Experts have designed these Class 8 Maths Notes and Chapter 7 Proportional Reasoning 1 Class 8 Notes for effective learning.
Class 8 Maths Chapter 7 Notes Proportional Reasoning 1
Class 8 Maths Notes Chapter 7 – Class 8 Proportional Reasoning 1 Notes
Comparison is something we do all the time, when we check who is taller, which bag is heavier, or which road is longer. We compare things in different ways, like by subtracting, measuring, or counting. Sometimes, we look at how many times one quantity fits into another, like how many cups of flour are needed for a cake. When we find that two different sets of things have the same kind of relationship, like a small recipe and a bigger one using the same mix of ingredients, it helps us understand how values grow or shrink in a balanced way. This idea is useful when we want to keep things fair, equal, or in the right order.
→ A ratio is a way of comparing two quantities of the same kind using the same unit.
→ A ratio is written in the form a: b, which means for every a units of the first quantity, there are b units of the second quantity. The numbers a and b are called the terms in the ratio.
→ Ratios can be written in its simplest form, like fractions, by dividing both terms by their HCF.
→ Multiplying or dividing both terms of a ratio by the same non-zero number does not change the ratio.
→ Adding or subtracting the same number to both terms changes the ratio and is not allowed for comparison.
→ When two ratios are same in their simplest forms, they are said to be in proportion.
→ If a : b :: c : d, then a × d = b × c
→ If a : b :: c : d, then the fourth term, d= \(\frac{(b×c)}{a}\).
→ Rule of Three : Two ratios are proportional if their terms are equal when cross multiplied. The fourth unknown quantity can be found through such cross-multiplication.
→ If x is divided into two parts in the ratio m: n, then the quantity of the first part = m × \(\frac{x}{m+n}\), and the quantity of second part = n × \(\frac{x}{m+n}\).
→ Unit Conversions
- 1 metre = 3.281 feet, 1 square metre = 10.764 square feet
- 1 acre = 43,560 square feet, 1 hectare = 10,000 square metres
- 1 hectare = 2.471 acres
- 1 millilitre (mL) = 1 cubic centimetre (cc), 1 litre = 1,000 mL or 1,000 cc,
- F = \(\frac{9}{5}\) × °C + 32,°C = \(\frac{5}{9}\) × (°F-32), where C = Celsius, F = Fahrenheit
Ratio
A ratio is a comparison of two or more quantities of the same kind taken in the same unit.
If x and y are two quantities of same kind taken in the same unit, then fraction x/y is known as ratio of x and y. It is written as x: y and read as ‘x is to y.
e.g. The ratio of 5l: 3l is \(\frac{5}{3}\) or 5 : 3.
In the expression x : y, the quantities x and y are called terms of ratio, where x is called the first term (or antecedent) and y is called the second term (or consequent).
Ratio is always expressed in numbers, that are coprimes.
It can only be calculated, if the quantities involved are in the same unit.
The value of a ratio (say x: y) remains unchanged, if both the antecedent (x) and consequent (y) are multiplied by the same non-zero number (say a)
i.e. x:y is the same as ax : ay. [∵ \(\frac{x}{y}=\frac{a x}{a y}\)]
The value of a ratio (say x : y) remains unchanged, if both the antecedent (x) and consequent (y) are divided by the same non-zero number (say a)
i.e. x : y is the same as \(\frac{x}{a}: \frac{y}{a}\) [∵ \(\frac{x}{y}=\frac{x \div a}{y \div a}\)]
Important Facts about Ratio
- The ratio between two or more unlike (or different) quantities does not exist.
- Ratio is taken only between positive quantities.
- The order of the terms in a ratio is important.
- The ratio is always expressed in the lowest or simplest form.
- A ratio is a number, so it has no units.
- Ratio x : y is not equal to ratio y: x i.e. x: y ≠ y: x.
- If the terms of the given ratio are in fractions, then convert the terms of the ratio in the whole numbers by multiplying each term by the LCM of their denominators.
e.g. \(\frac{2}{5}: \frac{3}{4}=\frac{2}{5}\) × 20: \(\frac{3}{4}\) × 20
= 2 × 4 : 3 × 5
= 8 :15
Ratio in their simplest form
To express a ratio in its simplest form, divide both the terms of the ratio by their highest common factor (HCF).
e.g. The simplest form of 12 : 16 is 3 : 4.
Sharing but not Equally
If x is divided into two parts in the ratio m: n then the quantity of the first part is m × \(\frac{x}{m+n}\) and the quantity of
the second part is n × \(\frac{n}{m+n}\).
Unit Conversions
Unit conversion is the process of changing a quantity expressed in one unit into an equivalent value in another unit of the same physical quantity. The conversion of units is required to solve various mathematical problems.
For example, if the length of a rectangle is given in metre whereas the breadth is given in centimetre then to determine the perimeter of the rectangle, we need to convert the units to make them uniform.
Given below is a unit conversion table depicting the relationship between different units.
| Quantity | Relationship |
| Length |
|
| Area |
|
| Volume |
|
| Temperature |
|
| Time |
|
Proportion
An equality of two ratios is called a proportion, i.e. if a, b, c and d are four quantities of same kind taken in same unit, then a:b::c:d is called the proportion, which means a:b = c:d
or \(\frac{a}{b}=\frac{c}{d}\) or ad = be b d
This is known as cross multiplication of terms.
The quantities a, b, c and d are called the terms of the proportion; a → first term, b → second term, c → third term and d → fourth term.
First and fourth terms are called extreme terms and second and third terms are called means or middle terms.
Note It If the quantities a, b, c and d are in proportion, then \(\frac{a}{b}=\frac{c}{d}\) or ad = bc. i.e. Product of extreme terms = Product of middle terms.
Trairasika-The Rule of Three
The ‘Rule of Three’ is a method used to find the fourth term when three terms of a proportion are known.
This helps us to solve proportional problems by applying the cross-multiplication method.
If a: b:: c: x, then x = \(\frac{b c}{a}\).