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Number - Ratio and Proportion

Grade 8IGCSE

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

🔑Concepts

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Definition of Ratio: A way to compare two or more quantities of the same kind, usually expressed in the form a:ba:b.

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Simplifying Ratios: Dividing all terms in a ratio by their Highest Common Factor (HCF) until they are in the simplest form.

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Dividing a Quantity in a Given Ratio: Finding the total number of parts and calculating the value of a single part.

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Direct Proportion: Two quantities are in direct proportion if one increases at the same rate as the other (y=kxy = kx).

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Inverse Proportion: Two quantities are in inverse proportion if one increases while the other decreases (y=kxy = \frac{k}{x}).

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Map Scales: Using ratios to represent real-life distances on a drawing, typically written as 1:n1 : n (where both sides are in the same units).

📐Formulae

Total Parts = a+b+ca + b + c

Value of One Part = Total QuantityTotal Parts\frac{\text{Total Quantity}}{\text{Total Parts}}

Share of x=Ratio share of x×Value of One Partx = \text{Ratio share of } x \times \text{Value of One Part}

Direct Proportion Equation: y=kxy = kx

Inverse Proportion Equation: y=kxy = \frac{k}{x}

Map Scale Ratio: 1:n=Map DistanceActual Distance1 : n = \frac{\text{Map Distance}}{\text{Actual Distance}}

💡Examples

Problem 1:

Divide 720intheratio720 in the ratio 3:5$.

Solution:

Part 1: 270,Part2:270, Part 2: 450

Explanation:

  1. Find total parts: 3+5=83 + 5 = 8. 2. Find value of one part: 720/8=90720 / 8 = 90. 3. Multiply each ratio share by the value of one part: 3×90=2703 \times 90 = 270 and 5×90=4505 \times 90 = 450.

Problem 2:

If 8 towels cost $56, how much do 13 towels cost?

Solution:

$91

Explanation:

This is direct proportion. First, find the cost of 1 towel (Unitary Method): 56/8=756 / 8 = 7. Then, multiply the cost per towel by 13: 7×13=917 \times 13 = 91.

Problem 3:

A map has a scale of 1:25,0001 : 25,000. If the distance between two villages on the map is 6 cm6\text{ cm}, calculate the actual distance in kilometers.

Solution:

1.5 km1.5\text{ km}

Explanation:

  1. Multiply map distance by scale: 6×25,000=150,000 cm6 \times 25,000 = 150,000\text{ cm}. 2. Convert cm to meters: 150,000/100=1,500 m150,000 / 100 = 1,500\text{ m}. 3. Convert meters to km: 1,500/1,000=1.5 km1,500 / 1,000 = 1.5\text{ km}.

Problem 4:

It takes 4 workers 6 hours to paint a fence. How long would it take 3 workers to paint the same fence, assuming they work at the same rate?

Solution:

8 hours

Explanation:

This is inverse proportion. Total 'man-hours' needed = 4×6=244 \times 6 = 24 hours. Divide the total man-hours by the new number of workers: 24/3=824 / 3 = 8 hours.