Proportional Reasoning 2 Class 8 Notes Maths Part 2 Chapter 3

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Class 8 Maths Chapter 3 Proportional Reasoning 2 Notes

Class 8 Proportional Reasoning 2 Notes

Ratio of Proportions

Ratio

• The ratio of a number a to another number b (where, b ≠ 0) is a fraction \(\frac{a}{b}\) and it is written as a: b. b
• In the ratio a: b, the first term a is called antecedent and the second term b is called consequent.
• Here, a and b should be the quantities of same kind and in the same units.

Ratios in Maps
Maps use a specific type of ratio called a Representative Fraction (RF) to show the relationship between distances on paper and actual ground distance.
e.g. A ratio of 1: 60,00,000 means 1 cm on the map equals 60,00,000 cm in reality. This ratio represents the “as the crow flies” geographical distance, not the distance along roads.
Note Large centimeter values are typically converted to kilometers (e.g. 60,00,000 cm = 60 km) to be more useful.

Ratios with Multiple Terms
A ratio with multiple terms is a comparision of three or more quantities of the same kind expressed in a single statement by separating the terms with colors (:), showing their relative magnitude simultaneously.

Properties

• Scaling Property Multiplying or dividing all terms in a ratio by the same non-zero number (k) does not change the ratio value.
• Simplest Form A ratio is in its lowest terms when the highest common factor (HCF) of all terms is 1.
• Order of Terms The order of terms is crucial; a: b: c is not the same as b:a:c.
• Unitless/Dimensionless Ratios compare quantities of the same kind (e.g. length, weight), so the unit cancel out, leaving a dimensionless number.
  e.g. If a spice mix uses a ratio of 8 : 4 : 2 : 1 and one ingredient is halved, all other ingredients must also be halved (becoming 4:2:1: 0.5) to keep the same flavor.

If a: b: c: d is proportional to p:q:r:s then
\(\frac{a}{p}=\frac{b}{q}=\frac{c}{r}=\frac{d}{s}\)
This involves splitting a total quantity into parts based on a specific ratio.

In this process,

• Add the terms of the ratio to find the ‘total parts’.
• Divide the whole quantity by the total parts to find the value of‘one part’.
• Multiply each term in the ratio by this ‘one part’ value to find the specific quantities.
• General Formula For a quantity x in ratio a : b : c + ………….
  the parts are x × \(\frac{a}{a+b+c+\ldots \ldots . .}\), x × \(\frac{b}{a+b+c+\ldots \ldots . .}\), x × \(\frac{c}{a+b+c+\ldots \ldots . .}\), ….
• Applications: This is used for mixing concrete (cement: sand : gravel), creating paint shades and calculating the internal angles of a triangle (180° total).

Direct Proportions
If the values of two quantities depend on each other in such a way that a increase in one, results in a corresponding increase in the other and vice-versa then the two quantities are said to be in direct proportion.
If two quantities a and b vary in direct proportion and if b₁ and b₂ are the values of b corresponding to the values a₁ and a₂ of a respectively then \(\frac{a_1}{b_1}=\frac{a_2}{b_2}\) = k (constant)

Note When two quantities x and yare in direct proportion (or vary directly), they are also written as x = y symbol ∝ stands for ‘is proportional to’.

Inverse Proportions
The two quantities may vary in such a way that if one increases, the other decreases and vice-versa then the two quantities are said to be in inverse proportion.
If two quantities a and b vary in inverse proportion to each other and b₁b₂ are the values of b corresponding to the values a₁, a₂ of a respectively then
a₁b₂ = k
and a₂b₂ = k
or \(\frac{a_1}{a_2}=\frac{b_2}{b_1}\)

Note When two quantities x and y are in inverse proportion
(or vary inversly), they are also written as x ∝ \(\frac{1}{y}\).

Pie Chart
A pie chart is a circular graph used to show how different proportions make up a whole. To create a pie chart, the angle of each sector must be proportional to the data it represents. Since, a circle has 360°, the angle of each sector is calculated as
Central angle = \(\frac{\text { Value }}{\text { Total value }}\) × 360°