EXAMPLE 1

The Magical Garden

In a magical garden, there are three types of flowers: roses, tulips, and daisies. Each rose has 5 petals, each tulip has 3 petals, and each daisy has 4 petals. If a magical butterfly lands on a flower, it doubles the number of petals on that flower. If a magical bee lands on a flower, it takes away 2 petals from that flower. One day, a magical butterfly landed on a rose, and two magical bees landed on two different tulips.

Question: How many total petals are there in the garden if there were originally 3 roses, 4 tulips, and 2 daisies?

Polya’s Problem-Solving Steps Applied to “The Magical Garden” Problem

1. Understanding the Problem:

  • We have three types of flowers: roses, tulips, and daisies.
  • Roses have 5 petals, tulips have 3, and daisies have 4.
  • A butterfly doubles the petals of a flower, while a bee reduces the petals by 2.
  • A butterfly landed on a rose, and two bees landed on two different tulips.
  • We need to find the total number of petals after these events, given the original number of each flower.

2. Devising a Plan:

  • First, calculate the total number of petals for each type of flower initially.
  • Account for the changes due to the butterfly and the bees.
  • Sum up the petals of all the flowers to get the total.

3. Carrying Out the Plan:

  • Initial Petal Count:
    • Roses: 3 roses × 5 petals/rose = 15 petals
    • Tulips: 4 tulips × 3 petals/tulip = 12 petals
    • Daisies: 2 daisies × 4 petals/daisy = 8 petals

    Total original petals = 15 + 12 + 8 = 35 petals

  • Changes due to Magical Creatures:
    • Butterfly on a rose: 1 rose × 5 petals/rose × 2 (because of the butterfly) = 10 petals. The increase is 10 – 5 = 5 petals.
    • Bees on two tulips: 2 bees × 2 petals/bee = 4 petals taken away.
  • Total Petals After the Magical Events: Original petals (35) + Additional petals due to butterfly (5) – Petals taken away by bees (4) = 35 + 5 – 4 = 36 petals

4. Look Back/Reflecting on the Solution:

  • We started with 35 petals and after the butterfly and bees did their magic, we ended up with 36 petals.
  • The solution seems logical as the butterfly added more petals to the rose than the bees took away from the tulips.
  • The method used provides a clear step-by-step breakdown of the problem, ensuring accuracy and understanding.

Answer: After the magical events, there are 36 petals in the garden.

EXAMPLE 2

The Mysterious Number Boxes

In the ancient city of Additonium, there’s a legend about three mysterious number boxes. Each box contains a set of unique numbers. When numbers from different boxes are combined, magical things happen!

  1. Box A contains numbers that, when added together, result in a sum between 1 and 20.
  2. Box B contains pairs of numbers. When a number from Box A is added to a number from Box B, the result is always an even number.
  3. Box C contains many numbers, but when three specific numbers from this box are added together, they equal the highest number in Box A.

One day, a curious traveler named Leo discovered that the highest number in Box A was 18. He also found a number 4 in Box B.

Question: If Leo wanted to use the number 4 from Box B and find two numbers from Box C to get a sum equal to the highest number in Box A, what could those two numbers from Box C be?

Polya’s Problem-Solving Steps Applied to “The Mysterious Number Boxes” Problem

1. Understanding the Problem:

  • We have three boxes: A, B, and C with different properties.
  • Box A’s highest number is 18.
  • Box B has a number 4.
  • We need to find two numbers from Box C that, when added to the number 4 from Box B, will equal 18.

2. Devising a Plan:

  • Since we know the highest number in Box A is 18 and we have a number 4 from Box B, we can subtract 4 from 18 to determine the combined value of the two numbers we need from Box C.
  • Once we have that combined value, we can think of possible pairs of numbers that would sum up to that value.

3. Carrying Out the Plan:

  • Subtract the number from Box B from the highest number in Box A: 18 (from Box A) – 4 (from Box B) = 14
  • Now, we need two numbers from Box C that sum up to 14. There can be multiple pairs, but some possible pairs are:
    • 7 and 7
    • 8 and 6
    • 9 and 5
    • 10 and 4
    • 11 and 3
    • 12 and 2
    • 13 and 1 (Note: Since the problem doesn’t specify unique numbers in Box C, repeated numbers like 7 and 7 are valid.)

4. Look Back/Reflecting on the Solution:

  • We’ve identified possible pairs of numbers from Box C that, when added to the number from Box B, equal the highest number in Box A.
  • The solution approach provides a systematic way to deduce the numbers in Box C based on the given information.
  • This method can be applied to other similar problems, showcasing its versatility and effectiveness.

Answer: Possible pairs of numbers from Box C that, when added to the number 4 from Box B, equal 18 are (7, 7), (8,6), (9,5), (10, 4), (11, 3), (12, 2), and (13, 1).

 

PRACTICE

The Tale of the Vanishing Apples

In the serene village of Subtracto, there’s a peculiar tree known as the “Whimsical Apple Tree.” This tree is unique because every morning, a certain number of apples appear on its branches, and by evening, a few of them mysteriously vanish, only to have new ones appear the next day.

One day, Lila, a young village mathematician, decided to study this tree. She noticed the following:

  1. On Monday morning, the tree had a certain number of apples.
  2. By Monday evening, half of the apples had vanished, and 3 more disappeared after that.
  3. On Tuesday morning, the number of apples on the tree was 10.

Lila was intrigued.

Question: How many apples were on the Whimsical Apple Tree on Monday morning?

Polya’s Problem-Solving Steps Applied to “The Tale of the Vanishing Apples” Problem

1. Understanding the Problem:

  • We have a tree that starts with a certain number of apples on Monday morning.
  • By Monday evening, half of those apples vanish, and an additional 3 apples disappear.
  • By Tuesday morning, there are 10 apples left on the tree.
  • We need to determine the initial number of apples on Monday morning.

2. Devising a Plan:

  • Let’s denote the initial number of apples on Monday morning as “x.”
  • We can set up an equation based on the information given: half of the apples vanish, and then 3 more disappear, leaving 10 apples on Tuesday morning.

3. Carrying Out the Plan:

  • Start with x apples.
  • Half of them vanish: x/2 apples remain.
  • 3 more disappear: x/2 – 3 apples remain.
  • This amount is equal to 10 apples on Tuesday morning.

Setting up the equation: x/2 – 3 = 10

Solving for x: x/2 = 13 x = 26

4. Look Back/Reflecting on the Solution:

  • We started with an unknown number of apples on Monday morning and used the given information to set up an equation.
  • Solving the equation, we found that there were 26 apples on the tree on Monday morning.
  • The solution seems logical, as losing half and then an additional 3 apples from 26 results in 10 apples by Tuesday morning.

Answer: There were 26 apples on the Whimsical Apple Tree on Monday morning.