LET Elementary Mathematics — Fractions, Decimals, Percent, Ratio and ProportionStudy Notes

Fractions represent parts of a whole. In Grade 3–4 mathematics, students first encounter fractions as equal parts of a shape or group. By Grades 5–6, they perform operations. As a teacher, you must be fluent in all four operations. **Adding and Subtracting Fractions:** The cardinal rule is that fractions can only be added or subtracted if they have the same denominator. You must find the Least Common Denominator (LCD). Step 1: Identify the denominators. Step 2: Find the LCD (the smallest number divisible by all denominators). Step 3: Rewrite each fraction with the LCD as the new denominator. Step 4: Add or subtract the numerators; keep the denominator. Step 5: Simplify to lowest terms by dividing both numerator and denominator by their GCF (Greatest Common Factor). **Multiplying Fractions:** Multiplication does not require a common denominator. The process is straightforward: Step 1: Simplify (cancel) common factors between any numerator and any denominator before multiplying. Step 2: Multiply all numerators together and all denominators together. Step 3: Simplify the result to lowest terms. Teaching tip: Canceling first keeps numbers manageable and helps students see the relationship between fractions. For example, in 3/4 × 8/9, notice that 3 and 9 share a factor of 3, and 4 and 8 share a factor of 4. Canceling gives (1/1) × (2/1) = 2, avoiding the larger intermediate result 24/36. **Dividing Fractions:** Division of fractions relies on the **reciprocal** (flipping the numerator and denominator of the divisor). The rule: a/b ÷ c/d = a/b × d/c. Step 1: Keep the first fraction as is. Step 2: Change the division sign to multiplication. Step 3: Flip (take the reciprocal of) the second fraction. Step 4: Multiply and simplify. This rule can be explained conceptually: dividing by a fraction answers the question "How many copies of this fraction fit into that quantity?" For instance, 3/5 ÷ 1/10 asks "How many tenths fit into three-fifths?" The answer is 6, because three-fifths = six tenths. **Mixed Numbers:** A mixed number combines a whole number and a proper fraction (e.g., 2 1/3). When performing multiplication or division, always convert mixed numbers to improper fractions first. To convert 2 1/3: multiply 2 × 3 = 6, add the numerator 1 to get 7, and place over the denominator: 7/3. This conversion is essential because the rules for multiplying and dividing apply only to fractions, not to mixed-number notation. **Lowest Terms:** A fraction is in lowest terms when the GCF of its numerator and denominator is 1. Always express final answers in lowest terms. For instance, 6/9 simplifies to 2/3 because GCF(6,9) = 3, and 6÷3=2, 9÷3=3. Teaching application: In a Grade 4 math lesson on sharing, if 12 students share 8 cookies equally, each gets 8/12 = 2/3 of a cookie. The reduction makes the answer clearer.

Decimals are another way to write fractions with denominators that are powers of 10 (10, 100, 1000, etc.). In the K–12 BEC, Grades 4–6 students learn decimal notation, comparing decimals, and operations. As a teacher, fluency with decimals is essential for translating real-world measurements and money. **Adding and Subtracting Decimals:** The key principle is to **align the decimal points** vertically before performing the operation. This ensures that tenths are added to tenths, hundredths to hundredths, and so on. Step 1: Write the numbers vertically, aligning decimal points. Step 2: Add or subtract as with whole numbers, moving from right to left. Step 3: Place the decimal point in the answer directly below the aligned points. Step 4: Add trailing zeros if needed to make the numbers look aligned (optional, but aids clarity). Teaching note: Many students forget the decimal point in the answer or misalign columns. Use grid paper or a decimal-point template to reinforce alignment. **Multiplying Decimals:** Unlike addition and subtraction, you do not align decimal points for multiplication. Instead: Step 1: Ignore the decimal points and multiply the numbers as if they were whole numbers. Step 2: Count the total number of decimal places (digits to the right of the decimal point) in both factors. Step 3: Place the decimal point in the product so it has that many decimal places. Rule: The number of decimal places in the product equals the **sum** of the decimal places in the factors. For example, 0.6 × 0.4: the first factor has 1 decimal place, the second has 1, so the product has 1+1=2 decimal places. 6×4=24, so the answer is 0.24. **Dividing Decimals:** When the divisor (the number you are dividing by) is a decimal, first convert it to a whole number: Step 1: Count decimal places in the divisor. Step 2: Move the decimal point in the divisor to the right until it becomes a whole number. Step 3: Move the decimal point in the dividend (the number being divided) the same number of places to the right. Step 4: Divide using standard long division. Step 5: Place the decimal point in the quotient directly above its position in the adjusted dividend. Example: 4.8 ÷ 0.6. The divisor 0.6 has 1 decimal place. Move both decimal points 1 place right: 48 ÷ 6 = 8. So 4.8 ÷ 0.6 = 8. Teaching application: In Grade 5, when dividing ₱48.50 among 5 students, students perform 48.50 ÷ 5 = 9.70, or ₱9.70 per student. This grounds abstract decimal division in currency. **Comparing and Ordering Decimals:** To compare decimals, align decimal points and compare digit by digit from left to right. It helps to append trailing zeros: 0.6 = 0.60, so comparing 0.6 and 0.55 becomes 0.60 vs. 0.55, making it clear that 0.60 > 0.55. **Decimal Place Value:** Understanding place value is foundational. In 3.456, the 3 is in the ones place, 4 in the tenths place (0.4), 5 in the hundredths place (0.05), and 6 in the thousandths place (0.006). This clarity helps students avoid errors like thinking 0.3 < 0.03. Teaching strategy: Use a place-value chart showing ones | tenths | hundredths | thousandths. Have students fill in the digits of decimals to reinforce the value of each position.

Fractions, decimals, and percents are three representations of the same mathematical relationship: a part compared to a whole. Fluency in converting among the three forms is essential for solving the majority of LET items, which mix notation styles and require translation. **Fundamental Relationship:** Percent means "per hundred," so any percent is fundamentally a fraction with denominator 100. For instance, 25% = 25/100 = 1/4 (simplified). **Converting from Fraction to Decimal:** Divide the numerator by the denominator using long division or a calculator. Example: 3/8 → Divide 3 by 8 to get 0.375. Teaching tip: This is the most intuitive conversion for students because it directly answers the question "How many parts of one whole?" For 3/8, you are asking "If I divide the whole into 8 equal parts, 3 of those parts is what decimal amount of the whole?" Answer: 0.375 or 37.5%. **Converting from Decimal to Percent:** Multiply by 100 (or equivalently, move the decimal point two places to the right) and append the percent symbol. Example: 0.25 → 0.25 × 100 = 25%. Alternative phrasing: 0.25 = 25/100 = 25%. Teaching note: Emphasize the visual alignment. The decimal 0.25 in a place-value chart shows 2 tenths and 5 hundredths. When you multiply by 100, the tenths place becomes the tens place and the hundredths place becomes the ones place in the percent: "2 tens and 5 ones" = 25 percent. **Converting from Percent to Decimal:** Divide by 100 (or move the decimal point two places to the left) and drop the percent symbol. Example: 35% → 35 ÷ 100 = 0.35. Teaching strategy: Use a visual arrow showing the decimal point moving left: 35.% → 3.5% → 0.35. This kinesthetic representation helps students internalize the magnitude of the shift. **Converting from Percent to Fraction:** Write the percent as a fraction with denominator 100, then simplify by dividing both numerator and denominator by their GCF. Example: 40% = 40/100. GCF(40,100) = 20, so 40/100 = 2/5. **Converting from Fraction to Percent:** First convert the fraction to a decimal (divide), then multiply by 100 to get the percent. Example: 3/8 → 0.375 → 37.5%. Alternatively, find an equivalent fraction with denominator 100 (if possible): not all fractions have a convenient equivalent with denominator 100, so the decimal method is more general. **Common Conversions (Memorization):** Teachers and students benefit from memorizing these frequent conversions: - 1/2 = 0.5 = 50% - 1/4 = 0.25 = 25% - 3/4 = 0.75 = 75% - 1/5 = 0.2 = 20% - 2/5 = 0.4 = 40% - 3/5 = 0.6 = 60% - 4/5 = 0.8 = 80% - 1/3 ≈ 0.333... ≈ 33 1/3% (or 33.3%) - 2/3 ≈ 0.667... ≈ 66 2/3% (or 66.7%) - 1/8 = 0.125 = 12.5% - 1/10 = 0.1 = 10% These conversions appear repeatedly in LET items on discounts, taxes, and class composition. **Application in Classroom Contexts:** In a Grade 5 math lesson, a teacher might say: "The class is 60% boys. That's the same as 3/5 of the class, or 0.6 of the class. If there are 30 students, that's 3/5 × 30 = 18 boys." This integration of fraction, decimal, and percent notation, grounded in a real classroom statistic, deepens understanding. **Repeating and Terminating Decimals:** Some fractions convert to terminating decimals (they end), while others convert to repeating decimals (a digit or block of digits repeats infinitely). Fractions with denominators containing only factors of 2 and 5 (e.g., 1/8, 3/20) terminate. Fractions with other prime factors (e.g., 1/3, 2/7) repeat. On the LET, you may encounter 1/3 ≈ 0.333 or 2/3 ≈ 0.667; knowing this distinction prevents confusion when rounding.

Every percentage problem, regardless of real-world context, falls into one of three cases built on the formula **Rate × Base = Percentage** (note: in this formula, "percentage" means the part, not the percent symbol). Mastering these three cases is perhaps the single most important skill for the LET mathematics section, as percentage problems appear in nearly every word-problem cluster. **Understanding the Terminology:** - **Rate** (or Percent): The percentage expressed as a decimal or fraction (e.g., 0.20 for 20%). - **Base**: The whole amount, the number you are taking a percent of. - **Percentage** (or Part): The result; the amount that is the given percent of the base. (Note: This is confusing terminology, but it is standard in LET questions. The "percentage" is not the percent symbol; it is the numerical part.) The formula: **Part = Rate × Base**, or equivalently, **P = R × B**. Familiarize yourself with rearrangements: - To find the part: Part = Rate × Base - To find the rate: Rate = Part ÷ Base - To find the base: Base = Part ÷ Rate **Case 1: Finding the Part (Find P given R and B)** Use the formula directly: Part = Rate × Base. Keyword clues: "What is...of?" "Find...of." "How much is...of?" Example: "What is 20% of 450?" Here, the rate is 20% = 0.20, the base is 450, and you are finding the part. Part = 0.20 × 450 = 90. Step-by-step reasoning: 1. Identify the rate (percent given). 2. Identify the base (the "of" amount). 3. Multiply: Rate × Base. 4. The result is the part. **Case 2: Finding the Rate (Find R given P and B)** Rearrange: Rate = Part ÷ Base. Convert the result to a percent by multiplying by 100. Keyword clues: "What percent...of?" "...is what percent of...?" "...as a percent." Example: "40 is what percent of 160?" Here, the part is 40, the base is 160, and you are finding the rate. Rate = 40 ÷ 160 = 0.25 = 25%. Step-by-step reasoning: 1. Identify the part (the number alone, not part of a phrase like "20% of"). 2. Identify the base (the "of" amount). 3. Divide: Part ÷ Base. 4. Multiply by 100 (or move decimal 2 places right) to express as a percent. **Case 3: Finding the Base (Find B given P and R)** Rearrange: Base = Part ÷ Rate. Keyword clues: "...is X% of what number?" "X% of what is...?" "Find the whole." Example: "15 is 25% of what number?" Here, the part is 15, the rate is 25% = 0.25, and you are finding the base. Base = 15 ÷ 0.25 = 60. Step-by-step reasoning: 1. Identify the part. 2. Identify the rate (percent given). 3. Divide: Part ÷ Rate (rate as a decimal). 4. The result is the base. **Common Mistakes and How to Avoid Them:** 1. **Confusing the base.** The base is the number immediately after "of," not necessarily the larger number. In "A class of 40 students, 5 of whom are absent, has what percent absenteeism?" the base is 40 (the whole class), not 5. 2. **Forgetting to convert the percent to a decimal.** When dividing or multiplying, use the decimal form (0.25 for 25%), not the percent form (25%). Multiply by 25 by mistake, and you will get a wildly incorrect answer. 3. **Dividing in the wrong direction.** In Case 2, divide Part by Base, not Base by Part. If you reverse it, your answer will be inverted (a reciprocal error). 4. **Misidentifying the case.** Read the problem carefully to determine what is unknown. If you are asked "What is 15% of 80?" (finding the part), but you instead divide 80 by 0.15 (as if finding the base), you will get 533.33, which is absurdly large and signals an error. **Percent Change and Successive Percents:** **Percent Increase/Decrease:** Formula: Percent Change = (Change ÷ Original) × 100, where Change = New – Original. - A positive result indicates an increase. - A negative result indicates a decrease. - **Critical:** The base is always the *original* value, not the new one. Example: A student's score rose from 40 to 50. What is the percent increase? Change = 50 – 40 = 10. Percent Increase = (10 ÷ 40) × 100 = 0.25 × 100 = 25%. Common mistake: A student might divide 10 by 50 (using the new value as the base), getting 20%. The correct answer is 25%, because the original score was 40. **Successive Percents (Two or More Percents Applied One After Another):** The trap the LET loves: two successive discounts or markups cannot simply be added or subtracted. Each percent is applied to the result of the previous step, not to the original value. Example: An appliance priced at ₱1,000 is marked down 10%, then a further 20% off the reduced price. What is the final price? Incorrect approach: 10% + 20% = 30% off, so pay 70%, getting 0.70 × 1,000 = ₱700. **This is wrong.** Correct approach: Step 1: First discount (10%). New price = 1,000 × (1 – 0.10) = 1,000 × 0.90 = ₱900. Step 2: Second discount (20% of the new price). Final price = 900 × (1 – 0.20) = 900 × 0.80 = ₱720. Overall percent discount = (1,000 – 720) ÷ 1,000 × 100 = 28%, **not 30%.** Why? Because the second 20% is applied to ₱900, not the original ₱1,000. The discounts "compound" or "cascade," reducing each other's effect. **Application in Philippine Contexts:** Filipino students encounter percentages daily: - **Store discounts:** "₱1,500 shirt, 25% off. How much do I pay?" (Case 1) - **VAT/Tax:** "The subtotal is ₱500. Add 12% VAT. Total?" (Case 1) - **Class composition:** "25 students, 15 passed. What percent passed?" (Case 2) - **Salary increases:** "Salary went from ₱20,000 to ₱22,000. What is the percent raise?" (Percent change) - **Compound discounts:** "Jeepney fare is ₱15 with a student discount of 20%, then another 10% off for a promo. What do I pay?" (Successive percents) As a teacher, frame percentage problems in these familiar scenarios to build student confidence and show the relevance of the mathematics.

A **ratio** compares two quantities of the same kind. A **proportion** states that two ratios are equal. These concepts are central to the K–12 BEC (Grade 5–6) and appear frequently on the LET in the context of scale drawings, recipe conversions, and part-to-whole relationships. **Ratio Basics:** A ratio can be written in three notations: **a : b** (colon notation), **a/b** (fraction notation), or **a to b** (word notation). All three represent the same comparison. Example: In a classroom with 12 boys and 18 girls, the ratio of boys to girls is 12 : 18 (colon), 12/18 (fraction), or "12 to 18" (words). **Simplifying Ratios:** A ratio should be expressed in **lowest terms**, just like a fraction. Divide both terms by their GCF. Example: 12 : 18. GCF(12, 18) = 6. So 12 : 18 = 2 : 3 (simplified). Teaching note: A ratio of 2 : 3 means for every 2 boys, there are 3 girls. This interpretation helps students understand that a simplified ratio preserves the relationship while using smaller, more manageable numbers. **Ratios Must Compare the Same Units:** Before writing a ratio, ensure both quantities are in the same units. If comparing 50 cm to 2 m, first convert: 2 m = 200 cm. Then the ratio is 50 : 200 = 1 : 4 (after simplifying). **Extended Ratios (More Than Two Terms):** A ratio can compare more than two quantities. For example, a recipe might call for flour : sugar : butter = 3 : 2 : 1, meaning for every 3 parts of flour, there are 2 parts of sugar and 1 part of butter. **Proportion Basics:** A **proportion** is an equation stating that two ratios are equal: a/b = c/d or a : b = c : d. Example: 2/3 = 6/9 is a proportion because both fractions equal the same value (both simplify to 2/3). **Cross-Multiplication (The Key Tool):** For a proportion a/b = c/d, cross-multiply to get ad = bc. This property is the foundation of solving proportions. Why does it work? If a/b = c/d, multiply both sides by bd: (a/b) × bd = (c/d) × bd, giving ad = bc. Teaching tip: Visualize the cross: the numerator of the left fraction multiplies the denominator of the right, and vice versa. **Solving a Proportion (Finding an Unknown):** Use cross-multiplication to set up an equation, then solve for the unknown. Example: Solve for x in 3/5 = x/20. Cross-multiply: 3 × 20 = 5 × x. 60 = 5x. x = 60 ÷ 5 = 12. Check: 3/5 = 0.6, and 12/20 = 0.6. ✓ **Direct Proportion:** When two quantities are **directly proportional**, they increase or decrease at the same rate. If one doubles, the other doubles. The ratios of corresponding values are equal. Example: A recipe for 6 servings needs 2 cups of rice. How many cups for 15 servings? Setting up the proportion: 2 cups / 6 servings = x cups / 15 servings. Cross-multiply: 2 × 15 = 6 × x. 30 = 6x. x = 5 cups. **Key principle for direct proportion:** Write the ratios in the same order (cups to servings on both sides). **Inverse Proportion:** When two quantities are **inversely proportional**, as one increases, the other decreases, such that their **product is constant**. Example: Four workers can finish repainting a classroom in 6 days. How long will 3 workers take, at the same pace? For inverse proportion: workers × days = constant. 4 × 6 = 3 × d. 24 = 3d. d = 8 days. Interpretation: Fewer workers (from 4 to 3) means more days (from 6 to 8), which is intuitive. The inverse relationship is preserved by the constant product. **Partitive Proportion (Dividing a Whole in a Given Ratio):** Partitive proportions divide a total into parts according to a ratio. Example: Divide ₱1,500 among three children in the ratio 2 : 3 : 5. Step 1: Find the total parts: 2 + 3 + 5 = 10. Step 2: Find the value of one part: ₱1,500 ÷ 10 = ₱150 per part. Step 3: Multiply each ratio term by the value per part: - Child 1: 2 × ₱150 = ₱300 - Child 2: 3 × ₱150 = ₱450 - Child 3: 5 × ₱150 = ₱750 Verification: ₱300 + ₱450 + ₱750 = ₱1,500. ✓ **Scale Drawings and Maps:** A **scale** is a ratio of a drawing's measurement to the actual measurement. For example, a map scale of 1 : 50,000 means 1 unit on the map represents 50,000 units in reality. Example: On a map with scale 1 : 50,000, two towns are 4 cm apart on the map. What is the actual distance? Using proportion: 1 cm (map) / 50,000 cm (actual) = 4 cm (map) / x cm (actual). Cross-multiply: 1 × x = 50,000 × 4. x = 200,000 cm = 2 km (converting: 200,000 cm ÷ 100,000 cm/km). Teaching note: Scale drawings appear in Grade 5–6 social studies (maps) and mathematics (geometric scaling). Understanding that a scale is a proportion helps students solve these problems systematically. **Common Mistakes:** 1. **Writing ratios in the wrong order.** The first quantity mentioned goes in the numerator. "The ratio of boys to girls" means boys in the numerator, girls in the denominator. Reversing this inverts the ratio. 2. **Forgetting to simplify.** Always reduce ratios to lowest terms. 12 : 18 should be 2 : 3, not left as 12 : 18. 3. **Confusing direct and inverse proportions.** Direct: "More input means more output" (e.g., more servings need more rice). Inverse: "More of one means less of the other" (e.g., more workers finish in less time). 4. **Misapplying cross-multiplication.** Cross-multiply only when the proportion is written as an equation a/b = c/d. For partitive proportions or other ratio problems, this method does not apply. 5. **Not maintaining units.** When setting up a proportion, ensure the units match on both sides. If one ratio compares cups to servings, the other must also. **Application in Filipino Contexts:** Ratios and proportions appear constantly in Filipino daily life: - **Class composition:** "Boys to girls ratio of 2 : 3 in a class of 30. How many boys?" (Partitive proportion.) - **Jeepney fares:** "If the fare increases in the ratio 4 : 5, how much will a ₱15 trip cost?" (Direct proportion.) - **Recipes:** "A recipe serves 4. How much of each ingredient for 10 servings?" (Direct proportion.) - **Map scales:** "Using a map with scale 1 : 25,000, calculate actual distances." (Proportion via cross-multiplication.) - **Mixture problems:** "Mixing paint in a ratio of 2 : 3 : 5 (red : blue : white) to make 500 ml." (Partitive proportion.)

A **rate** compares two quantities of different kinds, such as kilometers per hour, pesos per kilogram, or students per classroom. Rates are ubiquitous in Filipino contexts and appear frequently on the LET in word-problem clusters. A **unit rate** has a denominator of 1, making it ideal for comparisons. **Unit Rate Basics:** To find a unit rate, divide the first quantity by the second. Example: If 3 kg of rice costs ₱165, the unit rate is ₱165 ÷ 3 kg = ₱55 per kg. Teaching application: Unit rates settle "best buy" questions instantly. If one brand sells 500 g for ₱75 and another sells 750 g for ₱108, compute unit rates: ₱75 ÷ 500 g = ₱0.15 per gram vs. ₱108 ÷ 750 g = ₱0.144 per gram. The second is cheaper, so it is the better buy. **Speed, Distance, and Time:** The fundamental relationship is **Distance = Rate × Time**, or equivalently: - Rate = Distance ÷ Time - Time = Distance ÷ Rate These rearrangements form the basis of speed problems. Example: A jeepney travels 60 km in 3 hours. What is the average speed? Rate = 60 km ÷ 3 hours = 20 km/h. Example: If a jeepney travels at 20 km/h, how far will it go in 5 hours? Distance = 20 km/h × 5 hours = 100 km. Example: If a jeepney travels 100 km at 20 km/h, how long is the journey? Time = 100 km ÷ 20 km/h = 5 hours. **Teaching Note:** Students often confuse which quantity to divide. A helpful mnemonic is **"Distance = Rate × Time"** or **"D = RT."** To find one variable, isolate it algebraically. If you know D and R, then T = D ÷ R. **Mixture Problems:** Mixture problems combine two or more ingredients (or groups) with different rates (costs per unit, concentrations, percentages) to achieve a desired outcome. These problems are particularly common in Filipino contexts involving recipes, alloys, solutions, and blended commodities. **Type 1: Mixing at Different Unit Prices (Best Buy / Cost Mixture)** Example: How many kg of ₱40/kg rice should be mixed with 10 kg of ₱55/kg rice to produce a mixture worth ₱50/kg? Setup: - Let x = kg of ₱40/kg rice. - Cost of cheaper rice: 40x pesos. - Cost of expensive rice: 55 × 10 = ₱550. - Total weight: x + 10 kg. - Desired average price: ₱50/kg. Equation: Total cost / Total weight = Average price. (40x + 550) / (x + 10) = 50. Solve: 40x + 550 = 50(x + 10). 40x + 550 = 50x + 500. 550 – 500 = 50x – 40x. 50 = 10x. x = 5 kg. Answer: Mix 5 kg of ₱40/kg rice with 10 kg of ₱55/kg rice. Verification: Total cost = 40(5) + 55(10) = 200 + 550 = ₱750. Total weight = 5 + 10 = 15 kg. Average = 750 ÷ 15 = ₱50/kg. ✓ **Type 2: Mixing at Different Concentrations or Percentages (Solution Mixture)** Example: How much pure water should be added to 20 liters of a 25% salt solution to dilute it to 10%? Setup: - Let x = liters of water to add. - Pure water has 0% salt. - Original solution: 20 liters at 25% salt contains 0.25 × 20 = 5 liters of salt. - Final solution: (x + 20) liters at 10% salt contains 0.10(x + 20) liters of salt. - The amount of salt doesn't change; only the volume increases. Equation: Amount of salt is constant. 0.25 × 20 = 0.10 × (x + 20). 5 = 0.10x + 2. 3 = 0.10x. x = 30 liters. Answer: Add 30 liters of water. Verification: New solution has 5 liters of salt in 20 + 30 = 50 liters. Concentration = 5 ÷ 50 = 0.10 = 10%. ✓ **Type 3: Alloy or Blend Problems (Precious Metals, Textiles)** Example: Pure gold (100% gold) and 14-karat gold (14/24 = 58.3% gold) are mixed to produce 12 grams of 18-karat gold (18/24 = 75% gold). How much of each is needed? Setup: - Let x = grams of pure gold. - Then (12 – x) = grams of 14-karat gold. - Pure gold contributes: 1.00 × x = x grams of gold. - 14-karat gold contributes: (14/24) × (12 – x) grams of gold. - Final mixture: 0.75 × 12 = 9 grams of gold. Equation: x + (14/24)(12 – x) = 9. Multiply by 24 to clear fractions: 24x + 14(12 – x) = 216. 24x + 168 – 14x = 216. 10x = 48. x = 4.8 grams. Answer: 4.8 grams of pure gold and 12 – 4.8 = 7.2 grams of 14-karat gold. Verification: Gold content = 1.00(4.8) + (14/24)(7.2) = 4.8 + 4.2 = 9 grams. Percentage = 9 ÷ 12 = 0.75 = 75%. ✓ **General Approach to Mixture Problems:** 1. **Define a variable** for the unknown quantity (usually the amount of one ingredient or the final total). 2. **Set up a conservation equation.** The quantity being conserved is usually the amount of the pure substance (salt, gold, cost) or the total volume/weight. The key insight is that the total amount of pure substance before mixing equals the total after mixing (nothing is created or destroyed). 3. **Solve the equation** for the variable. 4. **Verify** by substituting back into the conservation equation. **Filipino Contexts for Rates and Mixtures:** - **Jeepney fares and speeds:** Calculating travel time or fuel consumption based on distance and speed. - **Grocery shopping (best buy):** Comparing unit prices of rice, sugar, and other staples to determine the cheapest option. - **Cooking and recipes:** Scaling ingredient quantities or mixing ingredients at different costs (e.g., cheaper flour with premium flour to achieve a target average cost). - **Alloy and textile industries:** Mixing metals or fabrics with different purities or compositions. - **Water or chemical mixing:** Diluting or concentrating solutions (e.g., sugar syrup, bleach solutions). As a teacher, frame rate and mixture problems in these realistic Filipino scenarios to help students see the mathematical concepts as tools for solving everyday problems, reinforcing the principle from RA 7836 that teaching should be competent, thoughtful, and relevant to students' lived experiences.