Ratio & Proportion
Key Concepts & Formulas
Provide 5-7 essential concepts for Ratio & Proportion:
| # | Concept | Quick Explanation |
|---|---|---|
| 1 | Basic Ratio | Comparison of two quantities in the form a:b, read as “a is to b” |
| 2 | Proportion | Equality of two ratios: a:b = c:d, also written as a:b::c:d |
| 3 | Mean Proportional | The middle term when three terms are in continued proportion: a:x = x:b, then x = √(ab) |
| 4 | Third Proportional | For numbers a and b, the third proportional x satisfies a:b = b:x, so x = b²/a |
| 5 | Compounded Ratio | Product of two or more ratios: (a:b) and (c:d) gives (ac:bd) |
| 6 | Duplicate Ratio | Square of a ratio: duplicate ratio of a:b is a²:b² |
Essential Formulas
| Formula | Usage |
|---|---|
| a:b = c:d means ad = bc | [When to use] Cross-multiplication to verify or solve proportion equations |
| Mean proportional of a,b = √(ab) | [When to use] Finding the middle term in continued proportion |
| Third proportional of a,b = b²/a | [When to use] Finding the fourth term when first two terms are given |
| If a:b = c:d, then a+b:b = c+d:d | [When to use] Componendo rule for ratio manipulation |
| If a:b = c:d, then a-b:b = c-d:d | [When to use] Dividendo rule for ratio manipulation |
10 Practice MCQs
Generate 10 MCQs with increasing difficulty (Q1-3: Easy, Q4-7: Medium, Q8-10: Hard)
Q1. Two trains have lengths in the ratio 3:4. If the shorter train is 180 meters long, what is the length of the longer train? A) 200m B) 240m C) 270m D) 320m
Answer: B) 240m
Solution: Let the lengths be 3x and 4x Given: 3x = 180 meters Therefore, x = 60 meters Length of longer train = 4x = 4 × 60 = 240 meters
Shortcut: 180 × (4/3) = 240 meters
Concept: Ratio & Proportion - Basic ratio application
Q2. The ratio of passenger coaches to goods wagons in a train is 5:8. If there are 65 passenger coaches, how many goods wagons are there? A) 91 B) 104 C) 117 D) 130
Answer: B) 104
Solution: Let passenger coaches = 5x, goods wagons = 8x Given: 5x = 65 Therefore, x = 13 Goods wagons = 8x = 8 × 13 = 104
Shortcut: 65 × (8/5) = 13 × 8 = 104
Concept: Ratio & Proportion - Finding unknown quantity
Q3. Find the mean proportional between 9 and 16. A) 12 B) 14 C) 15 D) 18
Answer: A) 12
Solution: Mean proportional = √(9 × 16) = √144 = 12
Shortcut: √(9 × 16) = √(3² × 4²) = 3 × 4 = 12
Concept: Ratio & Proportion - Mean proportional calculation
Q4. A train covers 240 km in 3 hours and 320 km in 4 hours. Are the speeds in proportion? A) Yes, they are in ratio 3:4 B) Yes, both are 80 km/h C) No, ratios differ D) Cannot determine
Answer: B) Yes, both are 80 km/h
Solution: First speed = 240/3 = 80 km/h Second speed = 320/4 = 80 km/h Since both speeds are equal (80:80 = 1:1), they are in proportion
Shortcut: Calculate speeds separately and compare
Concept: Ratio & Proportion - Verifying proportionality
Q5. If 3:5 = x:20, find the value of x. A) 9 B) 12 C) 15 D) 18
Answer: B) 12
Solution: Using cross-multiplication: 3 × 20 = 5 × x 60 = 5x x = 60/5 = 12
Shortcut: x = (3 × 20)/5 = 12
Concept: Ratio & Proportion - Solving proportion equations
Q6. The ratio of platform tickets to train tickets sold is 7:12. If 84 platform tickets were sold, how many total tickets were sold? A) 228 B) 252 C) 276 D) 300
Answer: C) 276
Solution: Let platform tickets = 7x = 84 Therefore, x = 12 Train tickets = 12x = 12 × 12 = 144 Total tickets = 84 + 144 = 228
Shortcut: Total ratio parts = 7 + 12 = 19 Total tickets = 84 × (19/7) = 228
Concept: Ratio & Proportion - Total quantity calculation
Q7. Find the third proportional to 8 and 12. A) 16 B) 18 C) 20 D) 24
Answer: B) 18
Solution: If 8:12 = 12:x, then x = 12²/8 = 144/8 = 18
Shortcut: x = b²/a = 12²/8 = 144/8 = 18
Concept: Ratio & Proportion - Third proportional
Q8. Two trains have speed ratio 4:5. If the faster train takes 6 hours to cover a distance, how long will the slower train take to cover the same distance? A) 7.5 hours B) 8 hours C) 8.5 hours D) 9 hours
Answer: A) 7.5 hours
Solution: Speed and time are inversely proportional If speed ratio = 4:5, then time ratio = 5:4 Let slower train time = 5x, faster train time = 4x Given: 4x = 6 hours Therefore, x = 1.5 hours Slower train time = 5x = 5 × 1.5 = 7.5 hours
Shortcut: Time = 6 × (5/4) = 7.5 hours
Concept: Ratio & Proportion - Inverse proportionality
Q9. In a railway workshop, the ratio of skilled to unskilled workers is 5:3. If 20 more skilled workers are recruited, the ratio becomes 3:1. Find the original number of unskilled workers. A) 24 B) 30 C) 36 D) 45
Answer: B) 30
Solution: Let original skilled = 5x, unskilled = 3x After recruitment: (5x + 20):3x = 3:1 Cross-multiplying: 5x + 20 = 9x 4x = 20 x = 5 Original unskilled workers = 3x = 3 × 5 = 30
Shortcut: Set up equation and solve for x
Concept: Ratio & Proportion - Changing ratios
Q10. A train's journey involves three sections with distance ratios 2:3:5. If the total journey is 600 km and the train takes total 10 hours, find the time ratio if speeds are in ratio 4:3:2 respectively. A) 2:3:5 B) 1:2:3 C) 2:4:5 D) 1:3:5
Answer: B) 1:2:3
Solution: Distance parts: 2x + 3x + 5x = 600 10x = 600, x = 60 Distances: 120 km, 180 km, 300 km Time = Distance/Speed Time ratio = (120/4):(180/3):(300/2) = 30:60:150 = 1:2:5
Shortcut: Time ratio = (Distance ratio)/(Speed ratio) for each part
Concept: Ratio & Proportion - Complex ratio relationships
5 Previous Year Questions
Generate PYQ-style questions with authentic exam references:
PYQ 1. The ratio of boys to girls in a railway school is 7:8. If there are 560 girls, find the number of boys. [RRB NTPC 2021 CBT-1]
Answer: 490
Solution: Girls = 8x = 560 x = 70 Boys = 7x = 7 × 70 = 490
Exam Tip: Always identify which quantity corresponds to which ratio part
PYQ 2. Find the mean proportional between 4 and 9. [RRB Group D 2022]
Answer: 6
Solution: Mean proportional = √(4 × 9) = √36 = 6
Exam Tip: Remember mean proportional is always the square root of the product
PYQ 3. If 2:7 = 6:x, then x equals [RRB ALP 2018]
Answer: 21
Solution: 2 × x = 7 × 6 2x = 42 x = 21
Exam Tip: Cross-multiplication is the fastest method for solving proportions
PYQ 4. Two trains have lengths in ratio 5:7. If the longer train is 350m, find the length of the shorter train. [RRB JE 2019]
Answer: 250m
Solution: Longer train = 7x = 350 x = 50 Shorter train = 5x = 5 × 50 = 250m
Exam Tip: Ensure you identify which ratio part matches the given quantity
PYQ 5. The ratio of ticketless travelers to ticket holders in a train is 1:15. If there are 480 passengers total, how many are ticketless? [RPF SI 2019]
Answer: 30
Solution: Total ratio parts = 1 + 15 = 16 Ticketless travelers = 480 × (1/16) = 30
Exam Tip: For total quantity questions, add all ratio parts first
Speed Tricks & Shortcuts
For Ratio & Proportion, provide exam-tested shortcuts:
| Situation | Shortcut | Example |
|---|---|---|
| Finding unknown in proportion | Cross-multiply directly | If 3:5 = x:20, then x = (3×20)/5 = 12 |
| Mean proportional | Multiply and take square root | Mean prop of 4,9 = √(4×9) = 6 |
| Third proportional | Square middle, divide by first | Third prop to 4,6 = 6²/4 = 36/4 = 9 |
| Combined ratios | Multiply corresponding terms | (2:3) and (4:5) → (2×4):(3×5) = 8:15 |
| Ratio of totals | Find one unit value first | If ratio 3:4 and first quantity is 45, then 1 unit = 15, second quantity = 4×15 = 60 |
Common Mistakes to Avoid
| Mistake | Why Students Make It | Correct Approach |
|---|---|---|
| Confusing mean and third proportional | Both involve three terms | Mean: a:x = x:b, Third: a:b = b:x |
| Adding ratios directly | Treating like fractions | Convert to same base or use unit method |
| Not simplifying ratios | Leaving in complex form | Always reduce to lowest terms (4:6 → 2:3) |
| Reversing ratio order | Writing b:a instead of a:b | Check which quantity comes first in question |
| Forgetting to add all parts | In total quantity questions | For ratio a:b:c, total parts = a+b+c |
Quick Revision Flashcards
| Front (Question/Term) | Back (Answer) |
|---|---|
| What is a ratio? | Comparison of two quantities of same kind |
| If a:b = c:d, then? | ad = bc (cross-multiplication) |
| Mean proportional formula | √(ab) |
| Third proportional to a,b | b²/a |
| Componendo rule | If a:b = c:d, then a+b:b = c+d:d |
| Dividendo rule | If a:b = c:d, then a-b:b = c-d:d |
| Duplicate ratio of 2:3 | 4:9 |
| Sub-duplicate ratio | Square root of given ratio |
| If 3:4 = x:12, find x | 9 |
| Ratio must have? | Same units for both quantities |
Topic Connections
How Ratio & Proportion connects to other RRB exam topics:
- Direct Link: Partnership (profit sharing ratios), Mixtures (concentration ratios), Time & Work (efficiency ratios)
- Combined Questions: Ages (age ratios with time), Speed-Distance-Time (speed ratios), Percentage (ratio to percentage conversion)
- Foundation For: Alligation method, Chain rule, Variation problems, Time & Work efficiency calculations
BETA
