MATH 100: Problem Solving via Polya's Method

Course Overview and Learning Objectives

• Course Title: MATH 100 - Mathematics in the Modern World

• Topic: Problem Solving (Polya’s Method)

• Instructor: Ronelo A. Fajardo, Ph.D.

• Institutional Affiliation: Department of Mathematics and Statistics, College of Science and Mathematics, Western Mindanao State University

• Date: October 08, 2025

• Learning Objectives:

  • Explain Polya’s four steps in problem solving.

  • Apply the steps in real-life (situational) and mathematical problems.

  • Develop systematic and logical thinking in solving problems.

Polya’s Four Steps in Problem Solving

• Step 1: Understand the Problem

  • Carefully read the problem statement to identify what is given and what needs to be determined.

  • Identify all explicit parameters, operational constraints, and target unknowns.

• Step 2: Devise a Plan

  • Establish the connection between the given information and the unknown target.

  • Formulate an appropriate strategy, pattern recognition method, or algebraic equation.

• Step 3: Carry Out the Plan

  • Execute the strategy by executing calculations or solving equations step-by-step.

  • Ensure accuracy at each algebraic or arithmetic stage.

• Step 4: Look Back

  • Review and verify the solution against the original problem criteria.

  • Confirm that the computed result is logically sound, mathematically consistent, and realistic.

Detailed Worked Examples

Daily Computation Problems

• Problem 1: Buying Snacks

  • Problem Statement: Ana buys $3$ hotdogs at $\text{₱}25$ each and a drink at $\text{₱}20$. How much does she pay?

  • Step 1: Understand the Problem:

  • Given data: Ana bought $3$ hotdogs costing $\text{₱}25$ each and $1$ drink costing $\text{₱}20$

  • Goal: Find the total amount she pays

  • Step 2: Devise a Plan:

  • Compute the total cost of the hotdogs by multiplying the quantity by the individual price

  • Add the cost of the drink to get the final total

  • Step 3: Carry Out the Plan:

  • Calculate hotdog cost:       $3 \times 25 = 75$

  • Calculate total payment:       $75 + 20 = 95$

  • Step 4: Look Back:

  • The computed total cost is reasonable based on the given individual item prices

  • Final Answer: $\text{₱}95$

• Problem 2: Travel Fare

  • Problem Statement: A tricycle charges $\text{₱}15$ for the first kilometer and $\text{₱}5$ for each additional kilometer. How much is the fare for $5$ kilometers?

  • Step 1: Understand the Problem:

  • Base fare: $\text{₱}15$ for the $1\text{st}$ kilometer

  • Additional fare rate: $\text{₱}5$ per kilometer for distance exceeding $1\text{ km}$

  • Target: Total fare for a $5\text{ km}$ trip

  • Step 2: Devise a Plan:

  • Compute the base fare for the first kilometer

  • Compute the distance beyond the first kilometer ($5 - 1$)

  • Multiply the additional distance by the rate per additional kilometer

  • Add the base fare to the additional distance cost

  • Step 3: Carry Out the Plan:

  • Cost for the first kilometer:       $\text{Base Fare} = 15$

  • Distance remaining:       $5 - 1 = 4$

  • Additional distance cost:       $4 \times 5 = 20$

  • Total fare:       $15 + 20 = 35$

  • Step 4: Look Back:

  • The computed fare is consistent with the flat rate plus variable distance charge

  • Final Answer: $\text{₱}35$

Sequence Problems

• Problem 3: Geometric Sequence with Common Ratio 2

  • Problem Statement: Find the next three terms of the sequence $5, 10, 20, 40$.

  • Step 1: Understand the Problem:

  • Given sequence: $5, 10, 20, 40$

  • Target: Identify the next $3$ consecutive terms

  • Step 2: Devise a Plan:

  • Analyze the ratio between consecutive terms to identify the underlying mathematical pattern

  • Step 3: Carry Out the Plan:

  • Identify pattern (each term is obtained by multiplying the previous term by $2$):       $5 \times 2 = 10$       $10 \times 2 = 20$       $20 \times 2 = 40$

  • Extend pattern to compute the next three terms:       $40 \times 2 = 80$       $80 \times 2 = 160$       $160 \times 2 = 320$

  • Step 4: Look Back:

  • The multiplication pattern of doubling each term remains uniform throughout

  • Final Answer: $80, 160, 320$

• Problem 4: Arithmetic Sequence with Common Difference 3

  • Problem Statement: Find the next three terms of the sequence $2, 5, 8, 11$.

  • Step 1: Understand the Problem:

  • Given sequence: $2, 5, 8, 11$

  • Target: Determine the next $3$ terms

  • Step 2: Devise a Plan:

  • Examine the difference between adjacent terms to find the common difference

  • Step 3: Carry Out the Plan:

  • Identify pattern (each term is obtained by adding $3$ to the previous term):       $2 + 3 = 5$       $5 + 3 = 8$       $8 + 3 = 11$

  • Extend pattern to calculate the next three terms:       $11 + 3 = 14$       $14 + 3 = 17$       $17 + 3 = 20$

  • Step 4: Look Back:

  • The difference between consecutive terms remains constant at $3$

  • Final Answer: $14, 17, 20$

• Problem 5: Geometric Sequence with Common Ratio 3

  • Problem Statement: Find the next three terms of the sequence $3, 9, 27, 81$.

  • Step 1: Understand the Problem:

  • Given sequence: $3, 9, 27, 81$

  • Target: Determine the next $3$ terms

  • Step 2: Devise a Plan:

  • Analyze the quotient between adjacent terms to determine the growth pattern

  • Step 3: Carry Out the Plan:

  • Identify pattern (each term is obtained by multiplying the previous term by $3$):       $3 \times 3 = 9$       $9 \times 3 = 27$       $27 \times 3 = 81$

  • Extend pattern to calculate the next three terms:       $81 \times 3 = 243$       $243 \times 3 = 729$       $729 \times 3 = 2187$

  • Step 4: Look Back:

  • The factor of $3$ multiplication holds consistently for all given terms

  • Final Answer: $243, 729, 2187$

Number Problems

• Problem 6: Consecutive Even Numbers

  • Problem Statement: Find two consecutive even numbers whose sum is 54$.\n - **Step 1: Understand the Problem**:\n - Given: Two consecutive even integers whose total sum is 54\n - Target: Find both integer values\n - **Step 2: Devise a Plan**:\n - Represent the first even number as 2x\n - Represent the next consecutive even number as 2x + 2\n - **Step 3: Carry Out the Plan**:\n - Set up the equation:\n      2x + (2x + 2) = 54\n - Combine like terms:\n      4x + 2 = 54\n - Isolate variable term:\n      4x = 52\n - Solve for x:\n      x = 13\n - Substitute x = 13 into representations:\n      2x = 2 imes 13 = 26\n      2x + 2 = 2 imes 13 + 2 = 28\n - **Step 4: Look Back**:\n - Verify sum:\n      26 + 28 = 54\n - **Final Answer**: 26$and$28\n\n- **Problem 7: Consecutive Odd Numbers**\n - **Problem Statement**: Find two consecutive odd numbers whose sum is 48$.

  • Step 1: Understand the Problem:

  • Given: Two consecutive odd numbers summing to $48$

  • Target: Find the two specific odd integers

  • Step 2: Devise a Plan:

  • Let the first odd number be $2x + 1$

  • Let the next consecutive odd number be $2x + 3$

  • Step 3: Carry Out the Plan:

  • Formulate the linear equation:       $(2x + 1) + (2x + 3) = 48$

  • Simplify:       $4x + 4 = 48$

  • Subtract $4$ from both sides:       $4x = 44$

  • Solve for $x$:       $x = 11$

  • Substitute $x = 11$:       $2x + 1 = 2 \times 11 + 1 = 23$       $2x + 3 = 2 \times 11 + 3 = 25$

  • Step 4: Look Back:

  • Verify sum:       $23 + 25 = 48$

  • Final Answer: $23$ and $25$

• Problem 8: Three Consecutive Integers

  • Problem Statement: Find three consecutive integers whose sum is 72$.\n - **Step 1: Understand the Problem**:\n - Given: Three consecutive integers summing to 72\n - Target: Identify all three integers\n - **Step 2: Devise a Plan**:\n - Let the three integers be x$,$x + 1$, and$x + 2\n - **Step 3: Carry Out the Plan**:\n - Set up algebraic equation:\n      x + (x + 1) + (x + 2) = 72\n - Combine terms:\n      3x + 3 = 72\n - Subtract 3:\n      3x = 69\n - Solve for x:\n      x = 23\n - Substitute x = 23:\n      x = 23\n      x + 1 = 23 + 1 = 24\n      x + 2 = 23 + 2 = 25\n - **Step 4: Look Back**:\n - Verify sum:\n      23 + 24 + 25 = 72\n - **Final Answer**: 23, 24, 25\n\n- **Problem 9: Product and Difference**\n - **Problem Statement**: Find two numbers whose product is 192$and one number is$4 more than the other.\n - **Step 1: Understand the Problem**:\n - Given: Product of two numbers is 192$; difference between them is$4\n - Target: Find the values of both numbers\n - **Step 2: Devise a Plan**:\n - Let the smaller number be x\n - Let the larger number be x + 4\n - **Step 3: Carry Out the Plan**:\n - Set up quadratic equation using product condition:\n      x(x + 4) = 192\n - Expand terms:\n      x^2 + 4x = 192\n - Rearrange to standard quadratic form:\n      x^2 + 4x - 192 = 0\n - Solve by factoring:\n      (x + 16)(x - 12) = 0\n - Roots:\n      x = -16 ext{ or } x = 12\n - Reject x = -16 since the requested numbers must be positive\n - Substitute x = 12:\n      x = 12\n      x + 4 = 12 + 4 = 16\n - **Step 4: Look Back**:\n - Verify product and difference:\n      12 imes 16 = 192\n      16 = 12 + 4\n - **Final Answer**: 12$and$16\n\n## Age Problems\n\n- **Problem 10: Present and Future Age**\n - **Problem Statement**: A father is currently 4$times as old as his son. In$10$years, the father will be only$2 times as old as his son. Find their present ages.\n - **Step 1: Understand the Problem**:\n - Given: Present father-son age multiplier is 4$; future father-son age multiplier in$10$years is$2\n - Target: Find present age of father and son\n - **Step 2: Devise a Plan**:\n - Let son's present age be x\n - Let father's present age be 4x\n - **Step 3: Carry Out the Plan**:\n - Model ages in 10 years:\n       ext{Son's age} = x + 10\n       ext{Father's age} = 4x + 10\n - Formulate linear equation based on future condition:\n      4x + 10 = 2(x + 10)\n - Solve equation:\n      4x + 10 = 2x + 20\n      4x - 2x = 20 - 10\n      2x = 10\n      x = 5\n - Substitute x = 5 for present ages:\n       ext{Son's present age} = 5\n       ext{Father's present age} = 4(5) = 20\n - **Step 4: Look Back**:\n - Check present condition:\n      20 = 4(5)\n - Check condition in 10 years:\n       ext{Father in } 10 ext{ years} = 30\n       ext{Son in } 10 ext{ years} = 15\n      30 = 2(15)\n - **Final Answer**: Father: 20$years old, Son:$5 years old\n\n- **Problem 11: Sum and Difference of Ages**\n - **Problem Statement**: The sum of a mother’s and daughter’s ages is 50$. The mother is$26 years older than the daughter. Find their ages.\n - **Step 1: Understand the Problem**:\n - Given: Sum of mother's and daughter's ages is 50$; difference between their ages is$26\n - Target: Determine present ages of both mother and daughter\n - **Step 2: Devise a Plan**:\n - Let daughter's age be x\n - Let mother's age be x + 26\n - **Step 3: Carry Out the Plan**:\n - Formulate sum equation:\n      x + (x + 26) = 50\n - Simplify:\n      2x + 26 = 50\n - Subtract 26:\n      2x = 24\n - Divide by 2:\n      x = 12\n - Compute individual ages:\n       ext{Daughter's age} = 12\n       ext{Mother's age} = 12 + 26 = 38\n - **Step 4: Look Back**:\n - Check sum:\n      12 + 38 = 50\n - **Final Answer**: Daughter = 12$years old, Mother =$38 years old\n\n# Exercises\n\n## Basic Daily Computation\n- **Exercise 1**: Maria buys 4$notebooks at$ ext{₱}18$each and a pen costing$ ext{₱}25. How much does she pay in total?\n- **Exercise 2**: A jeepney fare is ext{₱}13$for the first$4 ext{ km}$and$ ext{₱}2$for each additional kilometer. How much is the fare for a$9 ext{-km} trip?\n- **Exercise 3**: A student buys 2$burgers at$ ext{₱}55$each and a drink worth$ ext{₱}30$. How much change will she receive from$ ext{₱}200?\n- **Exercise 4**: A delivery service charges a base fee of ext{₱}40$plus$ ext{₱}10$per kilometer. How much is the delivery cost for$6 kilometers?\n\n## Sequences\n- **Exercise 1**: Find the next three terms of the sequence 4, 7, 10, 13, ext{…}\n- **Exercise 2**: Find the next three terms of the sequence 2, 6, 18, 54, ext{…}\n- **Exercise 3**: Find the next three terms of the sequence 100, 95, 90, 85, ext{…}\n- **Exercise 4**: Find the next three terms of the sequence 5, 15, 45, 135, ext{…}\n\n## Number Problems\n- **Exercise 1**: Translate into an equation and solve: Find two consecutive even numbers whose sum is 46$.

• Exercise 2: Translate into an equation and solve: Find two consecutive odd numbers whose sum is 64$.\n- **Exercise 3**: Translate into an equation and solve: Find three consecutive integers whose sum is ninety.\n- **Exercise 4**: Translate into an equation and solve: Find two numbers whose difference is 8$and whose sum is$42$.

Age Problems

• Exercise 1: Let $x$ represent the unknown age. Form an equation and solve: A father is $30$ years older than his son. The sum of their ages is $66$. Find their ages.

• Exercise 2: Let $x$ represent the unknown age. Form an equation and solve: A mother is $4$ times as old as her daughter. In $6$ years, the mother will be only $2$ times as old as her daughter. Find their present ages.

• Exercise 3: Let $x$ represent the unknown age. Form an equation and solve: The sum of a brother’s and sister’s ages is $28$. The brother is $6$ years older. Find their ages.

• Exercise 4: Let $x$ represent the unknown age. Form an equation and solve: Ten years ago, a man was twice as old as his son. The sum of their present ages is $50$. Find their present ages.

References

• Polya, G. (1945). How to Solve It. Princeton Univ. Press.

• Burton, D. (2017). Elementary Number Theory. McGraw-Hill.

• DepEd (2020). Most Essential Learning Competencies in Math.

• Posamentier, A. & Krulik, S. (2015). Problem-Solving Strategies. Corwin Press.

• Smith, R. T., & Minton, R. B. (2019). Mathematics: Its Power and Utility. Cengage.

Philosophical Reflection and Scripture

• Philosophical Quote:

  • "Mathematics does not just teach us how to add love or subtract hate, but it also gives us every reason to hope, to persevere, and to find solutions in life."

• Scripture:

  • Philippians 4:13: "I can do all things through Christ who strengthens me."