Finding Common Ground
To compare quantities fairly we put them on common ground. This Class 7 Ganita Prakash chapter compares quantities using ratios, checks when two ratios form a proportion, uses the unitary method to scale quantities, and expresses comparisons as percentages.
Learning objectives
- Compare quantities using ratios.
- Identify a proportion.
- Use the unitary method.
- Express comparisons as percentages.
Key concepts
Ratio
A ratio compares two quantities of the same kind by division, written a : b and read 'a to b'. The ratio of 6 boys to 4 girls is 6 : 4, which simplifies to 3 : 2 by dividing both parts by their common factor. A ratio has no units and stays the same when both parts are multiplied or divided by the same number.
Proportion
When two ratios are equal, the four quantities are in proportion, written a : b :: c : d. For example 2 : 3 :: 4 : 6, because both ratios equal 2/3. A quick check is cross-multiplication: the ratios are equal when a × d = b × c.
Unitary method
The unitary method first finds the value of one unit, then scales up. If 5 pens cost ₹40, one pen costs ₹40 ÷ 5 = ₹8, so 3 pens cost ₹8 × 3 = ₹24. Finding 'the value of one' is the common ground that solves many comparison problems.
Percentage
A percentage compares a quantity to 100, giving everything a common scale. To write a fraction as a percentage, multiply by 100, so 1/4 = 25%. To find a percentage of a quantity, multiply: 20% of 150 = (20/100) × 150 = 30. Percentages make different quantities easy to compare.
Important formulas
Proportion check
a : b :: c : d when a × d = b × c
Unitary method
value of one = total ÷ number; then × required
Percentage
p% of N = (p ÷ 100) × N
Key definitions
- Ratio
- A comparison of two like quantities by division, written a : b.
- Proportion
- A statement that two ratios are equal.
- Unitary method
- Finding the value of one unit, then scaling to the required number.
- Percentage
- A comparison to 100, written with the % sign.
Solved examples
Q1. Simplify the ratio 6 : 4.
Solution: Divide both by 2: 3 : 2.
Q2. Are 2 : 3 and 4 : 6 in proportion?
Solution: Yes, since 2 × 6 = 3 × 4 = 12.
Q3. If 5 pens cost ₹40, what do 3 cost?
Solution: One pen ₹8, so 3 cost ₹24.
Common mistakes to avoid
- Comparing quantities with different units as a ratio without converting.
- Forgetting to simplify a ratio to lowest terms.
- Mixing up the order of terms in a ratio (a : b is not b : a).
- Treating a percentage as a fraction without the denominator 100.
Finding Common Ground — MCQ Quiz
10 questions with instant feedback. Use number keys 1–4 to answer.
The ratio 8 : 12 in simplest form is:
Practice questions
Short answer
What is a ratio?
A comparison of two like quantities by division, written a : b.
When are two ratios in proportion?
When they are equal, i.e. a × d = b × c.
Write 1/4 as a percentage.
25%.
Long answer
Explain ratio and proportion with examples, including how to check a proportion.
A ratio compares two quantities of the same kind by division and is written a : b, read 'a to b'; for instance, 6 boys to 4 girls is 6 : 4, which simplifies to 3 : 2 by dividing both parts by 2. A proportion is a statement that two ratios are equal, written a : b :: c : d. To check whether two ratios form a proportion, we cross-multiply: the ratios are equal when a × d = b × c. For example, 2 : 3 and 4 : 6 are in proportion because 2 × 6 = 3 × 4 = 12. Ratios and proportions let us compare and scale quantities fairly.
Describe the unitary method and percentages as tools for comparing quantities.
The unitary method solves comparison problems by first finding the value of a single unit and then scaling to the required amount. For example, if 5 pens cost ₹40, one pen costs ₹40 ÷ 5 = ₹8, so 3 pens cost ₹8 × 3 = ₹24. Percentages give another common scale by comparing every quantity to 100: a fraction is turned into a percentage by multiplying by 100 (so 1/4 = 25%), and a percentage of a quantity is found by multiplying, as in 20% of 150 = 30. Both tools provide 'common ground', letting us compare and convert between quantities that start out in different forms.
HOTS (Higher Order Thinking)
In a class, the ratio of girls to boys is 3 : 2 and there are 20 boys. How many girls are there?
Each part is 20 ÷ 2 = 10, so girls = 3 × 10 = 30.
Which is the better deal: 4 pens for ₹100 or 6 pens for ₹138?
First is ₹25/pen, second is ₹23/pen; the 6-pen pack is cheaper per pen.
Quick revision
Revision notes
- Ratio a : b compares like quantities; simplify to lowest terms.
- Proportion: two equal ratios; check by a × d = b × c.
- Unitary method: find value of one, then scale.
- Percentage = comparison to 100; p% of N = (p/100) × N.
Key takeaways
- Ratios and percentages put quantities on common ground.
- A proportion is two equal ratios (cross-multiply to check).
- The unitary method scales via the value of one.
Frequently asked questions
What is the difference between a ratio and a proportion?
A ratio compares two quantities; a proportion states that two ratios are equal.
How do I write a fraction as a percentage?
Multiply it by 100.
What is the unitary method?
Find the value of one unit, then multiply for the number needed.