Key facts

  • Ratio compares quantities by division; a:b has meaning only after both terms are in the same unit.
  • A proportion a:b::c:d is true when ad = bc, so cross-multiplication is the fastest verification step.
  • Direct proportion keeps a quotient constant, while inverse proportion keeps a product constant.
  • Percentage means per hundred; every percentage change must be applied to the correct base.
  • Profit and loss percentages are calculated on cost price, while discount is calculated on marked price.

Key Points at a Glance

  1. 1

    Ratio compares quantities by division; a:b has meaning only after both terms are in the same unit.

  2. 2

    A proportion a:b::c:d is true when ad = bc, so cross-multiplication is the fastest verification step.

  3. 3

    Direct proportion keeps a quotient constant, while inverse proportion keeps a product constant.

  4. 4

    Percentage means per hundred; every percentage change must be applied to the correct base.

  5. 5

    Profit and loss percentages are calculated on cost price, while discount is calculated on marked price.

  6. 6

    Simple interest grows directly with principal, rate and time; compound interest adds each period's interest back to the principal.

  7. 7

    Average is total divided by count; weighted averages and partnership shares require multiplying by the relevant weight first.

  8. 8

    HCF handles common division, LCM handles common repetition, and alligation reverses a weighted average into a mixing ratio.

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Ratio as a comparison tool

Ratio is a compact way to compare two quantities of the same kind. If a class has 18 boys and 12 girls, the ratio of boys to girls is 18:12, which reduces to 3:2 after division by 6. The reduced form shows relative size, not the actual count. A 3:2 ratio may represent 3 and 2, 30 and 20, or 18 and 12; the multiplier changes but the comparison remains the same.

The first rule is unit discipline. A ratio between 2 hours and 30 minutes cannot be written as 2:30 until both are in the same unit. Since 2 hours = 120 minutes, the correct ratio is 120:30 = 4:1. In exam questions, units often shift between rupees and paise, kilometres and metres, or hours and minutes. Convert first, then reduce.

Ratios also work in three or more terms. If A:B = 2:3 and B:C = 6:5, make the B terms equal. The first ratio becomes A:B = 4:6, so A:B:C = 4:6:5. This linking method is safer than guessing from separate pairs.

Exam cue: a ratio stores relative comparison; absolute value appears only when a total, difference, or one part is given.

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