Hard Level Worksheet Questions

Interactive Comparing Quantities Using Proportion Worksheet

Proportion and ratio are fundamental concepts for comparing quantities, solving real-world problems involving percentage, finding unknown quantities, and understanding relationships between different measures. These skills are essential for advanced mathematics and practical applications.

First, let's explore basic ratio and proportion concepts with quick calculations.

1. Write the ratio of 2 hours to 150 minutes in simplest form.

Step 1: Convert to same units: 2 hours = 120150100180 minutes

Step 2: Ratio = 120 : 150 = 4:52:35:43:2 (dividing by 30)

Perfect! 2 hours = 120 minutes, so 120:150 = 4:5 in simplest form.

2. If the ratio of boys to girls in a class is 5 : 4, find the percentage of girls in the class.

Step 1: Total parts = 5 + 4 = 910811

Step 2: Girls' percentage = 44.44%40%45%50%

Excellent! Girls make up 4 out of 9 total parts = 44.44%.

3. Increase ₹1200 in the ratio 5 : 6.

Step 1: New amount = (6/5) × 1200 = ₹1440₹1200₹1000₹1500

Great! Increasing ₹1200 in ratio 5:6 gives ₹1440.

4. Express 25% as a ratio.

Step 1: 25% = 1/41/31/52/5

Step 2: As ratio: 1:44:11:33:1

Perfect! 25% = 1/4, which means 1 part out of 4 total parts.

5. If the cost of 8 pens is ₹96, find the cost of 5 pens using proportion.

Step 1: Cost per pen = ₹12₹10₹8₹15

Step 2: Cost of 5 pens = ₹60₹50₹40₹80

Excellent! Using proportion: 8:96 = 5:60.

6. In a ratio 7 : 9, what part is the first term of the sum of the terms?

Step 1: Sum of terms = 7 + 9 = 16151714

Step 2: First term's part = 7/16 = 7/169/167/99/7

Perfect! The first term (7) is 7/16 of the total.

7. Find the value of x: 8 : x = x : 18.

Step 1: Cross multiply: x² = 8 × 18 = 144164124140

Step 2: Therefore x = √144 = 12141016

Great! In continued proportion, x² = product of extremes.

8. A recipe uses sugar and flour in the ratio 3 : 5. If you have 240 g of flour, how much sugar is needed?

Step 1: If 5 parts = 240 g, then 1 part = 48 g50 g45 g40 g

Step 2: Sugar needed = 3 parts = 3 × 48 = 144 g150 g120 g160 g

Excellent! Using ratio 3:5, sugar = (3/5) × 240 = 144 g.

9. The price of a commodity increases from ₹500 to ₹550. Find the percentage increase.

Step 1: Increase = 550 - 500 = ₹50₹40₹60₹45

Step 2: Percentage = (50/500) × 100 = 10%8%12%15%

Perfect! (Increase/Original) × 100 = 10% increase.

Drag each concept to its correct category:

Part B: Short Answer Questions (2 Marks Each)

1. The weight of 12 books is 9.6 kg. Find the weight of 15 such books.

Step 1: Find weight per book

Weight per book = 9.6 ÷ 12 = 0.8 kg0.9 kg0.7 kg1.0 kg

Step 2: Calculate total weight

Weight of 15 books = 15 × 0.8 = 12 kg11 kg13 kg10 kg

Perfect! Using unit rate: 0.8 kg per book × 15 books = 12 kg.

2. A sum of ₹12,000 is divided between A, B, and C in the ratio 2 : 3 : 5. Find each share.

Step 1: Find total parts

Total parts = 2 + 3 + 5 = 109118

Step 2: Calculate each share

A's share = ₹2400₹2000₹2500₹3000

B's share = ₹3600₹3000₹4000₹3500

C's share = ₹6000₹5000₹7000₹5500

Excellent! Ratio division: A gets 2/10, B gets 3/10, C gets 5/10 of total.

3. The price of sugar was increased by 20%. If its original price was ₹25 per kg, find the new price.

Step 1: Calculate increase amount

Increase = 20% of 25 = ₹5₹4₹6₹7

Step 2: Find new price

New price = Original + Increase = 25 + 5 = ₹30₹29₹31₹32

Perfect! 20% increase means multiply by 1.2: 25 × 1.2 = ₹30.

4. The ratio of ages of A and B is 5 : 7. If sum of their ages is 72 years, find their present ages.

Step 1: Set up equation

Let ages be 5x and 7x. Sum: 5x + 7x = 72

12x = 72, so x = 6578

Step 2: Calculate actual ages

A's age = 5 × 6 = 30 years25 years35 years28 years

B's age = 7 × 6 = 42 years40 years45 years48 years

Great! Using ratio parts: A = 5×6 = 30, B = 7×6 = 42.

Part C: Long Answer Questions (4 Marks Each)

1. Monthly incomes of A, B, and C are in ratio 2 : 3 : 4. Monthly expenses in ratio 5 : 6 : 7. If A saves ₹3000, find savings of B and C.

Step 1: Set up variables

Let incomes be 2x, 3x, 4x and expenses be 5y, 6y, 7y

Step 2: Use A's saving condition

A's saving: 2x -+ 5y = (equation 1)

Step 3: Find relationship between x and y

From the ratios and given condition, we need another relationship.

Since all follow same expense pattern: x = 7500, y = 2400x = 5000, y = 1400x = 6000, y = 1800x = 8000, y = 3000

Step 4: Calculate other savings

B's saving = 3x - 6y = 3(7500) - 6(2400) = 22500 - 14400 = ₹8100₹7500₹9000₹7200

C's saving = 4x - 7y = 4(7500) - 7(2400) = 30000 - 16800 = ₹13200₹12000₹14000₹11500

Excellent! B saves ₹8100 and C saves ₹13200.

2. A sum of ₹15,000 is divided between P, Q, and R in ratio 1/2 : 1/3 : 1/4. Find each share.

Step 1: Convert to whole numbers

LCM of 2, 3, 4 =

1/2 : 1/3 : 1/4 = 6/12 : 4/12 : 3/12 = 6 : 4 : 33 : 2 : 112 : 8 : 62 : 3 : 4

Step 2: Find total parts

Total parts = 6 + 4 + 3 = 13121415

Step 3: Calculate shares

P's share = (6/13) × 15000 = ₹6923₹7000₹6500₹7500

Q's share = (4/13) × 15000 = ₹4615₹4500₹5000₹4200

R's share = (3/13) × 15000 = ₹3462₹3500₹3000₹4000

Perfect! Converting fractions to ratio 6 : 4 : 3 and dividing proportionally.

3. A school has boys and girls in ratio 7 : 5. If 50 more girls join, ratio becomes 7 : 6. Find initial numbers.

Step 1: Set up initial conditions

Let initial boys = 7x, initial girls =

Step 2: Set up new condition

After 50 girls join: boys = 7x, girls =

New ratio: 7x : (5x + 50) = 7 : 67 : 5

Step 3: Cross multiply and solve

× 7x = × (5x + 50)

So x = 50406045

Step 4: Find initial numbers

Initial boys = 7 × 50 = 350300400320

Initial girls = 5 × 50 = 250200300280

Outstanding! Initial: 350 boys, 250 girls. After: 350 boys, 300 girls (ratio 7:6).

4. A recipe requires ingredients in ratio 4 : 5 : 6. If you have 300 g of first ingredient, find quantities of other two.

Step 1: Find value of one part

If first ingredient = 4 parts = g

Then 1 part = 300 ÷ 4 = 75 g70 g80 g65 g

Step 2: Calculate other ingredients

Second ingredient = 5 parts = 5 × 75 = 375 g350 g400 g325 g

Third ingredient = 6 parts = 6 × 75 = 450 g400 g500 g420 g

Perfect! Using ratio scaling: 300g corresponds to 4 parts, so other ingredients follow proportionally.

Test your understanding with these multiple choice questions:

For each question, click on the correct answer:

1. The ratio of 50 paise to ₹2 is:

(a) 1 : 2 (b) 1 : 4 (c) 2 : 1 (d) 4 : 1

Correct! 50 paise : ₹2 = 50 paise : 200 paise = 1:4.

2. Which of the following is equal to 40%?

(a) 2/5 (b) 4/10 (c) 0.4 (d) All of these

Correct! 40% = 40/100 = 2/5 = 4/10 = 0.4. All are equivalent.

3. The ratio of 2 m to 80 cm is:

(a) 5 : 2 (b) 2 : 5 (c) 5 : 4 (d) 4 : 5

Correct! 2 m = 200 cm, so 200:80 = 5:2.

4. If a : b = 3 : 4, then b : a =

(a) 3 : 4 (b) 4 : 3 (c) 1 : 1 (d) None

Correct! Inverse ratio means flipping the terms: if a:b = 3:4, then b:a = 4:3.

5. If a quantity increases from 400 to 500, the percentage increase is:

(a) 10% (b) 20% (c) 25% (d) 50%

Correct! Increase = 100, Percentage = (100/400) × 100 = 25%.

🎉 Outstanding! You've Mastered Comparing Quantities Using Proportion! Here's what you accomplished:

• Understanding ratios and expressing them in simplest for

• Converting between ratios, percentages, and fraction

• Solving proportion problems using cross multiplicatio

• Calculating percentage increases and decrease

• Dividing quantities in given ratio

• Solving real-world problems involving ages, money, and measurement

• Working with complex ratios involving fraction

• Understanding continued proportions and mean proportional

• Applying unit rates and scaling in practical situations

Your solid foundation in proportion and ratios will help you excel in advanced topics like similar figures, trigonometry, probability, and real-world problem solving!