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Algebra - Ratio and Proportion

Grade 10ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Ratio is a mathematical comparison of two quantities of the same kind, expressed as ab\frac{a}{b} or a:ba:b. It has no units and exists only between similar quantities. Visually, think of a ratio as a dividing line or 'bar' that partitions a whole into relative parts.

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Proportion is the statement of equality between two ratios. If a:b=c:da:b = c:d, then a,b,c,da, b, c, d are said to be in proportion. This can be visualized as two balanced scales where the relationship between the first pair of weights is identical to the relationship between the second pair.

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In a proportion ab=cd\frac{a}{b} = \frac{c}{d}, the terms aa and dd are called 'Extremes', and bb and cc are called 'Means'. The fundamental rule of proportion is that the Product of Extremes equals the Product of Means (ad=bcad = bc), which can be visualized by drawing an 'X' through the equals sign for cross-multiplication.

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Continued Proportion occurs when the second term of the first ratio is the first term of the second ratio, such as ab=bc\frac{a}{b} = \frac{b}{c}. In this sequence, bb is known as the Mean Proportional between aa and cc. This creates a 'chain link' effect where terms are interconnected.

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Componendo and Dividendo is a crucial property used to simplify equations involving fractions. If ab=cd\frac{a}{b} = \frac{c}{d}, then a+ba−b=c+dc−d\frac{a+b}{a-b} = \frac{c+d}{c-d}. Visually, this operation involves 'merging' the denominator with the numerator through addition and subtraction simultaneously on both sides of the equation.

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The kk-Method is a substitution technique where equal ratios are set to a constant kk. For example, if ab=cd=k\frac{a}{b} = \frac{c}{d} = k, then a=bka = bk and c=dkc = dk. This visualizes variables as multiples of a common base, reducing the number of unknowns in complex algebraic proofs.

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Transformation properties like Invertendo (flipping the ratios to ba=dc\frac{b}{a} = \frac{d}{c}) and Alternendo (swapping the means to ac=bd\frac{a}{c} = \frac{b}{d}) allow for the flexible rearrangement of proportional terms while maintaining the overall equality.

📐Formulae

Ratio: a:b=aba:b = \frac{a}{b}

Proportion: ab=cd  ⟹  ad=bc\frac{a}{b} = \frac{c}{d} \implies ad = bc

Mean Proportional: b2=acb^2 = ac or b=acb = \sqrt{ac}

Invertendo: ab=cd  ⟹  ba=dc\frac{a}{b} = \frac{c}{d} \implies \frac{b}{a} = \frac{d}{c}

Alternendo: ab=cd  ⟹  ac=bd\frac{a}{b} = \frac{c}{d} \implies \frac{a}{c} = \frac{b}{d}

Componendo: ab=cd  ⟹  a+bb=c+dd\frac{a}{b} = \frac{c}{d} \implies \frac{a+b}{b} = \frac{c+d}{d}

Dividendo: ab=cd  ⟹  a−bb=c−dd\frac{a}{b} = \frac{c}{d} \implies \frac{a-b}{b} = \frac{c-d}{d}

Componendo and Dividendo: ab=cd  ⟹  a+ba−b=c+dc−d\frac{a}{b} = \frac{c}{d} \implies \frac{a+b}{a-b} = \frac{c+d}{c-d}

💡Examples

Problem 1:

Find the value of xx if x,12,16,24x, 12, 16, 24 are in proportion.

Solution:

  1. Since the numbers are in proportion, we can write: x12=1624\frac{x}{12} = \frac{16}{24} \ 2. Apply the Product of Extremes = Product of Means: x×24=12×16x \times 24 = 12 \times 16 \ 3. 24x=19224x = 192 \ 4. x=19224x = \frac{192}{24} \ 5. x=8x = 8

Explanation:

This problem uses the basic definition of proportion where the ratio of the first two terms is equal to the ratio of the last two terms.

Problem 2:

Solve for xx using the properties of proportion: x+5+x−16x+5−x−16=73\frac{\sqrt{x+5} + \sqrt{x-16}}{\sqrt{x+5} - \sqrt{x-16}} = \frac{7}{3}

Solution:

  1. Apply Componendo and Dividendo: (x+5+x−16)+(x+5−x−16)(x+5+x−16)−(x+5−x−16)=7+37−3\frac{(\sqrt{x+5} + \sqrt{x-16}) + (\sqrt{x+5} - \sqrt{x-16})}{(\sqrt{x+5} + \sqrt{x-16}) - (\sqrt{x+5} - \sqrt{x-16})} = \frac{7+3}{7-3} \ 2. Simplify the terms: 2x+52x−16=104\frac{2\sqrt{x+5}}{2\sqrt{x-16}} = \frac{10}{4} \ 3. x+5x−16=52\frac{\sqrt{x+5}}{\sqrt{x-16}} = \frac{5}{2} \ 4. Square both sides: x+5x−16=254\frac{x+5}{x-16} = \frac{25}{4} \ 5. Cross multiply: 4(x+5)=25(x−16)4(x+5) = 25(x-16) \ 6. 4x+20=25x−4004x + 20 = 25x - 400 \ 7. 420=21x  ⟹  x=42021420 = 21x \implies x = \frac{420}{21} \ 8. x=20x = 20

Explanation:

Componendo and Dividendo is the most efficient method here because it isolates the radical terms, making it easier to square and solve for the variable.