Ratio & Proportion - Flashcards
Master ratio and proportion with these problem-focused flashcards. Each card contains step-by-step solutions to help you understand the methods and avoid common mistakes in UPCAT-style problems.
Flashcards
Express the ratio 24:36 in its simplest form
Answer
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 12. 24÷12 = 2, 36÷12 = 3. Answer: 2:3
If 3 notebooks cost ₱45, how much do 7 notebooks cost?
Answer
Step 1: Set up proportion. 3 notebooks : ₱45 = 7 notebooks : x pesos. Step 2: Write as fractions: 3/45 = 7/x. Step 3: Cross multiply: 3x = 45 × 7 = 315. Step 4: Solve for x: x = 315 ÷ 3 = 105. Answer: ₱105
Divide ₱720 among A, B, and C in the ratio 2:3:4
Answer
Step 1: Add ratio parts: 2+3+4 = 9 total parts. Step 2: Find value of 1 part: ₱720 ÷ 9 = ₱80. Step 3: Calculate each share: A = 2 × ₱80 = ₱160, B = 3 × ₱80 = ₱240, C = 4 × ₱80 = ₱320. Step 4: Verify: ₱160+₱240+₱320 = ₱720 ✓
If 6 workers can finish a job in 10 days, how many days will 15 workers take?
Answer
Step 1: Identify inverse proportion (more workers = fewer days). Step 2: Use formula: Workers₁ × Days₁ = Workers₂ × Days₂. Step 3: Substitute: 6 × 10 = 15 × x. Step 4: Solve: 60 = 15x, so x = 60 ÷ 15 = 4. Answer: 4 days
Solve for x: 5:8 = x:24
Answer
Step 1: Write as equation: 5/8 = x/24. Step 2: Cross multiply: 5 × 24 = 8 × x. Step 3: Simplify: 120 = 8x. Step 4: Solve: x = 120 ÷ 8 = 15. Step 5: Check: 5/8 = 15/24 = 5/8 ✓. Answer: x = 15
In a class of 40 students, the ratio of boys to girls is 3:2. How many boys are there?
Answer
Step 1: Total ratio parts = 3+2 = 5 parts. Step 2: Value of 1 part = 40 students ÷ 5 parts = 8 students. Step 3: Boys = 3 parts × 8 students = 24 boys. Step 4: Check: Girls = 2 × 8 = 16, Total = 24+16 = 40 ✓. Answer: 24 boys
If a car travels 180 km in 3 hours, how far will it travel in 5 hours at the same speed?
Answer
Step 1: Identify direct proportion (more time = more distance at constant speed). Step 2: Set up proportion: 180 km : 3 hours = x km : 5 hours. Step 3: Write as equation: 180/3 = x/5. Step 4: Cross multiply: 180 × 5 = 3x, so 900 = 3x. Step 5: Solve: x = 300. Answer: 300 km
Express 0.75 as a ratio in simplest form
Answer
Step 1: Write as fraction: 0.75 = 75/100. Step 2: Find GCD of 75 and 100. Factors of 75: 1,3,5,15,25,75. Factors of 100: 1,2,4,5,10,20,25,50,100. GCD = 25. Step 3: Simplify: 75÷25 = 3, 100÷25 = 4. Step 4: Write as ratio: 3:4. Answer: 3:4
If 4 pipes can fill a tank in 6 hours, how long will 8 pipes take?
Answer
Step 1: Identify inverse proportion (more pipes = less time). Step 2: Use formula: Pipes₁ × Time₁ = Pipes₂ × Time₂. Step 3: Substitute: 4 × 6 = 8 × x. Step 4: Solve: 24 = 8x, so x = 24 ÷ 8 = 3. Answer: 3 hours
When should you use direct proportion vs inverse proportion?
Answer
Direct Proportion: When both quantities change in the SAME direction. If one doubles, the other doubles. Examples: cost and quantity, distance and time at constant speed. Formula: y = kx. Inverse Proportion: When quantities change in OPPOSITE directions. If one doubles, the other halves. Examples: workers and time, speed and time for fixed distance. Formula: xy = k.
Find x if x:12 = 15:20
Answer
Step 1: Write as equation: x/12 = 15/20. Step 2: Simplify right side: 15/20 = 3/4. Step 3: So x/12 = 3/4. Step 4: Cross multiply: 4x = 12 × 3 = 36. Step 5: Solve: x = 36 ÷ 4 = 9. Step 6: Check: 9/12 = 3/4 ✓. Answer: x = 9
A recipe for 4 people needs 200g sugar. How much sugar for 6 people?
Answer
Step 1: Identify direct proportion (more people = more sugar). Step 2: Set up proportion: 4 people : 200g = 6 people : x grams. Step 3: Write as equation: 4/200 = 6/x. Step 4: Cross multiply: 4x = 200 × 6 = 1200. Step 5: Solve: x = 1200 ÷ 4 = 300. Answer: 300g sugar
Two numbers are in ratio 5:7. If their sum is 84, find the numbers.
Answer
Step 1: Let the numbers be 5x and 7x. Step 2: Set up equation: 5x + 7x = 84. Step 3: Simplify: 12x = 84. Step 4: Solve: x = 84 ÷ 12 = 7. Step 5: Find numbers: First = 5 × 7 = 35, Second = 7 × 7 = 49. Step 6: Check: 35 + 49 = 84 ✓, 35:49 = 5:7 ✓. Answer: 35 and 49
Convert 25% to a ratio in simplest form
Answer
Step 1: Write percent as fraction: 25% = 25/100. Step 2: Find GCD of 25 and 100. Factors of 25: 1,5,25. Factors of 100: 1,2,4,5,10,20,25,50,100. GCD = 25. Step 3: Simplify: 25÷25 = 1, 100÷25 = 4. Step 4: Write as ratio: 1:4. Answer: 1:4
If 12 men can build a wall in 15 days, how many men are needed to build it in 10 days?
Answer
Step 1: Identify inverse proportion (fewer days = more men needed). Step 2: Use formula: Men₁ × Days₁ = Men₂ × Days₂. Step 3: Substitute: 12 × 15 = x × 10. Step 4: Solve: 180 = 10x, so x = 180 ÷ 10 = 18. Answer: 18 men
The ratio of length to width of a rectangle is 4:3. If the perimeter is 42 cm, find dimensions.
Answer
Step 1: Let length = 4x, width = 3x. Step 2: Perimeter formula: P = 2(length + width). Step 3: Substitute: 42 = 2(4x + 3x) = 2(7x) = 14x. Step 4: Solve: x = 42 ÷ 14 = 3. Step 5: Find dimensions: Length = 4 × 3 = 12 cm, Width = 3 × 3 = 9 cm. Step 6: Check: P = 2(12+9) = 42 ✓. Answer: 12 cm × 9 cm
What is the cross-product rule and when do you use it?
Answer
Cross-Product Rule: If a/b = c/d, then a × d = b × c. Use when: Solving proportions for unknown values. Example: If 3/4 = x/20, cross multiply: 3 × 20 = 4 × x, so 60 = 4x, therefore x = 15. This is the fastest way to solve most proportion problems.
A map scale is 1:50,000. If two cities are 8 cm apart on the map, what's the actual distance?
Answer
Step 1: Understand scale: 1 cm on map = 50,000 cm in reality. Step 2: Set up proportion: 1 cm : 50,000 cm = 8 cm : x cm. Step 3: Solve: x = 8 × 50,000 = 400,000 cm. Step 4: Convert to km: 400,000 cm = 4,000 m = 4 km. Answer: 4 km
Three partners invest ₱20,000, ₱30,000, and ₱40,000. If profit is ₱18,000, how is it shared proportionally?
Answer
Step 1: Find investment ratio: 20,000:30,000:40,000 = 2:3:4 (÷10,000). Step 2: Total parts = 2+3+4 = 9. Step 3: Value per part = ₱18,000 ÷ 9 = ₱2,000. Step 4: Calculate shares: Partner 1: 2 × ₱2,000 = ₱4,000, Partner 2: 3 × ₱2,000 = ₱6,000, Partner 3: 4 × ₱2,000 = ₱8,000. Answer: ₱4,000, ₱6,000, ₱8,000
Common mistake: Student says 'boys to girls is 3:2' with 30 students means 3 boys and 2 girls. What's wrong?
Answer
MISTAKE: Confusing ratio parts with actual numbers. CORRECT METHOD: Step 1: Ratio 3:2 means 3 parts boys + 2 parts girls = 5 total parts. Step 2: Each part = 30 ÷ 5 = 6 students. Step 3: Boys = 3 × 6 = 18, Girls = 2 × 6 = 12. The ratio gives PROPORTIONS, not actual counts. Always find the total parts first!
Diagrams
Ratio and Proportion Concept Overview
A comprehensive mind map showing all major concepts in ratio and proportion, including types, applications, and problem-solving strategies
Problem-Solving Decision Tree for Ratio and Proportion
A step-by-step decision tree to help students identify the type of ratio/proportion problem and choose the correct solving method
Distribution of Ratio and Proportion Problem Types in UPCAT
Shows the typical distribution of different ratio and proportion problem types that appear in UPCAT mathematics exams