LET Secondary Mathematics — Fractions, Decimals, Percent, Ratio and ProportionRevision Notes

Formulas

Exam Tips

• LET word problems often phrase fraction operations using Filipino contexts (allowance, cloth, rice). Identify the operation first before computing.
• Cancel common factors BEFORE multiplying to keep numbers small and avoid errors.
• For LCD with large denominators, use prime factorization — it is faster and more reliable.
• Check reasonableness: if you added two positive fractions and got something smaller than either, you made an error.
• Mixed-number addition/subtraction can be done by keeping whole and fractional parts separate — no need to convert if adding only.

Key Points

• A fraction a/b represents a part of a whole; 'a' is the numerator (part) and 'b' is the denominator (whole).
• A fraction is in lowest terms when GCF(numerator, denominator) = 1.
• Adding and subtracting fractions REQUIRES a common denominator (LCD).
• Multiplying fractions: multiply numerators together AND denominators together, then simplify.
• Dividing fractions: multiply the dividend by the RECIPROCAL of the divisor (KFC — Keep, Flip, Change).
• Mixed numbers must be converted to improper fractions before multiplying or dividing.
• To convert a mixed number to an improper fraction: (whole × denominator + numerator) / denominator.
• Always simplify your final answer to lowest terms.

Definitions

Section Title

Operations on Fractions

Common Mistakes

• Adding or subtracting fractions without finding the LCD first (e.g., wrongly writing 1/2 + 1/3 = 2/5).
• Forgetting to convert mixed numbers to improper fractions before multiplying or dividing.
• Not simplifying the final answer to lowest terms.
• When dividing, flipping the WRONG fraction — always flip the DIVISOR (the second fraction), not the dividend.
• Canceling across an addition or subtraction sign (canceling is only valid in multiplication).

Formulas

Exam Tips

• Always write a zero before the decimal point for clarity: write 0.25, not .25.
• When the LET gives peso amounts, set up as decimals and align carefully.
• Double-check your decimal multiplication by estimating: 0.6 × 0.4 ≈ 0.5 × 0.4 = 0.2, confirming 0.24 is reasonable.
• Memorize common decimal-fraction equivalents (see Conversions section) to speed up computation.

Key Points

• Adding and subtracting decimals: ALIGN the decimal points, then operate as with whole numbers.
• Multiplying decimals: ignore the points, multiply as whole numbers, then count decimal places.
• The number of decimal places in the PRODUCT equals the SUM of the decimal places in both factors.
• Dividing decimals: move the divisor's decimal point to make it a whole number; move the dividend's point the same number of places.
• Zeros may be annexed to the right of a decimal without changing its value (0.5 = 0.50 = 0.500).
• Comparing decimals: compare digit by digit from left to right after aligning decimal points.

Definitions

Section Title

Operations on Decimals

Common Mistakes

• Failing to align decimal points when adding or subtracting (e.g., 2.5 + 0.35 written as 2.5 + 0.35 misaligned giving 2.85 correct but 2.6 if aligned wrongly).
• Forgetting to count decimal places correctly in multiplication.
• Moving only one decimal point when dividing instead of moving BOTH.
• Dropping a zero placeholder (e.g., 0.06 × 0.3: 6×3=18, 3 decimal places → 0.018, NOT 0.18).

Formulas

Exam Tips

• Write the conversion rules on your scratch paper at the start of the exam: ×100 (decimal→%), ÷100 (%→decimal).
• LET often mixes forms in one problem (e.g., '55%, 3/5, and 0.6 — arrange from least to greatest'). Convert ALL to decimals first.
• For 'of' problems, the moment you see 'of', write a multiplication sign.

Key Points

• Percent means 'per hundred' — it is a fraction with denominator 100.
• Fraction to Decimal: DIVIDE the numerator by the denominator.
• Decimal to Percent: MULTIPLY by 100 (move decimal point TWO places to the RIGHT).
• Percent to Decimal: DIVIDE by 100 (move decimal point TWO places to the LEFT).
• Percent to Fraction: Write the percent over 100, then SIMPLIFY.
• Fraction to Percent: First convert to decimal, then multiply by 100.
• Memorize the table of benchmark fractions, decimals, and percents for speed.
• The word 'OF' in problems means MULTIPLY (e.g., 3/4 of 20 = 3/4 × 20 = 15).

Definitions

Section Title

Converting Among Fractions, Decimals, and Percent

Common Mistakes

• Moving the decimal in the WRONG direction when converting (e.g., writing 25% as 25.0 instead of 0.25).
• Confusing 'multiply by 100' and 'divide by 100' for the two conversion directions.
• Not simplifying percent-to-fraction conversions (leaving 20/100 instead of 1/5).
• Treating 33% as exactly 1/3 — the exact value is 33 1/3% or 0.3333...; using 33% introduces a rounding error.

Formulas

Exam Tips

• Memorize the three formulas as a triangle: Part on top, Rate and Base on the bottom. Cover the unknown to see the operation.
• The word 'is' signals the PART; the word 'of' signals the BASE; the percent sign signals the RATE.
• For VAT problems: 12% VAT is added, so multiply by 1.12. For a discount: subtract the percent, so multiply by (1 − rate).
• Percent increase trap: 'A salary was increased by 20% to ₱12,000. What was the original salary?' → Base = 12,000 ÷ 1.20 = ₱10,000.
• Successive discounts: NEVER add the percentages. Always compute step by step on the running total.

Key Points

• Every percentage problem fits one of THREE cases based on the formula: Rate × Base = Part (Percentage).
• CASE 1 — Find the Part (Percentage): Part = Rate × Base. The rate and base are given.
• CASE 2 — Find the Rate: Rate = Part ÷ Base. Express as a percent by multiplying by 100.
• CASE 3 — Find the Base: Base = Part ÷ Rate. The part and rate are given.
• The BASE is always the original whole (100%) — the number that follows 'of' or 'out of'.
• The RATE is the percent (convert to decimal before computing).
• The PART is the result — the portion of the base indicated by the rate.
• Percent increase/decrease: (change ÷ original) × 100. The ORIGINAL is always the base.
• Successive percents act on DIFFERENT bases — you cannot simply add the two percents.

Definitions

Section Title

The Three Percentage Cases

Common Mistakes

• Using the NEW value instead of the ORIGINAL as the base in percent-change problems.
• Adding successive percents (e.g., 10% + 20% = 30% discount) instead of computing step by step.
• Forgetting to convert the percent to a decimal before multiplying (using 25 instead of 0.25).
• Confusing which case applies — draw a ratio box: identify which of Part/Rate/Base is missing.
• In Case 3, dividing the base by the part instead of the part by the rate.

Formulas

Exam Tips

• ALWAYS check: does more of A mean more of B (direct) or less of B (inverse)? The answer determines which method to use.
• For map scales, always convert to the SAME UNIT before computing, then convert back to the required unit at the end.
• In partitive proportion, ALWAYS verify: the shares must add up to the total. Use this as your check.
• Set up proportions in the SAME ORDER: if the left side is 'servings/cups', the right side must also be 'servings/cups'.
• If a ratio has THREE terms (e.g., 2:3:5), the sum of parts = 2+3+5 = 10, not 3.

Key Points

• A RATIO compares two quantities of the same kind; written as a:b or a/b.
• A ratio must always be simplified to LOWEST TERMS (12:18 = 2:3).
• Ratios must compare the SAME UNITS — convert before writing the ratio.
• A PROPORTION states that two ratios are equal: a/b = c/d.
• Cross-multiplication: if a/b = c/d, then a×d = b×c (the cross-products are equal).
• DIRECT proportion: as one quantity increases, the other also increases (same-direction change).
• INVERSE proportion: as one quantity increases, the other decreases (opposite-direction change); the PRODUCT is constant.
• PARTITIVE proportion: a whole is divided into parts according to a given ratio.
• Scale drawings and maps are proportions: scale = drawing length : actual length.
• The key cue word 'OF' means multiply; set up ratios in the SAME ORDER on both sides.

Definitions

Section Title

Ratio and Proportion

Common Mistakes

• Setting up the ratios in DIFFERENT orders on the two sides of the proportion (e.g., putting 'old:new' on the left and 'new:old' on the right).
• Treating an inverse proportion as direct proportion (setting up ratios instead of products).
• Comparing quantities in DIFFERENT UNITS without converting first (e.g., 50 cm to 2 m without converting 2 m to 200 cm first).
• In partitive proportion, forgetting to divide the total by the SUM of ratio terms.
• Leaving a ratio in non-simplified form (e.g., 12:18 instead of 2:3).

Formulas

Exam Tips

• Memorize D = R × T as a triangle: cover the unknown (D, R, or T) to see the formula.
• For best-buy problems, ALWAYS state your answer with units: 'the 750g pack at ₱0.144/g is the better buy'.
• Average speed trap: if you travel equal DISTANCES (not equal times), you CANNOT average the speeds arithmetically.
• In time problems: convert minutes to hours before using the formula (30 minutes = 0.5 hours).

Key Points

• A RATE compares quantities of DIFFERENT kinds (km per hour, pesos per kilo, pupils per teacher).
• A UNIT RATE has a denominator of 1; find it by dividing (₱165 for 3 kg → ₱55/kg).
• For best-buy comparisons, find the UNIT RATE for each option; the smaller unit cost is the better buy.
• Speed problems use the formula: Distance = Rate × Time (D = R × T).
• Rearrangements: R = D ÷ T and T = D ÷ R.
• For mixture problems, find the unit rate (cost per unit) and compare or blend proportionally.
• Average speed (when same distance is covered at two different speeds): Average Speed = 2 × r1 × r2 / (r1 + r2).

Definitions

Section Title

Rate and Mixture Applications

Common Mistakes

• Using the TOTAL distance instead of the distance for each leg in average-speed problems.
• Simply averaging two speeds arithmetically (e.g., (40+60)/2 = 50) when the harmonic mean formula is needed for equal distances.
• Comparing total prices instead of unit prices when packages have different sizes.
• Confusing RATE with RATIO — rate compares different units; ratio compares same units.

Formulas

Exam Tips

• For sets of 3+ fractions, convert all to decimals — it is faster than finding the LCD of multiple denominators.
• Use benchmark 1/2 = 0.5: fractions below 0.5 are less than half; above 0.5 are greater than half. Use this to eliminate wrong options.
• In ordering problems (least to greatest), write the decimal equivalents on your scratch paper, order them, then match back to the original form.

Key Points

• Three reliable methods to compare fractions: (1) convert to DECIMAL, (2) rewrite with a COMMON DENOMINATOR, (3) CROSS-MULTIPLY.
• Cross-multiply to compare a/b and c/d: compute a×d and b×c; the LARGER product is over the LARGER fraction.
• When ordering a mix of fractions, decimals, and percents: convert ALL to DECIMALS first.
• Use benchmark values (0, 1/2, 1) for quick checks — is the fraction closer to 0, to 1/2, or to 1?
• Denominators alone do NOT determine size: 1/8 < 1/3 even though 8 > 3.

Definitions

Section Title

Comparing and Ordering Fractions and Decimals

Common Mistakes

• Comparing fractions by numerators alone or denominators alone (1/3 vs 1/4 — students wrongly say 1/4 is bigger because 4>3).
• In cross-multiplication for comparison, associating the larger product with the WRONG fraction (the product b×c goes with a/b, and a×d goes with c/d).
• Not converting all values to the same form before ordering mixed sets.