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Proportional Reasoning-2 - A Slice of the Pie

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 such that the ratio xy\frac{x}{y} remains constant. This is represented as x1y1=x2y2=k\frac{x_1}{y_1} = \frac{x_2}{y_2} = k.

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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. This is represented as x1y1=x2y2=kx_1 y_1 = x_2 y_2 = k.

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Proportionality in Pie Charts: In a pie chart, the 'slice' or sector area is directly proportional to the value it represents. The central angle of each sector is calculated based on the ratio of the component value to the total value.

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Ratio Division: To divide a total quantity WW into a ratio a:ba:b, the first part is aa+b×W\frac{a}{a+b} \times W and the second part is ba+b×W\frac{b}{a+b} \times W.

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Unitary Method: This involves finding the value of a single unit first (by division) and then finding the value of the required number of units (by multiplication).

📐Formulae

x1y1=x2y2 (Direct Proportion)\frac{x_1}{y_1} = \frac{x_2}{y_2} \text{ (Direct Proportion)}

x1y1=x2y2 (Inverse Proportion)x_1 y_1 = x_2 y_2 \text{ (Inverse Proportion)}

Central Angle=(Component ValueTotal Value×360∘)\text{Central Angle} = \left( \frac{\text{Component Value}}{\text{Total Value}} \times 360^\circ \right)

Percentage of a Component=(Component ValueTotal Value×100)%\text{Percentage of a Component} = \left( \frac{\text{Component Value}}{\text{Total Value}} \times 100 \right) \%

💡Examples

Problem 1:

If 1515 workers can build a wall in 4848 hours, how many workers will be required to do the same work in 3030 hours?

Solution:

This is a case of inverse proportion because as the number of workers increases, the time taken decreases. Let the required number of workers be x2x_2. Given: x1=15x_1 = 15, y1=48y_1 = 48 hours, y2=30y_2 = 30 hours. Using x1y1=x2y2x_1 y_1 = x_2 y_2: 15×48=x2×3015 \times 48 = x_2 \times 30 x2=15×4830x_2 = \frac{15 \times 48}{30} x2=1×482=24x_2 = \frac{1 \times 48}{2} = 24 So, 2424 workers are required.

Explanation:

Inverse proportion applies here because the total 'work-hours' needed remains constant. We set the product of the initial state equal to the product of the final state.

Problem 2:

In a survey, 40%40\% of students chose 'Pizza' as their favorite food. If the total number of students is 900900, calculate the central angle for the 'Pizza' sector in a pie chart.

Solution:

Method 1: Using percentage Central Angle=40% of 360∘\text{Central Angle} = 40\% \text{ of } 360^\circ Central Angle=40100×360∘=144∘\text{Central Angle} = \frac{40}{100} \times 360^\circ = 144^\circ

Method 2: Finding the value first Number of students who like Pizza: 40100×900=360\frac{40}{100} \times 900 = 360 Using the formula: Central Angle=360900×360∘=144∘\text{Central Angle} = \frac{360}{900} \times 360^\circ = 144^\circ

Explanation:

The central angle of a pie chart is directly proportional to the percentage. Since 100%100\% corresponds to 360∘360^\circ, 1%1\% corresponds to 3.6∘3.6^\circ.

Problem 3:

A baker uses 33 cups of flour to make 1212 large cookies. How many cups of flour are needed to make 2020 large cookies?

Solution:

This is a case of direct proportion. Let xx be the cups of flour and yy be the number of cookies. x1y1=x2y2\frac{x_1}{y_1} = \frac{x_2}{y_2} 312=x220\frac{3}{12} = \frac{x_2}{20} 14=x220\frac{1}{4} = \frac{x_2}{20} x2=204=5x_2 = \frac{20}{4} = 5 Vertical check for simple addition of parts if needed: 3+25\begin{array}{r} 3 \\ + 2 \\ \hline 5 \end{array} He needs 55 cups of flour.

Explanation:

Since the size of the cookies is constant, the amount of flour increases linearly with the number of cookies (Direct Proportion).