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Proportional Reasoning-1 - Problem Solving with Proportional Reasoning

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 xy\frac{x}{y} remains constant (kk).

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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 their product x×yx \times y remains constant (kk).

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The Unitary Method is a technique where we first find the value of a single unit and then multiply it by the required number of units to find the total value.

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Scale drawings and maps use proportional reasoning where the Scale is the ratio of the distance on the drawing to the actual distance: Scale=Distance on MapActual Distance\text{Scale} = \frac{\text{Distance on Map}}{\text{Actual Distance}}.

📐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

k=yx (Constant of Variation for Direct Proportion)k = \frac{y}{x} \text{ (Constant of Variation for Direct Proportion)}

k=x×y (Constant of Variation for Inverse Proportion)k = x \times y \text{ (Constant of Variation for Inverse Proportion)}

💡Examples

Problem 1:

A car travels 432 km432 \text{ km} on 48 litres48 \text{ litres} of petrol. How far will it travel on 20 litres20 \text{ litres} of petrol?

Solution:

Let the distance traveled on 20 litres20 \text{ litres} be x kmx \text{ km}. Since distance and petrol consumed are in direct proportion: x1y1=x2y2\frac{x_1}{y_1} = \frac{x_2}{y_2} 43248=x20\frac{432}{48} = \frac{x}{20} x=432×2048x = \frac{432 \times 20}{48} x=9×20=180 kmx = 9 \times 20 = 180 \text{ km} The car will travel 180 km180 \text{ km}.

Explanation:

As the amount of petrol decreases, the distance covered will also decrease proportionally. This is a case of direct proportion.

Problem 2:

If 3636 men can finish a piece of work in 2525 days, how many days will 1515 men take to do the same work?

Solution:

Let the number of days be yy. Since the number of men and time taken are in inverse proportion: x1y1=x2y2x_1 y_1 = x_2 y_2 36×25=15×y36 \times 25 = 15 \times y y=36×2515y = \frac{36 \times 25}{15} y=36×53=12×5=60 daysy = \frac{36 \times 5}{3} = 12 \times 5 = 60 \text{ days} It will take 60 days60 \text{ days} for 1515 men to complete the work.

Explanation:

Fewer men will take more time to complete the same task, which indicates an inverse relationship.

Problem 3:

Calculate the difference in cost if 55 pens cost ₹125₹125 and the price of 11 pen is reduced by ₹3₹3. Use vertical subtraction to show the change for 88 pens.

Solution:

Cost of 11 pen = 1255=₹25\frac{125}{5} = ₹25. New cost of 11 pen = 25−3=₹2225 - 3 = ₹22. Original cost for 88 pens = 8×25=₹2008 \times 25 = ₹200. New cost for 88 pens = 8×22=₹1768 \times 22 = ₹176. Difference in cost: 200−17624\begin{array}{r} 200 \\ -176 \\ \hline 24 \end{array} The difference is ₹24₹24.

Explanation:

First, find the unit price using the unitary method, adjust the price, and then calculate the total for the new quantity.