CEUET Mathematics — Ratio & ProportionStudy Notes

A ratio compares two quantities of the same kind, telling us how many times one quantity contains another. Think of it as a mathematical way to express relationships between numbers. **Three Ways to Write Ratios:** 1. Using a colon: 3:5 (read as "3 to 5") 2. As a fraction: 3/5 3. With words: 3 to 5 **Simplifying Ratios:** Just like fractions, ratios can be simplified by dividing both terms by their greatest common divisor (GCD). **Step-by-Step Example:** Simplify the ratio 24:36 Step 1: Find the GCD of 24 and 36 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 GCD = 12 Step 2: Divide both terms by the GCD 24 ÷ 12 = 2 36 ÷ 12 = 3 Step 3: Write the simplified ratio 24:36 = 2:3 **Continued Ratios:** When comparing three or more quantities, we use continued ratios like 2:3:5. These are simplified by dividing all terms by the same number. Example: Simplify 6:9:15 Step 1: Find GCD of all three numbers (GCD = 3) Step 2: Divide each term: 6÷3 : 9÷3 : 15÷3 = 2:3:5

A proportion is a mathematical statement that two ratios are equal. This is where ratios become powerful problem-solving tools. **Proportion Format:** a/b = c/d or a:b = c:d **The Cross-Product Rule (Most Important Tool):** If a/b = c/d, then a × d = b × c This cross-multiplication method solves most proportion problems. **Step-by-Step Solution Process:** **Example 1:** Solve for x in 3/4 = x/20 Step 1: Write the proportion clearly 3/4 = x/20 Step 2: Cross-multiply 3 × 20 = 4 × x 60 = 4x Step 3: Solve for x x = 60 ÷ 4 = 15 Step 4: Verify (always check!) 3/4 = 15/20 3/4 = 3/4 ✓ **Example 2:** If 5 notebooks cost ₱125, how much do 8 notebooks cost? Step 1: Set up the proportion 5 notebooks / ₱125 = 8 notebooks / x pesos Step 2: Cross-multiply 5x = 125 × 8 5x = 1000 Step 3: Solve x = 1000 ÷ 5 = ₱200 Step 4: Verify Cost per notebook = ₱125 ÷ 5 = ₱25 8 notebooks = 8 × ₱25 = ₱200 ✓

Understanding whether quantities have a direct or inverse relationship is crucial for setting up correct proportions. **Direct Proportion:** As one quantity increases, the other increases at the same rate. Formula: y = kx (where k is constant) Symbol: y ∝ x **Inverse Proportion:** As one quantity increases, the other decreases at the same rate. Formula: xy = k (where k is constant) Symbol: y ∝ 1/x **Recognition Strategy:** Ask: "If I double the first quantity, what happens to the second?" - If it doubles → Direct proportion - If it halves → Inverse proportion **Direct Proportion Examples with Solutions:** **Example 1:** Speed and Distance (time constant) If a car traveling at 60 km/h covers 180 km, how far will it travel at 80 km/h? Step 1: Recognize direct relationship (faster speed = more distance) Step 2: Set up proportion: 60/180 = 80/x Step 3: Cross-multiply: 60x = 180 × 80 = 14,400 Step 4: Solve: x = 14,400 ÷ 60 = 240 km **Inverse Proportion Examples with Solutions:** **Example 2:** Workers and Time If 6 workers complete a job in 12 days, how long will 9 workers take? Step 1: Recognize inverse relationship (more workers = less time) Step 2: Use inverse formula: workers × time = constant 6 × 12 = 9 × x Step 3: Solve: 72 = 9x, so x = 8 days **Example 3:** Speed and Time (distance constant) A journey takes 4 hours at 75 km/h. How long at 100 km/h? Step 1: Recognize inverse relationship (faster speed = less time) Step 2: Speed × time = constant distance 75 × 4 = 100 × x Step 3: Solve: 300 = 100x, so x = 3 hours

Partitive proportion problems ask you to divide a total amount into parts that follow a given ratio. This is extremely common in UPCAT word problems. **The Four-Step Method:** 1. Add up all parts of the ratio 2. Divide the total by the sum to find one 'share' 3. Multiply each ratio part by the share value 4. Verify that parts add up to the original total **Detailed Examples:** **Example 1:** Divide ₱840 among three friends A, B, and C in the ratio 2:3:7 Step 1: Add ratio parts 2 + 3 + 7 = 12 parts total Step 2: Find value of one part ₱840 ÷ 12 = ₱70 per part Step 3: Calculate each person's share A gets: 2 × ₱70 = ₱140 B gets: 3 × ₱70 = ₱210 C gets: 7 × ₱70 = ₱490 Step 4: Verify ₱140 + ₱210 + ₱490 = ₱840 ✓ **Example 2:** A school has 450 students. The ratio of Grade 11 to Grade 12 is 4:5. How many students are in each grade? Step 1: Add ratio parts 4 + 5 = 9 parts total Step 2: Find students per part 450 ÷ 9 = 50 students per part Step 3: Calculate each grade Grade 11: 4 × 50 = 200 students Grade 12: 5 × 50 = 250 students Step 4: Verify 200 + 250 = 450 ✓ **Example 3:** An inheritance of ₱1,200,000 is divided among three heirs in the ratio 3:5:4. Find each heir's share. Step 1: Total parts = 3 + 5 + 4 = 12 Step 2: Value per part = ₱1,200,000 ÷ 12 = ₱100,000 Step 3: Individual shares: First heir: 3 × ₱100,000 = ₱300,000 Second heir: 5 × ₱100,000 = ₱500,000 Third heir: 4 × ₱100,000 = ₱400,000 Step 4: Check: ₱300,000 + ₱500,000 + ₱400,000 = ₱1,200,000 ✓

Every percentage problem is actually a ratio problem in disguise. Understanding this connection makes percentage calculations much easier. **Key Insight:** Percent means 'per hundred', so any percent is a ratio with 100 as the second term. 25% = 25:100 = 25/100 = 1/4 **Converting Between Forms:** - Percent to ratio: 30% = 30:100 = 3:10 - Ratio to percent: 3:20 = 15:100 = 15% - Percent to fraction: 40% = 40/100 = 2/5 **Solving Percentage Problems Using Proportions:** **Example 1:** What is 35% of ₱800? Method 1 (Direct): 35% × ₱800 = 0.35 × ₱800 = ₱280 Method 2 (Proportion): 35/100 = x/800 Cross-multiply: 35 × 800 = 100x 28,000 = 100x x = ₱280 **Example 2:** ₱150 is what percent of ₱500? Set up proportion: 150/500 = x/100 Cross-multiply: 150 × 100 = 500x 15,000 = 500x x = 30% **Example 3:** If 40% of a number is 120, what is the number? Set up: 40/100 = 120/x Cross-multiply: 40x = 120 × 100 = 12,000 x = 12,000 ÷ 40 = 300 **Percentage Change Problems:** **Example 4:** A price increases from ₱200 to ₱250. What is the percentage increase? Step 1: Find the increase Increase = ₱250 - ₱200 = ₱50 Step 2: Set up proportion (increase compared to original) 50/200 = x/100 Step 3: Solve x = (50 × 100) ÷ 200 = 25% Formula: Percentage change = (New value - Old value)/Old value × 100%

UPCAT test makers deliberately include traps in ratio and proportion problems. Recognizing these common mistakes can save you valuable points. **Trap 1: Unit Mismatch** Problem: Different units in the same ratio Example: If a car travels 2 hours at 60 km/h, how far in 30 minutes at the same speed? Wrong approach: 2/distance = 30/x (mixing hours and minutes) Correct approach: Convert first 30 minutes = 0.5 hours Then: 2 hours/120 km = 0.5 hours/x km x = 30 km **Trap 2: Ratio vs Total Confusion** Problem: Students think the ratio numbers are the actual quantities Example: In a class, the ratio of boys to girls is 3:2. If there are 40 students total, how many boys? Wrong thinking: 3 boys (using ratio number directly) Correct solution: Step 1: Total parts = 3 + 2 = 5 Step 2: Students per part = 40 ÷ 5 = 8 Step 3: Boys = 3 × 8 = 24 **Trap 3: Direct vs Inverse Misidentification** Problem: Assuming wrong relationship type Example: If 4 pipes fill a tank in 6 hours, how long for 6 pipes? Wrong (direct): 4/6 = 6/x → x = 9 hours Correct (inverse): More pipes = less time 4 × 6 = 6 × x → x = 4 hours **Trap 4: Forgetting to Verify** Always plug your answer back into the original problem Example: Solve 3/7 = x/21 Solution: x = 9 Verification: 3/7 = 9/21 = 3/7 ✓ **Trap 5: Percentage Base Confusion** Problem: Using wrong base for percentage calculations Example: A shirt costs ₱500. After 20% discount, VAT of 12% is added. Final price? Wrong: ₱500 × 0.8 × 1.12 = ₱448 (applying VAT to original price) Correct: Discounted price = ₱500 × 0.8 = ₱400 Final price = ₱400 × 1.12 = ₱448 **Problem-Solving Checklist:** 1. Read the problem completely before writing anything 2. Identify if it's direct or inverse proportion 3. Check units and convert if necessary 4. Set up the proportion carefully 5. Cross-multiply and solve 6. Verify your answer makes sense 7. Check units in your final answer