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LET Elementary Mathematics — Numbers, Number Sense and the Four OperationsDetailed Explanation

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A detailed, step-by-step explanation of Numbers, Number Sense and the Four Operations for LET Elementary aspirants. This page goes deeper than the summary and study notes, walking through the reasoning behind each concept so you understand why Professional Regulation Commission (PRC) tests it the way it does in the LET Elementary Mathematics subtest.

Exam context

The Licensure Examination for Professional Teachers — Elementary is conducted by Professional Regulation Commission (PRC) and is scheduled for Bi-annual. The Mathematics subtest is marked as "Core" in the official pattern, and Numbers, Number Sense and the Four Operations appears in position 1st of 7 in the LET Elementary Mathematics review rotation. Passing mark: Weighted average of 75% with no grade below 50%. Recent LET Elementary 2026 papers have drawn roughly a meaningful share of questions from this subject.

Numbers, Number Sense and the Four Operations - Detailed Explanation

Number sense is the cornerstone of mathematical competency for any elementary school teacher. As a future licensed professional teacher under RA 7836 (Philippine Teachers Professionalization Act of 1994), you must not only solve numerical problems correctly but also understand WHY the rules work so you can teach them effectively to Grades 1-6 pupils. In the LET General Education Mathematics cluster, which comprises 40% of your Gen-Ed score, questions on numbers and operations test your ability to classify numbers accurately, apply the order of operations without error, operate on signed numbers, and use GCF and LCM to solve real-world problems. This chapter covers every sub-topic in this cluster: the real number system, properties of real numbers, GEMDAS/PEMDAS, integers, factors and multiples, GCF and LCM, estimation, absolute value, scientific notation, place value, and Roman numerals. Mastering this chapter gives you a solid foundation for every other Mathematics topic on the LET.

Concepts

The Real Number System

The real number system is organized as a series of nested sets, where each set is completely contained inside the next larger set. Think of it like a set of Russian dolls: the smallest doll (Natural Numbers) fits inside the next (Whole Numbers), which fits inside Integers, which fits inside Rationals, and all of these together with the Irrationals form the Real Numbers. NATURAL NUMBERS (N): Also called counting numbers. These are the numbers we use to count objects in the classroom: 1, 2, 3, 4, 5, ... Zero is NOT a natural number. WHOLE NUMBERS (W): Natural numbers plus zero: 0, 1, 2, 3, 4, ... When you count 'zero pupils absent today,' you are using a whole number. INTEGERS (Z): All whole numbers and their negatives: ..., -3, -2, -1, 0, 1, 2, 3, ... Temperature readings in Baguio City often involve negative integers during December and January. RATIONAL NUMBERS (Q): Any number that can be written as a fraction a/b where both a and b are integers and b ≠ 0. Rational numbers include: all integers (since any integer n = n/1), terminating decimals (0.25 = 1/4), and repeating decimals (0.333... = 1/3, 0.666... = 2/3). The key test: CAN it be written as a fraction of two integers? IRRATIONAL NUMBERS: Numbers that CANNOT be written as a fraction. Their decimal representations neither terminate nor repeat. Examples: π (pi ≈ 3.14159...), √2 ≈ 1.41421..., √3, √5, e. CRITICAL LET POINT: √9 = 3 is RATIONAL (it equals 3/1), but √2 is IRRATIONAL. Always check whether the number under the radical is a perfect square. REAL NUMBERS (R): The union of all rational and irrational numbers. Every number on the number line is a real number. CRITICAL SUBSET RELATIONSHIPS: - Every natural number is also a whole number - Every whole number is also an integer - Every integer is also a rational number (n = n/1) - No irrational number is rational - All rationals and irrationals together make up the real numbers

Examples

The key decision point is always: can this number be expressed as a ratio of two integers? If YES → Rational. If NO → Irrational. For square roots, check if the radicand is a perfect square. For decimals, check if they terminate or repeat. Note that √16 = 4, which is actually a Natural Number — it belongs to ALL the subsets above it.

Scenario

Classify each number: -5, 0, 3/4, √16, √7, 0.272727..., π, 2.5

Solution

-5: Integer, Rational, Real 0: Whole, Integer, Rational, Real 3/4: Rational, Real √16 = 4: Natural, Whole, Integer, Rational, Real √7: Irrational, Real 0.272727... (repeating): Rational, Real (= 3/11) π: Irrational, Real 2.5: Rational, Real (= 5/2)

A) 0.141414... is a repeating decimal = 14/99, so it is RATIONAL. B) √25 = 5, a natural number, so RATIONAL. C) 22/7 is already written as a fraction, so RATIONAL (note: 22/7 is an APPROXIMATION of π, but 22/7 itself is rational). D) √11 — since 11 is not a perfect square (3²=9, 4²=16), √11 is irrational. This is one of the most common LET traps — confusing 22/7 with π.

Scenario

A LET item states: 'Which of the following is an irrational number? A) 0.141414... B) √25 C) 22/7 D) √11'

Solution

The answer is D) √11

Applications

  • Teaching Grade 3 pupils to classify numbers on the number line
  • Explaining to Grade 4 pupils why 0.333... and 1/3 represent the same quantity
  • Differentiating between exact values (π) and approximations (3.14 or 22/7) in problem-solving
  • Helping Grade 6 pupils understand that the square root of a non-perfect square is a non-repeating, non-terminating decimal

Misconceptions

  • WRONG: '22/7 is irrational because it approximates π.' CORRECT: 22/7 is perfectly rational — it is a ratio of two integers. Only π itself is irrational.
  • WRONG: 'All decimals are irrational.' CORRECT: Terminating and repeating decimals are rational.
  • WRONG: 'Negative numbers cannot be rational.' CORRECT: -3/4, -2, -0.5 are all rational.
  • WRONG: '0 is not a real number.' CORRECT: 0 is a whole number, integer, rational, and real number.
  • WRONG: '√4 is irrational because it has a radical sign.' CORRECT: Always evaluate the root first. √4 = 2, which is natural.

Related Concepts

  • Properties of Real Numbers
  • Absolute Value
  • Scientific Notation
  • Place Value
  • Operations on Integers

Common Exam Questions

Example

Which set does -√4 belong to? Step 1: √4 = 2. Step 2: -√4 = -2. Step 3: -2 is a negative integer. Answer: Integer, Rational, and Real — but NOT whole or natural.

Approach

List the nested sets from smallest to largest: N, W, Z, Q, Irrational, R. Determine the SMALLEST set the number belongs to, then list all sets above it.

Question Type

Classification question

Example

'Every integer is a rational number.' TRUE — because any integer n can be written as n/1. 'Every rational number is an integer.' FALSE — 1/2 is rational but not an integer.

Approach

Use the nesting rule: check if the statement claims a number in a smaller set belongs to a larger set (TRUE) or vice versa (possibly FALSE).

Question Type

True or False on subset relationships

Example

Is 0.101001000100001... rational or irrational? The pattern is adding one more zero each time — it does NOT repeat a fixed block, so it is IRRATIONAL.

Approach

Ask two questions: Does it terminate? Does it repeat? If EITHER is yes, it is rational. If BOTH are no, it is irrational.

Question Type

Identifying rational vs. irrational from decimal form

Key Points To Remember

  • Natural numbers start at 1; whole numbers start at 0
  • All integers are rational (any integer n = n/1), but NOT all rationals are integers
  • √(perfect square) is rational: √4=2, √9=3, √16=4, √25=5, √36=6, √49=7, √64=8, √81=9, √100=10
  • √(non-perfect square) is irrational: √2, √3, √5, √6, √7, √8, etc.
  • Repeating decimals ARE rational — they can be written as fractions
  • Terminating decimals ARE rational — they can be written as fractions
  • Pi (π) is irrational — its decimal never terminates nor repeats
  • The number 0 is a whole number, integer, and rational number (0 = 0/1)
  • The nesting order is: N ⊂ W ⊂ Z ⊂ Q ⊂ R

Properties of Real Numbers

Properties of real numbers are the mathematical rules that justify EVERY algebraic step. On the LET, you are often asked to NAME the property that justifies a given step, or to identify which property is demonstrated by a given equation. There are six major properties to master. 1. COMMUTATIVE PROPERTY (from Latin 'commutare' = to exchange) Addition: a + b = b + a → Example: 7 + 3 = 3 + 7 = 10 Multiplication: a × b = b × a → Example: 4 × 9 = 9 × 4 = 36 ⚠ DOES NOT apply to subtraction: 8 - 3 ≠ 3 - 8 ⚠ DOES NOT apply to division: 12 ÷ 4 ≠ 4 ÷ 12 2. ASSOCIATIVE PROPERTY (from 'associate' = to group) Addition: (a + b) + c = a + (b + c) → (2+3)+4 = 2+(3+4) = 9 Multiplication: (a × b) × c = a × (b × c) → (2×3)×4 = 2×(3×4) = 24 ⚠ DOES NOT apply to subtraction or division Key distinction: Commutative changes ORDER; Associative changes GROUPING. 3. DISTRIBUTIVE PROPERTY (MOST TESTED on LET) a(b + c) = ab + ac → Example: 5(3 + 4) = 5×3 + 5×4 = 15 + 20 = 35 Also works with subtraction: a(b - c) = ab - ac → 6(10 - 3) = 60 - 18 = 42 This property 'distributes' the multiplication across the terms inside the parenthesis. 4. IDENTITY PROPERTY Additive Identity: a + 0 = a → 15 + 0 = 15 (Zero is the additive identity) Multiplicative Identity: a × 1 = a → 15 × 1 = 15 (One is the multiplicative identity) 5. INVERSE PROPERTY Additive Inverse: a + (-a) = 0 → 7 + (-7) = 0 (Opposites sum to zero) Multiplicative Inverse (Reciprocal): a × (1/a) = 1, where a ≠ 0 → 5 × (1/5) = 1 6. CLOSURE PROPERTY A set is CLOSED under an operation if performing that operation on members of the set always produces a result that is also in the set. Integers are CLOSED under: addition (3 + (-5) = -2, still an integer), subtraction, multiplication Integers are NOT CLOSED under: division (3 ÷ 2 = 1.5, which is NOT an integer) Natural numbers are NOT CLOSED under subtraction (3 - 5 = -2, NOT a natural number)

Examples

The multiplier 4 is distributed to each addend inside the parenthesis. The result: 4 × 8 = 32, and 20 + 12 = 32. Both sides are equal. This is the most tested property on the LET because it is used in mental math, simplification, and factoring.

Scenario

Name the property illustrated: 4 × (5 + 3) = 4 × 5 + 4 × 3

Solution

Distributive Property of Multiplication over Addition

The numbers themselves did not change order (3, 7, 5 remain in the same left-to-right sequence). What changed is how they are GROUPED — the parentheses moved. This is the Associative (grouping) property, not Commutative (order). Compare with: 3 + 7 = 7 + 3, which is Commutative because the ORDER of 3 and 7 swapped.

Scenario

Name the property: (3 + 7) + 5 = 3 + (7 + 5)

Solution

Associative Property of Addition

-8 and 8 are additive inverses of each other — they are equal in absolute value but opposite in sign. Their sum equals zero, which is the additive identity. This property is the foundation for solving equations: when you 'add the same number to both sides,' you are using this property.

Scenario

Which property states that -8 + 8 = 0?

Solution

Additive Inverse Property (also called Property of Additive Inverses or Inverse Property of Addition)

Applications

  • Justifying steps in solving algebraic equations (foundational for Grade 6 Mathematics)
  • Mental math shortcuts: 4 × 25 = 4 × (20 + 5) = 80 + 20 = 100 (Distributive)
  • Checking computation: 17 × 6 = 17 × 6 (Commutative confirms reversibility)
  • Explaining to pupils why the order of addends does not matter in real-world counting problems
  • Factoring expressions by applying the distributive property in reverse: ab + ac = a(b+c)

Misconceptions

  • WRONG: 'Subtraction is commutative because 5 - 3 = 3 - 5.' CORRECT: 5 - 3 = 2 but 3 - 5 = -2. They are NOT equal.
  • WRONG: 'Associative and Commutative are the same.' CORRECT: Commutative changes ORDER; Associative changes GROUPING without changing order.
  • WRONG: 'Division is associative: (12÷4)÷3 = 12÷(4÷3).' CORRECT: (12÷4)÷3 = 3÷3 = 1, but 12÷(4÷3) = 12÷(4/3) = 9. NOT equal.
  • WRONG: 'The distributive property only works with addition.' CORRECT: a(b-c) = ab - ac also uses the distributive property.

Related Concepts

  • Real Number System
  • Order of Operations
  • Operations on Integers
  • Algebraic Expressions (Chapter 2 link)

Common Exam Questions

Example

Identify the property: 5 × (2 × 9) = (5 × 2) × 9. The numbers are in the same order (5, 2, 9) but the grouping changed → Associative Property of Multiplication.

Approach

Look for four signals: (1) Did the ORDER change? → Commutative. (2) Did the GROUPING change without reordering? → Associative. (3) Was a number distributed across a sum/difference? → Distributive. (4) Was 0 added or 1 multiplied? → Identity.

Question Type

Name-the-property question

Example

Compute 8 × 97 using the Distributive Property: 8 × (100 - 3) = 800 - 24 = 776.

Approach

Recognize when rearranging or regrouping makes computation easier. If given 13 + 47 + 27, regroup as 13 + 27 + 47 = 40 + 47 = 87 (Commutative then Associative).

Question Type

Apply properties for mental math

Key Points To Remember

  • Commutative = change ORDER (a+b = b+a); only addition and multiplication
  • Associative = change GROUPING [(a+b)+c = a+(b+c)]; only addition and multiplication
  • Distributive = MOST TESTED: a(b+c) = ab+ac; can be used forward or in reverse (factoring)
  • Additive identity is 0; multiplicative identity is 1
  • Additive inverse of a number is its NEGATIVE (opposite); their sum = 0
  • Multiplicative inverse of a number is its RECIPROCAL; their product = 1
  • Zero has no multiplicative inverse (cannot divide by zero)
  • Closure: integers are closed under +, -, × but NOT ÷

Order of Operations (GEMDAS / PEMDAS)

The Order of Operations is a universally agreed-upon set of rules that tells us in which sequence to evaluate a mathematical expression. Without this agreement, the same expression could produce different answers depending on how it is read. The LET tests this topic almost every year because it has very specific 'trap' scenarios. The Philippine K-12 curriculum uses the acronym GEMDAS: G - Grouping symbols (parentheses (), brackets [], braces {}, fraction bars, radical signs) E - Exponents (and roots, since roots are fractional exponents) M - Multiplication } D - Division } Done LEFT TO RIGHT, whichever comes FIRST A - Addition } S - Subtraction } Done LEFT TO RIGHT, whichever comes FIRST The international equivalent is PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) or BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction). CRITICAL RULE ABOUT SAME-RANK OPERATIONS: Multiplication and Division have the SAME rank — neither comes before the other. Whichever appears FIRST reading left to right is done first. Similarly, Addition and Subtraction have the SAME rank. THE MOST COMMON LET TRAP: '12 ÷ 4 × 3 = ?' Many test-takers do 4 × 3 first = 12, then 12 ÷ 12 = 1. WRONG. Correct: Read left to right. 12 ÷ 4 = 3 first, then 3 × 3 = 9. NESTED GROUPING SYMBOLS: Work from the INNERMOST grouping outward. Example: 3 × [2 + (4 - 1)] = 3 × [2 + 3] = 3 × 5 = 15 FRACTION BARS as grouping: The fraction bar in (3 + 7)/(2 × 5) means evaluate numerator AND denominator separately before dividing: (3+7)/(2×5) = 10/10 = 1

Examples

This is the exact example from the LET reference material. Notice that after handling the exponent in Step 1, we scan left-to-right for multiplication/division. We encounter division (6÷2) before multiplication (×3), so division is done first. Then multiplication. Then we handle addition and subtraction left to right.

Scenario

Evaluate: 12 + 6 ÷ 2 × 3 - 4²

Solution

Step 1 (Exponents): 4² = 16 → 12 + 6 ÷ 2 × 3 - 16 Step 2 (Mult/Div, left to right): 6 ÷ 2 = 3 → 12 + 3 × 3 - 16 Step 3 (Mult/Div continued): 3 × 3 = 9 → 12 + 9 - 16 Step 4 (Add/Sub, left to right): 12 + 9 = 21 → 21 - 16 Step 5: 21 - 16 = 5 FINAL ANSWER: 5

This is the classic LET trap. Without parentheses, division and multiplication are done left-to-right. If the problem wanted multiplication first, it would use parentheses: 20 ÷ (4 × 5). As written, we must work left to right.

Scenario

Evaluate: 20 ÷ 4 × 5

Solution

Step 1: Read left to right. 20 ÷ 4 = 5 Step 2: 5 × 5 = 25 FINAL ANSWER: 25 COMMON WRONG ANSWER: 20 ÷ (4 × 5) = 20 ÷ 20 = 1 ← This is INCORRECT because there are no parentheses.

Inside the grouping (2²-1), we still apply GEMDAS: exponent before subtraction. Then we exit the grouping and continue with left-to-right multiplication/division before addition.

Scenario

Evaluate: 3 + 4 × (2² - 1) ÷ 5

Solution

Step 1 (Grouping - innermost first): Evaluate (2² - 1) Sub-step: 2² = 4 Sub-step: 4 - 1 = 3 Expression: 3 + 4 × 3 ÷ 5 Step 2 (Mult/Div, left to right): 4 × 3 = 12 → 3 + 12 ÷ 5 Step 3 (Mult/Div continued): 12 ÷ 5 = 2.4 → 3 + 2.4 Step 4 (Addition): 3 + 2.4 = 5.4 FINAL ANSWER: 5.4

Applications

  • Teaching Grade 4 pupils the correct sequence for multi-step word problems
  • Using order of operations to correctly compute complex expressions in science and math
  • Programming and technology literacy: computers follow order of operations strictly in formulas
  • Financial computations: calculating discounts, taxes, and totals in the correct order

Misconceptions

  • WRONG: 'Always multiply before dividing.' CORRECT: Multiplication and division are same-rank; work left to right.
  • WRONG: 'Always add before subtracting.' CORRECT: Addition and subtraction are same-rank; work left to right.
  • WRONG: '-3² = 9.' CORRECT: -3² = -(3²) = -9 because the exponent applies only to 3, not to the negative sign. To get 9, write (-3)².
  • WRONG: 'PEMDAS means P, then E, then M, then D, then A, then S — six separate steps.' CORRECT: MD is one step (same rank, L to R) and AS is one step (same rank, L to R). There are really four stages, not six.

Related Concepts

  • Properties of Real Numbers
  • Operations on Integers
  • Absolute Value
  • Exponents and Scientific Notation

Common Exam Questions

Example

5 + 2 × 3² - (4-1). Work: (4-1)=3; 3²=9; 2×9=18; 5+18-3 = 20. Common wrong answer: computing left to right without GEMDAS gives 5+2=7, ×3=21, ²=441, -(4-1) wrong.

Approach

Apply GEMDAS strictly. Write out each step. The wrong answer choices are carefully designed to match the answer you get if you skip a step or apply operations in the wrong order.

Question Type

Direct evaluation — which answer is correct?

Example

Make 3 + 5 × 2 = 16 true by inserting parentheses. (3+5)×2 = 8×2 = 16. ✓

Approach

Try placing parentheses in different positions and evaluate. The parentheses will force a different operation to happen first.

Question Type

Insert parentheses to make the equation true

Key Points To Remember

  • GEMDAS: Grouping → Exponents → Multiplication/Division (L to R) → Addition/Subtraction (L to R)
  • Multiplication does NOT always come before Division — they are equal rank, done left to right
  • Addition does NOT always come before Subtraction — they are equal rank, done left to right
  • Work INNERMOST grouping symbols first, then outward
  • Fraction bars and radical signs are also grouping symbols
  • Exponents are done BEFORE multiplication, division, addition, or subtraction
  • Without parentheses, -3² means -(3²) = -9, NOT (-3)² = 9
  • A negative sign in front of parentheses distributes: -(3+5) = -3-5 = -8

Operations on Integers (Signed Numbers)

Integers are the set of all whole numbers and their negatives: {..., -3, -2, -1, 0, 1, 2, 3, ...}. Operations on signed numbers follow specific rules that pupils in Grades 5 and 6 are expected to master. As a teacher-candidate, you must know these rules perfectly AND be able to explain them conceptually. ADDITION OF INTEGERS: Rule 1 — SAME SIGNS: Add the absolute values and KEEP the common sign. Example: -5 + (-3) = -(5+3) = -8 (both negative: add and keep negative) Example: 4 + 9 = 13 (both positive: ordinary addition) Rule 2 — DIFFERENT SIGNS: Subtract the smaller absolute value from the larger, and take the sign of the number with the LARGER absolute value. Example: -8 + 3: |−8|=8, |3|=3; 8−3=5; sign of −8 (larger absolute value) → −5 Example: -8 + 12: |−8|=8, |12|=12; 12−8=4; sign of 12 (larger absolute value) → +4 SUBTRACTION OF INTEGERS: Rule: ADD THE OPPOSITE. Convert the subtraction to addition: a − b = a + (−b). Example: 5 − (−3) = 5 + 3 = 8 (subtracting a negative = adding a positive) Example: −6 − 4 = −6 + (−4) = −10 Example: −7 − (−2) = −7 + 2 = −5 MULTIPLICATION AND DIVISION OF INTEGERS: Sign rules (based on the SIGNS of the factors, not their values): POSITIVE × POSITIVE = POSITIVE: 6 × 4 = 24 NEGATIVE × NEGATIVE = POSITIVE: (−6)(−4) = 24 POSITIVE × NEGATIVE = NEGATIVE: 6 × (−4) = −24 NEGATIVE × POSITIVE = NEGATIVE: (−6) × 4 = −24 Same rule for division: (−24) ÷ (−6) = +4; (−24) ÷ 6 = −4 MULTIPLE NEGATIVE FACTORS: Count the number of negative factors: EVEN number of negative factors → POSITIVE product ODD number of negative factors → NEGATIVE product Example: (−2)(−3)(−4) — three negatives (odd) → NEGATIVE: = −24 Example: (−2)(−3)(−4)(−1) — four negatives (even) → POSITIVE: = 24 POWERS OF NEGATIVE NUMBERS: (−3)² = (−3)(−3) = 9 (POSITIVE — even exponent) (−3)³ = (−3)(−3)(−3) = −27 (NEGATIVE — odd exponent) ⚠ TRAP: −3² ≠ (−3)². Without parentheses, −3² = −(3²) = −9

Examples

This is a classic LET-style word problem that uses integer operations. The key is recognizing that 'dropped 3°C per hour for 4 hours' means multiplying a negative rate by a positive time, giving a negative change. Then we add the negative change to the starting temperature.

Scenario

Temperature problem: At noon in Baguio City, temperature was 14°C. It dropped 3°C per hour for 4 hours. What was the temperature at 4:00 PM?

Solution

Drop per hour = −3°C Total drop over 4 hours = −3 × 4 = −12°C Temperature at 4 PM = 14 + (−12) = 2°C FINAL ANSWER: 2°C

This example tests both the power rules for negative numbers AND subtraction of integers. In Step 3, subtracting negative 27 becomes adding 27. A common error is writing (−3)² = −9, confusing it with −3²= −9 (no parentheses).

Scenario

Evaluate: (−3)² − (−3)³

Solution

Step 1: (−3)² = (−3)(−3) = 9 (negative × negative = positive; even exponent) Step 2: (−3)³ = (−3)(−3)(−3) = (9)(−3) = −27 (odd exponent) Step 3: 9 − (−27) = 9 + 27 = 36 FINAL ANSWER: 36

When multiplying several integers, the sign of the answer depends solely on counting how many negative factors there are. Even count → positive, odd count → negative. This shortcut saves time on LET items with long chains of multiplication.

Scenario

Compute the product: (−2)(+3)(−1)(+4)(−2)

Solution

Count negative factors: (−2), (−1), (−2) → 3 negative factors (ODD) Odd number of negatives → product is NEGATIVE Absolute value of product: 2 × 3 × 1 × 4 × 2 = 48 FINAL ANSWER: −48

Applications

  • Teaching Grade 6 pupils about negative numbers in real-life contexts: temperature, sea level, bank balances, football yardage
  • Computing net gains/losses in financial literacy lessons
  • Science applications: pH levels, direction of forces, below-zero temperatures
  • Understanding number lines as a visual tool for integer operations

Misconceptions

  • WRONG: '(−5) + (−3) = −2.' CORRECT: Same signs → add values: 5+3=8, keep the sign → −8.
  • WRONG: 'A negative times a negative is negative.' CORRECT: Negative × Negative = POSITIVE. Only unlike signs give a negative product.
  • WRONG: '−4² = 16.' CORRECT: −4² = −(4²) = −16. For 16, you need (−4)² = 16.
  • WRONG: 'Subtracting makes smaller.' CORRECT: Subtracting a NEGATIVE number makes the result larger. 5 − (−3) = 8 > 5.

Related Concepts

  • Absolute Value
  • Order of Operations
  • Properties of Real Numbers
  • Number Line

Common Exam Questions

Example

What is −15 − (−8) + (−3)? Step 1: −15 + 8 + (−3) [change subtraction to addition]. Step 2: −15 + 8 = −7. Step 3: −7 + (−3) = −10. Answer: −10.

Approach

Apply the sign rules step by step. Never rush the sign determination. Write the sign first, then the value.

Question Type

Direct computation with signed numbers

Example

A submarine is at −120 meters. It rises 45 meters, then descends 70 meters. Final depth: −120 + 45 − 70 = −120 + 45 + (−70) = −75 + (−70) = −145 meters.

Approach

Assign positive/negative values based on context (gain=positive, loss=negative; above=positive, below=negative). Set up the arithmetic expression, then compute.

Question Type

Word problem with signed numbers

Key Points To Remember

  • Same signs in addition: add values, keep the sign
  • Different signs in addition: subtract values, take sign of larger absolute value
  • Subtraction = add the opposite: a − b = a + (−b)
  • Like signs in multiplication/division → Positive result
  • Unlike signs in multiplication/division → Negative result
  • Even count of negative factors → Positive product
  • Odd count of negative factors → Negative product
  • Even exponent on a negative base → Positive result
  • Odd exponent on a negative base → Negative result
  • −3² = −9 (no parentheses, exponent applies to 3 only); (−3)² = +9

Factors, Multiples, and Divisibility

Understanding factors and multiples is essential not only for the LET but for teaching elementary mathematics where these concepts are introduced beginning in Grade 4. Mastery here directly prepares you for the GCF/LCM topic. KEY DEFINITIONS: FACTOR (Divisor): A number that divides another number exactly (no remainder). Example: Factors of 12 are 1, 2, 3, 4, 6, and 12 because each divides 12 exactly. MULTIPLE: The result of multiplying a number by a counting number. Multiples of 4: 4, 8, 12, 16, 20, ... (4×1, 4×2, 4×3, ...) DIVISIBILITY RULES (must be memorized for LET speed): ÷ 2: Last digit is even (0, 2, 4, 6, 8) → 138 ✓ (ends in 8) ÷ 3: Sum of all digits is divisible by 3 → 231: 2+3+1=6, 6÷3=2 ✓ ÷ 4: Last TWO digits form a number divisible by 4 → 316: 16÷4=4 ✓ ÷ 5: Last digit is 0 or 5 → 245 ✓ (ends in 5) ÷ 6: Divisible by BOTH 2 AND 3 → 132: even ✓ and 1+3+2=6÷3=2 ✓ ÷ 8: Last THREE digits divisible by 8 → 1,240: 240÷8=30 ✓ ÷ 9: Sum of all digits is divisible by 9 → 738: 7+3+8=18, 18÷9=2 ✓ ÷ 10: Last digit is 0 → 4,590 ✓ ÷ 11: Alternating sum of digits (odd positions minus even positions) is divisible by 11 → 121: 1−2+1=0 ✓ PRIME AND COMPOSITE NUMBERS: PRIME: Has EXACTLY TWO factors — 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29... ⚠ KEY FACT: 2 is the ONLY even prime number ⚠ KEY FACT: 1 is NEITHER prime NOR composite (only ONE factor: itself) COMPOSITE: Has MORE THAN TWO factors. Examples: 4, 6, 8, 9, 10, 12... 4 has factors: 1, 2, 4 (three factors → composite) PRIME FACTORIZATION: Every composite number can be written as a unique product of prime numbers. This is the Fundamental Theorem of Arithmetic. Method — Factor Tree: 60 → 4 × 15 → (2×2) × (3×5) = 2² × 3 × 5 PRIMES TO MEMORIZE (for quick factoring): 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Examples

This method is efficient because we only test primes up to the square root. If a number has a factor larger than its square root, it must have a corresponding factor smaller than the square root. So if no small factor is found, the number is prime.

Scenario

Is 127 prime or composite?

Solution

Step 1: Find √127 ≈ 11.27. We only need to check prime divisors up to 11. Step 2: Check 2 → 127 is odd, NOT divisible. Step 3: Check 3 → 1+2+7=10, not divisible by 3. Step 4: Check 5 → does not end in 0 or 5. Step 5: Check 7 → 127 ÷ 7 = 18.14... NOT divisible. Step 6: Check 11 → 127 ÷ 11 = 11.5... NOT divisible. Since no prime ≤ √127 divides 127, it is PRIME.

Divisibility rules let us answer in seconds without performing long division. For LET multiple-choice items, eliminate options using the quickest rule first. Here, testing divisibility by 4 is faster (just last 2 digits), so we eliminate D first, then test 9 on the remaining options.

Scenario

Use divisibility rules to determine which of the following is divisible by both 4 and 9: A) 2,736 B) 3,204 C) 5,184 D) 7,218

Solution

Check divisibility by 4 (last two digits ÷ 4): A) 36 ÷ 4 = 9 ✓ B) 04 ÷ 4 = 1 ✓ C) 84 ÷ 4 = 21 ✓ D) 18 ÷ 4 = 4.5 ✗ (eliminate D) Check divisibility by 9 (digit sum ÷ 9) for A, B, C: A) 2+7+3+6 = 18; 18÷9 = 2 ✓ B) 3+2+0+4 = 9; 9÷9 = 1 ✓ C) 5+1+8+4 = 18; 18÷9 = 2 ✓ All three (A, B, C) are divisible by both 4 and 9. ANSWER: A, B, and C.

Applications

  • Teaching Grade 4 and 5 pupils how to find factors using divisibility rules
  • Using prime factorization as the foundation for computing GCF and LCM
  • Identifying prime numbers in cryptography and number theory
  • Creating fair groups in the classroom: 'Can 24 pupils be divided into equal groups of 4? Yes — 4 is a factor of 24.'

Misconceptions

  • WRONG: '1 is a prime number.' CORRECT: 1 has only ONE factor (itself). Prime numbers must have EXACTLY TWO distinct factors. 1 is neither prime nor composite.
  • WRONG: 'Even numbers are composite.' CORRECT: 2 is even AND prime.
  • WRONG: 'Divisibility by 6 just means checking the last digit.' CORRECT: Divisibility by 6 requires divisibility by BOTH 2 (even last digit) AND 3 (digit sum divisible by 3).
  • WRONG: 'Testing division by 2, 4, 6, 8 checks more cases.' CORRECT: You only need to test PRIME divisors (2, 3, 5, 7, 11, ...) because if a number is divisible by a composite, it is already divisible by that composite's prime factors.

Related Concepts

  • GCF and LCM
  • Real Number System (Integers)
  • Fractions (simplification uses GCF)
  • Estimation

Common Exam Questions

Example

Express 360 as a product of prime factors: 360 = 8×45 = (2³)×(9×5) = 2³ × 3² × 5.

Approach

For prime/composite: test divisibility by primes up to the square root. For prime factorization: use a factor tree and write in exponential form.

Question Type

Identify prime or composite, OR find prime factorization

Example

Which number is divisible by 9? A) 513 B) 524 C) 531 D) 542. Digit sums: A)9✓ B)11✗ C)9✓ D)11✗. Both A and C qualify; test against other choices.

Approach

Memorize the rules exactly. For multi-part conditions (like divisibility by 6), check each component separately.

Question Type

Divisibility rule application

Key Points To Remember

  • 1 is NEITHER prime nor composite — this is the single most tested fact about primes on the LET
  • 2 is the ONLY even prime number
  • Divisibility by 3 or 9: use DIGIT SUM test
  • Divisibility by 4: test LAST TWO digits only
  • Divisibility by 6 = divisible by BOTH 2 and 3
  • Divisibility by 8: test LAST THREE digits
  • Every composite number has a unique prime factorization (Fundamental Theorem of Arithmetic)
  • A number is prime if it has no prime factor less than or equal to its square root
  • To find all factors of n: systematically pair them (1×n, 2×?, 3×?, etc.)
  • Perfect squares have an ODD number of factors; all other composites have an even number

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

GCF and LCM are among the most heavily tested topics in LET Mathematics because they appear in both direct computation questions AND in real-world word problems. The critical skill is knowing WHICH one to use in a given situation. GREATEST COMMON FACTOR (GCF): Definition: The LARGEST factor that is shared by all given numbers. When to use: Any problem involving EQUAL GROUPS or SPLITTING with NOTHING LEFT OVER. 'Greatest number of groups,' 'largest equal groups,' 'distributing evenly' → GCF METHOD 1 — Prime Factorization: 1. Write prime factorization of each number 2. Identify COMMON prime factors 3. Use the LOWEST (smallest) exponent for each common prime 4. Multiply these together Example: GCF(18, 24) 18 = 2 × 3² 24 = 2³ × 3 Common primes: 2 (lowest exponent: 2¹) and 3 (lowest exponent: 3¹) GCF = 2¹ × 3¹ = 6 METHOD 2 — Listing Factors: Factors of 18: 1, 2, 3, 6, 9, 18 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Common factors: 1, 2, 3, 6 → Greatest: 6 ✓ LEAST COMMON MULTIPLE (LCM): Definition: The SMALLEST multiple that is shared by all given numbers. When to use: Any problem involving REPEATING EVENTS that must COINCIDE, or when ADDING/SUBTRACTING FRACTIONS. 'When will they meet again,' 'next time they ring together,' 'adding fractions' → LCM METHOD 1 — Prime Factorization: 1. Write prime factorization of each number 2. Identify ALL primes that appear in ANY number 3. Use the HIGHEST exponent for each prime 4. Multiply these together Example: LCM(18, 24) 18 = 2 × 3² 24 = 2³ × 3 All primes: 2 (highest exponent: 2³) and 3 (highest exponent: 3²) LCM = 2³ × 3² = 8 × 9 = 72 METHOD 2 — Listing Multiples: Multiples of 18: 18, 36, 54, 72, 90... Multiples of 24: 24, 48, 72, 96... First common multiple: 72 ✓ USEFUL CHECK FORMULA: For any TWO numbers: GCF × LCM = Product of the two numbers Verification: 6 × 72 = 432 = 18 × 24 = 432 ✓ This formula lets you find LCM if you know GCF (and vice versa), and to check your work. GCF vs. LCM DECISION TABLE: 'Greatest number of students to share items equally' → GCF 'Largest equal groups with nothing left over' → GCF 'Next time two events coincide' → LCM 'When will they meet again?' → LCM 'Least number of tiles to cover both rooms' → LCM 'Adding fractions' (finding LCD) → LCM

Examples

The key word 'greatest number of students' combined with 'nothing left over' signals GCF. GCF gives the largest equal grouping that divides both quantities exactly. This is a standard LET problem type used in practical school scenarios.

Scenario

GCF problem: A teacher has 48 pencils and 36 notebooks to distribute so that every student receives the same number of pencils and the same number of notebooks, with none left over. What is the greatest number of students who can share the items?

Solution

This is a GCF problem ('greatest number' + 'nothing left over') Step 1: Prime factorize both numbers 48 = 2⁴ × 3 36 = 2² × 3² Step 2: Common primes are 2 and 3. Take LOWEST exponents: 2 → min(4,2) = 2² and 3 → min(1,2) = 3¹ Step 3: GCF = 2² × 3 = 4 × 3 = 12 Answer: 12 students can receive items. Verification: 48 ÷ 12 = 4 pencils each; 36 ÷ 12 = 3 notebooks each. ✓

This is the classic 'repeating events' problem type. The LCM gives the first moment when both events coincide again after a common starting point. The barangay church bells context is a culturally relevant example that could appear in any LET examination.

Scenario

LCM problem: Two church bells in a barangay toll at intervals of 18 minutes and 24 minutes. If they toll together at 8:00 AM, at what time will they next toll together?

Solution

This is an LCM problem ('when will they next coincide') Step 1: Prime factorize both numbers 18 = 2 × 3² 24 = 2³ × 3 Step 2: ALL primes, HIGHEST exponents: 2 → max(1,3) = 2³ and 3 → max(2,1) = 3² Step 3: LCM = 2³ × 3² = 8 × 9 = 72 minutes Step 4: Convert 72 minutes to hours: 72 ÷ 60 = 1 hour and 12 minutes 8:00 AM + 1 hour 12 minutes = 9:12 AM Answer: 9:12 AM

This problem tests the formula GCF × LCM = product of two numbers. Knowing this formula converts a seemingly complex problem into a simple equation. Always verify your answer by computing the GCF and LCM of the two numbers you found.

Scenario

Mixed: The GCF of two numbers is 6 and their LCM is 60. If one number is 12, what is the other?

Solution

Use the formula: GCF × LCM = Product of two numbers 6 × 60 = 12 × second number 360 = 12 × second number Second number = 360 ÷ 12 = 30 Answer: 30 Verification: GCF(12, 30): 12 = 2² × 3; 30 = 2 × 3 × 5 GCF = 2 × 3 = 6 ✓ LCM = 2² × 3 × 5 = 60 ✓

Applications

  • Simplifying fractions: divide numerator and denominator by their GCF
  • Finding the LCD for adding/subtracting fractions: the LCD is the LCM of the denominators
  • Resource distribution problems in school settings (dividing supplies equally)
  • Scheduling problems: when do two repeating events next occur simultaneously
  • Grade 5-6 mathematics teaching: real-world problem-solving using GCF and LCM

Misconceptions

  • WRONG: 'GCF is always found by multiplying; LCM by dividing.' CORRECT: The distinction is based on the words 'common' (GCF) vs. 'multiple' (LCM), not multiplication vs. division.
  • WRONG: 'LCM is the larger number.' CORRECT: LCM is the smallest COMMON MULTIPLE, but it can be equal to one of the given numbers. LCM(4,8) = 8.
  • WRONG: 'GCF and LCM are always the same type of problem.' CORRECT: Always identify whether it is a splitting/grouping (GCF) or coinciding events/fractions (LCM) problem.
  • WRONG: 'GCF × LCM = sum of the numbers.' CORRECT: GCF × LCM = PRODUCT (multiplication) of the two numbers.

Related Concepts

  • Prime Factorization
  • Fractions (simplification, LCD)
  • Factors and Multiples
  • Word Problem Strategies

Common Exam Questions

Example

Find GCF and LCM of 90 and 126. 90=2×3²×5; 126=2×3²×7. GCF=2×3²=18. LCM=2×3²×5×7=630. Check: 18×630=11,340=90×126=11,340 ✓

Approach

Always use prime factorization for numbers larger than 20. For GCF: common primes, lowest powers. For LCM: all primes, highest powers. Write the factorizations side by side for easy comparison.

Question Type

Direct computation of GCF or LCM

Example

A janitor mops Room A every 4 days and Room B every 6 days. If he mops both today, in how many days will he mop both rooms on the same day again? LCM(4,6)=12 days.

Approach

Read the problem carefully. 'Largest equal groups / distributing evenly / nothing left over' → GCF. 'When do repeated events coincide again / next meeting time' → LCM.

Question Type

Word problem requiring correct identification of GCF or LCM

Key Points To Remember

  • GCF = common primes with LOWEST exponents; LCM = all primes with HIGHEST exponents
  • GCF is for EQUAL SPLITTING problems (grouping, distributing); LCM is for REPEATING EVENTS problems (coinciding, meeting again)
  • GCF × LCM = product of the two numbers (verification tool)
  • GCF is always LESS THAN OR EQUAL TO both numbers
  • LCM is always GREATER THAN OR EQUAL TO both numbers
  • If two numbers share no common factors (they are relatively prime), their GCF = 1
  • If two numbers are relatively prime, their LCM = their product
  • GCF is used when SIMPLIFYING FRACTIONS to lowest terms
  • LCM is used as the Least Common Denominator (LCD) when adding/subtracting fractions

Estimation and Mental Computation

Estimation is the ability to make a reasonably accurate guess at an answer without performing exact computation. On the LET, estimation questions test whether you can arrive at a quick, reasonable answer AND whether you can check if a given answer is reasonable (catching gross errors). METHOD 1 — ROUNDING TO THE LEADING DIGIT (FRONT-END ESTIMATION): Round each number to its leading (leftmost non-zero) digit, then compute. Example: Estimate 4,987 × 21 Round: 5,000 × 20 = 100,000 Exact: 104,727 This confirms the exact answer is in the correct range. METHOD 2 — COMPATIBLE NUMBERS: Replace numbers with nearby numbers that are easy to compute mentally. Example: Estimate 487 ÷ 52 Replace with: 500 ÷ 50 = 10 This is easy to compute and gives a reasonable estimate. METHOD 3 — CLUSTERING: When several numbers are close to a common value, multiply that common value by the count. Example: 48 + 51 + 53 + 47 + 50 ≈ 50 × 5 = 250 MENTAL COMPUTATION SHORTCUTS: × 10: Move decimal one place right → 3.45 × 10 = 34.5 × 100: Move decimal two places right → 3.45 × 100 = 345 × 5: Multiply by 10, then halve → 48 × 5 = 480 ÷ 2 = 240 × 25: Multiply by 100, then divide by 4 → 36 × 25 = 3,600 ÷ 4 = 900 × 9: Multiply by 10, then subtract the original → 67 × 9 = 670 − 67 = 603 × 11: Write the number, insert the digit sum in between → 23 × 11: sum=5, answer=253 ROUNDING RULES (for the LET): Look at the digit IMMEDIATELY TO THE RIGHT of the place you are rounding to. 5 or more → round UP (increase the target digit by 1) 4 or less → round DOWN (keep the target digit, truncate the rest) Examples: 3,748 rounded to the nearest hundred: look at tens digit (4) → round DOWN → 3,700 3,748 rounded to the nearest ten: look at ones digit (8) → round UP → 3,750 3,748 rounded to the nearest thousand: look at hundreds digit (7) → round UP → 4,000

Examples

This example shows how estimation serves as a check: if the exact answer were ₱1,047,270 (misplaced decimal) or ₱10,472.70 (missing a zero), the estimate of ₱100,000 would catch the error immediately. Teaching pupils to estimate before computing is a key elementary mathematics strategy.

Scenario

A school orders 21 boxes of tiles at ₱4,987 per box. Estimate the total cost, then find the exact cost.

Solution

ESTIMATION: Round 4,987 ≈ 5,000 and 21 ≈ 20 Estimate: 5,000 × 20 = ₱100,000 EXACT CALCULATION: 4,987 × 21 = 4,987 × 20 + 4,987 × 1 = 99,740 + 4,987 = ₱104,727 Reasonableness check: ₱104,727 is close to ₱100,000 estimate. ✓ The exact answer is in the same ballpark as the estimate, confirming no computation error.

The place values from right to left in 5,672,348 are: ones(8), tens(4), hundreds(3), thousands(2), ten-thousands(7), hundred-thousands(6), millions(5). The ten-thousands digit is 7. The digit immediately to its right is 2 (thousands). Since 2 < 5, we keep 7 and replace the rest with zeros.

Scenario

Round 5,672,348 to the nearest ten-thousand.

Solution

Step 1: Identify the ten-thousands digit: 5,6[7]2,348 — the ten-thousands digit is 7 (in 5,670,000's place). Step 2: Look at the digit immediately to its right: the thousands digit is 2. Step 3: Since 2 < 5, round DOWN — keep the 7, replace all digits to the right with zeros. Answer: 5,670,000

Applications

  • Teaching Grade 3-4 pupils to check the reasonableness of answers using estimation
  • Mental math in daily life: computing approximate costs when shopping
  • Scientific calculations: using scientific notation and estimation together
  • DepEd classroom strategy: 'Estimate before you calculate' as a problem-solving habit

Misconceptions

  • WRONG: 'Round 35 to the nearest ten gives 30.' CORRECT: The ones digit is 5 (≥5), so round UP → 40.
  • WRONG: 'Estimation means any rough guess.' CORRECT: Estimation uses specific strategies (rounding, compatible numbers) to produce a mathematically reasonable approximation.
  • WRONG: 'After rounding, keep the original digits to the right of the rounded place.' CORRECT: Replace all digits to the right of the rounded place with ZEROS.

Related Concepts

  • Place Value
  • Rounding
  • Scientific Notation
  • Mental Computation Strategies

Common Exam Questions

Example

Estimate 3,914 + 2,073 + 4,156. Round to thousands: 4,000 + 2,000 + 4,000 = 10,000. Exact: 10,143. Best estimate is 10,000.

Approach

Round each number to its leading digit and compute. Choose the option closest to your rounded answer.

Question Type

Which estimate is closest / most reasonable?

Example

Round 47,856 to the nearest thousand: thousands digit=7, look right: 8≥5, so round up 7→8: answer=48,000.

Approach

Identify the target place. Find the digit to its immediate right. Apply the rule: ≥5 round up, <5 round down (keep). Replace all digits to the right with zeros.

Question Type

Rounding to a specific place value

Key Points To Remember

  • When rounding, look at the digit IMMEDIATELY TO THE RIGHT of the target place
  • 5 or more → round up; 4 or less → keep (round down)
  • Leading digit rounding: round to the first significant (non-zero) digit
  • Compatible numbers: replace with nearby round numbers for easy mental calculation
  • × 5 shortcut: multiply by 10 then divide by 2
  • × 25 shortcut: multiply by 100 then divide by 4
  • Estimation is used to check reasonableness — a key mathematical habit of mind
  • On the LET, 'approximately equal to' or 'estimate' questions expect rounded, not exact, values
  • Front-end estimation may underestimate or overestimate — that is acceptable for checking purposes

Absolute Value, Place Value, Scientific Notation, and Roman Numerals

ABSOLUTE VALUE: The absolute value of a number is its distance from zero on the number line. Distance is always non-negative, so absolute value is NEVER negative. |a| = a if a ≥ 0, and |a| = −a if a < 0 Examples: |7| = 7; |−7| = 7; |0| = 0 Apply GEMDAS INSIDE absolute value bars before taking the absolute value: |3 − 8| = |−5| = 5 (NOT |3| − |8| = 3 − 8 = −5) COMPARING INTEGERS using absolute value: On the number line, numbers increase to the RIGHT. So: −3 > −8 (because −3 is to the RIGHT of −8 on the number line) |−8| > |−3| (because 8 > 3), but in INTEGER comparison, −3 > −8 PLACE VALUE: In our base-10 number system, the value of a digit depends on its POSITION. From the decimal point: LEFT: ones, tens, hundreds, thousands, ten-thousands, hundred-thousands, millions... RIGHT: tenths, hundredths, thousandths... In 4,725: the 7 represents 700 (sevens in the hundreds place) In 27.4: the 4 represents 4-tenths (0.4) SCIENTIFIC NOTATION: A way of writing very large or very small numbers as: N × 10ⁿ where 1 ≤ N < 10 and n is an integer. To convert TO scientific notation: 1. Move the decimal point until ONE non-zero digit is to the left 2. Count how many places you moved 3. Moving LEFT → positive exponent (large numbers) 4. Moving RIGHT → negative exponent (small numbers < 1) Examples: 45,000 → decimal moves 4 places left → 4.5 × 10⁴ 0.0032 → decimal moves 3 places right → 3.2 × 10⁻³ ROMAN NUMERALS: I=1, V=5, X=10, L=50, C=100, D=500, M=1,000 ADDITION rule: same or decreasing symbols left to right → add: VIII = 5+1+1+1 = 8 SUBTRACTION rule: smaller symbol BEFORE larger → subtract: IV = 5−1 = 4 STANDARD SUBTRACTIVE PAIRS: IV=4, IX=9, XL=40, XC=90, CD=400, CM=900 Examples: 49 = 40 + 9 = XL + IX = XLIX 2024 = 2000 + 20 + 4 = MM + XX + IV = MMXXIV

Examples

Absolute value simply removes the negative sign. Once all absolute values are computed, treat the expression as ordinary arithmetic. The answer is 8, not −6 (which would result from incorrectly writing the expression without taking absolute value first).

Scenario

Evaluate: |−7| + |3| − |−2|

Solution

Step 1: Compute each absolute value: |−7| = 7, |3| = 3, |−2| = 2 Step 2: Substitute: 7 + 3 − 2 Step 3: Left to right: 7 + 3 = 10; 10 − 2 = 8 FINAL ANSWER: 8

The rule is: count how many places the decimal moves. Right = negative exponent (small number). Left = positive exponent (large number). The coefficient (7.86) must satisfy 1 ≤ coefficient < 10.

Scenario

Write 0.0000786 in scientific notation.

Solution

Step 1: Move decimal point to get one non-zero digit to the left. 0.0000786 → 7.86 (moved 5 places to the RIGHT) Step 2: Moving right means the original number is SMALL (< 1), so the exponent is NEGATIVE. Answer: 7.86 × 10⁻⁵ Verification: 7.86 × 10⁻⁵ = 7.86 × 0.00001 = 0.0000786 ✓

Work left to right. When a smaller-value symbol appears BEFORE a larger-value symbol, subtract. When equal or decreasing, add. MMCMXCIX is the Roman numeral for 2,999. Notice that CM (not DC) is used for 900, and XC (not LXXXX) is used for 90.

Scenario

Convert MMCMXCIX to Hindu-Arabic numeral.

Solution

Break down each symbol group: MM = 1000 + 1000 = 2000 CM = 1000 − 100 = 900 (C before M = subtract) XC = 100 − 10 = 90 (X before C = subtract) IX = 10 − 1 = 9 (I before X = subtract) Total = 2000 + 900 + 90 + 9 = 2,999 FINAL ANSWER: 2,999

Applications

  • Absolute value: computing difference between temperatures, scores, or positions (always a positive distance)
  • Place value: foundation for all arithmetic operations taught in Grades 1-4
  • Scientific notation: connecting to science lessons (speed of light = 3×10⁸ m/s, size of a bacterium = 10⁻⁶ m)
  • Roman numerals: used in chapter numbers, clock faces, and formal documents; Grade 2-3 curriculum topic

Misconceptions

  • WRONG: '|3 − 8| = |3| − |8| = 3 − 8 = −5.' CORRECT: Evaluate inside first: 3−8=−5, then |−5|=5. Absolute value cannot be distributed across subtraction.
  • WRONG: 'A negative exponent in scientific notation means a negative number.' CORRECT: 3.2 × 10⁻³ = 0.0032 (a POSITIVE small number). Negative exponent means SMALL, not negative.
  • WRONG: 'IC = 99 in Roman numerals.' CORRECT: I can only be subtracted from V and X. For 99, use XCIX (90 + 9). Standard pairs must be used.
  • WRONG: 'IIII = 4 in Roman numerals.' CORRECT: Modern standard uses IV = 4. Repeating a symbol more than 3 times is non-standard.

Related Concepts

  • Order of Operations
  • Scientific Notation and Metric Prefixes
  • Place Value and Rounding
  • Number Line and Integers

Common Exam Questions

Example

|5 − 12| − |3 − 7| = |−7| − |−4| = 7 − 4 = 3.

Approach

ALWAYS evaluate the expression inside the absolute value bars FIRST. Then apply the absolute value (make it positive). Then continue with outside operations.

Question Type

Absolute value computation with operations inside

Example

6.02 × 10²³ → move decimal 23 places right → 602,000,000,000,000,000,000,000 (Avogadro's number).

Approach

Standard to scientific: count decimal moves, determine sign of exponent. Scientific to standard: positive exponent → move right (larger); negative exponent → move left (smaller).

Question Type

Convert between scientific and standard notation

Example

CDXLIV: CD=400, XL=40, IV=4 → 444.

Approach

Read left to right. If current value is less than next value → subtract. Otherwise → add. Build a table from M to I.

Question Type

Roman numeral conversion

Key Points To Remember

  • Absolute value is always NON-NEGATIVE: |number| ≥ 0 always
  • Evaluate expressions INSIDE absolute value bars before taking absolute value
  • Comparing negative integers: the one CLOSER to zero is LARGER (−3 > −8)
  • Scientific notation: coefficient is between 1 and 10 (1 ≤ N < 10)
  • Large number (moved decimal left) → POSITIVE exponent
  • Small number less than 1 (moved decimal right) → NEGATIVE exponent
  • Roman numeral subtraction: ONLY specific pairs are valid (IV, IX, XL, XC, CD, CM)
  • In Roman numerals, a symbol can be repeated at most 3 times (III = 3, but IIII is non-standard)
  • Place value: digit's value = digit × position value

Practice Problems

The key skill tested is recognizing perfect squares (4, 9, 16, 25, 36, 49, 64, 81, 100) and understanding that only their square roots are rational. For repeating decimals, remember they are always rational. The answer D is correct because 50 is not a perfect square.

Problem

Problem 1 (Real Number System): Which of the following is classified as an irrational number? A) 0.252525... B) √36 C) −7/3 D) √50

Solution

ANSWER: D) √50 Step 1: Analyze each option: A) 0.252525... is a REPEATING decimal → Rational (= 25/99) B) √36 = 6 → Natural number, hence Rational C) −7/3 is already a fraction of two integers → Rational D) √50 = √(25 × 2) = 5√2. Since √2 is irrational, 5√2 is irrational. 50 is NOT a perfect square (7²=49, 8²=64), so √50 is irrational.

Apply GEMDAS strictly: G first (parentheses), then E (exponent), then MD left-to-right (4×3=12 then 12÷6=2), then AS left-to-right (3+2=5 then 5-1=4). A common error is computing 2² after the multiplication, or doing subtraction before addition.

Problem

Problem 2 (Order of Operations): Evaluate: 3 + 2² × (8 − 5) ÷ 6 − 1

Solution

ANSWER: 5 Step 1 (Grouping): (8 − 5) = 3 Expression becomes: 3 + 2² × 3 ÷ 6 − 1 Step 2 (Exponent): 2² = 4 Expression becomes: 3 + 4 × 3 ÷ 6 − 1 Step 3 (Mult/Div, left to right): 4 × 3 = 12 → 3 + 12 ÷ 6 − 1 12 ÷ 6 = 2 → 3 + 2 − 1 Step 4 (Add/Sub, left to right): 3 + 2 = 5 → 5 − 1 = 4 FINAL ANSWER: 4

This integer word problem uses negative numbers to represent a decrease in temperature. Multiply the negative rate (−2) by the positive time (6) to get total change (−12). Then add the change to the starting value (8). This type of problem is common in LET Mathematics.

Problem

Problem 3 (Integers): The temperature in a storage room was 8°C. The temperature dropped 2°C per hour for 6 hours. What was the temperature after 6 hours?

Solution

ANSWER: −4°C Step 1: Temperature drop per hour = −2°C Step 2: Total drop after 6 hours = −2 × 6 = −12°C Step 3: Final temperature = 8 + (−12) = 8 − 12 = −4°C

The phrase 'both buy on the same day again' signals LCM. They both buy today (day 0), then next coincidence is at day 36. Verify: multiples of 12 include 12, 24, 36; multiples of 18 include 18, 36. First common multiple after 0 is 36 ✓

Problem

Problem 4 (GCF/LCM): Two teachers are buying school supplies. Teacher A buys a set of pens every 12 days and Teacher B every 18 days. If they both bought pens today, in how many days will they both buy pens on the same day again?

Solution

ANSWER: 36 days This is an LCM problem (two repeating events coinciding again). Step 1: Prime factorize: 12 = 2² × 3 18 = 2 × 3² Step 2: LCM = all primes with HIGHEST exponents: 2 → 2² (highest); 3 → 3² (highest) LCM = 2² × 3² = 4 × 9 = 36 Answer: 36 days

The key phrases 'equal rows,' 'same number per row,' and 'greatest number' all point to GCF. The principal wants as few rows as possible (largest groups), which is achieved by using the GCF.

Problem

Problem 5 (GCF Application): A school principal wants to arrange 72 Grade 5 pupils and 48 Grade 6 pupils into equal rows for a program, with each row having only one grade level and the same number of pupils in every row. What is the greatest number of pupils per row?

Solution

ANSWER: 24 pupils per row This is a GCF problem (greatest equal groups). Step 1: Prime factorize: 72 = 2³ × 3² 48 = 2⁴ × 3 Step 2: GCF = common primes with LOWEST exponents: 2 → 2³ (lowest of 3,4); 3 → 3¹ (lowest of 2,1) GCF = 2³ × 3 = 8 × 3 = 24 Answer: 24 pupils per row Verification: 72 ÷ 24 = 3 rows of Grade 5; 48 ÷ 24 = 2 rows of Grade 6 ✓

For each: (a) 7 is multiplied to each term inside parentheses → Distributive. (b) Numbers 2, 5, 8 are in the same left-to-right order, but the GROUPING moved → Associative (not Commutative). (c) Product equals 1 → Inverse (multiplicative). (d) Sum equals 0 → Inverse (additive).

Problem

Problem 6 (Properties): Identify the property illustrated by each equation: a) 7 × (4 + 3) = 7 × 4 + 7 × 3 b) (2 + 5) + 8 = 2 + (5 + 8) c) 9 × (1/9) = 1 d) −15 + 15 = 0

Solution

a) Distributive Property of Multiplication over Addition b) Associative Property of Addition (grouping changed; order stayed the same: 2, 5, 8) c) Multiplicative Inverse Property (9 and 1/9 are reciprocals; their product = 1) d) Additive Inverse Property (−15 and 15 are opposites; their sum = 0)

This problem demonstrates the importance of the divisibility rules. Divisibility by 6 = div by 2 AND div by 3. Divisibility by 9 requires digit sum divisible by 9 (a stricter requirement than divisibility by 3). A number can be divisible by 3 (digit sum divisible by 3) but not by 9 (digit sum not divisible by 9). Example: 24 has digit sum 6 (÷3 yes, ÷9 no) and is even → divisible by 6 but not 9.

Problem

Problem 7 (Divisibility): Which of the following numbers is divisible by 6 but NOT by 9? A) 162 B) 216 C) 198 D) 126

Solution

ANSWER: D) 126 Divisible by 6 requires: divisible by BOTH 2 (even) AND 3 (digit sum ÷ 3) Divisible by 9 requires: digit sum ÷ 9 Check each: A) 162: even ✓; 1+6+2=9, ÷3 ✓, ÷9 ✓ → Div by 6 AND 9. Eliminate. B) 216: even ✓; 2+1+6=9, ÷3 ✓, ÷9 ✓ → Div by 6 AND 9. Eliminate. C) 198: even ✓; 1+9+8=18, ÷3 ✓, ÷9 ✓ → Div by 6 AND 9. Eliminate. D) 126: even ✓; 1+2+6=9, ÷3 ✓ → Div by 6 ✓; 9 ÷ 9 = 1... wait: digit sum is 9, which IS divisible by 9. Re-check all options looking for divisible by 6 but NOT 9: Let us try 132: 1+3+2=6, ÷3✓, ÷9? No. But 132 is not in choices. Among given options, let us recheck D) 126: 1+2+6=9, 9÷9=1 → IS divisible by 9. Actually ALL of A, B, C, D have digit sums divisible by 9. Let's use the correct answer from digit sums: A: 1+6+2=9 ✓ by 9 B: 2+1+6=9 ✓ by 9 C: 1+9+8=18 ✓ by 9 D: 1+2+6=9 ✓ by 9 This problem as stated would need different numbers for a clear answer. The correct teaching point: to be divisible by 6 but not 9, find a number divisible by 2 and 3, where digit sum is divisible by 3 but NOT by 9. Example: 24 (digit sum=6, divisible by 3 but not 9, and even) → divisible by 6 but not 9.

Front-end estimation rounds to the leading digit. 498 rounds to 500 (leading digit 5 in hundreds place); 32 rounds to 30 (leading digit 3 in tens place). The exact computation uses the distributive property: 498 × 32 = 498 × (30 + 2). The estimate correctly predicted the magnitude (five-digit number around 15,000).

Problem

Problem 8 (Estimation): Estimate 498 × 32 using front-end estimation. Then verify by computing the exact answer.

Solution

ESTIMATION (front-end / leading digit): Round 498 ≈ 500 and 32 ≈ 30 Estimate = 500 × 30 = 15,000 EXACT CALCULATION: 498 × 32 = 498 × 30 + 498 × 2 = 14,940 + 996 = 15,936 Reasonableness check: 15,936 is close to 15,000 estimate. ✓ The estimate is within about 6% of the exact answer — reasonable.

XL is one of the six standard subtractive pairs (XL = 40). V = 5 and II = 2. Adding: 40 + 5 + 2 = 47. This is a common LET item type since Roman numerals appear in the Grade 2-3 elementary curriculum. Memorize the six subtractive pairs: IV=4, IX=9, XL=40, XC=90, CD=400, CM=900.

Problem

Problem 9 (Roman Numerals): A textbook chapter heading reads 'CHAPTER XLVII.' In Hindu-Arabic numerals, this chapter number is:

Solution

ANSWER: 47 Break down XLVII: XL: X before L means subtract: 50 − 10 = 40 V: 5 II: 1 + 1 = 2 Total: 40 + 5 + 2 = 47

For numbers smaller than 1, moving the decimal to the right requires a NEGATIVE exponent. Count carefully: 0.000000000027 — there are 10 zeros after the decimal before the 2, so the decimal moves 11 places right (from original position past the 1, 0 before the 2... count the zeros: 0.0(1)0(2)0(3)0(4)0(5)0(6)0(7)0(8)0(9)0(10)2(11)7). The exponent is −11.

Problem

Problem 10 (Scientific Notation): The mass of a red blood cell is approximately 0.000000000027 kg. Express this in scientific notation.

Solution

ANSWER: 2.7 × 10⁻¹¹ Step 1: Move the decimal point to get one non-zero digit to the left of the decimal: 0.000000000027 → 2.7 Step 2: Count how many places the decimal moved: Moving from the original position to after '2': 11 places to the RIGHT Step 3: Moving right → negative exponent Answer: 2.7 × 10⁻¹¹

Exam Preparation Tips

  • MEMORIZE the nested sets of the real number system: N ⊂ W ⊂ Z ⊂ Q ⊂ R. Know which set is the 'home base' of each type of number. Perfect squares have rational square roots; all others are irrational.
  • NEVER forget: 1 is neither prime nor composite; 2 is the only even prime. These are the top two tested facts about prime numbers on the LET.
  • For ORDER OF OPERATIONS, always write out every step — never try to do multiple steps mentally in one jump. The traps are always in the Multiplication/Division step (they are equal rank, not M before D) and the Addition/Subtraction step (equal rank, not A before S).
  • SIGNED NUMBERS: Before computing, determine the SIGN of your answer. Wrong signs cause more errors than wrong calculations. Even count of negative factors → positive; odd count → negative.
  • For GCF/LCM word problems, ALWAYS identify which one is needed BEFORE computing: 'split into equal groups/nothing left over' → GCF; 'next coincidence/repeating events' → LCM. Memorize the formula GCF × LCM = product of two numbers as a verification tool.
  • DIVISIBILITY RULES: Practice the digit-sum rule for 3 and 9 until it is automatic. Test the last-two-digits for divisibility by 4 on any number. These are the two rules most frequently bypassed by test-takers who do long division instead.
  • PRACTICE PRIME FACTORIZATION using factor trees until you can factorize numbers up to 200 in under 30 seconds. This single skill unlocks GCF, LCM, and fraction simplification simultaneously.
  • For ABSOLUTE VALUE, remember: evaluate the INSIDE first (using GEMDAS), THEN take the absolute value. Never distribute absolute value across addition or subtraction.
  • SCIENTIFIC NOTATION: The simple rule is 'large number → positive exponent (decimal moved left), small number less than 1 → negative exponent (decimal moved right).' Practice by converting the metric prefixes: kilo=10³, milli=10⁻³, micro=10⁻⁶.
  • ESTIMATION: Before submitting any computed answer, do a quick estimation to check reasonableness. If your exact answer is far from your estimate, you made a computation error somewhere. This habit alone can save 2-3 points on the LET.
  • For the LET, allocate your time wisely: aim for 45 seconds to 1 minute per Mathematics item. If a computation is taking too long, estimate and move on — an approximate answer in multiple-choice can still be selected correctly.
  • ROMAN NUMERALS: Memorize the six subtractive pairs (IV, IX, XL, XC, CD, CM) and the seven basic symbols (I, V, X, L, C, D, M). Convert by breaking the Roman numeral from left to right, subtracting when a smaller precedes a larger.
  • PROPERTIES: When the LET asks 'What property is illustrated?', look for these signals: order changed → Commutative; grouping changed (order same) → Associative; multiplication across a sum/difference → Distributive; adding 0 or multiplying by 1 → Identity; multiplying to get 1 or adding to get 0 → Inverse.
  • REVIEW DepEd K-12 curriculum guides for Grade 4-6 Mathematics: the LET tests you on content you will TEACH, so understanding the elementary curriculum's scope and sequence helps you contextualize which concepts are most important.
  • TAKE TIMED PRACTICE TESTS: The LET Mathematics section requires accuracy under time pressure. Practice with LET reviewer books (Rex, Lorimar, MSA) and focus on items from this cluster since Number Sense is consistently tested every year.
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In summary

Mastery of Numbers, Number Sense, and the Four Operations is your foundation for success in the LET Mathematics section of the General Education component. As a future licensed professional teacher governed by RA 7836 and the Code of Ethics for Professional Teachers, you are expected not only to solve these problems correctly but also to understand them deeply enough to teach them with clarity, accuracy, and enthusiasm to Grades 1-6 pupils. The seven major areas of this chapter — the real number system, properties of real numbers, GEMDAS, operations on integers, factors/multiples/divisibility, GCF and LCM, and estimation — are all interconnected. Your ability to classify numbers helps you apply the right operation rules. Your fluency with divisibility rules accelerates prime factorization. Your precision with prime factorization enables fast GCF and LCM computation. And your habit of estimation protects against costly errors. For the LET specifically, keep these non-negotiable facts sharp: 1 is NEITHER prime nor composite; 2 is the ONLY even prime; GEMDAS means multiplication and division are SAME rank (left to right); GCF is for equal-grouping problems while LCM is for repeating-event problems; and GCF × LCM = product of two numbers. Approach your LET review with the mindset of a professional educator: understanding the 'WHY' behind every rule makes the 'HOW' easier to remember, apply, and one day teach. Consistent, daily practice with worked examples, timed drill problems, and self-checking through estimation will build the automaticity you need for exam-day performance. Kaya mo ito — you have the training, and with systematic preparation, you will demonstrate the mathematical competency the PRC and the Filipino education system require of every licensed elementary school teacher.

Next chapter

Fractions, Decimals, Percent, Ratio and Proportion

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