Proportional Reasoning 1 Class 8 Notes

Proportional Reasoning 1 Class 8 Notes

Proportional Reasoning 1 Class 8 Notes

Proportionality

A relationship between two quantities where the ratio remains constant.

Key Concept: If one quantity changes, the other changes by the same factor.

Ratios

A way to compare two quantities using division.

Forms of Ratios :

As a fraction : [latex]\frac{a}{b}[/latex]

With a colon : a : b

In words : a to b

Simplest Form of Ratios

The smallest whole number representation of a ratio.

Finding Simplest Form :

Step 1 : Find the Highest Common Factor (HCF).

Step 2 : Divide both terms by the HCF.

Example : Ratio (72 : 96) HCF : 24

Simplification : [latex]\frac{72}{24}[/latex] : [latex]\frac{96}{24}[/latex] = 3 : 4

Problem Solving with Proportional Reasoning

Rule of Three (Trairasika):

Concept: If two ratios are equal, we can find an unknown quantity.

Notation : a: b :: c : d

Cross Multiplication a.d = b.c

Finding Unknown : d = b. [latex]\frac{c}{a}[/latex]

Sharing Quantities in Ratios

Dividing a Quantity:

Ratio m : n means for every m parts of one quantity, there are n parts of another.

Steps :

Total Parts : m + n

Size of Each Group : [latex]\frac{x}{\{m+n\}}[/latex]

Calculating Each Part :

First part : m × [latex]\frac{x}{\{m+n\}}[/latex]

Second part : n × [latex]\frac{x}{\{m+n\}}[/latex]

Example : Dividing ₹ 800 in the ratio (3 : 1)

Total Parts : (3 + 1 = 4)

Size of Each Group : [latex]\frac{800}{4}[/latex] = 200

Calculating Each Share:

A’s share : 3 × 200 = 600

B’s share : 1 × 200 = 200

Unit Conversions

Importance : Helps in understanding measurements in different contexts.

Common Conversions :

Length : 1 metre = 3.281 feet

Area : 1 square metre = 10.764 square feet

1 acre = 43,560 square feet

1 hectare = 10,000 square metres

1 hectare = 2.471 acres

Volume : 1 millilitre (mL)

= 1 cubic centimetre (cc)

1 litre = 1,000 mL = 1,000 cc

Temperature : Celsius to Fahrenheit:

F = [latex]\frac{9}{5}[/latex]C + 32

Fahrenheit to Celsius : C = [latex]\frac{9}{5}[/latex](F - 32)

Example :

25°C = [latex]\frac{9}{5}[/latex] × 25 + 32 = 77°F

Observing Similarity in Change

Proportionality refers to a relationship between two quantities where they change in such a way that the ratio between them remains constant. This means that if one quantity increases or decreases, the other does sd by the same factor.

Example: Imagine you are making lemonade. If you use 10 spoons of sugar for 6 glasses of lemonade, and you want to make 18 glasses, you need to find out how much sugar to use to keep the same sweetness.

To do this, we can set up a proportion :

6 glasses : 10 spoons : : 18 glasses : x spoons

Solve for x :

Divide both sides by 4 :

x = [latex]\frac{20}{4}[/latex] = 5

So, you need 5 cups of flour to make 10 cookies.

Pramana, Ichchha, and Ichchhaphala

In ancient Indian mathematics, particularly as described by Aryabhata, the terms pramana, ichchha, and ichchhaphala are used in the context of proportional reasoning:

1. Pramana : This refers to the measure or the known quantity (in our example, the amount of flour).
2. Ichchha : This is the requisition or the desired quantity (the number of cookies you want to make).
3. Ichchhaphala : This is the yield or the resulting quantity (the amount of flour needed for the desired number of cookies).

The relationship can be expressed as :

Pramana × Ichchhaphala = phala × Ichchha This means that if you multiply the known quantity by the yield, it should equal the product of the known result and the desired quantity.

Example of Pramana and Ichchha

Example 2 : If a cook uses 10 kg of rice for 100 students (pramana), how much rice is needed for 80 students (ichchha)?

1. Set up the proportion :

Known : 10 kg for 100 students.

Unknown : x kg for 80 students.

We can write :

100 : 10 :: 80 : x

Cross multiply :

This gives us :

100x = 10 . 80

Simplifying :

100x = 800

Solve for x :

Dividing both sides by 100 :

x= [latex]\frac{800}{100}[/latex] = 8

Thus, the cook should use 8 kg of rice for 80 students.

Sharing, but Not Equally!

When we want to divide a quantity x in the ratio m : n, we follow these steps :

Understanding the Ratio : The ratio m : n, means for every m parts of one quantity, there are n parts of another.

Example : A ratio of 3 :1 means 3 parts for the first and 1 part for the second.

Total parts:

Total parts = m + n

Size of each part:

Size of each part = [latex]\frac{x}{m+n}[/latex]

Calculating each share:

First part = m × [latex]\frac{x}{m+n}[/latex]

Second part = n × [latex]\frac{x}{m+n}[/latex]

Final Representation:

(m × [latex]\frac{x}{m+n}[/latex], n × [latex]\frac{x}{m+n}[/latex])

Example:

Total amount = ₹ 800

Ratio = 3 : 1

Total parts = 3 + 1 = 4

Each part = [latex]\frac{800}{4}[/latex] = 200

A’s share :

3 × 200 = 600

B’s share:

1 × 200 = 200

So, A gets ₹ 600 and B gets ₹ 200.

Unit Conversions

Length 1 metre = 3.281 feet

Area 1 square metre = 10.764 square feet

1 acre = 43,560 square feet

1 hectare = 10,000 square metres

1 hectare = 2.471 acres

Volume

1 millilitre (mL) = 1 cubic centimetre (cc)

1 litre = 1,000 mL = 1,000 cc

Temperature

Celsius to Fahrenheit:

F = [latex]\frac{9}{5}[/latex] C + 32

Fahrenheit to Celsius :

C = [latex]\frac{9}{5}[/latex](F - 32)

Example:

25°C = [latex]\frac{9}{5}[/latex] × 25 + 32 = 77°F