LET Secondary Mathematics — Fractions, Decimals, Percent, Ratio and ProportionSummary

A fraction represents a part-to-whole relationship, written as numerator/denominator. To add or subtract fractions, identify the Least Common Denominator (LCD), rewrite each fraction with that denominator, then combine numerators. For example, 2/3 + 1/4 becomes 8/12 + 3/12 = 11/12 (LCD = 12). Multiplying fractions requires multiplying numerators and denominators, then simplifying: 3/4 × 8/9 = 24/36 = 2/3. Dividing fractions uses the reciprocal rule: 3/5 ÷ 2/7 = 3/5 × 7/2 = 21/10. Mixed numbers (e.g., 2 1/3) must be converted to improper fractions (7/3) before multiplying or dividing. A fraction is in lowest terms when the greatest common factor (GCF) of numerator and denominator is 1.

Decimals represent fractions with denominators that are powers of 10. To add or subtract decimals, align decimal points and proceed as with whole numbers. For multiplication, ignore decimal points during multiplication, then place the decimal point in the product so that the total number of decimal places equals the sum of decimal places in the factors: 0.6 × 0.4 = 0.24 (1 + 1 = 2 decimal places). To divide decimals, move the decimal point in the divisor to make it a whole number, move the dividend's point the same number of places, then divide normally: 4.8 ÷ 0.6 becomes 48 ÷ 6 = 8.

The percent means 'per hundred,' so a percent is a fraction with denominator 100. Conversions follow systematic rules: (1) Fraction to decimal: divide numerator by denominator. 1/4 = 0.25. (2) Decimal to percent: multiply by 100 (move decimal point two places right). 0.25 = 25%. (3) Percent to decimal: divide by 100 (move decimal point two places left). 25% = 0.25. (4) Percent to fraction: write the percent over 100 and simplify. 25% = 25/100 = 1/4. Key equivalences for the LET: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/5 = 0.2 = 20%, 1/3 ≈ 0.333 ≈ 33.3%, 1/10 = 0.1 = 10%.

Every percentage problem fits one of three cases built on the formula Rate × Base = Part (where 'part' denotes the percentage, not a percentage sign). (1) Find the part: part = rate × base. 'What is 20% of 450?' → 0.20 × 450 = 90. (2) Find the rate: rate = part ÷ base. '40 is what percent of 160?' → 40 ÷ 160 = 0.25 = 25%. (3) Find the base: base = part ÷ rate. '15 is 25% of what number?' → 15 ÷ 0.25 = 60. The formula can be rearranged as a proportion: part/base = rate/100, solved by cross-multiplication. A critical insight: the base is always the 'whole' or 'original' quantity; failing to identify it correctly is a common trap. For example, in 'A class passed 45 students, which was 90% of those who took the test,' the base is the total who took the test (unknown), not 45.

Percent change measures how much a quantity has increased or decreased relative to its original value. The formula is: Percent change = (new value - original value) ÷ original value × 100%. A positive result indicates an increase; a negative result indicates a decrease. The critical point: the base (denominator) is always the original value, not the new one. Example: A score rises from 40 to 50. Change = 50 - 40 = 10. Percent increase = (10 ÷ 40) × 100% = 25%. A common trap: a 20% price decrease followed by a 20% increase does not return to the original price. If the original price is ₱100, a 20% decrease gives ₱80. A 20% increase on ₱80 gives 80 × 1.20 = ₱96, not ₱100. The two 20%s act on different bases (₱100 vs. ₱80), so they do not cancel.

A ratio compares two quantities of the same kind, written as a : b or a/b. A proportion states that two ratios are equal: a/b = c/d. The defining computational tool is cross-multiplication: if a/b = c/d, then a × d = b × c. Ratios must compare quantities in the same units; if comparing 50 cm to 2 m, convert to 50 cm : 200 cm = 1 : 4. A ratio should be simplified to lowest terms by dividing both parts by their GCF. For example, 12 : 18 simplifies to 2 : 3. Ratios can have more than two terms (a recipe ratio of 1 : 2 : 3). To solve a proportion, cross-multiply and isolate the unknown: if 2/6 = x/15, then 2 × 15 = 6 × x, so 30 = 6x, giving x = 5.

Direct proportion describes a relationship in which two quantities increase or decrease together at the same rate. If one is multiplied by k, the other is also multiplied by k. The two ratios can be set in the same order and cross-multiplied. Example: A recipe for 6 servings needs 2 cups of rice. For 15 servings: 2/6 = x/15, so x = 5 cups. Inverse proportion describes a relationship in which as one quantity increases, the other decreases, and their product remains constant. If workers increase, time to complete a job decreases: workers × days = constant. Example: 4 workers finish a job in 6 days; 3 workers take 4 × 6 ÷ 3 = 8 days. Partitive proportion divides a whole into parts that maintain a given ratio. To divide ₱1,500 in the ratio 2 : 3 : 5, sum the ratio parts (2 + 3 + 5 = 10), calculate the value of one part (₱1,500 ÷ 10 = ₱150), then multiply by each ratio component: shares are ₱300, ₱450, and ₱750.

A rate compares quantities of different kinds, such as kilometers per hour, pesos per kilogram, or students per teacher. A unit rate has a denominator of 1 and is found by dividing the first quantity by the second. Example: If 3 kg of rice costs ₱165, the unit rate is ₱165 ÷ 3 = ₱55 per kilogram. Unit rates enable quick comparison (best-buy problems) and problem-solving (calculating total cost or time). Speed is a rate: distance ÷ time = speed; rearranged, distance = speed × time and time = distance ÷ speed. Example: A jeepney traveling at 40 km/h for 2 hours covers 40 × 2 = 80 km.

When comparing fractions, three reliable methods exist. (1) Convert each to a decimal by dividing: 3/8 = 0.375 and 5/12 ≈ 0.417, so 5/12 > 3/8. (2) Rewrite over a common denominator: 3/8 = 9/24 and 5/12 = 10/24, so 5/12 > 3/8. (3) Cross-multiply: 3 × 12 = 36 vs. 5 × 8 = 40; the larger product (40) pairs with 5/12, confirming 5/12 > 3/8. When ordering mixed number types (fractions, decimals, percents), convert all to one form (usually decimals): 0.6, 55%, and 3/5 become 0.60, 0.55, and 0.60, showing 55% is smallest. Benchmark comparison also helps: compare each value to 0, 1/2, and 1 to gain intuition.

Percent appears in daily financial and commercial contexts: (1) VAT (Value-Added Tax) at 12% is added to a price: total = price × 1.12. A ₱500 meal with 12% VAT costs ₱500 × 1.12 = ₱560. (2) Discount subtracts a percent from the price: sale price = original price × (1 - discount rate). A ₱450 shirt at 20% off costs ₱450 × 0.80 = ₱360. (3) Commission is a percent of sales earned by an agent: commission = sales × commission rate. (4) Simple interest on savings or loans: interest = principal × rate × time. A ₱5,000 loan at 8% annual interest for 2 years accrues ₱5,000 × 0.08 × 2 = ₱800 in interest. (5) Successive percents apply one after another to changing bases, not to the original. A ₱1,000 item discounted 10% then 20% costs 1,000 × 0.90 × 0.80 = ₱720, an overall 28% discount.

Mixture problems combine items of different values or quantities. A typical context: 'How many kilograms of rice costing ₱50/kg must be mixed with 5 kg of rice costing ₱60/kg to achieve an average price of ₱55/kg?' Set up: let x = kg of ₱50 rice. Total value of cheaper rice + total value of costlier rice = total value of mixture. 50x + 60(5) = 55(x + 5). Solving: 50x + 300 = 55x + 275, so 25 = 5x, giving x = 5 kg. Verification: 5 kg at ₱50 (₱250) + 5 kg at ₱60 (₱300) = ₱550 for 10 kg at ₱55/kg. ✓ Best-buy problems compare unit prices of different package sizes. If 500 g of rice costs ₱75 and 750 g costs ₱108, unit rates are ₱0.15/g and ₱0.144/g respectively. The 750 g pack is the better value. Both problem types reduce to proportional reasoning and unit-rate comparison.

All three represent the same part-to-whole relationship. A fraction (e.g., 1/4) expresses parts of a divided whole. A decimal (0.25) is a fraction whose denominator is a power of 10. A percent (25%) is a fraction with denominator 100, expressed as parts per hundred. Understanding this trinity—that 1/4 = 0.25 = 25%—allows translation between forms, enabling problem-solvers to pick the form most convenient for calculation. The LET tests this equivalence by mixing all three forms in a single problem.

Addition and subtraction of fractions and decimals both follow the principle of aligning units (common denominators for fractions; aligned decimal points for decimals). Multiplication of fractions (multiply numerators and denominators) and decimals (count total decimal places) obey scaling principles. Division of both types uses a reciprocal concept: fraction division uses the reciprocal operation, and decimal division uses place-value shifting. Mastering one operation facilitates understanding of the other.

All three cases spring from the foundational equation Rate × Base = Part. Depending on which variable is unknown, the formula rearranges: to find the part, multiply rate by base; to find the rate, divide part by base; to find the base, divide part by rate. Recognizing this unifying structure—rather than memorizing three separate rules—deepens conceptual understanding and accelerates problem-solving. A proportion format (part/base = rate/100) enables cross-multiplication, offering an alternative solving pathway.

A ratio (a : b) is a comparison between two quantities. A proportion (a/b = c/d) asserts that two ratios are equal, enabling scaling. Direct proportion (same-order ratios) scales quantities that move together. Inverse proportion (constant product) scales quantities that move oppositely (workers × days = constant). Partitive proportion divides a whole into parts maintaining a ratio. All three applications use the same cross-multiplication solving technique, but the conceptual context differs. Recognizing which type applies requires reading comprehension and contextual reasoning, not just procedural computation.

A rate (e.g., ₱55 per kilogram) compares quantities of different kinds. A unit rate (₱55/1 kg) is the rate with denominator 1, obtained by division. Unit rates enable 'best-buy' comparisons, speed-distance-time calculations, and mixture problem setups. The formula distance = rate × time exemplifies how rates structure applied problems. Fluency with unit-rate calculation—dividing total quantity by total value to find the 'per 1' value—unlocks rapid problem-solving across contexts.

Percent change (e.g., 'A score increased from 40 to 50; what is the percent increase?') is solved using the Three Cases framework. The change (50 - 40 = 10) is the 'part'; the original value (40) is the 'base'; the percent change is the 'rate.' Using the formula rate = part ÷ base, percent change = 10 ÷ 40 = 0.25 = 25%. The critical conceptual point: the base is always the original value, not the new one. Successive percent changes apply the percentage cases sequentially, each to a new base, which is why successive percents do not add or cancel.

A fraction in lowest terms has numerator and denominator sharing no common factors other than 1 (GCF = 1). Similarly, a ratio in lowest terms is fully reduced: 12 : 18 simplifies to 2 : 3. Both concepts embody the principle that a relationship can be expressed in infinitely many equivalent forms (e.g., 1/2 = 2/4 = 3/6), but the simplest form is most elegant and least prone to error. Teaching simplification builds mathematical maturity: students learn that equivalent forms express the same relationship and that the choice of form affects computational convenience.

Whether the problem is a recipe scale (2/6 = x/15), a map scale (1/50,000 = 4cm/x km), or a direct proportion (students/class = 25/30), cross-multiplication (a/b = c/d implies ad = bc) solves for the unknown instantly. This algebraic technique—central to the LET—transcends context, making it a powerful, generalizable strategy. Repeated application builds fluency and confidence in handling scaled, proportional situations.