3.1 Fundamental Operations, Number Theory, and Business Math

Order of Operations (PEMDAS / GEMDAS)

In the Philippine Licensure Examination for Professional Teachers (LET), arithmetic competency begins with the strict application of the standard order of operations. The acronym GEMDAS (Grouping symbols, Exponents, Multiplication and Division, Addition and Subtraction) or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) dictates the sequence in which mathematical operations must be evaluated.

Operational Rules and Left-to-Right Hierarchy

1. Grouping Symbols / Parentheses (G / P): Evaluate expressions inside parentheses (), brackets [], braces {}, absolute value bars ||, or radical signs from the innermost set outward.
2. Exponents and Roots (E): Calculate powers and roots (e.g., $3^3 = 27$, $\sqrt{64} = 8$).
3. Multiplication and Division (M & D): Perform multiplication and division in the order they appear from left to right. Multiplication does not take precedence over division; they hold equal rank.
4. Addition and Subtraction (A & S): Perform addition and subtraction in the order they appear from left to right. Addition does not take precedence over subtraction; they hold equal rank.

Worked Example 1: Evaluating Multi-Operation Expressions

Problem: Simplify the expression: $18 \div 3 \times 2 + (8 - 3)^2 - 4 \times 3$.

Step 1: Evaluate grouping symbols.
$8 - 3 = 5 \implies 18 \div 3 \times 2 + 5^2 - 4 \times 3$

Step 2: Evaluate exponents.
$5^2 = 25 \implies 18 \div 3 \times 2 + 25 - 4 \times 3$

Step 3: Perform Multiplication and Division from left to right.
First, $18 \div 3 = 6$.
Next, $6 \times 2 = 12$.
Then, $4 \times 3 = 12$.
Replacing these in the expression yields:
$12 + 25 - 12$

Step 4: Perform Addition and Subtraction from left to right.
$12 + 25 = 37 \implies 37 - 12 = 25$

Final Answer: $25$

LET Exam Scenario Trap: Left-to-Right Equal Rank

A classic distractor on the LET takes advantage of the misconception that Multiplication precedes Division or Addition precedes Subtraction.
Consider $12 \div 4 \times 3$.

• Incorrect (PEMDAS Trap): Doing multiplication first gives $4 \times 3 = 12$, then $12 \div 12 = 1$.
• Correct: Working left to right gives $12 \div 4 = 3$, then $3 \times 3 = 9$.

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Number Theory Fundamentals

Number theory explores the properties of integers, prime numbers, factors, and multiples. Mastery of these principles is critical for solving word problems involving LCM and GCF.

Prime and Composite Numbers

• Prime Number: A positive integer greater than 1 that has exactly two distinct positive divisors: 1 and itself (e.g., 2, 3, 5, 7, 11, 13, 17, 19, 23, 29). Note that 2 is the only even prime number and the smallest prime number.
• Composite Number: A positive integer greater than 1 that has more than two positive divisors (e.g., 4, 6, 8, 9, 10, 12, 14, 15). The number 1 is neither prime nor composite.
• Prime Factorization: Expressing a composite number as a unique product of prime factors (Fundamental Theorem of Arithmetic).

Divisibility Rules

Rapid divisibility testing saves critical time during the exam. The table below outlines key divisibility rules tested on the LET:

Greatest Common Factor (GCF) and Least Common Multiple (LCM)

• Greatest Common Factor (GCF): The largest positive integer that divides two or more integers without a remainder. Used for simplifying fractions and dividing items into equal groups.
• Least Common Multiple (LCM): The smallest positive integer that is a multiple of two or more integers. Used for finding common denominators and determining when periodic events coincide.

Worked Example 2: Finding GCF and LCM via Prime Factorization

Problem: Find the GCF and LCM of $48$, $72$, and $108$.

Step 1: Write down the prime factorization of each number.
$48 = 2^4 \times 3^1$ $72 = 2^3 \times 3^2$ $108 = 2^2 \times 3^3$

Step 2: Calculate GCF by taking the smallest power of common prime factors.
Common prime bases are 2 and 3.
Smallest power of 2 is $2^2 = 4$.
Smallest power of 3 is $3^1 = 3$.
$\text{GCF} = 2^2 \times 3^1 = 4 \times 3 = 12$

Step 3: Calculate LCM by taking the highest power of all prime factors present.
Highest power of 2 is $2^4 = 16$.
Highest power of 3 is $3^3 = 27$.
$\text{LCM} = 2^4 \times 3^3 = 16 \times 27 = 432$

Final Answer: $\text{GCF} = 12$, $\text{LCM} = 432$

Business Mathematics and Practical Applications

Business mathematics on the LET assesses your capability to solve real-world financial problems involving percentages, simple interest, compound interest, discounts, markups, commissions, and proportions.

Percentages, Markups, and Discounts

• Percentage Formula: $\text{Part} = \text{Base} \times \text{Rate}$
• Percentage Increase / Decrease:
  $\text{Percent Change} = \frac{|\text{New Value} - \text{Old Value}|}{\text{Old Value}} \times 100\%$
• Discount Amount: $\text{Discount} = \text{Original Price} \times \text{Discount Rate}$
• Sale Price: $\text{Sale Price} = \text{Original Price} - \text{Discount}$
• Markup Amount: $\text{Markup} = \text{Cost} \times \text{Markup Rate}$
• Selling Price: $\text{Selling Price} = \text{Cost} + \text{Markup}$

Worked Example 3: Consecutive Discounts

Problem: A laptop originally priced at ₱35,000 is offered with consecutive discounts of 20% and 10%. What is the final selling price of the laptop?

Step 1: Apply the first discount of 20%.
$\text{Discount}_1 = 35,000 \times 0.20 = \text{₱}7,000$ $\text{Price after 1st discount} = 35,000 - 7,000 = \text{₱}28,000$

Step 2: Apply the second discount of 10% to the reduced price.
$\text{Discount}_2 = 28,000 \times 0.10 = \text{₱}2,800$ $\text{Final Sale Price} = 28,000 - 2,800 = \text{₱}25,200$

LET Exam Scenario Trap: Do not add consecutive discount rates ($20% + 10% = 30%$). Taking 30% off ₱35,000 gives ₱24,500, which is incorrect because the second discount applies only to the already reduced price!

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Simple Interest vs. Compound Interest

Interest is the fee paid for using borrowed money or the earnings accrued from financial investments.

Simple Interest

Simple interest is calculated solely on the principal amount for the entire duration of the loan or investment.

$\text{Formula: } I = P \times R \times T$ $\text{Total Accumulated Amount: } A = P + I = P(1 + RT)$

Where:

• $I$ = Interest earned or owed
• $P$ = Principal amount (initial investment or loan)
• $R$ = Annual interest rate (expressed as a decimal)
• $T$ = Time in years

Worked Example 4: Simple Interest Calculation

Problem: Teacher Maria borrowed ₱60,000 from a credit union at a simple interest rate of 6.5% per annum for 3 years and 6 months. How much interest will she pay, and what is the total amount due?

Step 1: Convert time into years.
3 years and 6 months = $3.5$ years.

Step 2: Calculate Interest ($I$).
$I = 60,000 \times 0.065 \times 3.5 = \text{₱}13,650$

Step 3: Calculate Total Amount ($A$).
$A = 60,000 + 13,650 = \text{₱}73,650$

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Compound Interest

Compound interest is interest calculated on both the initial principal and the accumulated interest from previous periods ("interest on interest").

$\text{Formula: } A = P \left(1 + \frac{r}{n}\right)^{nt}$ $\text{Compound Interest Earned: } I = A - P$

Where:

• $A$ = Final compound amount
• $P$ = Principal amount
• $r$ = Annual interest rate (as a decimal)
• $n$ = Number of compounding periods per year
• $t$ = Time in years

Worked Example 5: Quarterly Compound Interest

Problem: An amount of ₱20,000 is invested in a bank account paying 8% per annum compounded quarterly. What will be the balance after 2 years?

Step 1: Identify values.
$P = 20,000$, $r = 0.08$, $n = 4$, $t = 2$.

Step 2: Substitute into compound interest formula.
$A = 20,000 \left(1 + \frac{0.08}{4}\right)^{4 \times 2} = 20,000 (1 + 0.02)^8 = 20,000 (1.02)^8$

Step 3: Evaluate.
$(1.02)^8 \approx 1.171659$ $A = 20,000 \times 1.171659 = \text{₱}23,433.18$

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Ratio and Proportion Applications

Proportions are statements expressing the equality of two ratios: $\frac{a}{b} = \frac{c}{d}$ or $a:b = c:d$.

Types of Proportions

1. Direct Proportion: As one quantity increases, the other quantity increases proportionally ($\frac{x_1}{y_1} = \frac{x_2}{y_2}$).
   Example: If 5 books cost ₱1,250, how much do 12 books cost?
2. Inverse (Indirect) Proportion: As one quantity increases, the other quantity decreases proportionally ($x_1 y_1 = x_2 y_2$).
   Example: If 6 workers can construct a fence in 10 days, how many days will 15 workers take working at the same rate?
3. Partitive Proportion: A whole quantity is partitioned into parts based on a given ratio ($a:b:c$).
   Example: Dividing ₱45,000 among three siblings in the ratio $2:3:4$.

Worked Example 6: Partitive Proportion

Problem: Three school departments receive a total budget of ₱180,000 distributed in the ratio $3:4:5$. How much budget does the largest recipient receive?

Step 1: Sum the ratio parts.
$3 + 4 + 5 = 12\text{ parts}$

Step 2: Determine the value of one part.
$\text{Value per part} = \frac{\text{₱}180,000}{12} = \text{₱}15,000$

Step 3: Calculate the largest portion (5 parts).
$\text{Largest Portion} = 5 \times 15,000 = \text{₱}75,000$

Final Answer: ₱75,000