1 / 9

Solving the Apple Problem: A Step-by-Step Algebraic Approach

In this problem-solving activity, we analyze a scenario involving three hungry individuals and a bag of apples. As each person awakens, they consume one-third of the remaining apples, until only eight apples are left in the bag. Through algebraic reasoning and backward calculation, we determine the original quantity of apples. This engaging exercise challenges students to write steps in general terms, solve known variables, and check their work through algebra or visual methods. Ultimately, the original number of apples was found to be 27, demonstrating the effectiveness of structured problem-solving strategies.

ronat
Télécharger la présentation

Solving the Apple Problem: A Step-by-Step Algebraic Approach

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Problem-Solving Activity AlyciaScarpelli and Stefanie Del Rosso

  2. Apple Problem • Three tired and hungry people had a bag of apples. While the other two were asleep, one of the three awoke, ate one-third of the apples, and went back to sleep. Later a second person awoke, ate one-third of the remaining apples and went back to sleep. Finally, the third person awoke and ate one-third of the remaining apples, leaving 8 apples in the bag. How many apples were in the bag originally?

  3. Hints • Write each step in general terms • Solve for what is known • Work backwards • Check your answer through algebra or a picture

  4. Solution Algebraically: Solve for X 1) We have three people, one woke up and ate 1/3 of the apples. So, 1- (1/3)= 2/3 left. So, we can write Y= (2/3)X, for X is the total number of apples in the bag and Y is the number of apples left in the bag. 2) Next, the second person woke up and ate 1/3 of what was left in the bag. So, we can write Z = (2/3) Y, for Z is the number of apples left in the bag after the second person eats 1/3 of the previous remainder.

  5. Solution (continued) 3) Then, the third person wakes up and eats 1/3 of the remaining apples, leaving 8 apples in the bag. So, we have (2/3)Z - 8 = 0. • Solving for Z: (2/3)Z – 8 = 0, we get (2/3)Z = 8, where Z = 12. • Working backwards, we can plug Z = 12 into (2/3)Y = Z. So, we get (2/3)Y = 12. Solving for Y, we get Y = 18. • Continuing to work backwards, we can plug Y = 18 into (2/3)X = Y. Solving for X, we get X = 27. Thus, the original number of apples in the bag was 27.

  6. Solution: Summed up Y= (2/3)X, Z = (2/3) Y (2/3)Z - 8 = 0 Now Solve For Z: (2/3)Z = 8 Z = 12 Now work backwards by plugging Z into the equation Z= (2/3) Y, and solve for Y 12 =(2/3) Y Y= 18 Continue to work backwards by plugging Y into the equation Y=(2/3)X, and solve for X 18=(2/3) X X= 27 Therefore there are 27 apples originally in the bag.

  7. Algebraic Check • We found the original number of apples in the bag to be 27. To check: (2/3)X = Y (2/3)(27) = 18 = Y (2/3)Y = Z (2/3)(18) = 12 = Z (2/3)Z – 8 = 0 (2/3) (12) – 8 = 0 0 = 0

  8. Check 27/3 = 9, the 1st person ate 9 27- 9= 18, after the 1st person ate 9, there are 18 apples left in the bag. 18/3= 6, the 2nd person ate 6 18-6= 12, after the 2nd person ate 6, there are 12 apples left in the bag. 12/3=4, the 3rd person ate 4 12-4= 8, after the 3rd person ate 4, there are 8 apples left in the bag.

  9. Visual Check There are 8 apples left in the bag

More Related