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Number - Ratio, Proportion, and Rate

Grade 10IGCSE

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

🔑Concepts

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Simplifying Ratios: Expressing ratios in their simplest form by dividing by the highest common factor.

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Dividing a Quantity in a Given Ratio: Finding the value of one part by dividing the total by the sum of the ratio parts.

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Direct Proportion: Two quantities increase or decrease at the same rate (y=kxy = kx).

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Inverse Proportion: As one quantity increases, the other decreases (y=k/xy = k/x).

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Map Scales: Representing real-world distances using a ratio (1:n), where 1 unit on the map equals nn units in reality.

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Scale Factors for Area and Volume: If the length scale factor is kk, the area scale factor is k2k^2 and the volume scale factor is k3k^3.

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Compound Measures: Calculating rates such as Speed (Distance/Time) and Density (Mass/Volume).

📐Formulae

y=kxy = kx (Direct Proportion)

y=kxy = \frac{k}{x} (Inverse Proportion)

Average Speed=Total DistanceTotal Time\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}

Density=MassVolume\text{Density} = \frac{\text{Mass}}{\text{Volume}}

(L1L2)2=A1A2(\frac{L_1}{L_2})^2 = \frac{A_1}{A_2} (Area ratio of similar shapes)

(L1L2)3=V1V2(\frac{L_1}{L_2})^3 = \frac{V_1}{V_2} (Volume ratio of similar shapes)

💡Examples

Problem 1:

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

Solution:

120,120, 180, $420

Explanation:

First, find the total number of parts: 2+3+7=122 + 3 + 7 = 12. Divide the total amount by the number of parts to find the value of one part: 720/12=60720 / 12 = 60. Multiply each part of the ratio by 60: 2×60=1202 \times 60 = 120, 3×60=1803 \times 60 = 180, and 7×60=4207 \times 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:

Set up the equation y=k/x2y = k/x^2. Substitute the known values to find k: 4=k/32⇒4=k/9⇒k=364 = k/3^2 \Rightarrow 4 = k/9 \Rightarrow k = 36. Now use the new x value in the equation y=36/x2y = 36/x^2: y=36/62=36/36=1y = 36/6^2 = 36/36 = 1.

Problem 3:

Two similar cylinders have heights of 5 cm and 10 cm. If the volume of the smaller cylinder is 40 cm³, find the volume of the larger cylinder.

Solution:

320 cm³

Explanation:

Find the length scale factor k=10/5=2k = 10 / 5 = 2. Since volume is proportional to the cube of the length scale factor, the volume scale factor is k3=23=8k^3 = 2^3 = 8. Multiply the smaller volume by the volume scale factor: 40×8=32040 \times 8 = 320 cm³.

Problem 4:

A map has a scale of 1:25,000. A forest on the map has an area of 4 cm². Calculate the actual area of the forest in square kilometers.

Solution:

0.25 km²

Explanation:

The length scale factor is 1 cm : 25,000 cm. Convert 25,000 cm to km: 25,000/100,000=0.2525,000 / 100,000 = 0.25 km. So, 1 cm : 0.25 km. The area scale factor is (1 cm)2:(0.25 km)2(1 \text{ cm})^2 : (0.25 \text{ km})^2, which is 1 cm² : 0.0625 km². Multiply the map area by the area scale factor: 4×0.0625=0.254 \times 0.0625 = 0.25 km².