NCERT Class 8 Mathematics Ganita Prakash Part 2 Chapter 03 Proportional Reasoning 2 PDF Download

Chapter 3: Proportional Reasoning-2

3.1 Proportionality - A Quick Recap

In an earlier chapter, we explored proportional relationships between quantities and we used the ratio notation to represent such relationships. When two or more related quantities change by the same factor, we call that relationship a proportional relationship. For example, idli batter is made by mixing rice and urad dal. The proportion of these two can have regional variations. One of the proportions used is: for 2 cups of rice, we add 1 cup of urad dal. We represent this relationship using the ratio notation 2 : 1.

Viswanath made idlis by mixing 6 cups of rice with 3 cups of urad dal, while Puneet made idlis by mixing 4 cups of rice with 2 cups of urad dal. If cooked in the same way, would their idlis taste the same?

Viswanath's mixture can be represented as 6 : 3 and Puneet's mixture as 4 : 2.

Recall that, to verify that these two ratios are proportional, we can use the cross-multiplication method:

[Diagram showing: 6 : 3 and 4 : 2 with cross-multiplication arrows, See in your textbook]

The two products are the same (12). So, the two ratios are proportional. It is likely that the idlis would taste the same, if all the other ingredients are proportional too!

In general, we can say that two ratios \( a : b \) and \( c : d \) are proportional if

\[ a \times d = b \times c, \text{ or} \]

\[ \frac{a}{c} = \frac{b}{d} \]

Teacher's Note

When checking if two ratios are proportional, cross-multiplication is faster than converting to fractions. Remember: multiply the outer terms and the inner terms. If \( 6 \times 2 = 3 \times 4 \), both equal 12, so the ratios are proportional. Always check both products match before saying yes.

3.2 Ratios in Maps

Have you noticed that in many maps there is a ratio given, usually in the lower right corner of the map? It usually contains 1 and a very large number, such as 1 : 60,00,000. What does RF 1 : 60,00,000 mean? What does it indicate? Can you guess?

A Representative Fraction (RF) is an expression that shows the ratio between a distance on the map and the corresponding actual distance on the ground.

For example, if the ratio on a map is 1 : 60,00,000, that means a distance of 1 cm on the map is equivalent to a geographical distance of 60,00,000 cm. Remember, this is geographical distance and not road distance!

[Figure 3.2: Map of South India showing cities including Bengaluru, Chennai, Mangaluru, Mysuru, Kannur, Tirupati, Puducherry, Coimbatore, Kochi, Madurai, and Thiruvananthapuram, with RF = 1 : 60,00,000, See in your textbook]

Convert 60,00,000 cm to kilometres. It is 60 km. Verify this.

Using the map above, can you find the geographical distance between Bengaluru and Chennai? Also, find the geographical distance between Mangaluru and Chennai.

[Hint: Use a ruler to find the distance between the cities on the map. Then, use the ratio given on the map to find the actual geographical distance.]

Try to find the distances between the same two pairs of cities with different maps that have different scales (ratios). Do they all give the same geographical distance, approximately?

Map Making Activity

Guide students to make a sketch of their classroom with an accurate scale (ratio of 1: 50). They should mark the location of various objects in the classroom like the teacher's desk, blackboard, fans and lights, according to scale. Students can use appropriate symbols to represent different objects like fans, lights, tables, chairs, and so on.

Teacher's Note

When you use a map with RF 1 : 60,00,000, remember that 1 unit on the map (say 1 cm) stands for 60,00,000 of those same units on the ground (60,00,000 cm = 600 km). If you measure 2 cm on the map, multiply it by the RF to get 2 \( \times \) 60,00,000 cm on the ground. Always use a ruler carefully, and convert the final answer to a sensible unit like kilometres.

3.3 Ratios with More than 2 Terms

Viswanath is experimenting with a spice mix powder. He makes the powder by grinding 8 spoons of coriander seeds, 4 red chillies, 2 spoons of toor dal, and 1 spoon of fenugreek (methi) seeds. For his spice mix powder, the ratio of coriander seeds to red chillies to toor dal to fenugreek seeds is

\[ 8 : 4 : 2 : 1 \]

[Figure: Illustration of two children making spice mix powder, See in your textbook]

Notice that the ratio has 4 terms. Ratios can have many terms if each of the quantities change by the same factor to maintain the proportional relationship.

Puneet has only 2 red chillies in his kitchen. But he wants to make spice mix powder that tastes the same as Viswanath's spice mix powder. How much of the other ingredients should Puneet use to make his spice mix powder?

For Puneet's spice mix powder to be similar to Viswanath's, the ratio of all the ingredients should be the same as Viswanath's spice mix powder. Puneet has only 2 red chillies. He has half the number of chillies that Viswanath used in his mixture. So, the quantity of the other ingredients should also be reduced to half.

Thus, Puneet should add 4 spoons of coriander seeds, 2 red chillies, 1 spoon of toor dal, and half a spoon of fenugreek seeds. The ratio is

\[ 4 : 2 : 1 : 0.5 \]

Key Points

• Two ratios are proportional if the cross products are equal: \( a \times d = b \times c \) for ratios \( a : b \) and \( c : d \).
• A Representative Fraction (RF) on a map shows the relationship between distance on the map and actual ground distance; for example, RF 1 : 60,00,000 means 1 cm on the map represents 60,00,000 cm or 600 km on the ground.
• Ratios can have more than two terms, and all quantities must change by the same factor to keep the ratio proportional.

This is a preview of the first 3 pages. To get the complete chapter, click below.