LET Secondary Mathematics — Numbers, Number Sense and the Four OperationsRevision Notes

Exam Tips

• Draw the nested set diagram from memory: N inside W, W inside Z, Z inside Q, Q plus irrationals inside R.
• When asked 'which set does this number belong to?', identify the MOST SPECIFIC (smallest) set first, then list all larger sets that also contain it.
• For any square root: test if it is a perfect square. If yes → rational; if no → irrational.
• Repeating decimals on the LET are ALWAYS rational — convert them to fractions to confirm.
• LET items sometimes use the phrase 'non-real' as a distractor — remember that all the numbers above are real; 'imaginary' or 'complex' numbers are NOT in the real number system.

Key Points

• The real number system is built in nested sets: Natural ⊂ Whole ⊂ Integer ⊂ Rational ⊂ Real. Irrational numbers are also Real but are NOT Rational.
• Natural numbers (N): counting numbers 1, 2, 3, … They exclude 0.
• Whole numbers (W): 0, 1, 2, 3, … They include 0 but exclude negatives.
• Integers (Z): …, -3, -2, -1, 0, 1, 2, 3, … They include all whole numbers and their negatives.
• Rational numbers (Q): any number expressible as a/b where b ≠ 0. Their decimals either TERMINATE (e.g., 0.25) or REPEAT (e.g., 0.333…). Every integer is rational because n = n/1.
• Irrational numbers: non-terminating, non-repeating decimals. Examples: π ≈ 3.14159…, √2 ≈ 1.41421…, √5, e.
• √(perfect square) is RATIONAL — for example, √9 = 3 and √25 = 5.
• √(non-perfect square) is IRRATIONAL — for example, √2 and √7.
• A repeating decimal such as 0.666… IS rational because it equals 2/3.
• Every natural number is also a whole number, integer, rational, and real number simultaneously.
• In the K–12 Mathematics curriculum, Grade 4 pupils begin classifying fractions and mixed numbers; Grade 7 introduces the full real number system.

Definitions

Section Title

The Real Number System and Classification of Numbers

Common Mistakes

• Saying 0 is a natural number — it is NOT; 0 is the first whole number.
• Thinking that a fraction like -3/4 is NOT an integer — it is correct that it is not, but be careful: -3 itself IS an integer.
• Classifying √4 as irrational — √4 = 2, which is a natural number and therefore rational.
• Assuming all square roots are irrational — only square roots of non-perfect squares are irrational.
• Forgetting that every integer is a rational number (because n = n/1).
• Thinking 0.999… is not equal to 1 — mathematically, 0.999… = 1 exactly, and both are rational.

Formulas

Exam Tips

• Memorize the pattern: COMMUTATIVE = order (C for Change order); ASSOCIATIVE = grouping (A for Arrangement of groups); DISTRIBUTIVE = spreading (D for Distribute to each).
• If a LET item shows a(b + c) = ab + ac, the answer is always DISTRIBUTIVE PROPERTY — no exceptions.
• For identity elements: additive identity = 0 (adding zero changes nothing); multiplicative identity = 1 (multiplying by one changes nothing).
• When asked about CLOSURE, check: does the operation always stay within the set? Integers under division: NO.

Key Points

• The six properties — Commutative, Associative, Distributive, Identity, Inverse, and Closure — are named on the LET. You must identify WHICH property justifies a given step.
• Commutative Property: order of operands can be switched. a + b = b + a; a × b = b × a. ONLY for addition and multiplication, NOT subtraction or division.
• Associative Property: grouping of operands can be changed. (a + b) + c = a + (b + c); (a × b) × c = a × (b × c). NOT for subtraction or division.
• Distributive Property: a(b + c) = ab + ac. This is the MOST TESTED property on the LET. It 'distributes' multiplication over addition or subtraction.
• Identity Property: Adding 0 leaves a number unchanged (additive identity = 0). Multiplying by 1 leaves a number unchanged (multiplicative identity = 1).
• Inverse Property: Every number has an additive inverse (opposite): a + (-a) = 0. Every non-zero number has a multiplicative inverse (reciprocal): a × (1/a) = 1.
• Closure Property: Performing an operation on members of a set always produces a member of the same set. Integers are closed under +, -, and ×, but NOT under ÷ (e.g., 3 ÷ 2 = 1.5, not an integer).
• The Distributive Property is used to justify mental computation: 7 × 98 = 7 × (100 - 2) = 700 - 14 = 686.

Definitions

Section Title

Properties of Real Numbers

Common Mistakes

• Applying the Commutative Property to subtraction: 8 - 3 ≠ 3 - 8. Commutative works ONLY for + and ×.
• Confusing Commutative (order) with Associative (grouping) — they are different properties.
• Forgetting that the additive identity is 0 and the multiplicative identity is 1 — do not mix them up.
• Thinking that 0 has a multiplicative inverse — it does NOT (you cannot divide by zero).
• Saying integers are closed under division — 7 ÷ 2 = 3.5, which is NOT an integer.

Exam Tips

• After computing, ALWAYS re-read the expression and trace your steps to catch sign or sequence errors.
• When you see 'evaluate' or 'simplify' on the LET, immediately plan your GEMDAS sequence before touching a number.
• For nested parentheses like {3 × [2 + (5 - 1)]}, circle the innermost pair first and work outward.
• A common LET item format: 'Which of the following equals [expression]?' — eliminate wrong answers by checking GEMDAS compliance.
• On the fraction bar: treat the top and bottom as if each were in parentheses. Solve each side completely before dividing.

Key Points

• GEMDAS is the Philippine K–12 version of PEMDAS/BODMAS. The order is: Grouping → Exponents → Multiplication/Division (left to right) → Addition/Subtraction (left to right).
• G — Grouping symbols: solve everything inside parentheses (), brackets [], braces {}, fraction bars, and radical signs first, working from the innermost outward.
• E — Exponents and roots: evaluate all powers and radicals next.
• M/D — Multiplication and Division are the SAME rank. Work from LEFT TO RIGHT — whichever appears first.
• A/S — Addition and Subtraction are the SAME rank. Work from LEFT TO RIGHT — whichever appears first.
• CRITICAL RULE: 20 ÷ 4 × 5 = 5 × 5 = 25 (left to right). Many test-takers wrongly get 20 ÷ 20 = 1 by doing multiplication first.
• A fraction bar is a grouping symbol: (6 + 2) ÷ 4 means the entire numerator 6 + 2 must be computed before dividing.
• Nested parentheses: solve the innermost set first, then work outward.
• In the K–12 curriculum, GEMDAS is formally introduced in Grade 4 (DepEd Mathematics Curriculum Guide).

Definitions

Section Title

Order of Operations — GEMDAS

Common Mistakes

• Doing multiplication BEFORE division regardless of position — M and D share equal rank; go left to right.
• Doing addition BEFORE subtraction regardless of position — A and S share equal rank; go left to right.
• Forgetting that a fraction bar is a grouping symbol — always simplify numerator and denominator separately before dividing.
• Ignoring the innermost parentheses first in nested expressions.
• Treating -3² as (-3)² = 9 — without parentheses, only the base 3 is squared, giving -9. With parentheses (-3)² = +9.

Formulas

Exam Tips

• Temperature problems (Baguio City dropping 3°C per hour) are standard LET contexts for signed numbers. Translate the problem into an integer expression first.
• When multiplying a chain of integers, count the number of negative signs: odd count = negative answer, even count = positive answer.
• Use the 'Keep-Change-Change' (KCC) mnemonic for subtraction: KEEP the first number, CHANGE the subtraction sign to addition, CHANGE the sign of the second number.
• For absolute value expressions, treat the bars like parentheses — solve the inside first.

Key Points

• ADDITION of integers: same signs → add the absolute values and KEEP the sign. Different signs → subtract the smaller absolute value from the larger and take the sign of the number with the LARGER absolute value.
• Example (same signs): -8 + (-5) = -(8 + 5) = -13.
• Example (different signs): -8 + 3 = -(8 - 3) = -5 (negative wins because |-8| > |3|). And -8 + 12 = +(12 - 8) = +4 (positive wins).
• SUBTRACTION of integers: ADD THE OPPOSITE. a - b = a + (-b). Change the sign of the second number, then add.
• Example: 5 - (-3) = 5 + 3 = 8. And -6 - 4 = -6 + (-4) = -10.
• MULTIPLICATION and DIVISION of integers: LIKE signs (both + or both -) → POSITIVE result. UNLIKE signs (one +, one -) → NEGATIVE result.
• Examples: (-6)(-4) = +24; (-24) ÷ 6 = -4; (3)(-7) = -21; (-30) ÷ (-6) = +5.
• Count of negative signs rule: EVEN number of negative factors → POSITIVE product; ODD number → NEGATIVE product.
• Example: (-2)(-3)(-1) = 6 × (-1) = -6 (3 negative signs = odd = negative).
• ABSOLUTE VALUE |n|: the distance from 0 on the number line; always non-negative. |-7| = 7; |7| = 7; |0| = 0.
• Evaluate inside absolute value bars FIRST (they act as grouping symbols): |3 - 8| = |-5| = 5.
• On the number line, numbers increase to the RIGHT. Among negatives, the one closer to 0 is LARGER: -3 > -8.

Definitions

Section Title

Operations on Integers (Signed Numbers)

Common Mistakes

• Forgetting 'add the opposite' in subtraction: writing 5 - (-3) = 5 - 3 = 2 instead of 5 + 3 = 8.
• Misapplying the double-negative rule: -(-3) = +3, not -3.
• Confusing (-3)² = 9 with -3² = -9. The parentheses make a critical difference.
• Wrongly saying -3 < -8 because 3 < 8 — on the number line, -3 is to the RIGHT of -8, so -3 > -8.
• Evaluating |3 - 8| as |3| - |8| = 3 - 8 = -5 (wrong) instead of |3 - 8| = |-5| = 5 (correct).

Exam Tips

• Memorize the first 15 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
• In LET multiple-choice items about prime or composite, quickly compute: does any number besides 1 and itself divide it? Use divisibility rules to check 2, 3, 5, 7 as trial divisors.
• For large numbers, applying digit-sum rules (÷3 or ÷9) is faster than actual division.
• Prime factorization using a factor tree: write the number at the top, split into any two factors, circle the primes as you find them, keep splitting composites until all branches end in primes.

Key Points

• A FACTOR (divisor) of a number divides it exactly (no remainder). A MULTIPLE of a number is obtained by multiplying it by any integer.
• PRIME NUMBER: exactly 2 factors — 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, …
• COMPOSITE NUMBER: more than 2 factors. Examples: 4, 6, 8, 9, 10, …
• CRITICAL: 1 is NEITHER prime NOR composite (it has only one factor — itself).
• CRITICAL: 2 is the ONLY EVEN prime number.
• PRIME FACTORIZATION: every composite number can be written as a unique product of primes (Fundamental Theorem of Arithmetic). Example: 60 = 2² × 3 × 5.
• Use a FACTOR TREE or repeated division to find prime factorization.
• DIVISIBILITY RULES (must memorize for speed): By 2 — last digit is even. By 3 — digit sum divisible by 3. By 4 — last two digits form a number divisible by 4. By 5 — ends in 0 or 5. By 6 — divisible by BOTH 2 and 3. By 8 — last three digits divisible by 8. By 9 — digit sum divisible by 9. By 10 — ends in 0.
• Divisibility by 6 requires checking BOTH rules (2 and 3) simultaneously.
• Digit sum shortcut: for 738, digit sum = 7 + 3 + 8 = 18. Since 18 ÷ 9 = 2, the number 738 is divisible by 9 (and also by 3 since 18 ÷ 3 = 6).

Definitions

Section Title

Factors, Multiples, Primes, and Divisibility Rules

Common Mistakes

• Saying 1 is a prime number — it is NOT. The PRC consistently uses 1 as a distractor in prime/composite items.
• Forgetting that 2 is the only even prime — every other even number is composite.
• Applying the divisibility-by-3 rule (digit sum) to test divisibility by 9 incorrectly — digit sum must be divisible by 9 for the number itself to be divisible by 9.
• Testing only one condition for divisibility by 6 — you must verify BOTH divisibility by 2 AND by 3.
• Making errors in the factor tree by including a non-prime factor at the end — always stop ONLY at prime factors.

Formulas

Exam Tips

• In every word problem, ask: 'Am I SPLITTING INTO GROUPS (GCF)? Or am I finding when events MEET AGAIN (LCM)?'
• Classic GCF problem template: 'A teacher has X items and Y items to distribute equally with none left over. What is the greatest number of students?' → Find GCF(X, Y).
• Classic LCM problem template: 'Two events happen every X and Y minutes. If they both happen now, when is the next time they happen together?' → Find LCM(X, Y), then add to the starting time.
• Use GCF × LCM = a × b as a 5-second verification after every GCF/LCM calculation.
• For three or more numbers: compute GCF or LCM two at a time — GCF(a, b, c) = GCF(GCF(a, b), c).

Key Points

• GCF (Greatest Common Factor): the LARGEST number that divides all given numbers exactly. Also called Greatest Common Divisor (GCD).
• GCF is used when: splitting items into EQUAL GROUPS with NOTHING LEFT OVER (equal distribution problems).
• LCM (Least Common Multiple): the SMALLEST number that all given numbers divide into exactly.
• LCM is used when: finding when REPEATING EVENTS COINCIDE again, and for ADDING/SUBTRACTING FRACTIONS with different denominators.
• METHOD — Prime Factorization for GCF: write the prime factorization of each number. Multiply the COMMON primes raised to the SMALLEST (lowest) exponent.
• METHOD — Prime Factorization for LCM: write the prime factorization of each number. Multiply EVERY prime that appears, each raised to the HIGHEST exponent.
• EXAMPLE: 18 = 2 × 3² and 24 = 2³ × 3. GCF = 2¹ × 3¹ = 6 (common primes 2 and 3, take lowest powers: 1 and 1). LCM = 2³ × 3² = 8 × 9 = 72 (all primes, take highest powers: 3 and 2).
• KEY VERIFICATION: GCF × LCM = product of the two numbers. Check: 6 × 72 = 432 = 18 × 24. ✓
• This relationship GCF × LCM = a × b allows you to find the LCM if you know the GCF, and vice versa.
• Listing method (for small numbers): list factors of each number, find the largest common one (GCF). List multiples, find the smallest common one (LCM).

Definitions

Section Title

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

Common Mistakes

• Swapping GCF and LCM in word problems — GCF = equal splitting; LCM = event coincidence. Read the problem carefully.
• In GCF: using the HIGHEST exponent instead of the LOWEST (that gives LCM, not GCF).
• In LCM: using only the COMMON primes (that gives GCF, not LCM) — LCM uses EVERY prime.
• Forgetting to include non-common primes in LCM. For 18 = 2 × 3² and 25 = 5², LCM = 2 × 3² × 5² = 450.
• Not verifying: GCF × LCM = a × b. This check catches most errors in 10 seconds.

Formulas

Exam Tips

• For estimation questions on the LET, always round to the LEADING (first) digit of each number before computing — this gives the fastest, cleanest estimate.
• Use estimation to CHECK your exact calculation: if your exact answer is far from your estimate, you likely made an error.
• Scientific notation: count the digits from the original decimal point to its new position. Moving right = negative exponent (small number); moving left = positive exponent (large number).
• For Roman numerals: read left to right. If a letter is followed by a LARGER letter, SUBTRACT it. Otherwise, ADD it.
• Place value items on the LET often ask: 'The digit 7 in 4,725 represents ___.' The answer is 700 or 7 hundreds — know how to express both.

Key Points

• PLACE VALUE: the value a digit holds because of its position. In 4,725: 4 is in the thousands (4,000); 7 is in the hundreds (700); 2 is in the tens (20); 5 is in the ones (5).
• Decimal places to the RIGHT of the decimal point: tenths (0.1), hundredths (0.01), thousandths (0.001).
• ROUNDING RULE: look at the digit IMMEDIATELY TO THE RIGHT of the target place. If it is 5 or more → round UP (add 1 to the target digit, replace following digits with 0). If it is 4 or less → round DOWN (keep the target digit, replace following digits with 0).
• Example: round 3,748 to nearest hundred. Target: hundreds digit is 7. Digit to the right: 4 (less than 5) → keep 7. Answer: 3,700.
• Example: round 3,748 to nearest ten. Target: tens digit is 4. Digit to the right: 8 (5 or more) → round up to 5. Answer: 3,750.
• ESTIMATION STRATEGIES: (1) ROUNDING to the leading digit — round each number to its first digit, then compute. (2) COMPATIBLE NUMBERS — choose nearby numbers that are easy to compute mentally. (3) FRONT-END ESTIMATION — add or multiply the leading digits first, then adjust.
• MENTAL MATH SHORTCUTS: × 10, 100, 1,000 → shift decimal right. ÷ 10, 100, 1,000 → shift decimal left. × 5 = × 10 then ÷ 2. × 25 = × 100 then ÷ 4. 25 × 4 = 100.
• SCIENTIFIC NOTATION: write as a × 10ⁿ where 1 ≤ a < 10. Large numbers: positive exponent (45,000 = 4.5 × 10⁴). Small numbers: negative exponent (0.0032 = 3.2 × 10⁻³).
• ROMAN NUMERALS: I=1, V=5, X=10, L=50, C=100, D=500, M=1,000. Subtractive pairs: IV=4, IX=9, XL=40, XC=90, CD=400, CM=900.
• Roman numeral rule: a smaller symbol BEFORE a larger one is SUBTRACTED. A smaller or equal symbol AFTER is ADDED.
• Example: XLIX = XL + IX = 40 + 9 = 49. MMXXIV = 1000 + 1000 + 20 + 4 = 2024.

Definitions

Section Title

Estimation, Mental Computation, Place Value, and Rounding

Common Mistakes

• Rounding to the wrong place: 'round to the nearest hundred' means the hundreds digit changes, not the tens digit.
• Forgetting to look at the digit to the RIGHT of the target, not the target digit itself.
• In scientific notation: writing 12.5 × 10³ (wrong — coefficient must be less than 10); should be 1.25 × 10⁴.
• Roman numerals: writing IIIX for 7 (wrong — you cannot subtract a smaller from a much larger like that). Correct: VII = 7.
• Confusing mental math with estimation — mental computation gives the EXACT answer quickly; estimation gives an APPROXIMATE answer.