cookies.
3 * (x - 2) = 5 + 2 * 3
3 * x - 6 = 11
3 * x = 17
x = <<17=17>>17
The original batch contained 2 * 17 - 3 = <<2*17-3=31>>31 cookies. Answer: \boxed{31}.
Let's solve the problem step-by-step:
1. Let's assume that Melanie ate x number of cookies. So, Felipe ate 2x cookies, Hillary ate (2x - 3) cookies, and Nicholas ate 3 * (2x - 3) cookies.
2. We know that Nicholas left 2 cookies, so we can write the following equation:
3 * (2x - 3) - 2 = 5
3. Let's solve the equation:
6x - 9 - 2 = 5
6x - 11 = 5
6x = 5 + 11
6x = 16
x = 16/6
x ≈ 2.67
4. We assumed that Melanie ate x cookies, and since we cannot have a fraction of a cookie, we can conclude that Melanie ate 3 cookies.
5. Now, we can find the number of cookies in the original batch. Since Felipe ate twice as many as Melanie, Felipe ate 2 * 3 = 6 cookies.
6. Hillary ate 3 fewer cookies than Felipe, so she ate 6 - 3 = 3 cookies.
7. Finally, Nicholas ate 3 times as many cookies as Hillary, which is 3 * 3 = 9 cookies.
8. To find the total number of cookies in the original batch, we add up the number of cookies each person ate: 3 + 6 + 3 + 9 = 21.
Therefore, there were 21 cookies in the original batch.
To solve this problem, we can use a step-by-step approach. Let's go through it together:
1. Let's start by assigning variables to the number of cookies each child ate:
- Melanie ate M cookies.
- Felipe ate 2M cookies (twice as many as Melanie).
- Hillary ate (2M - 3) cookies (3 fewer than Felipe).
- Nicholas ate 3(2M - 3) cookies (3 times as many as Hillary).
2. We are given that Nicholas left 2 cookies for his parents. So, we can set up an equation:
Nicholas' total cookies - Nicholas' parents' cookies = 2.
3(2M - 3) - 2 = 2.
Simplifying this equation gives us:
6M - 9 - 2 = 2.
6M - 11 = 2.
3. Now, let's solve the equation to find the value of M (the number of cookies Melanie ate):
6M - 11 = 2.
Adding 11 to both sides:
6M = 13.
Dividing both sides by 6:
M = 13/6.
4. However, M represents the number of cookies Melanie ate, which needs to be a whole number. So, let's check the fractional value of M. In this case, since the result is not a whole number, the given information is not consistent.
Hence, the given scenario does not provide a valid solution.