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Direct and Inverse Proportions - Problem Solving

Grade 8CBSE

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

🔑Concepts

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Direct Proportion: Two quantities xx and yy are said to be in direct proportion if they increase or decrease together in such a way that the ratio of their corresponding values remains constant. Visually, if you plot values of yy against xx on a coordinate plane, the points will form a straight line passing through the origin (0,0)(0,0).

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Constant of Variation in Direct Proportion: For any two quantities in direct proportion, the relationship is expressed as xy=k\frac{x}{y} = k or x=kyx = ky, where kk is a positive constant. This means if xx increases, yy increases by the same factor.

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Inverse Proportion: Two quantities xx and yy are in inverse proportion if an increase in xx causes a proportional decrease in yy (and vice versa) such that their product remains constant. Visually, the graph of an inverse proportion relationship is a smooth curve called a hyperbola that approaches the axes but never touches them.

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Constant of Variation in Inverse Proportion: The mathematical relationship for inverse variation is x×y=kx \times y = k. This implies that as one variable grows larger, the other must grow smaller to keep the product kk the same.

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Identifying Direct Variation: Real-world examples include the relationship between the distance traveled and fuel consumed by a vehicle at a constant speed, or the total cost of items and the number of items purchased. In these cases, more of one leads to more of the other.

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Identifying Inverse Variation: Common examples include the relationship between the number of workers and the time taken to complete a task, or the speed of a vehicle and the time taken to cover a fixed distance. In these scenarios, increasing the 'effort' or 'speed' reduces the 'time'.

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The Ratio Table Method: To solve proportion problems, it is helpful to organize data in a table with two rows. For direct proportion, we equate the ratios of the columns: x1y1=x2y2\frac{x_1}{y_1} = \frac{x_2}{y_2}. For inverse proportion, we equate the products of the columns: x1y1=x2y2x_1 y_1 = x_2 y_2.

📐Formulae

Direct Proportion Equation: x1y1=x2y2\frac{x_1}{y_1} = \frac{x_2}{y_2}

Inverse Proportion Equation: x1y1=x2y2x_1 y_1 = x_2 y_2

Constant of Proportion (Direct): k=yxk = \frac{y}{x}

Constant of Proportion (Inverse): k=x×yk = x \times y

💡Examples

Problem 1:

A car travels 432432 km on 4848 litres of petrol. How far would it travel on 2020 litres of petrol?

Solution:

Let the distance traveled on 2020 litres be xx km. Since the amount of petrol and distance covered are in direct proportion, we have: 43248=x20\frac{432}{48} = \frac{x}{20} Cross-multiplying gives: 48×x=432×2048 \times x = 432 \times 20 x=432×2048x = \frac{432 \times 20}{48} x=9×20=180x = 9 \times 20 = 180 So, the car will travel 180180 km.

Explanation:

This is a direct proportion problem because less petrol will result in less distance covered. We set up a ratio of distance to petrol and solve for the unknown value xx.

Problem 2:

If 1212 workers can build a wall in 4848 hours, how many workers will be required to do the same work in 3232 hours?

Solution:

Let the required number of workers be xx. As the number of workers increases, the time taken to finish the work decreases. Therefore, this is an inverse proportion. Using the formula x1y1=x2y2x_1 y_1 = x_2 y_2: 12×48=x×3212 \times 48 = x \times 32 x=12×4832x = \frac{12 \times 48}{32} Dividing 4848 and 3232 by 1616: x=12×32x = \frac{12 \times 3}{2} x=6×3=18x = 6 \times 3 = 18 So, 1818 workers are required.

Explanation:

Since the task (building the wall) remains the same, reducing the time requires increasing the workforce. This inverse relationship requires the product of workers and hours to remain constant.