Ganita Prakash | Grade 8 | Proportional Reasoning-1: Solutions to Questions 6-12 (Page 177)

📚 NCERT Maths Class 8

Chapter 7: Proportional Reasoning - Part 1

Questions 6-12 (Page 177) - Complete Solutions

📖 About This Chapter

This chapter covers Proportional Reasoning - understanding ratios, proportions, and how to solve real-world problems using the Rule of Three (Trairasika). Below are detailed solutions for Questions 6-12 from the "Figure it Out" exercise on Page 177.

Question 6

Harmain's Age Problem 👧👦

📋 Problem: Harmain is a 1-year-old girl. Her elder brother is 5 years old. What will be Harmain's age when the ratio of her age to her brother's age is 1 : 2?

🔍 Given Information:

• Harmain's current age = 1 year
• Brother's current age = 5 years
• Required ratio = 1 : 2

💡 Key Insight:

The age gap between siblings never changes! It will always be:
5 - 1 = 4 years

✏️ Solution:

Let Harmain's age = x years when ratio becomes 1:2

Then brother's age = (x + 4) years

Setting up the proportion:

x : (x + 4) = 1 : 2

x / (x + 4) = 1 / 2

Cross multiplying:

2 × x = 1 × (x + 4)
2x = x + 4
2x - x = 4
x = 4

✅ Answer: Harmain will be 4 years old
(Her brother will be 8 years old)

🔎 Verification: 4 : 8 = 1 : 2 ✓ Correct!

Question 7

Gold and Water Mass Ratio 🥇💧

📋 Problem: The mass of equal volumes of gold and water are in the ratio 37 : 2. If 1 litre of water is 1 kg in mass, what is the mass of 1 litre of gold?

🔍 Given Information:

• Gold : Water = 37 : 2 (by mass for equal volumes)
• 1 litre of water = 1 kg

✏️ Solution:

Setting up the proportion:

Gold : Water = 37 : 2

If Water = 1 kg, then:

Gold : 1 = 37 : 2

Solving:

2 parts = 1 kg

1 part = 1/2 kg = 0.5 kg

37 parts = 37 × 0.5 = 18.5 kg

✅ Answer: Mass of 1 litre of gold = 18.5 kg

💡 Fun Fact: Gold is extremely dense! That's why even a small amount of gold feels very heavy.

Question 8

Manure for Farming 🌱🚜

📋 Problem: It is good farming practice to apply 10 tonnes of cow manure for 1 acre of land. A farmer is planning to grow tomatoes in a plot of size 200 ft by 500 ft. How much manure should he buy?

🔍 Given Information:

• Manure needed = 10 tonnes per acre
• Plot size = 200 ft × 500 ft
• Unit conversion: 1 acre = 43,560 sq ft

✏️ Solution:

Step 1: Calculate area of the plot

Area = 200 × 500 = 1,00,000 sq ft

Step 2: Convert to acres

Area in acres = 1,00,000 ÷ 43,560

= 2.296 acres ≈ 2.3 acres

Step 3: Calculate manure needed

Manure = 10 tonnes × 2.3

= 23 tonnes

✅ Answer: The farmer should buy approximately 23 tonnes of manure

Question 9

Filling a Bucket 🚰🪣

📋 Problem: A tap takes 15 seconds to fill a mug of water. The volume of the mug is 500 mL. How much time does the same tap take to fill a bucket of water if the bucket has a 10-litre capacity?

🔍 Given Information:

• Time for 500 mL = 15 seconds
• Bucket capacity = 10 litres = 10,000 mL

✏️ Solution:

Using Proportional Reasoning:

Volume : Time :: Volume : Time

500 : 15 :: 10,000 : ?

Cross multiplication:

Time = (15 × 10,000) ÷ 500

Time = 1,50,000 ÷ 500

Time = 300 seconds

Converting to minutes:
300 seconds ÷ 60 = 5 minutes

✅ Answer: 300 seconds = 5 minutes

Question 10

Cost of Land 🏡💰

📋 Problem: One acre of land costs ₹15,00,000. What is the cost of 2,400 square feet of the same land?

🔍 Given Information:

• 1 acre = ₹15,00,000
• 1 acre = 43,560 sq ft
• Required area = 2,400 sq ft

✏️ Solution:

Setting up the proportion:

Area : Cost :: Area : Cost

43,560 : 15,00,000 :: 2,400 : ?

Using cross multiplication:

Cost = (2,400 × 15,00,000) ÷ 43,560

Cost = 36,00,00,000 ÷ 43,560

Cost = ₹82,644.63

✅ Answer: Approximately ₹82,645

Question 11

Ploughing the Field 🐂🚜

📋 Problem: A tractor can plough the same area of a field 4 times faster than a pair of oxen. A farmer wants to plough his 20-acre field. A pair of oxen takes 6 hours to plough an acre of land. How much time would it take if the farmer used a pair of oxen to plough the field? How much time would it take him if he decides to use a tractor instead?

🔍 Given Information:

• Oxen: 6 hours per acre
• Tractor: 4 times faster than oxen
• Total field = 20 acres

✏️ Solution:

🐂 Using Oxen

Time for 1 acre = 6 hours

Time for 20 acres = 6 × 20

= 120 hours

🚜 Using Tractor

Tractor is 4× faster

Time = 120 ÷ 4

= 30 hours

✅ Answer:
Oxen: 120 hours | Tractor: 30 hours

Question 12

Cost of Metals in ₹10 Coin 🪙

📋 Problem: The ₹10 coin is an alloy of copper and nickel called 'cupro-nickel'. Copper and nickel are mixed in a 3 : 1 ratio to get this alloy. The mass of the coin is 7.74 grams. If the cost of copper is ₹906 per kg and the cost of nickel is ₹1,341 per kg, what is the cost of these metals in a ₹10 coin?

🔍 Given Information:

• Copper : Nickel = 3 : 1
• Total mass = 7.74 grams
• Copper cost = ₹906 per kg
• Nickel cost = ₹1,341 per kg

✏️ Solution:

Step 1: Find mass of each metal

Total parts = 3 + 1 = 4 parts

Copper	= (3/4) × 7.74 = 5.805 grams
Nickel	= (1/4) × 7.74 = 1.935 grams

Step 2: Convert cost to per gram

Copper	= ₹906 ÷ 1000 = ₹0.906 per gram
Nickel	= ₹1,341 ÷ 1000 = ₹1.341 per gram

Step 3: Calculate total cost

Cost of Copper	= 5.805 × 0.906 = ₹5.26
Cost of Nickel	= 1.935 × 1.341 = ₹2.59
Total Cost	₹5.26 + ₹2.59 = ₹7.85

✅ Answer: Cost of metals in a ₹10 coin ≈ ₹7.85

💡 Interesting Observation: The actual metal in the coin costs less than its face value. The remaining value represents the manufacturing cost and the coin's purchasing power given by the government!

📝 Key Takeaways from This Chapter

📌 Ratio Basics

A ratio a : b means for every 'a' units of first quantity, there are 'b' units of second quantity.

📌 Proportional Ratios

Two ratios a : b :: c : d are proportional if ad = bc (cross multiplication)

📌 Dividing in Ratio

To divide x in ratio m : n:
First part = m × x/(m+n)
Second part = n × x/(m+n)

📐 Important Formulas & Conversions

1 acre = 43,560 sq ft

1 litre = 1,000 mL

1 kg = 1,000 grams

1 metre = 3.281 feet

📚 NCERT Maths Class 8 - Chapter 7: Proportional Reasoning

Solutions prepared for easy understanding | Questions from Page 177

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