Hard Level Worksheet
Very Short Answer Questions (1 Mark Each)
(1) Simplify the ratio 144:192.
Correct! Dividing both by 48: 144 : 192 = 3 : 4.
(2) If the ratio of two numbers is 11:13 and their difference is 24, find the numbers.
Smaller:
Perfect! Let numbers be 11x and 13x. Difference = 2x = 24, so x = 12. Numbers are 132 and 156.
(3) Find the fourth proportional of 7, 21, 49.
Excellent! If 7 : 21 = 49 : x, then x = (21 × 49) ÷ 7 = 147.
(4) A shopkeeper mixes two varieties of rice in the ratio 5:7. If he wants 96 kg of the mixture, how much of each variety should he use?
First variety:
(5) Convert the ratio 9:16 into a fraction.
Great! Ratio 9:16 as a fraction =
Short Answer Questions (2 Marks Each)
Answer each question clearly
(1) The ratio of the speeds of two cars is 7:9. If the faster car travels 270 km, find the distance traveled by the slower car.
Distance by slower car:
Excellent! If speeds are 7:9, then distances are also 7:9. Slower car distance =
(2) Divide ₹4200 among 3 persons such that their shares are in the ratio 5 : 6 : 7.
First person: ₹
Perfect! Total parts = 18. Shares:
(3) A container has a mixture of milk and water in the ratio 4:5. If there are 36 liters of milk, find the quantity of water.
Quantity of water:
Correct! If milk : water = 4 : 5, then water =
(4) The ratio of the areas of two squares is 16:25. Find the ratio of their sides.
Ratio of sides:
Great! Since area =
(5) The ratio of ages of two sisters is 7:9. Four years hence, their ages will be in the ratio 4:5. Find their present ages.
Younger sister:
Perfect! Let ages be 7x and 9x. After 4 years: (7x+4):(9x+4) = 4:5. Solving: 5(7x+4) = 4(9x+4), x = 4.
Long Answer Questions (4 Marks Each)
Note: Answer each question with complete steps and clear explanations.
(1) A sum of ₹7800 is divided among A, B, and C in the ratio of their ages. If their ages are in the ratio 3 : 5 : 7, find each person's share.
A's share: ₹
Correct! Total parts = 15. A gets
(2) A map is drawn at a scale of 1:75,000. The distance between two towns on the map is 12 cm. Find the actual distance in km.
Actual distance:
Perfect! Actual distance = 12 × 75,000 = 9,00,000 cm = 9 km.
(3) The ratio of the length to the breadth of a rectangle is 5:3. If the perimeter of the rectangle is 64 m, find its length and breadth.
Length:
Excellent! Let length = 5x, breadth = 3x. Perimeter = 2(5x + 3x) = 16x = 64, so x = 4.
(4) A person mixes two types of sugar costing ₹40/kg and ₹50/kg in the ratio 2:3. Find the cost of 1 kg of the mixture.
Cost per kg: ₹
Great! Total cost = (2×40 + 3×50) = 230. Total quantity = 5 kg. Cost per kg =
(5) Two numbers are in the ratio 9:14. If 18 is added to each number, the new ratio becomes 3:5. Find the original numbers.
Smaller number:
Correct! Let numbers be 9x and 14x. After adding 18: (9x+18):(14x+18) = 3:5. Solving: 5(9x+18) = 3(14x+18), x = 5.
Part B: Objective Questions (1 Mark Each)
Choose the correct answer and write the option (a/b/c/d)
(1) The ratio 225:300 in simplest form is:
(a) 3:4 (b) 5:6 (c) 7:8 (d) 9:12
Correct! Dividing both by 75: 225:300 = 3:4.
(2) The fourth proportional to 8, 12, 18 is:
(a) 24 (b) 27 (c) 28 (d) 30
Correct! If 8:12 = 18:x, then x = (12 × 18) ÷ 8 = 27.
(3) The ratio of the areas of two squares is 49:64. The ratio of their sides is:
(a) 7:8 (b) 49:64 (c) 1:1 (d) 8:7
Correct! Side ratio =
(4) A mixture contains milk and water in the ratio 5:3. If there are 20 liters of water, find the quantity of milk:
(a) 25 liters (b) 30 liters (c) 32 liters (d) 35 liters
Correct! If milk:water = 5:3, then milk =
(5) Two numbers are in the ratio 7:11. Their sum is 162. The larger number is:
(a) 98 (b) 99 (c) 100 (d) 101
Correct! Let numbers be 7x and 11x. Sum = 18x = 162, so x = 9. Larger = 11×9 = 99.
(6) A person mixes 4 kg of sugar costing ₹40/kg with 6 kg costing ₹50/kg. The cost per kg of the mixture is:
(a) ₹45 (b) ₹46 (c) ₹48 (d) ₹50
Correct! Total cost = (4×40 + 6×50) = 460. Total weight = 10 kg. Cost per kg =
(7) If 7 pens cost ₹84, the cost of 15 pens is:
(a) ₹180 (b) ₹175 (c) ₹160 (d) ₹150
Correct! Cost per pen = ₹12. Cost of 15 pens = 15 × 12 = ₹180.
(8) The ratio of the ages of two sisters is 5:7. If the sum of their ages is 96, the age of the younger sister is:
(a) 30 (b) 35 (c) 40 (d) 42
Correct! Let ages be 5x and 7x. Sum = 12x = 96, so x = 8. Younger = 5×8 = 40.
(9) Two numbers are in the ratio 9:14. If 18 is added to each, the new ratio is 3:5. The smaller number is:
(a) 36 (b) 42 (c) 45 (d) 48
Correct! Let numbers be 9x and 14x. After adding 18: (9x+18):(14x+18) = 3:5. Solving gives x = 5, so smaller = 45.
(10) A map is drawn with a scale of 1:60,000. The actual distance for 10 cm on the map is:
(a) 6 km (b) 60 km (c) 600 m (d) 0.6 km
Correct! Actual distance = 10 × 60,000 = 6,00,000 cm = 6 km.
Expert Ratio Challenge
Determine whether these statements are True or False:
Expert Proportion Mastery Quiz
🎉 Congratulations! What You've Mastered:
You have successfully completed the "Expert Ratio and Proportion" worksheet and learned:
(1) Complex Ratio Simplification: Mastering large number ratios and finding optimal common factors
(2) Advanced Algebraic Ratios: Solving changing ratio problems with future conditions
(3) Area and Square Root Relationships: Understanding how area ratios relate to linear dimension ratios
(4) Mixture and Weighted Average Problems: Calculating costs and quantities in complex mixing scenarios
(5) Multi-variable Proportional Systems: Handling problems with multiple interdependent ratios
(6) Age Progression Problems: Solving complex age-related ratio changes over time
(7) Advanced Map Scale Applications: Converting between units with precision in large-scale problems
(8) Complex Sharing and Division: Managing multi-person divisions with intricate ratio conditions
(9) Changing Ratio Analysis: Understanding how adding constants affects proportional relationships
(10) Expert Problem-solving Strategies: Developing systematic approaches to complex multi-step problems
Exceptional achievement! You have mastered the most advanced concepts in ratio and proportion and can tackle professional-level mathematical problems!