FEUCAT Mathematics — Ratio & ProportionFlash Cards

Express the ratio 24:36 in its simplest form

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?

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

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?

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

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?

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?

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

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?

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?

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

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?

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.

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

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?

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.

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?

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?

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?

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?

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!