Introduction to Ratios and Proportion
Ratios and proportion, finding increase or decrease percent.
Introduction — Comparing Quantities Using Proportion
Class 8 Mathematics · CBSE, Telangana & Andhra Pradesh Syllabus · Chapter 5
This is the introduction to Chapter 5, Comparing Quantities Using Proportion, in Class 8 Mathematics (CBSE, Telangana & Andhra Pradesh syllabus). Before solving the exercises in this chapter, it's essential to be completely comfortable with four foundational ideas: ratio, proportion, compound ratio, and percentage. Every later exercise in this chapter — whether it's about sharing money in a given ratio, scaling up a recipe, or calculating profit and loss — is really just one of these four ideas applied to a real-life situation.
Below, each concept is explained from first principles with its formula and a simple worked example, so that you walk into the exercises of this chapter already confident with the underlying mathematics.
What Is a Ratio?
A ratio is an ordered comparison of two quantities of the same kind, written as a : b (read "a is to b"). The first term, a, is called the antecedent, and the second term, b, is called the consequent. Order matters — the ratio 3 : 5 is not the same comparison as 5 : 3.
a : b → a = antecedent, b = consequent
- Multiplying or dividing both terms of a ratio by the same non-zero number does not change its value — exactly like simplifying a fraction.
- A ratio should always be written in its simplest form, by dividing both terms by their highest common factor (HCF).
- Both quantities being compared must be measured in the same unit before the ratio is formed.
A ratio of 3 : 4 simply means "for every 3 units of one quantity, there are 4 matching units of the other."
A basket has 15 oranges and 20 apples. Express the number of oranges to apples as a ratio in simplest form.
What Is a Proportion?
When two ratios are equal, the four quantities involved are said to be in proportion. If a : b = c : d, then a, b, c, and d are in proportion, and this is often written as a : b :: c : d.
If a : b = c : d, then a × d = b × c
Here, a and d are called the extremes, while b and c are called the means. This gives the single most useful rule in the whole chapter:
The "cross-multiplication" rule: product of extremes (a × d) always equals product of means (b × c) in a true proportion.
Continued Proportion (Mean Proportional)
A special case occurs when the means of a proportion are the same quantity: if a : b = b : c, then a, b, and c are said to be in continued proportion, and b is called the mean proportional between a and c.
If a : b = b : c, then b² = a × c
Check whether 4, 6, 10, and 15 are in proportion.
Find x if 4, x, and 9 are in continued proportion (i.e. 4 : x = x : 9).
What Is a Compound Ratio?
Sometimes two separate ratios need to be combined into a single one — for example, when a quantity depends on two different factors at once. The compound ratio of two simple ratios is found by multiplying their antecedents together and their consequents together.
Compound ratio of a : b and c : d = (a × c) : (b × d)
Find the compound ratio of 2 : 3 and 5 : 7.
What Is a Percentage?
A percentage compares a number to 100 — the word "percent" literally means "per every hundred." A percentage is simply a special kind of ratio (or fraction) whose consequent is fixed at 100, which makes it very easy to compare different quantities on a common scale.
A 10×10 grid is the easiest way to picture any percentage — 25% is exactly 25 of the 100 small squares shaded in.
x% of y = y × (x/100) = xy/100
x is what % of y → (x/y) × 100 %
Find 25% of 80.
15 is what percentage of 60?
Percentage Increase and Decrease
When a quantity x changes by y%, the new value can be found directly with these two formulas — no need to calculate the change separately and then add or subtract it:
Increased value = x × (100+y)/100 | Decreased value = x × (100−y)/100
A quantity of 200 is increased by 10%. Find the new value.
A quantity of 500 is decreased by 20%. Find the new value.
Quick Reference — All Formulas at a Glance
| Concept | Formula |
|---|---|
| Simplest form of a ratio | Divide both terms by their HCF |
| Proportion test | a : b = c : d ⇔ a × d = b × c |
| Mean proportional | a : b = b : c ⇒ b² = a × c |
| Compound ratio | a:b and c:d ⇒ ac : bd |
| x% of y | xy / 100 |
| x as a % of y | (x/y) × 100 % |
| Increase x by y% | x × (100+y)/100 |
| Decrease x by y% | x × (100−y)/100 |
Common Mistakes to Avoid
- Comparing quantities in different units: Always convert both quantities to the same unit (e.g. both in cm, or both in minutes) before writing a ratio — comparing 2 m to 50 cm directly as "2 : 50" is incorrect; it should first become 200 cm : 50 cm = 4 : 1.
- Forgetting to simplify a ratio: A ratio like 15 : 20 is technically correct but should always be reduced to its simplest form, 3 : 4, especially before comparing it with another ratio.
- Mixing up means and extremes: In a : b = c : d, it's easy to accidentally multiply a × c instead of a × d when checking a proportion — always cross-multiply diagonally, not straight across.
- Treating "increase by 10%" and "increase to 10%" as the same thing: "Increased by 10%" means add 10% of the original value; "increased to 110" is a completely different statement — always re-read the wording carefully.
- Forgetting the percentage increase/decrease shortcut: Calculating the percentage change separately and then adding/subtracting it works, but it's slower and more error-prone than directly using x × (100±y)/100.
- Confusing compound ratio with simply adding two ratios: A compound ratio is found by multiplying antecedents together and consequents together — not by adding the two ratios term by term.
What This Introduction Prepares You For
With ratio, proportion, compound ratio, and percentage now defined, the exercises that follow in this chapter put each idea to work — including Exercise 5.1 on direct applications of ratio and proportion and Exercise 5.2 on real-life percentage problems such as discounts, profit and loss, and simple interest.
These ideas also connect closely to Direct and Inverse Proportions, the next major chapter in Class 8, where the concept of a proportion is extended to describe how two changing quantities relate to each other.