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Proportional Reasoning-2 - Proportionality — A Quick Recap

Grade 8CBSE

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

🔑Concepts

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Two quantities xx and yy are said to be in Direct Proportion if they increase or decrease together such that the ratio of their corresponding values remains constant. This is expressed as xy=k\frac{x}{y} = k or x=kyx = ky, where kk is a positive constant.

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Two quantities xx and yy are said to be in Inverse Proportion if an increase in xx causes a proportional decrease in yy (and vice-versa) such that the product of their corresponding values remains constant. This is expressed as xy=kxy = k.

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In direct proportion, if y1y_1 is the value of yy corresponding to x1x_1, and y2y_2 is the value of yy corresponding to x2x_2, then x1y1=x2y2\frac{x_1}{y_1} = \frac{x_2}{y_2}.

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In inverse proportion, if y1y_1 is the value of yy corresponding to x1x_1, and y2y_2 is the value of yy corresponding to x2x_2, then x1y1=x2y2x_1 y_1 = x_2 y_2.

📐Formulae

x1y1=x2y2\frac{x_1}{y_1} = \frac{x_2}{y_2}

x1×y1=x2×y2x_1 \times y_1 = x_2 \times y_2

x=ky (Direct Proportion)x = ky \text{ (Direct Proportion)}

xy=k (Inverse Proportion)xy = k \text{ (Inverse Proportion)}

💡Examples

Problem 1:

If the cost of 1515 kg of sugar is ₹600600, what is the cost of 2525 kg of sugar?

Solution:

Let the weight of sugar be xx and the cost be yy. Since weight and cost are in direct proportion: x1y1=x2y2\frac{x_1}{y_1} = \frac{x_2}{y_2} Given: x1=15x_1 = 15, y1=600y_1 = 600, x2=25x_2 = 25. 15600=25y2\frac{15}{600} = \frac{25}{y_2} y2=25×60015y_2 = \frac{25 \times 600}{15} y2=25×40=1000y_2 = 25 \times 40 = 1000 To find the total after adding a service tax of ₹5050: 1000+501050\begin{array}{r} 1000 \\ + 50 \\ \hline 1050 \end{array} The cost of 2525 kg of sugar is ₹10001000.

Explanation:

As the quantity of sugar increases, the cost increases proportionally. This is a case of direct proportion where we cross-multiply to find the unknown value.

Problem 2:

If 2020 workers can build a wall in 4848 hours, how many workers will be required to do the same work in 3030 hours?

Solution:

Let the number of workers be xx and the number of hours be yy. This is a case of inverse proportion because more workers will take less time. x1y1=x2y2x_1 y_1 = x_2 y_2 Given: x1=20x_1 = 20, y1=48y_1 = 48, y2=30y_2 = 30. 20×48=x2×3020 \times 48 = x_2 \times 30 x2=20×4830x_2 = \frac{20 \times 48}{30} x2=2×483=2×16=32x_2 = \frac{2 \times 48}{3} = 2 \times 16 = 32 So, 3232 workers are required.

Explanation:

In inverse proportion, the product of the two variables remains constant (x×y=kx \times y = k). Reducing the time requires increasing the number of workers.