NCERT MCQ Solutions for Class 8 Maths Ganita Prakash Chapter 7 Proportional Reasoning-1

The “Puzzle Time!” section states, “The 10-coin triangle can be flipped with just 3 moves; did you figure out how?”. This explicitly provides the answer for the 10-coin triangle.

Images A, C, and D look similar because their widths and heights have changed by the same factor. In contrast, for image B, the width and height have not changed by the same factor from image A, causing distortion.

“We use the notion of a ratio to represent such proportional relationships in mathematics”. This introduces the fundamental concept for discussing proportionality in the context of varying quantities.

“The numbers 60 and 40 are called the terms of the ratio”. In a general ratio a:b, ‘a’ and ‘b’ are individual quantities that form the comparison.

“We can reduce ratios to their simplest form by dividing the terms by their HCF”. This method allows for easy comparison of different ratios to check for proportionality.

“When two ratios are the same in their simplest forms, we say that the ratios are in proportion, or that the ratios are proportional”. The symbol ‘::’ is used to indicate this relationship.

The ratio is 6:10. The first term (6) increased to 18, a factor of 3 (18 divided by 6). So, the second term (10) must also be multiplied by 3, resulting in 30 spoons of sugar.

Nitin’s ratio of wall length to cement is 60:3, which simplifies to 20:1. Hari’s ratio is 40:2, also simplifying to 20:1. Since the ratios are proportional, the walls are equally strong. 

The example with Neelima’s age shows that adding 9 years to both ages changes the ratio from 3:30 (1:10) to 12:39 (4:13). Therefore, “When we add (or subtract) the same number from the terms of a ratio, the ratio changes and is not necessarily proportional to the original ratio”.

The chapter states, “The situation above is a typical example of a problem where we need to use proportional reasoning to find a solution”. This highlights the core skill applied in Rule of Three problems.

The chapter derives this relationship: “Therefore, c/a = d/b. Multiplying both sides by ab, we get, ab × (c/a) = ab × (d/b), leading to bc = ad or ad = bc”. This is known as cross multiplication.

“In ancient India, Aryabhata (199 CE) and others called such problems of proportionality Rule of Three problems”. They involved three given quantities to find a fourth unknown.

Aryabhata says, “Multiply the phala by the ichchhā and divide the resulting product by the pramāņa.” This translates to ichchhāphala = (phala × ichchhā) ÷ pramāņa.

First, convert 4 hours to 240 minutes. The proportion is 150:90::240:x. Cross-multiplying gives 150x = 240 × 90, so x = (240 × 90) ÷ 150 = 144.

To compare, convert Himachal tea price to 1 kg: 200g costs ₹200, so 1kg (1000g) costs ₹1000 (₹200 × 5). Meghalaya tea costs ₹800 per 1 kg. Therefore, Himachal tea is more expensive.

The chapter provides the general formula: “If we want to divide a quantity x in the ratio of m: n, then the parts will be m × (x⁄(m+n)) and n × (x⁄(m+n))”.
This formula correctly distributes the total quantity based on the ratio.

The investment ratio 75000:25000 simplifies to 3:1. The total parts are 3 + 1 = 4. Each part is ₹4000 ÷ 4 = ₹1000. Prashanti gets 3 parts, so 3 × ₹1000 = ₹3,000.

Original mixture: Sand = 30 kg, Cement = 10 kg. New ratio 5:2 with sand 30 kg implies cement = (2⁄5) × 30 = 12 kg. Since 10 kg of cement is already present, add 2 kg (12 – 10).

The unit conversion table states directly: “1 square metre = 10.764 square feet”. This is a specific conversion factor provided in the chapter.

The unit conversion section clearly provides the formula: “Celsius = (5⁄9) × (Fahrenheit – 32)”. This formula correctly accounts for the offset and scaling between the two temperature scales.