Mrs. Carlyle bought a bag of peanuts for her children. When Phillip, Joy, Brent, and Preston came home from school, they each took some peanuts from the bag.
- Phillip took [tex]$\frac{1}{3}$[/tex] of the peanuts from the bag.
- Joy took [tex]$\frac{1}{4}$[/tex] of the remaining peanuts.
- Brent took [tex]$\frac{1}{2}$[/tex] of the remaining peanuts.
Preston took 10 peanuts.
There were 71 peanuts remaining in the bag.
1. How many peanuts were originally in the bag?
2. How many peanuts did each child take?
Answer (2)
Let x be the original number of peanuts. Phillip took 3 x , Joy took 4 1 × 3 2 x = 6 x , and Brent took 2 1 × 2 x = 4 x .
After Preston took 10 peanuts, 71 remained, so 4 x − 10 = 71 .
Solving for x , we get x = 324 .
Phillip took 108, Joy took 54, Brent took 81, and Preston took 10 peanuts. The original number of peanuts was 324 .
Explanation
Problem Analysis Let's analyze the problem. We know that Phillip, Joy, Brent, and Preston took peanuts from a bag. We are given the fractions of peanuts taken by Phillip, Joy, and Brent, the number of peanuts Preston took, and the number of peanuts remaining. We need to find the original number of peanuts and how many each child took.
Phillip's Peanuts Let x be the original number of peanuts in the bag.
Phillip took 3 1 of the peanuts, so he took 3 x peanuts. The remaining peanuts are x − 3 x = 3 2 x .
Joy's Peanuts Joy took 4 1 of the remaining peanuts, so she took 4 1 × 3 2 x = 6 1 x peanuts. The remaining peanuts are 3 2 x − 6 1 x = 6 4 x − 6 1 x = 6 3 x = 2 1 x .
Brent's Peanuts Brent took 2 1 of the remaining peanuts, so he took 2 1 × 2 1 x = 4 1 x peanuts. The remaining peanuts are 2 1 x − 4 1 x = 4 1 x .
Preston's Peanuts Preston took 10 peanuts, so the remaining peanuts after Preston took his share are 4 1 x − 10 . We know that 71 peanuts remained, so we have the equation 4 1 x − 10 = 71 .
Finding the Original Number of Peanuts Now, we solve for x :
4 1 x − 10 = 71 4 1 x = 71 + 10 4 1 x = 81 x = 81 × 4 x = 324 So, there were originally 324 peanuts in the bag.
Calculating Each Child's Share Now we calculate how many peanuts each child took:
Phillip took 3 1 x = 3 1 ( 324 ) = 108 peanuts. Joy took 6 1 x = 6 1 ( 324 ) = 54 peanuts. Brent took 4 1 x = 4 1 ( 324 ) = 81 peanuts. Preston took 10 peanuts.
Final Answer Therefore, the original number of peanuts in the bag was 324. Phillip took 108 peanuts, Joy took 54 peanuts, Brent took 81 peanuts, and Preston took 10 peanuts.
Examples
Imagine you're dividing a pizza among friends. One person takes a third, another takes a quarter of what's left, and so on. This problem helps you understand how fractions work when you're sharing something, like pizza, cookies, or even deciding how to split chores at home. By understanding fractions, you can make sure everyone gets their fair share!
Mrs. Carlyle originally had 324 peanuts in the bag. Each child took the following amounts: Phillip took 108 peanuts, Joy took 54 peanuts, Brent took 81 peanuts, and Preston took 10 peanuts. After all took their share, 71 peanuts remained in the bag.
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