ICSE Class 10 Maths Ratio and Proportion Solutions
ICSE Class 10 Maths ratio and proportion in Selina Concise Mathematics Chapter 7 teaches you to compare quantities, form proportions, apply componendo and dividendo, and solve algebraic ratio questions step by step. The method is simple: convert the statement into a fraction or scale factor, cross-multiply carefully, simplify, and write the final ratio in lowest terms.
Chapter snapshot for Selina Concise Chapter 7
Concept snapshot: A ratio is a scale, not always the actual value. If A:B=7:5, write A=7k and B=5k. The common factor k keeps the comparison unchanged and helps solve word problems.
Concise Mathematics Selina Solutions Class 10 ICSE Chapter 7 Ratio and Proportion (Including Properties and Uses) commonly tests ratio chaining, duplicate ratio, sub-duplicate ratio, triplicate ratio, mean proportional, and algebraic proof using properties of proportion.
Formula reference for Ratio and Proportion
| Rule | Form | Use |
|---|---|---|
| Proportion | \frac{a}{b}=\frac{c}{d}\Rightarrow ad=bc | Cross-multiply |
| Compound ratio | \frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd} | Combine ratios |
| Duplicate ratio | a:b\Rightarrow a^2:b^2 | Square both terms |
| Triplicate ratio | a:b\Rightarrow a^3:b^3 | Cube both terms |
| Sub-duplicate ratio | a:b\Rightarrow \sqrt a:\sqrt b | Take square roots |
| Mean proportional | If a:x=x:b, then x^2=ab | Find the middle term |
Worked solutions from Chapter 7 question types
Question 1(a): If A:B=7:5 and B:C=7:5, find A:C
Step 1: Write both ratios as fractions.
\frac{A}{B}=\frac{7}{5},\qquad \frac{B}{C}=\frac{7}{5}
Step 2: Multiply the two equations so that B cancels.
\frac{A}{B}\times\frac{B}{C}=\frac{7}{5}\times\frac{7}{5}
\frac{A}{C}=\frac{49}{25}
Final answer: A:C=49:25.
Question 1(b): If \frac{x^2-1}{x^2+1}=\frac{3}{5}, find x
Step 1: Cross-multiply.
5(x^2-1)=3(x^2+1)
Step 2: Expand and simplify.
5x^2-5=3x^2+3
2x^2=8
x^2=4
Final answer: x=\pm2.
Question 1(c): If \((2x+3y):(3x+2y)=4:3\), find x:y
Step 1: Convert the ratio into a proportion.
\frac{2x+3y}{3x+2y}=\frac{4}{3}
Step 2: Cross-multiply and simplify.
3(2x+3y)=4(3x+2y)
6x+9y=12x+8y
y=6x
Final answer: x:y=1:6.
Question: Divide \text{₹}1290 among A, B, C, where A=\frac{2}{5}B and B:C=4:3
Step 1: Let B=4k and C=3k.
Step 2: Then A=\frac{2}{5}\times4k=\frac{8k}{5}.
A:B:C=\frac{8k}{5}:4k:3k=8:20:15
Step 3: Total parts =8+20+15=43.
\frac{1290}{43}=30
A=240,\qquad B=600,\qquad C=450
Final answer: A=\text{₹}240, B=\text{₹}600, C=\text{₹}450.
Question: If \((x+3):(4x+1)\) is the duplicate ratio of 3:5, find x
Step 1: Duplicate ratio of 3:5 is 9:25.
\frac{x+3}{4x+1}=\frac{9}{25}
Step 2: Cross-multiply and solve.
25(x+3)=9(4x+1)
25x+75=36x+9
11x=66
Final answer: x=6.
Question: If m:n is the duplicate ratio of m+x:n+x, prove x^2=mn
Step 1: Translate the statement.
\frac{m}{n}=\frac{(m+x)^2}{(n+x)^2}
Step 2: Cross-multiply and expand.
m(n+x)^2=n(m+x)^2
mn^2+2mnx+mx^2=nm^2+2mnx+nx^2
Step 3: Cancel and factor.
mx^2-nx^2=nm^2-mn^2
x^2(m-n)=mn(m-n)
Hence proved: x^2=mn, for m\ne n.
Additional worked examples
Worked example 1: Changed ratio
Question: Two numbers are in the ratio 3:5. If 8 is added to each, the ratio becomes 5:7.
\frac{3k+8}{5k+8}=\frac{5}{7}
21k+56=25k+40
k=4
Final answer: The numbers are 12 and 20.
Worked example 2: Mean proportional
Question: Find the mean proportional between 9 and 16.
\frac{9}{x}=\frac{x}{16}
x^2=144
Final answer: x=12.
Worked example 3: Componendo and dividendo idea
Question: If a:b=5:3, find \((a+b):(a-b)\).
(a+b):(a-b)=(5k+3k):(5k-3k)
=8k:2k=4:1
Final answer: 4:1.
Examiner’s mindset for ratio and proportion
Marks are usually earned for the correct equation, valid algebraic steps, and the final answer in the asked form. Do not jump from the given ratio to the answer without showing the proportion used.
Common mistakes students make
- Do not cancel x from x+y; cancel common factors only.
- Do not forget x=\pm2 when x^2=4, unless the context restricts the value.
- Duplicate ratio means a^2:b^2; sub-duplicate ratio means \sqrt a:\sqrt b.
- In \(7(x-y)\), the expansion is 7x-7y.
Quick answer index
| Question type | Answer |
|---|---|
| A:B=7:5,\ B:C=7:5 | A:C=49:25 |
| \frac{x^2-1}{x^2+1}=\frac{3}{5} | x=\pm2 |
| \((2x+3y):(3x+2y)=4:3\) | x:y=1:6 |
| Duplicate ratio of 3:5 | 9:25 |
| Mean proportional of 9 and 16 | 12 |
Sources and syllabus alignment
This page is aligned with the standard ICSE Class 10 Mathematics treatment of ratio and proportion and the Selina Concise Mathematics Chapter 7 topic. For official syllabus context, use the CISCE website. For overlapping fundamentals, use NCERT resources.
Frequently Asked Questions
How do I start ICSE Class 10 Maths ratio questions?
Start by writing the ratio as a fraction or using a scale factor, such as A=7k,\ B=5k.
What is duplicate ratio?
The duplicate ratio of a:b is a^2:b^2. For example, the duplicate ratio of 3:5 is 9:25.
What is sub-duplicate ratio?
The sub-duplicate ratio of a:b is \sqrt a:\sqrt b, when the square roots are meaningful.
Why is x=\pm2 when x^2=4?
Both 2 and -2 square to 4, so both values satisfy the equation unless a practical context rejects one value.