Ratio Class 6 ICSE — Mathematics Chapter 8
This Ratio Class 6 ICSE chapter, Chapter 8 of the ICSE Mathematics book, builds the rules you need for ratios and proportions. It starts with the product property of a proportion, then defines continued proportion, the mean proportional and the third proportional. It proves that equal ratios share one value, and closes with the five properties of ratio and proportion. Each section builds on the last, so the chapter reads as one connected argument.
Download Ratio Class 6 ICSE Mathematics Chapter 8 PDF
Download the Ratio Class 6 ICSE Mathematics chapter PDF to revise offline or print for practice. It is the full chapter as printed in the book — 21 pages, 6 sections, 59 questions.
What is inside Chapter 8
21 pages · 6 sections · 59 exercise questions
| Section | Title |
|---|---|
| 1 | An Important Property |
| 2 | 6. Continued Proportion |
| 3 | 7. Mean Proportional (or Geometric Mean) |
| 4 | 8. Third Proportional |
| 5 | 9. An Important Property |
| 6 | 10. Properties of Ratio and Proportion |
- EXERCISE 8(A): 49 questions
- EXERCISE 8 (B): 10 questions
1 An Important Property
This section states the rule that holds every proportion together. In a proportion a:b::c:d, the product of the extremes equals the product of the means. The extremes are the first and last terms, a and d. The means are the two middle terms, b and c.
The reason is algebraic. The notation a:b::c:d means the two ratios are equal, so \frac{a}{b} = \frac{c}{d}. Multiply both sides by bd. The result is ad = bc, which is exactly the product rule.
a:b::c:d \Leftrightarrow \frac{a}{b} = \frac{c}{d} \Leftrightarrow ad = bc
Check the pattern with 4:5::12:15. The product of the extremes is 4 \times 15 = 60. The product of the means is 5 \times 12 = 60. The products match, so the proportion is true.
A common mistake is multiplying the wrong pairs. Students multiply the first two terms, a \times b, or the last two, c \times d. Both give a false check. The extremes are the outer pair, a and d. The means are the inner pair, b and c.
2 6. Continued Proportion
This section defines a chain of equal ratios. Numbers a, b, c, d, e, f are in continued proportion when each ratio equals the next:
\frac{a}{b} = \frac{b}{c} = \frac{c}{d} = \frac{d}{e} = \frac{e}{f}
Each adjacent pair shares the same common ratio. The chain can stop at three numbers or extend further, but the rule stays the same: every ratio in the chain must simplify to one value. The common ratio is the multiplier that takes each term to the next. This setup feeds the next two sections, which name the middle term and the last term of the chain.
Check the pattern with 2, 4, 8, 16:
\frac{2}{4} = \frac{4}{8} = \frac{8}{16} = \frac{1}{2}
The common ratio is \frac{1}{2} for every pair. Because the value never changes, the four numbers are in continued proportion. To test any set, simplify each adjacent ratio. If all give the same fraction, the set is in continued proportion.
3 7. Mean Proportional (or Geometric Mean)
This section finds the middle term of a continued proportion. If a, b, c are in continued proportion, then \frac{a}{b} = \frac{b}{c}. Cross multiply to get b^2 = ac, which gives:
b = \sqrt{ac}
This value of b is the mean proportional between a and c. The book also calls it the geometric mean, so expect both names in questions. The geometric mean is one of the key terms of this ICSE chapter.
Work one with original numbers. Find the mean proportional between 9 and 25. Multiply the two numbers, then take the square root:
b = \sqrt{9 \times 25} = \sqrt{225} = 15
Check it: \frac{9}{15} = \frac{3}{5} and \frac{15}{25} = \frac{3}{5}. The ratios match, so 15 is correct.
The common mistake is using the average instead of the square root. The average of 9 and 25 is \frac{9+25}{2} = 17. The mean proportional is 15. Always take the square root of the product, never the sum divided by two.
4 8. Third Proportional
This section names the last term of a special proportion. If a:b = b:c, then c is the third proportional to a and b. The same proportion also makes c the fourth proportional to a, b, b, because the four terms read a, b, b, c.
Work an example with original numbers. Find the third proportional to 3 and 6. Write 3:6 = 6:c, so \frac{3}{6} = \frac{6}{c}. Then cross multiply:
3c = 36
So c = 12. The third proportional to 3 and 6 is 12. Check it: \frac{3}{6} = \frac{1}{2} and \frac{6}{12} = \frac{1}{2}, so the proportion holds.
Keep two similar terms separate. The mean proportional is the middle term between two numbers. The third proportional is the last term that completes a:b = b:c. The third proportional is also the fourth proportional to a, b, b, an edge case that trips many students. Name the term you need before you start computing.
5 9. An Important Property
This section proves a property that saves work later. If several ratios are equal, each ratio equals the sum of the antecedents divided by the sum of the consequents. In symbols, if \frac{a}{b} = \frac{c}{d} = \frac{e}{f}, then each ratio
= \frac{a+c+e}{b+d+f}
The proof uses one letter for the shared value. Let \frac{a}{b} = \frac{c}{d} = \frac{e}{f} = k. Then a = bk, c = dk, and e = fk. Add the numerators and denominators:
\frac{a+c+e}{b+d+f} = \frac{bk+dk+fk}{b+d+f} = \frac{k(b+d+f)}{b+d+f} = k
Because k was the value of every ratio, the property holds. Each ratio equals \frac{\text{sum of antecedents}}{\text{sum of consequents}}.
Verify with original numbers \frac{1}{2} = \frac{2}{4} = \frac{3}{6}. The antecedent sum is 1+2+3 = 6. The consequent sum is 2+4+6 = 12. Their quotient is \frac{6}{12} = \frac{1}{2}, exactly the common ratio.
6 10. Properties of Ratio and Proportion
This section collects five transformations of a proportion. Each one is a legal algebraic move on the equality \frac{a}{b} = \frac{c}{d}. They appear across both exercise sets, so learn which terms stay and which swap.
Here is the comparison of the five properties with their starting and transformed statements.
| Property | Starting statement | Transformed statement |
|---|---|---|
| Invertendo | a:b::c:d | b:a::d:c |
| Alternendo | a:b::c:d | a:c::b:d |
| Componendo | a:b::c:d | (a+b):b::(c+d):d |
| Dividendo | a:b::c:d | (a-b):b::(c-d):d |
| Componendo and dividendo | a:b::c:d | (a+b):(a-b)::(c+d):(c-d) |
Invertendo takes the reciprocal of each ratio. Alternendo swaps the means, so the first terms of both ratios pair up. Componendo adds 1 to both sides of \frac{a}{b} = \frac{c}{d}, giving \frac{a+b}{b} = \frac{c+d}{d}. Dividendo subtracts 1 instead. Componendo and dividendo divides those two results, one by the other.
Watch the pattern with original numbers. Start with 4:6::10:15. Apply alternendo, so the means swap places:
4:10::6:15
Check it: \frac{4}{10} = \frac{2}{5} and \frac{6}{15} = \frac{2}{5}. The new proportion is true.
Two mistakes trip students here.
- Applying a property too early. Every transformation only works if a:b::c:d is already true. If the four terms are not in proportion, the result is false.
- Swapping the wrong terms. Invertendo swaps the two terms of each ratio. Alternendo swaps only the two middle terms.
The five properties stay useful across later chapters. You will meet them again in the other Class 6 ICSE Maths chapters.
What to practise from Ratio Class 6 ICSE
This chapter carries two exercise blocks. Work through both in order, because 8(B) builds on the skills from 8(A). The Class 6 syllabus is published on the official CISCE website.
- EXERCISE 8(A) — 49 questions. This block drills finding and simplifying ratios, converting units before comparing, combining ratios, dividing a total in a given ratio, ages and mixture problems, and comparing ratios to find which is larger.
- EXERCISE 8(B) — 10 questions. This block drills the named ratio types: duplicate, triplicate, sub-duplicate, sub-triplicate, reciprocal and compounded ratios. Later questions combine several types in one ratio.
The mapping is clear. 8(A) builds the method of setting up a proportion and solving it. 8(B) tests whether you can recognise ratio operations by name. Master continued proportion, the mean proportional and the third proportional first, because 8(A) uses them throughout. The chapter summary restates the key definitions and is a quick check before you start.
The full Class 6 ICSE book list organises every chapter in one place. The percentage chapter is a natural next step after ratio.
Ratio Class 6 ICSE frequently asked questions
How do I find the mean proportional between two numbers in ratio class 6 icse?
Multiply the two numbers, then take the square root. The mean proportional between a and c is \sqrt{ac}. For example, between 9 and 25 it is \sqrt{225} = 15.
What is the third proportional of a and b in Class 6 ICSE ratio?
It is the value c that makes a:b = b:c. Cross multiply to get c = \frac{b^2}{a}. For a = 3 and b = 6, the third proportional is \frac{36}{3} = 12.
When should I use componendo and dividendo in this ICSE chapter?
Use it when a problem asks for a new ratio built from sums and differences of the original terms. It turns a:b::c:d into (a+b):(a-b)::(c+d):(c-d). It works only when the four terms are already in proportion.
How do I compare two ratios to decide which is larger in Class 6 ICSE Maths?
Write each ratio as a fraction, then compare the fractions. Convert to equal denominators or cross multiply. The larger fraction belongs to the larger ratio.
What is the difference between a duplicate ratio and a triplicate ratio in Class 6 ICSE Maths?
The duplicate ratio of a:b is a^2:b^2, formed by squaring each term. The triplicate ratio is a^3:b^3, formed by cubing each term. Exercise 8(B) drills both.
Reference: ICSE Class 6 Mathematics textbook material, chapter 8: Ratio — CISCE-aligned.