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Number - Ratio, Proportion, and Rate

Grade 12IGCSE

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

🔑Concepts

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Ratio: A way to compare quantities of the same kind, expressed as a:b. It can be simplified by dividing both sides by the same factor.

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Dividing a Quantity: To divide a total into a given ratio, find the total number of parts, divide the total quantity by this sum to find the value of one part, then multiply by the specific parts of the ratio.

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Direct Proportion: Two quantities increase or decrease at the same rate. If y is proportional to x, then y = kx, where k is the constant of proportionality.

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Inverse Proportion: As one quantity increases, the other decreases. If y is inversely proportional to x, then y = k/x.

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Rates: A comparison of two different quantities (e.g., Speed = Distance/Time, Density = Mass/Volume).

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Scale Factors: For similar shapes, if the linear scale factor is k, the area scale factor is k² and the volume scale factor is k³.

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Map Scales: Represented as 1:n, meaning 1 unit on the map represents n units in reality.

📐Formulae

y=kxy = kx (Direct Proportion)

y = rac{k}{x} (Inverse Proportion)

y=kxny = kx^n (Proportion to nthn^{th} power)

ext{Average Speed} = rac{ ext{Total Distance}}{ ext{Total Time}}

ext{Scale Factor (k)} = rac{ ext{New Length}}{ ext{Original Length}}

rac{ ext{Area}_1}{ ext{Area}_2} = ( rac{L_1}{L_2})^2

rac{ ext{Volume}_1}{ ext{Volume}_2} = ( rac{L_1}{L_2})^3

💡Examples

Problem 1:

Divide $720 in the ratio 2 : 3 : 7.

Solution:

120,120, 180, $420

Explanation:

Total parts = 2 + 3 + 7 = 12. Value of one part = 720/12=720 / 12 = 60. Multiply each part: 2 × 60 = 120, 3 × 60 = 180, 7 × 60 = 420.

Problem 2:

y is inversely proportional to the square of x. When x = 3, y = 4. Find y when x = 6.

Solution:

y = 1

Explanation:

Equation: y=k/x2y = k/x^2. Substitute knowns: 4=k/32⇒k=4×9=364 = k/3^2 \Rightarrow k = 4 \times 9 = 36. The formula is y=36/x2y = 36/x^2. When x = 6, y=36/62=36/36=1y = 36/6^2 = 36/36 = 1.

Problem 3:

A map has a scale of 1 : 50,000. If the area of a lake on the map is 4 cm², calculate the actual area in km².

Solution:

1 km²

Explanation:

Linear scale k=50,000k = 50,000. Area scale k2=(50,000)2k^2 = (50,000)^2. Actual area = 4 cm2×2,500,000,000=10,000,000,000 cm24 \text{ cm}^2 \times 2,500,000,000 = 10,000,000,000 \text{ cm}^2. Convert to km²: 10,000,000,000/100210,000,000,000 / 100^2 (to m²) / 100021000^2 (to km²) = 10,000,000,000/10,000,000,000=1 km210,000,000,000 / 10,000,000,000 = 1 \text{ km}^2.