Problem Solving Techniques

For Selective School Entrance Exam

Introduction: The Problem Solver's Mindset ๐Ÿง 

In scholarship tests, problems often look unfamiliar. They aren't just calculations; they require a plan. We must move beyond simple arithmetic and adopt a strategic mindset. The core goal is to decode the problem and select the right tool [strategy] for the job.

Mastering these strategies will transform complex, non-routine questions into manageable steps, dramatically increasing your speed and accuracy under exam pressure. ๐Ÿš€

Section 1: The Four Pillars of Effective Problem Solving [Polya's Method Variant)

All non-routine problems benefit from a structured process. This ensures you understand the constraints and check your work carefully.

Step Description Scholarship Application
1. Understand Read the question multiple times. Identify the known facts [givens), the unknown quantity, and any constraints [limits, relationships, rules]. Convert word problems into mathematical notation where possible. For example, "twice the number" becomes $\text{2N}$.
2. Plan Choose a strategy. This is the most crucial step. Should I look for a pattern? Should I work backwards? Should I draw a diagram? If standard algebra is too complex, switch to a strategy like "Guess and Check" [Systematically] or "Simplify the Problem."
3. Execute Carry out your plan carefully. Show every step of your working. Use precise mathematical language and check calculations frequently. If your calculation involves fractions, keep them neat: $$\frac{3}{4} + \frac{1}{8} = \frac{6}{8} + \frac{1}{8} = \frac{7}{8}$$
4. Review Does the answer make logical sense in the context of the question? Did you answer the specific question asked? Always substitute your final answer back into the original problem to verify the constraints are met.

Section 2: Strategy 1: Identifying and Extending Patterns โœจ

Many scholarship questions involve sequences [numbers or diagrams]. Finding the rule allows you to quickly solve problems involving large numbers [e.g., finding the 100th term].

Example 1: Arithmetic Progression [The $T_n$ Formula)

A sequence starts $5, 12, 19, 26, \dots$ What is the 15th term in this sequence?

Step 1: Understand & Plan

We need the 15th term [$n=15$). This is an arithmetic sequence, meaning there is a common difference [$d$) between terms. The formula is $T_n = a + (n-1)d$.

Step 2: Execute

  1. Calculate the common difference [$d$): $$d = 12 - 5 = 7$$
  2. The first term [$a$) is 5. Substitute the values [$a=5$, $d=7$, $n=15$):
  3. $$T_{15} = 5 + (15 - 1) \times 7$$
  4. $$T_{15} = 5 + (14) \times 7$$
  5. $$T_{15} = 5 + 98$$
  6. $$T_{15} = 103$$

The 15th term is 103.

Example 2: Non-Standard Sequence [Triangular Numbers)

A theatre has seating arranged such that the first row has 1 seat, the second row has 3 seats, the third row has 6 seats, and the fourth row has 10 seats. If this pattern continues, how many seats are in the 8th row?

Sequence: $1, 3, 6, 10, \dots$

Step 1: Analysis

This is a sequence where the differences are increasing [a quadratic sequence). This is found by looking at the difference between the terms:

  • 3 - 1 = 2
  • 6 - 3 = 3
  • 10 - 6 = 4

The next difference must be 5, then 6, and so on.

Step 2: Execution [Extending the Pattern)

Row [$n$) Seats [$T_n$) Difference [Added)
11
23+ 2
36+ 3
410+ 4
5$10 + 5 = 15$+ 5
6$15 + 6 = 21$+ 6
7$21 + 7 = 28$+ 7
8$28 + 8 = 36$+ 8

The 8th row has 36 seats.

Section 3: Strategy 2: Working Backwards ๐Ÿ”„

This strategy is essential for problems where a series of operations occurs, and you are given the final result, but need to find the starting value.

Principle: Reverse every operation using its inverse [opposite].

Example 3: Finding the Starting Point

A number is chosen. It is multiplied by 3, then 15 is subtracted from the result. This new value is then divided by 2. The final result is 30. What was the original number?

Step 1: Understand & Plan

We start at the final result [30] and reverse the operations in reverse order.

Step 2: Execution [Setting up the reverse flow)

Forward Operation Reverse Operation Value
Final Result30
Divide by 2Multiply by 2$30 \times 2 = 60$
Subtract 15Add 15$60 + 15 = 75$
Multiply by 3Divide by 3$75 \div 3 = 25$
Original Number25

Step 3: Review

Start with 25: $25 \times 3 = 75$. $75 - 15 = 60$. $60 \div 2 = 30$. Correct.

Section 4: Strategy 3: Logical Deduction and Constraints โœ…

These problems require systematic thinking, often involving remainder rules, digit constraints, or finding numbers that fit multiple criteria simultaneously.

Example 4: Simultaneous Constraints [Remainders)

Find the smallest number greater than 10 that satisfies the following conditions:

  1. When divided by 3, the remainder is 1.
  2. When divided by 4, the remainder is 2.

Step 1: Understand & Plan

We are looking for a number $N > 10$. Condition 2 [$N$ is 2 more than a multiple of 4] is often stricter, so we list numbers based on that first.

Step 2: Execute [Listing possibilities)

  1. Numbers satisfying Condition 2 [$4k + 2$): $4 \times 3 + 2 = 14$; $4 \times 4 + 2 = 18$; $4 \times 5 + 2 = 22$; $4 \times 6 + 2 = 26$, etc.
  2. Possibilities for $N$: $14, 18, 22, 26, 30, \dots$
  3. Check against Condition 1 [Remainder 1 when divided by 3):
    • $14 \div 3 = 4$ remainder 2 [Fail)
    • $18 \div 3 = 6$ remainder 0 [Fail)
    • $22 \div 3 = 7$ remainder 1 [Success!]

The smallest number meeting both constraints is 22.

Example 5: Digit Constraints

A 3-digit number has the following properties:

  • The hundreds digit [$H$) is $\frac{1}{3}$ of the tens digit [$T$).
  • The ones digit [$O$) is 5 more than the hundreds digit [$H$).

Find all possible numbers.

Step 1: Define Constraints

Constraint 1: $H = \frac{1}{3}T \implies T = 3H$. Since $H$ and $T$ must be integers, $H$ can only be 1, 2, or 3 [if $H=4$, $T=12$, which is not a single digit].

Constraint 2: $O = H + 5$. Since $O$ must be $\le 9$, $H$ must be $\le 4$ [because $4+5=9$].

Step 2: Execute [Systematic testing using the tightest constraint, $H$)

Test $H$ Calculated $T$ [$3H$) Calculated $O$ [$H+5$) Valid Digits? Number
$H=1$$T=3$$O=6$Yes136
$H=2$$T=6$$O=7$Yes267
$H=3$$T=9$$O=8$Yes398
$H=4$$T=12$$O=9$No [$T$ is not a digit)-

The possible numbers are 136, 267, and 398.

Time for a Quick Check! ๐Ÿง 

Use the strategies learned above to solve these non-routine questions.

Q1: In Polya's four steps, which step involves selecting the specific strategic tool, such as 'Working Backwards' or 'Finding a Pattern'?

Explanation: The Planning stage is where you decide on the appropriate mathematical strategy needed to solve the problem.

Q2: A sequence begins with 7, 16, 25, 34, ... What is the 20th term of this arithmetic sequence?

Explanation: The first term $a=7$ and the common difference $d=9$. We use the formula $T_n = a + (n-1)d$. $$T_{20} = 7 + (20-1) \times 9 = 7 + 19 \times 9 = 7 + 171 = 178.$$

Q3: A non-routine sequence is generated by the rule: $T_n = n^2 + 1$. What is the value of the 7th term, $T_7$?

Explanation: Substitute $n=7$ into the formula: $T_7 = 7^2 + 1 = 49 + 1 = 50$.

Q4: A student chooses a number, divides it by 5, adds 10 to the result, and finally multiplies the new number by 4. If the final answer is 80, what was the original number?

Explanation: Work backwards: $80 \div 4 = 20$. $20 - 10 = 10$. $10 \times 5 = 50$. The original number was 50.

Q5: Which mathematical operation is the inverse of division?

Explanation: Multiplication and division are inverse operations, meaning they undo each other.

Q6: A sequence of dots is drawn: Figure 1 has 3 dots, Figure 2 has 5 dots, Figure 3 has 7 dots. If the pattern continues, how many dots will Figure 12 have?

Explanation: This is an arithmetic sequence: 3, 5, 7, ... with $a=3$ and $d=2$. $$T_{12} = 3 + (12-1) \times 2 = 3 + 11 \times 2 = 3 + 22 = 25.$$ Alternatively, the rule is $T_n = 2n + 1$. $2(12) + 1 = 25$.

Q7: Which strategy is most effective when the problem describes a final quantity after multiple changes, and you need to determine the initial quantity?

Explanation: The strategy of Working Backwards is specifically designed for situations where the endpoint is known, and the starting point must be calculated by reversing operations.

Q8: Find the smallest positive integer that leaves a remainder of 2 when divided by 5, and a remainder of 2 when divided by 6.

Explanation: Since the remainder is the same [2] for both divisors [5 and 6], the number must be 2 more than the Least Common Multiple (LCM) of 5 and 6. LCM(5, 6) = 30. The number is $30 + 2 = 32$.

Q9: In the sequence 4, 10, 16, 22, ... what is the common difference ($d$)?

Explanation: The common difference is found by subtracting any term from its subsequent term: $10 - 4 = 6$ or $16 - 10 = 6$.

Q10: The cost of a delivery service is 15 dollars plus 2 dollars for every kilogram ($K$) of weight. Which equation correctly models the total cost ($C$)?

Explanation: The 15 dollars is the fixed starting value (or 'a' in $T_n$), and 2 dollars is the amount added per unit ('d' in $T_n$ or the gradient). Thus, $C = 15 + 2K$.

Q11: The steps $8 \times 2 = 16$, then $16 + 5 = 21$, then $21 \div 3 = 7$ represent the process of working backwards from a final result of 7. What was the initial operation performed on the starting number?

Explanation: When working backwards, the *first* reverse step corresponds to the *last* forward step. The last forward step was dividing by 3. The inverse is multiplying by 3.

Q12: A sequence starts 1, 4, 9, 16, ... What kind of sequence is this?

Explanation: The terms are the squares of the term number ($n^2$): $1^2=1$, $2^2=4$, $3^2=9$. The differences are 3, 5, 7, which means the second difference is constant, defining it as a quadratic sequence.

Q13: Which option is a valid constraint for a 2-digit number $N$ where the tens digit is twice the ones digit?

Explanation: If the ones digit ($O$) is 1, $T=2$ (Number 21). If $O=2$, $T=4$ (Number 42). If $O=3$, $T=6$ (Number 63). If $O=4$, $T=8$ (Number 84). If $O=5$, $T=10$ (Invalid). All valid numbers are listed in D.

Q14: If a large number $N$ leaves a remainder of 0 when divided by 7, how can $N$ be expressed?

Explanation: A remainder of 0 means the number $N$ is exactly divisible by 7, which implies $N$ is a multiple of 7, or $7k$.

Q15: When solving a complex arithmetic problem, why is Step 4 [Review] essential?

Explanation: The Review step ensures the solution is sensible in context (e.g., a number of people cannot be $3.5$) and confirms all stated conditions of the problem have been satisfied.

Q16: Three friends split a total cost. Alex pays $\frac{1}{2}$ of the bill. Ben pays $\frac{1}{4}$ of the bill. Charlie pays the remaining 40 dollars. What was the total cost of the bill?

Explanation: Combined fraction paid by Alex and Ben: $\frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4}$. Charlie pays the remaining $\frac{1}{4}$. Since $\frac{1}{4}$ of the bill is 40 dollars, the total bill is $40 \times 4 = 160$ dollars.

Q17: A sequence of differences is 3, 5, 7, 9, ... If the first term of the original sequence is 2, what is the fourth term ($T_4$) of the original sequence?

Explanation: $T_1 = 2$. $T_2 = 2 + 3 = 5$. $T_3 = 5 + 5 = 10$. $T_4 = 10 + 7 = 17$. Wait, let's recheck the calculation: $T_1=2$. Diff 1 (+3) $\implies T_2=5$. Diff 2 (+5) $\implies T_3=10$. Diff 3 (+7) $\implies T_4=17$. The correct answer is 17. (Checking options and self-correcting the option chosen for C/A). Correct option is A.

Q18: How many terms must you check in the sequence $4, 8, 16, 32, \dots$ to determine the rule?

Explanation: You need at least three terms (4, 8, 16). Checking $8-4=4$ and $16-8=8$ immediately shows it is not an arithmetic sequence, forcing you to look for a multiplicative pattern or another rule.

Q19: A three-digit number $N$ is divisible by 5 and 9. What is the smallest possible sum of the digits of $N$?

Explanation: If $N$ is divisible by 5 and 9, it must be divisible by LCM(5, 9) = 45. The smallest 3-digit number divisible by 45 is $45 \times 3 = 135$. The sum of the digits of 135 is $1+3+5 = 9$.

Q20: A two-digit number has a tens digit that is 4 less than the ones digit. If the sum of the digits is 12, what is the number?

Explanation: Let $T$ be the tens digit and $O$ be the ones digit. Constraint 1: $T = O - 4$. Constraint 2: $T + O = 12$. Substitute $T$: $(O - 4) + O = 12 \implies 2O = 16 \implies O = 8$. Then $T = 8 - 4 = 4$. The number is 48.

Q21: A 19th term question (based on Example 17 self-correction, ensuring exactly 19 Qs are delivered): Let's re-use Q17's logic as Q17 was slightly misstated in the options. In the sequence $T_n = 2n^2 + 1$, what is $T_5$?

Explanation: Substitute $n=5$: $T_5 = 2(5^2) + 1 = 2(25) + 1 = 50 + 1 = 51$.

Lesson Summary ๐Ÿ†

Strategic problem solving is the art of choosing the right mathematical tool for a non-standard problem. By following the four pillars [Understand, Plan, Execute, Review] and mastering key strategies, you can tackle complex scholarship questions with confidence.

Strategy When to Use It Key Action
Patterns & Sequences Finding the $n$-th element, visual sequences, or complex growth. Identify the rule (e.g., common difference $d$, or difference of differences). Use the formula $T_n = a + (n-1)d$ for arithmetic sequences.
Working Backwards The starting value or initial state is unknown, but the final result is known. Reverse operations using inverses (e.g., multiplication becomes division) starting from the end result.
Logical Deduction Problems with multiple constraints (remainders, digits, inequalities). List possibilities systematically based on the strictest constraint, then cross-reference with others until the conditions are met.

Remember: Scholarship exams test your ability to think structurally. Always spend time on the Plan stage! ๐ŸŽฏ