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Class 8 Maths Chapter 3 Proportional Reasoning 2 Notes
Class 8 Proportional Reasoning 2 Notes
→ Ratios in the form a : b : c : d : … indicate that for every a units of the first quantity, there are b units of the second quantity, c units of the third quantity, and so on.
→ If x is divided into many parts in the ratio p : q : r : s : …, then the quantity of the first part is x × \(\frac{p}{(p+q+r+s+\cdots)}\), the quantity of the second part is x × \(\frac{q}{(p+q+r+s+\cdots)}\), and so on.
→ Two quantities are directly proportional when they both change by the same factor, and their quotient remains the same.
For example, if x and y are two directly proportional quantities, and (x1, x2, x3, …) and ( y1, y2, y3, …) are the corresponding values of x and y, then \(\frac{x_1}{y_1}=\frac{x_2}{y_2}=\frac{x_3}{y_3}\) = …. = k, where k is a constant.
→ Quantities are inversely proportional if, when one quantity changes by a factor n, the other quantity changes by the inverse \(\frac {1}{n}\).
For example, if x and y are two inversely proportional quantities, and (x1, x2, x3, …) and (y1, y2, y3, …) are the corresponding values of x and y, then x1y1 = x2y2 = x3y3 = … = n, where n is a constant.
Proportionality — A Quick Recap
An equality of two ratios is called a proportion, that is, if ratio a : b is equal to ratio c : d, that is, if
\(\frac{a}{b}=\frac{c}{d}\) …..(1)
Then these ratios are said to be in proportion and are written as a : b :: c : d, which is read as a is to b as c is to d.
a is called the first proportional, b is called the second proportional, c is called the third proportional, and d is called the fourth proportional.
From (1), by definition of proportion \(\frac{a}{b}=\frac{c}{d}\),
cross-multiplying ad = bc.
That is, Ist proportion × IVth proportion = IInd proportion × IIIrd proportion
We can remember it as ad = bc, that is product of extremes = the product of means
Remark: Second test for two ratios \(\frac {a}{b}\) and \(\frac {c}{d}\) to be equivalent that is \(\frac{a}{b}=\frac{c}{d}\) that is \(\frac {a}{b}\) and \(\frac {c}{d}\) are in proportion.
⇒ ad = bc
⇒ Product of extremes = product of means.
Ratios in Maps
A map scale is fundamentally the ratio of a distance on the map to the corresponding distance on the ground. This ratio can be expressed in several ways, each serving a different purpose and user need. There are three primary types of map scales: fractional or ratio scales, linear scales, and verbal scales.
A fractional or ratio scale uses a representative fraction to indicate the scale. For example, a scale of 1 : 100,000 means that 1 unit on the map represents 100,000 units in real life. This type of scale is often used for its precision and clarity.
Representative Fraction (R.F.): It shows the relationship between the map distance and the corresponding ground distance in units of length. The use of units to express the scale makes it the most versatile method.
Systems of Measurements
Metric System of Measurement
- 1 km = 1000 m
- 1 m = 100 cm
- 1 cm = 10 mm
English System of Measurement
- 1 Mile = 8 Furlongs
- 1 Furlong = 220 Yards
- 1 Yard = 3 feet
- 1 Foot = 12 Inches
Example: Find the distance from Delhi to Mumbai in the following map.
Solution:
We are given 1 cm : 200 km
Therefore, 6 cm would be 6 × 200 km
6 × 200 km = 1,200 km
Hence, the distance from Delhi to Mumbai is 1,200 km.
Ratios with More Than 2 Terms
Ratios with more than two terms (such as a : b : c) represent the relative sizes of three or more quantities simultaneously. While a standard two-term ratio compares two parts, a multi-term ratio establishes a fixed relationship between every included quantity.
To simplify a ratio with multiple terms, we must divide all terms by their Greatest Common Factor (GCF).
For example, the ratio 12 : 18 : 24 simplifies to 2 + 3 + 4 after dividing each part by 6.
Note: We cannot simplify only two terms of a three-term ratio; the same operation must be applied to every part.
Total Parts: To find the share of each quantity, add all the ratio terms together to find the “total parts”.
Example: For a 3 : 1 : 2 ratio, the total parts are 3 + 1 + 2 = 6.
Proportions with Multiple Terms: In a proportion like a : b : c = x : y : z, the ratio of any two terms on the left must equal the ratio of the corresponding two terms on the right (e.g., \(\frac{a}{x}=\frac{b}{y}=\frac{c}{z}\))
Example: If the ratio of boys to girls to teachers in a classroom is 3 : 4 : 1 and there are 12 boys, find the number of girls and teachers.
Solution:
Given ratio = 3 : 4 : 1
Number of boys = 12
Let the common multiplier be k.
3k = 12
⇒ k = 4
Number of girls = 4k
= 4 × 4
= 16
Number of teachers = 1k
= 1 × 4
= 4
Dividing A Whole in A Given Ratio
Dividing a whole quantity into a given ratio (e.g., a : b) involves distributing the total amount into unequal parts that maintain a specific relative size. This process, often referred to as “sharing in a ratio,” can be solved using either the Unitary Method or the Fraction Method.
Unitary Method
This is the most common method for dividing quantities:
1. Find the Total Number of Parts: Add all the numbers in the ratio together.
Example: If the ratio is 3 : 4, the total parts are 3 + 4 = 7.
2. Calculate the Value of One Part: Divide the total quantity (the “whole”) by the total number of parts.
Value of one part = \(\frac{\text { Total Quantity }}{\text { Total Number of Parts }}\)
Multiply Each Ratio Part: Multiply the value of one part by each number in the ratio to find the individual shares.
Share of A = a × value of one part
Share of B = 6 × value of one part
Alternative Method (Fraction Method)
We can treat each part of the ratio as a fraction of the whole.
For a ratio a : b, Share A is \(\frac{a}{a+b}\) of the total, and Share B is \(\frac{b}{a+b}\) of the total.
Example: Divide 180 in the ratio 3 : 4 : 5.
Total Parts: 3 + 4 + 5 = 12
Value of One Part: 180 ÷ 12 = 15
Individual Shares:
3 × 15 = 45
4 × 15 = 60
5 × 15 = 75
Check: 45 + 60 + 75 = 180
The sum must always equal the original whole.
A Slice of the Pie
Pie Chart or Circle Graph
A pie chart is a simple way of presenting data in a circle (diagram) for easy understanding. This circle is divided into non-intersecting adjacent sectors.
The number of sectors = Number of components in the data = Number of class intervals in the given grouped frequency distribution.
The size (area) of each sector is proportional to the activity or information it represents.
Sector of a Circle
If O is the centre of a circle and AB is an arc of the circle, then the region OAB (shaded portion) bounded by arc AB of the circle and the two bounding radii OA and OB of the circle is called a sector.
In fact, a pie chart shows the relationship between a whole and its parts (components).
Each component can be expressed as a fraction of the whole.
Below, let us learn the concept of a pie chart through the following diagram.
Pie Chart for the time spent by a child during a day
In this figure, there are five components, namely: play, school, sleep, homework, and others; each is represented by a sector.
We can also observe that the size of each sector is proportional to the number of hours given for each component.
In the above diagram,
the component ‘play’ = \(\frac{3 \text { hours }}{24 \text { hours }}\left(=\frac{1}{8}\right)\)th part of a day
Component ‘School’ = \(\frac{6 \text { hours }}{24 \text { hours }}\left(=\frac{1}{4}\right)\)
Component ‘Sleep’ = \(\frac{8 \text { hours }}{24 \text { hours }}\left(=\frac{1}{3}\right)\)
Component ‘Home work’ = \(\frac{4 \text { hours }}{24 \text { hours }}\left(=\frac{1}{6}\right)\)
Component ‘Others’ = \(\frac{3 \text { hours }}{24 \text { hours }}\left(=\frac{1}{8}\right)\)
Adding all five fractions for five components, we have
Sum = \(\frac{3}{24}+\frac{6}{24}+\frac{8}{24}+\frac{4}{24}+\frac{3}{24}\)
= \(\frac{3+6+8+4+3}{24}\)
= \(\frac {24}{24}\)
= 1
= Whole (day)
Note:
1. Therefore whole (i.e., the entire circle) = the sum of all components.
2. A pie chart is a way of showing how something is shared or divided.
Remark: Dictionary meaning of the word ‘Pie’ is a baked food (like pastry) having a sweet filling and topped with a pastry crust.
The pie graph (chart) looks like a pastry (stated above), and the components resemble slices cut from a pie.
Example: Each of the following pie charts gives a different piece of information about a class. Find the fraction of the circle representing each of this information.
Solution:
P% as a fraction = \(\frac {P}{100}\)
(i) From the pie chart, the fraction of the circle representing the ‘girls’ = 50%
= \(\frac {50}{100}\)
= \(\frac {1}{2}\)
and fraction of the circle representing the ‘boys’ = 50%
= \(\frac {50}{100}\)
= \(\frac {1}{2}\)
(iii) From the pie chart, the fraction of the circle representing ‘Walk’ = 40%
= \(\frac {40}{100}\)
= \(\frac {2}{5}\)
fraction of the circle representing ‘Cycle’ = 20%
= \(\frac {40}{100}\)
= \(\frac {1}{5}\)
and fraction of the circle representing (Bus or Car) = 40%
= \(\frac {40}{100}\)
= \(\frac {2}{5}\)
(iii) From the pie chart, the fraction of the circle representing those who hate Mathematics = 15%
= \(\frac {15}{100}\)
= \(\frac {2}{5}\)
Fraction of the circle representing those who love Mathematics = (100 – 15)%
= 85%
= \(\frac {85}{100}\)
= \(\frac {17}{20}\)
Construction of a Pie Chart
In this section, we shall learn ‘How to construct a pie chart’.
We have learnt in a pie chart that the size (or area) of each sector representing a component of a pie diagram is proportional to the component value. …(1)
But we can observe that the area of a sector is proportional to the angle subtended by the arc (AB) of the sector at the centre O of the circle. …(2)
This angle ∠AOB at the centre is called the sector angle or central angle.
From (1) and (2), we can say that
The component values (different sectors) of a pie diagram are proportional to the corresponding sector angles at the centre.
Please observe and learn that the total angle at the centre of a circle is 360°.
The central angles of sectors are fractions of 360°.
Central angle of a component = \(\left(\frac{\text { Value of the component }}{\text { Total sum of component values }}\right) \times 360^{\circ}\)
Note: For the pie chart given in example, the fractions obtained were \(\frac{1}{8}, \frac{1}{4}, \frac{1}{3}, \frac{1}{6}, \frac{1}{8}\)
Let us multiply each of them by 360°, and we get the central (sector) angles as
\(\frac {1}{8}\) × 360° = 45°
\(\frac {1}{4}\) × 360° = 90°
\(\frac {1}{3}\) × 360° = 120°
\(\frac {1}{6}\) × 360° = 60°
\(\frac {1}{8}\) × 360° = 45°
Sum of all these central (sector) angles = 45° + 90° + 120° + 60° + 45° = 360°
∴ In the figure of Example in the pie chart, the angle subtended by the sector representing ‘play’ is 45°, the angle for sector ‘school’ is 90°, and so on.
Algorithm for Construction of a Pie Chart
Step 1. Find the sum of all the component values. In case of a frequency distribution, component values are class frequencies, and the sum (= Total frequency).
Let the component values be 10, 20, 40, 50
Sum of these values = 120
Step 2. Form a table to find the sector (central) angles corresponding to different components (or class frequencies) by using the formula stated above, namely.
Sector angle = \(\frac{\text { Value (or frequency) of one component }}{\text { Total sum of all components }} \times 360^{\circ}\)
Step 3. Draw a circle of any convenient radius.
(Take the radius a little larger so that the diagram looks good and pleasing, and labels (names) of the components can be easily written in the sectors)
Step 4. Draw a horizontal radius.
Step 5. Starting with this horizontal radius, draw the sector angles obtained in Step 2 to get the sectors (See sector of a circle) for different components.
(Of course, these sectors should be non-intersecting and adjacent as stated in the definition of a pie chart)
Step 6. Shade the sectors obtained by different patterns or designs and label each one of them.
Step 7. Give the heading for the pie chart.
Inverse Proportions
We come across many situations in our day-to-day life where we can easily observe that proportion (i.e., change) in one quantity brings proportion (change) in the other quantity.
For example:
(i) If the number of notebooks purchased increases, then their cost also increases.
We find that these two items are changing: the number of notebooks and the cost of notebooks. These two items are called variables.
(ii) If the number of people contributing to flood victims increases, the fund’s collection increases.
(iii) As the speed of a vehicle increases, the time taken to cover the same distance decreases.
(iv) For a given construction work, the fewer workers, the less time it will take to complete the work.
Conversely less the number of workers, the more the time for completion.
Proportion: If two quantities depend on each other in such a way that a change in one results in a corresponding change in the other, then the two quantities are said to be in proportion.
We know that the ratio of two quantities a and b (b ≠ 0) of the same kind and in the same units is a ÷ b or \(\frac {a}{b}\) and is denoted by a : b.
Proportion: An equality of two ratios is called a proportion.
Types of Proportions: There are two types of proportions.
(i) Direct Proportion: Two quantities x andy are said to be in direct proportion if whenever the value of x increases (or decreases), then the value of y also increases (or decreases) in such a way that the ratio \(\frac {x}{y}\) remains constant (= k say) …(1)
This constant k is called the constant of (direct) proportion.
If x1 and y1 are the corresponding values of two quantities x and y in direct proportion at one point, then from (1)
\(\frac{x_1}{y_1}\) = k …(2)
Similarly, if x2 andy2 are their (x and y) corresponding values at another point, then again from (1),
\(\frac{x_2}{y_2}\) = k …(3)
From (2) and (3), \(\frac{x_1}{y_1}=\frac{x_2}{y_2}\)
Cross-multiplying, x1y2= x2y1
⇒ \(\frac{x_1}{x_2}=\frac{y_1}{y_2}\)
⇒ x1 : x2 = y1 : y2 (By definition of ratio in Note 1)
Hence, the rule: If two quantities x and y are in direct proportion, the ratio of any two values of x is equal to the ratio of the corresponding values of y.
Remark: When x and y are in direct proportion, we also say that x and y are directly proportional to each other.
(ii) Inverse Proportion: Two quantities x and y are said to be in inverse proportion if whenever the value of x increases (or decreases), then the value of y decreases (or increases).
Remark: If two quantities are in inverse proportion, then we also say that they are inversely proportional to each other.
The post Proportional Reasoning 2 Class 8 Notes Maths Part 2 Chapter 3 appeared first on Learn CBSE.
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