Hard Level Worksheet Questions

Ratio applications involve complex problem-solving scenarios including business partnerships, investment calculations, mixture problems, and proportional reasoning. These advanced concepts are essential for understanding financial mathematics, engineering calculations, and real-world analytical thinking.

Let's explore advanced ratio applications and proportional reasoning.

1. Write the ratio of 2 hours to 75 minutes in simplest form.

Step 1: Convert to same units: 2 hours = 120100150180 minutes

Step 2: Ratio = 120:75, simplified by dividing by 1510525

Step 3: Simplest form = 8:56:54:35:3

Perfect! 2 hours = 120 minutes, so 120:75 = 8:5.

2. If ₹360 is divided between A and B in the ratio 5:7, find B's share.

Step 1: Total parts = 5 + 7 = 12101514

Step 2: B's share = (7/12) × 360 = ₹210₹200₹220₹240

Excellent! B gets 7 parts out of 12 total parts.

3. Find the mean proportional between 18 and 50.

Step 1: Mean proportional = √(18 × 50) = √9008001000850

Step 2: Mean proportional = 30253520

Great! Mean proportional between a and b is √(a×b) = √900 = 30.

4. Express 5 km to 250 m as a ratio in simplest form.

Step 1: Convert to same units: 5 km = 5000500500005500 m

Step 2: Ratio = 5000:250, simplified by dividing by 2501005001000

Step 3: Simplest form = 20:125:115:110:1

Perfect! 5000:250 = 20:1 after dividing by 250.

5. Write the third proportional to 8 and 12.

Step 1: If a:b = b:c, then c is third proportional

Step 2: 8:12 = 12:c, so c = (12 × 12) ÷ 8 = 18162015

Excellent! Third proportional: 8:12 = 12:18.

6. Find the fourth proportional to 3, 5, and 15.

Step 1: If a:b = c:d, then d is fourth proportional

Step 2: 3:5 = 15:d, so d = (5 × 15) ÷ 3 = 25203035

Great! Fourth proportional: 3:5 = 15:25.

7. If 8 workers complete a job in 15 days, how many days will 12 workers take?

Step 1: Total work = 8 × 15 = 120100140160 worker-days

Step 2: Time for 12 workers = 120 ÷ 12 = 10 days8 days12 days15 days

Perfect! Inverse proportion: more workers means less time.

8. Find the cost of 15 pens if the cost of 8 pens is ₹120.

Step 1: Cost per pen = ₹120 ÷ 8 = ₹15₹12₹18₹10

Step 2: Cost of 15 pens = 15 × ₹15 = ₹225₹200₹250₹275

Excellent! Using unit rate method: ₹15 per pen.

Drag each problem to its correct application type:

Part B: Short Answer Questions (2 Marks Each)

1. Monthly incomes of A and B are in ratio 4:5, expenses in ratio 7:9. If each saves ₹1,500, find their incomes.

Step 1: Set up variables

Let incomes be 4x and 5x, expenses be 7y and 9y

Step 2: Set up savings equations

A's savings: 4x 7y =

B's savings: 5x 9y =

Step 3: Solve the system

From the ratio of savings and given information: x = 3000250035004000

Step 4: Calculate incomes

A's income = 4 × 3000 = ₹12000₹10000₹15000₹8000

B's income = 5 × 3000 = ₹15000₹12000₹18000₹10000

Perfect! A earns ₹12,000 and B earns ₹15,000 monthly.

2. Sides of two triangles are in ratio 3:4. If smaller triangle's perimeter is 60 cm, find larger triangle's perimeter.

Step 1: Understand proportional relationship

Since corresponding sides are in ratio 3:4, perimeters are also in ratio

Step 2: Calculate larger perimeter

If smaller perimeter : larger perimeter = 3:4

Then larger perimeter = () × = 80 cm70 cm90 cm75 cm

Excellent! Similar figures have proportional linear measurements.

3. A train covers 120 km in 2 hours. How long to cover 300 km at same speed?

Step 1: Calculate speed

Speed = Distance ÷ Time = 60 km/h50 km/h70 km/h80 km/h

Step 2: Calculate time for 300 km

Time = 5 hours4 hours6 hours7 hours

Great! Using speed = distance/time formula consistently.

4. Mixture contains milk and water in ratio 7:3. How much water to add to 40 litres to make ratio 3:2?

Step 1: Find initial quantities

Initial milk = () × 40 = 28 litres25 litres30 litres35 litres

Initial water = () × 40 = 12 litres10 litres15 litres8 litres

Step 2: Calculate required water for ratio 3:2

For ratio 3:2, We add x litres of water.

Milk = 28 litres.

Water becomes 12+x12-x litres.

Milk/Water =

28/12+x = 3/2, Where x = (Upto 2 decimal places)

Outstanding! Mixture problems require careful ratio calculations.

Part C: Long Answer Questions (4 Marks Each)

1. A and B together have ₹2,400. If 3/8 of A's amount equals 1/4 of B's amount, find each amount.

Step 1: Set up variables

Let A have ₹x, then B has ₹(2400 - x2400 + x)

Step 2: Set up equation from given condition

()x = ()(2400 - x)

Where x = 96080012001000

Step 3: Find both amounts

A has ₹960, B has ₹(2400 - 960) = ₹1440₹1400₹1500₹1200

Perfect! A has ₹960 and B has ₹1,440.

2. Daily wages of two workers are in ratio 7:9. If each gets ₹9 increase, ratio becomes 8:9. Find original wages.

Step 1: Set up initial conditions

Let original wages be 7x and 9x

Step 2: Set up equation after increase

After ₹9 increase: (7x + 97x - 9) and (9x + 99x - 9)

New ratio: (7x + 9):(9x + 9) = 8:99:8

Step 3: Cross multiply and solve

(7x + 9) = (9x + 9)

Solving x = 1230.5

Step 4: Find original wages

Therefore, Original wages are ₹ and ₹

Great work! You carefully used ratios to trace back the original daily wages.

3. ₹84,000 distributed among P, Q, R such that P gets 4/7 of Q's share, Q gets 3/5 of R's share.

Step 1: Express in terms of R's share

Let R = x

Then Q = ()x

And P = (4/7) × (3/5)x = ()x

Step 2: Set up total equation

P + Q + R =

(12/35)x + (3/5)x + x = 84000

Solving x = 43235450004000050000

Step 3: Calculate individual shares

R gets ₹43,235₹43,325

Q gets = ₹25941₹26000₹25000₹27000

P gets = ₹14824₹15000₹14000₹16000

Excellent! Complex ratio problem solved systematically.

4. A, B, C invest in ratio 3 : 4 : 5. After 6 months, A invests ₹12,000 more, B withdraws ₹8,000. Find profit shares if total profit is ₹2,10,000.

Step 1: Calculate time-weighted investments

Let initial investments be 3x, 4x, 5x

Step 2: Calculate effective capital × time

A's effective investment = 3x × + (3x +- ) × 6

= (x + 12000)

B's effective investment = 4x × + (4x -+ ) × 6

= (x − 8000)

C's effective investment = 5x × = x

So the profit-sharing ratio is proportional to

(6x + 12000) : (8x − 8000) : (10x)

Step 3: (with x = ₹8,000): Compute the ratio

A =

B =

C:

Ratio = 60,000 : 56,000 : 80,000 = 15 : 14 : 2014 : 15 : 20

Step 4: Split the profit ₹2,10,000 in 15 : 14 : 20

Through complex calculations involving the profit distribution

Total parts = 15 + 14 + 20 =

One part = ₹2,10,000 ÷ 49 ≈ ₹4,285.71₹4,394.81 (rounded to 2 decimals)

A’s share = ₹ (rounded to 2 decimals)

B’s share = ₹

C’s share = ₹ (rounded to 2 decimals)

Outstanding! Partnership profit sharing with mid-year changes.

Test your understanding with these multiple choice questions:

For each question, click on the correct answer:

1. The mean proportional between 25 and 100 is:

(a) 50 (b) 75 (c) 60 (d) 40

Correct! Mean proportional = √(25 × 100) = √2500 = 50.

2. The ratio 0.75 : 1.25 in simplest form is:

(a) 3:5 (b) 4:5 (c) 5:3 (d) 5:4

Correct! 0.75:1.25 = 75:125 = 3:5 (dividing by 25).

3. The third proportional to 6 and 9 is:

(a) 12 (b) 13.5 (c) 18 (d) 27

Correct! If 6:9 = 9:x, then x = (9×9)/6 = 81/6 = 13.5.

4. If 12 workers can build a wall in 15 days, 20 workers will take:

(a) 9 days (b) 10 days (c) 8 days (d) 12 days

Correct! Total work = 12×15 = 180 worker-days. So 20 workers take 180÷20 = 9 days.

5. The scale of a map is 1:2,00,000. A distance of 3 cm on the map represents:

(a) 4 km (b) 5 km (c) 6 km (d) 7 km

Correct! 3 cm × 200000 = 600000 cm = 6000 m = 6 km.

🎉 Outstanding! You've Mastered Hard Level Ratio Applications! Here's what you accomplished:

✓ Advanced Proportion Calculations: Mean proportionals, third and fourth proportionals

✓ Complex Division Problems: Multi-variable ratio distribution with constraints

✓ Investment and Partnership: Time-weighted capital calculations and profit sharing

✓ Mixture and Alligation: Changing ratios and component calculations

✓ Inverse Proportion Mastery: Work-time, speed-distance relationships

✓ Chain Ratio Problems: Multiple dependent relationships and systematic solving

✓ Scale and Map Applications: Real-world measurement and representation

✓ Financial Mathematics: Income-expense ratios, wage calculations, business scenarios

Your expertise in ratio applications prepares you for advanced business mathematics, engineering calculations, and complex analytical problem-solving!