6.1 Arithmetic Fundamentals, Fractions & Decimals

Arithmetic Fundamentals, Fractions & Decimals

NAPOLCOM publishes Basic Concepts and Problem-Solving Skills under Quantitative Reasoning but does not publish an official percentage weight or item count for the area. Unlike academic examinations where electronic aids are permitted, the PNPE evaluates a candidate's mental agility, computational precision, and analytical clarity under strict calculator-free conditions. Mastery of foundational arithmetic, fraction mechanics, and decimal operations provides the speed and accuracy necessary to solve complex word problems efficiently.

Independent Preparation Notice: This study module is independently developed by OpenExamPrep to assist prospective applicants in mastering the quantitative reasoning concepts required for the examination. OpenExamPrep is an independent educational publisher and is not affiliated with, endorsed by, or partnered with the National Police Commission (NAPOLCOM) or the Philippine National Police (PNP).

---

1. The Strict Calculator Prohibition on the NAPOLCOM PNPE

Under NAPOLCOM examination rules and civil service testing standards across the Philippines, calculators, smartwatches, digital organizers, and mobile devices are strictly prohibited inside the examination room. Possession of unauthorized electronic devices during the examination is grounds for immediate confiscation, test invalidation, and administrative disqualification.

Time Management and Computational Demands

The examination has a 3-hour (180-minute) overall limit, but the current official announcement does not state a fixed total item count. Establish a working pace from the actual instructions and number of questions, including time to calculate, choose an answer, and review the official answer sheet.

To succeed without a calculator, candidates must cultivate three core test-taking disciplines:

• Scratchpad Space Economy: Organize scratch calculations into neat, labeled columns to avoid misreading one's own handwriting or losing track of intermediate numerical values.
• Precision in Basic Operations: Eliminate common arithmetic errors in multi-digit addition, borrowing across zeros in subtraction, and manual long division.
• Elimination via Estimation: Round intermediate numbers to benchmark values to eliminate unreasonable multiple-choice distractors before conducting detailed pencil-and-paper computation.

---

2. Order of Operations: Mastering Nested PEMDAS / BODMAS

Mathematical expressions containing multiple operations must be evaluated according to a standardized hierarchy to yield a unique, correct value. In Philippine and American curricula, this convention is known as PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) or BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction).

The Operational Hierarchy

1. Grouping Symbols (Innermost to Outermost): Evaluate expressions inside parentheses ( ), brackets [ ], and braces { }. If grouping symbols are nested, always evaluate the innermost set first and work outward.
2. Exponents and Roots: Evaluate powers, squared terms, cubed terms, and radical expressions.
3. Multiplication and Division (Equal Priority, Left to Right): A frequent source of candidate error is assuming multiplication always precedes division. Multiplication and division have identical operational priority and must be performed in the exact sequence in which they appear from left to right.
4. Addition and Subtraction (Equal Priority, Left to Right): Like multiplication and division, addition and subtraction have identical operational priority and must be performed strictly in order from left to right.

Step-by-Step Worked Drill with Nested Grouping

Consider the evaluation of the following complex operational expression typical of PNPE quantitative items:

$42 - 3 \times [16 - (5 + 2^3 \div 4) + 18 \div 6 \times 2]$

• Step 1 (Innermost Parentheses): Inside (5 + 2³ ÷ 4), evaluate the exponent first: $2^3 = 8$. Then evaluate division: $8 \div 4 = 2$. Complete the addition: $5 + 2 = 7$. The expression simplifies to: $42 - 3 \times [16 - 7 + 18 \div 6 \times 2]$.
• Step 2 (Operations Inside Brackets): Evaluate division and multiplication from left to right: $18 \div 6 = 3$, then $3 \times 2 = 6$. Now resolve addition and subtraction from left to right: $16 - 7 = 9$, then $9 + 6 = 15$. The expression simplifies to: $42 - 3 \times 15$.
• Step 3 (Outer Multiplication): Multiply before subtracting: $3 \times 15 = 45$.
• Step 4 (Final Subtraction): $42 - 45 = -3$.

---

3. Number Theory: Primes, Integers & Fast Divisibility Rules

Number theory principles allow candidates to inspect numbers and determine factors instantly without resorting to cumbersome trial division.

Properties of Integers

• Parity Rules (Even and Odd Numbers):
  • $\text{Even} \pm \text{Even} = \text{Even}$; $\text{Odd} \pm \text{Odd} = \text{Even}$; $\text{Even} \pm \text{Odd} = \text{Odd}$.
  • $\text{Even} \times \text{Even} = \text{Even}$; $\text{Odd} \times \text{Odd} = \text{Odd}$; $\text{Even} \times \text{Odd} = \text{Even}$.
• Prime Numbers: An integer greater than 1 that possesses exactly two distinct positive divisors: 1 and itself. The number 0 and 1 are neither prime nor composite. The number 2 is the only even prime number and the smallest prime.
• Primes under 50: Candidates should memorize the fifteen primes under 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

Fast Divisibility Rules

Mastering divisibility tests enables rapid reduction of large fractions and instant verification of multiple-choice options:

---

4. Greatest Common Factor (GCF) & Least Common Multiple (LCM)

In administrative and law enforcement contexts, GCF and LCM solve fundamental logistical challenges regarding resource allocation, group distribution, and duty synchronization.

Definitions and Core Distinction

• Greatest Common Factor (GCF): The largest positive integer that divides two or more integers without leaving a remainder. Used to divide items into the largest possible equal groups with nothing left over.
• Least Common Multiple (LCM): The smallest positive integer that is a common multiple of two or more integers. Used to find the earliest point in time when recurring events synchronize.

Computation Techniques

Method 1: Prime Factorization

Express each number as a product of prime powers:

• GCF: Take the lowest power of common prime factors.
• LCM: Take the highest power of all prime factors present.

Example: Determine the GCF and LCM of $72$, $108$, and $180$.

• $72 = 2^3 \times 3^2$
• $108 = 2^2 \times 3^3$
• $180 = 2^2 \times 3^2 \times 5^1$
• $\text{GCF} = 2^{\min(3,2,2)} \times 3^{\min(2,3,2)} = 2^2 \times 3^2 = 4 \times 9 = 36$.
• $\text{LCM} = 2^{\max(3,2,2)} \times 3^{\max(2,3,2)} \times 5^{\max(0,0,1)} = 2^3 \times 3^3 \times 5^1 = 8 \times 27 \times 5 = 1,080$.

Method 2: Continuous Division (The Ladder Method)

Write the numbers in a row and divide simultaneously by prime factors that divide all terms until no common factor remains. The product of the common divisors on the left gives the GCF. Multiplying the GCF by all remaining quotients at the bottom gives the LCM.

Practical Law Enforcement Applications

• GCF Scenario (Resource Distribution): A station logistics officer receives 96 tactical flashlights, 144 personal medical kits, and 216 boxes of ammunition. The supplies must be distributed equally among the maximum number of patrol precincts without any leftover items. How many precincts can be supplied?
  • Calculate $\text{GCF}(96, 144, 216) = 24$ precincts. Each precinct receives $4$ flashlights, $6$ medical kits, and $9$ boxes of ammunition.
• LCM Scenario (Patrol Cycle Synchronization): Three regional border checkpoints conduct perimeter patrols. Checkpoint 1 dispatches a patrol every $20$ minutes, Checkpoint 2 every $30$ minutes, and Checkpoint 3 every $45$ minutes. If all three checkpoints dispatch patrols simultaneously at 08:00 AM, at what time will they next dispatch patrols at the exact same moment?
  • Calculate $\text{LCM}(20, 30, 45)$: $20 = 2^2 \times 5^1$; $30 = 2^1 \times 3^1 \times 5^1$; $45 = 3^2 \times 5^1$. $\text{LCM} = 2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5 = 180\text{ minutes} = 3\text{ hours}$.
  • Adding $3$ hours to 08:00 AM gives 11:00 AM.

---

5. Fractions: Operations, Classifications & Comparisons

A fraction represents a part of a whole or a quotient of two integers: $\frac{a}{b}$, where $a$ is the numerator and $b$ is the denominator ($b \neq 0$).

Classification of Fractions

• Proper Fraction: Numerator is strictly less than the denominator ($a < b$), such as $\frac{3}{7}$. The value is strictly between $0$ and $1$.
• Improper Fraction: Numerator is greater than or equal to the denominator ($a \ge b$), such as $\frac{11}{4}$. The value is $1$ or greater.
• Mixed Number: An integer combined with a proper fraction, such as $2 \frac{3}{4}$.
  • To convert a mixed number to an improper fraction: Multiply the whole number by the denominator, add the numerator, and place the result over the original denominator: $2 \frac{3}{4} = \frac{(2 \times 4) + 3}{4} = \frac{11}{4}$.
  • To convert an improper fraction to a mixed number: Divide numerator by denominator; the quotient is the whole number and the remainder is the new numerator: $17 \div 5 = 3$ remainder $2 \implies 3 \frac{2}{5}$.

Arithmetic Operations on Fractions

Addition and Subtraction

Fractions must possess a common denominator before their numerators can be combined:

1. Determine the Least Common Denominator (LCD), which is the LCM of the denominators.
2. Convert each fraction to an equivalent fraction with the LCD.
3. Add or subtract the numerators while keeping the common denominator unchanged.
4. Reduce the resulting fraction to simplest form.

$\frac{5}{6} - \frac{3}{8} + \frac{1}{4}$

• Denominators are $6, 8, 4$. Their LCM is $24$.
• Convert: $\frac{5 \times 4}{6 \times 4} = \frac{20}{24}$; $\frac{3 \times 3}{8 \times 3} = \frac{9}{24}$; $\frac{1 \times 6}{4 \times 6} = \frac{6}{24}$.
• Combine: $\frac{20 - 9 + 6}{24} = \frac{17}{24}$.

Multiplication

Multiply numerators together and denominators together. Simplify before multiplying by canceling common factors between any numerator and any denominator (cross-cancellation):

$\frac{14}{25} \times \frac{15}{28} = \frac{\cancel{14}^1}{\cancel{25}_5} \times \frac{\cancel{15}^3}{\cancel{28}_2} = \frac{1 \times 3}{5 \times 2} = \frac{3}{10}$

Division (The Invert-and-Multiply Reciprocal Rule)

Dividing by a fraction is mathematically identical to multiplying by its reciprocal:

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Example: A precinct has $6 \frac{2}{3}$ liters of forensic fingerprint developing solution. Each latent print lift kit requires $\frac{4}{9}$ liter of solution. How many complete kits can be prepared?

• Convert mixed number: $6 \frac{2}{3} = \frac{20}{3}$.
• Divide by $\frac{4}{9}$: $\frac{20}{3} \div \frac{4}{9} = \frac{20}{3} \times \frac{9}{4} = \frac{\cancel{20}^5}{\cancel{3}_1} \times \frac{\cancel{9}^3}{\cancel{4}_1} = 5 \times 3 = 15\text{ kits}$.

Comparing and Ordering Fractions

When comparing fractions on a calculator-free test, avoid converting every fraction into decimals via long division. Use these rapid techniques:

• Cross-Multiplication Method (Butterfly Method): To compare $\frac{a}{b}$ and $\frac{c}{d}$:
  • Multiply $a \times d$ and $b \times c$.
  • If $a \times d > b \times c$, then $\frac{a}{b} > \frac{c}{d}$.
  • Example: Compare $\frac{7}{11}$ and $\frac{5}{8}$. Compute $7 \times 8 = 56$ and $11 \times 5 = 55$. Since $56 > 55$, it follows that $\frac{7}{11} > \frac{5}{8}$.
• Benchmark Comparison: Compare fractions against standard benchmarks like $1/2$:
  • In $\frac{9}{19}$, half of $19$ is $9.5$, so $\frac{9}{19} < \frac{1}{2}$.
  • In $\frac{11}{20}$, half of $20$ is $10$, so $\frac{11}{20} > \frac{1}{2}$.
  • Therefore, $\frac{11}{20} > \frac{9}{19}$ without performing long division.

---

6. Decimals & Mental Math Estimation Strategies

Decimals represent fractions whose denominators are powers of ten ($10, 100, 1,000$). Mastery of decimal alignment and rapid estimation prevents costly procedural errors.

Decimal Terminology & Conversions

• Terminating Decimals: A simplified fraction $\frac{a}{b}$ produces a terminating decimal if and only if the prime factorization of the denominator $b$ contains no prime factors other than 2 or 5 (e.g., $\frac{3}{8} = \frac{3}{2^3} = 0.375$; $\frac{7}{20} = \frac{7}{2^2 \times 5} = 0.35$).
• Repeating Decimals: If the denominator contains any prime factor other than 2 or 5, the decimal repeats indefinitely (e.g., $\frac{1}{3} = 0.\overline{3}$; $\frac{5}{6} = 0.8\overline{3}$; $\frac{4}{11} = 0.\overline{36}$).
• Converting Repeating Decimals to Fractions:
  • Pure repeating decimals: $0.\overline{d} = \frac{d}{9}$; $0.\overline{ab} = \frac{ab}{99}$; $0.\overline{abc} = \frac{abc}{999}$.
  • Example: $0.\overline{27} = \frac{27}{99} = \frac{3}{11}$.

Decimal Arithmetic Rules

• Addition and Subtraction: Align the numbers vertically by their decimal points. Fill empty trailing spaces with zeros to prevent columnar misalignment.
• Multiplication: Multiply the numbers as if they were whole integers, ignoring decimal points. Count the total number of decimal digits across all factors. Place the decimal point in the product so that it has the same total number of decimal places.
  • Example: $0.045 \times 1.2$. Multiply $45 \times 12 = 540$. Factor $0.045$ has 3 decimal places; factor $1.2$ has 1 decimal place. Total decimal places = $3 + 1 = 4$. Product is $0.0540 = 0.054$.
• Division: If the divisor is a decimal, move its decimal point to the right until it becomes an integer. Move the decimal point in the dividend to the right by the exact same number of places, adding zeros if necessary. Then execute standard long division.
  • Example: $14.4 \div 0.12 \implies 1,440 \div 12 = 120$.

Rapid Mental Math Shortcuts for Test Day

1. Compatible Number Rounding: In multiple-choice questions where the four options are spaced widely apart, round awkward numbers to friendly benchmarks.
   • Example: Calculate $198.6 \times 0.49$. Round to $200 \times 0.50 = 100$. Look for the option closest to $100$ ($97.314$).
2. Units Digit Verification: When performing multi-digit addition or multiplication of whole numbers, calculate only the units digit to discard three out of four distractors immediately.
   • Example: $3,458 \times 2,197$ must end in the units digit of $8 \times 7 = 56$ (ends in $6$). Any option ending in $2, 4,$ or $8$ is eliminated instantly.