Ratio, Proportion & Commercial Maths (VI–VIII) • Topic 2 of 5

Percentage

Percent means 'per hundred', so x% is simply the fraction x/100 — a ratio with a fixed second term of 100, which is exactly why percentages make comparison easy (everything is rescaled to a common base). To find x% of a number, multiply by x/100; to express one quantity as a percentage of another, divide and multiply by 100. Fluent teachers carry the everyday equivalents in their head: 1/2 = 50%, 1/4 = 25%, 1/5 = 20%, 1/10 = 10%, 3/4 = 75%. PEDAGOGY: the cleanest way to introduce percentage in Class VII is through the 100-grid (a 10 × 10 square) where children shade squares and literally see 'so many out of a hundred', linking fraction, decimal and percent as three names for the same quantity. COMMON MISCONCEPTIONS: the biggest is reading the number off without the 'per hundred' idea — a child writes 1/4 = 25 (dropping the % sign) or thinks 25% of a number is always 25. Children also believe a percentage can never exceed 100, and that a 20% rise followed by a 20% fall returns the original amount (it does not, because the two percentages are taken on different bases). HOW TESTED: direct 'find the percentage' computation, converting between fraction/decimal/percent, and a misconception-spotting pedagogy item.

✅ Solved examples

1. Find 15% of 200.

15% of 200 = (15 / 100) × 200 = 15 × 2 = 30.

2. What percent of 80 is 20?

(20 / 80) × 100 = (1/4) × 100 = 25%.

3. A student scored 45 marks out of 60. Express this as a percentage.

(45 / 60) × 100 = 0.75 × 100 = 75%.

4. Convert 3/5 into a percentage.

(3 / 5) × 100 = 60%. (Equivalently 3/5 = 0.6 = 60%.)

✏️ Practice — try these, take hints as needed

1. Find 30% of 150.

30% = 30/100.

Multiply by 150.

2. What percent of 50 is 35?

Divide the part by the whole, then × 100.

35 / 50 = 0.7.

70%

3. Convert 0.08 into a percentage.

Multiply the decimal by 100.

4. In a class of 40 students, 24 are girls. What percentage of the class are girls?

Girls out of total, × 100.

24 / 40 simplifies to 3/5.

60%

📝 Topic test — 8 questions

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Key Concepts — Quick Reference

This chapter

Ratio, proportion & percentage

Ratio in simplest form	a : b = (a ÷ HCF) : (b ÷ HCF)
Proportion	a : b :: c : d ⇔ a × d = b × c (product of extremes = product of means)
Unitary method	value of 1 unit = total ÷ number of units
Percentage	x% of N = (x / 100) × N ; what % is a of b = (a / b) × 100

Commercial maths

Profit / Loss	Profit = SP − CP ; Loss = CP − SP (when SP < CP)
Profit % / Loss %	Profit% = (Profit / CP) × 100 ; Loss% = (Loss / CP) × 100 (always on CP)
Discount	Discount = MP − SP ; Discount% = (Discount / MP) × 100 (always on MP)
Simple Interest	SI = (P × R × T) / 100 ; Amount = P + SI
Direct / Inverse variation	Direct: x / y = k (constant); Inverse: x × y = k (constant)