Ratio and Proportion

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A ratio is a comparison between two quantities of the same kind. It is written as a:b or a/b, where 'a' is the first term (antecedent) and 'b' is the second term (consequent). Example: The ratio of 2

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1) Ratios are unitless 2) Both quantities must be in the same unit 3) Ratio remains unchanged when both terms are multiplied or divided by the same non-zero number 4) If second term is 100, it represe

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Compare ratios by cross multiplication: If ad > bc, then a/b > c/d. If ad < bc, then a/b < c/d. If ad = bc, then a/b = c/d. Example: Compare 3/4 and 5/7: 3×7 = 21, 4×5 = 20. Since 21 > 20, so 3/4 > 5/

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Two quantities are in direct proportion when their ratio remains constant. As one increases, the other increases proportionally. Formula: x/y = k (constant). Example: A car covers 10 km per liter. For

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Two quantities are in inverse proportion when their product remains constant. As one increases, the other decreases. Formula: x × y = k (constant). Example: Speed × Time = Distance. A car at 50 km/hr

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Invertendo: If a/b = c/d, then b/a = d/c. We flip both ratios. Example: If 3/4 = 6/8, then 4/3 = 8/6. Proof: From a/b = c/d, we get ad = bc, which gives us bd/ac = 1, so b/a = d/c.

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Alternando: If a/b = c/d, then a/c = b/d. We alternate the terms. Example: If 6/8 = 9/12, then 6/9 = 8/12 = 2/3. Proof: From a/b = c/d, we get ad = bc, dividing by cd gives a/c = b/d.

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Componendo: If a/b = c/d, then (a+b)/b = (c+d)/d. We add numerator to denominator. Example: If 3/5 = 6/10, then (3+5)/5 = (6+10)/10, i.e., 8/5 = 16/10 = 8/5. Proof: Add 1 to both sides of a/b = c/d.