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Direct and Inverse Proportions - Time and Work Problems

Grade 8ICSE

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 expressed as xy=k\frac{x}{y} = k. Visually, a graph of direct proportion is a straight line passing through the origin (0,0)(0,0), where the slope of the line represents the constant kk.

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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. Their product remains constant, expressed as x×y=kx \times y = k. Visually, the graph of an inverse proportion is a downward-sloping curve known as a rectangular hyperbola that approaches the xx and yy axes but never intersects them.

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Work and Time Relationship: The time taken to complete a piece of work is inversely proportional to the number of persons working on it. If the number of workers increases, the time taken decreases. Visually, this can be imagined as a fixed 'area' of work that is divided among more people, making each person's share (and thus the total time) smaller.

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One Day's Work: If a person can complete a task in nn days, then the part of the work finished in 11 day is 1n\frac{1}{n}. This concept is the foundation for solving complex 'Time and Work' problems by converting total time into rates of work.

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Combined Work: When multiple individuals work together, their combined rate of work is the sum of their individual one-day work rates. For instance, if person A completes 1x\frac{1}{x} of the work in a day and person B completes 1y\frac{1}{y}, their combined daily work is (1x+1y)(\frac{1}{x} + \frac{1}{y}).

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Work and Efficiency: Efficiency is the rate at which work is done. A more efficient person takes less time to complete a task. Efficiency and Time are inversely proportional. If A is twice as efficient as B, A will take 12\frac{1}{2} the time B takes to finish the same work.

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Total Work as a Unit: In mathematical calculations, the total work to be completed is usually represented as the whole number 11. Any part of the work done is represented as a fraction of this 11 (e.g., half the work is 12\frac{1}{2}).

📐Formulae

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

Inverse Proportion: x1×y1=x2×y2x_1 \times y_1 = x_2 \times y_2

Work done in 11 day = 1Total time to complete the work\frac{1}{\text{Total time to complete the work}}

Total time to complete work = 1Work done in 1 day\frac{1}{\text{Work done in 1 day}}

If A can do work in xx days and B in yy days, time taken together: T=x×yx+yT = \frac{x \times y}{x + y}

General Chain Rule: M1×D1×H1W1=M2×D2×H2W2\frac{M_1 \times D_1 \times H_1}{W_1} = \frac{M_2 \times D_2 \times H_2}{W_2} where MM=Men, DD=Days, HH=Hours, and WW=Work.

💡Examples

Problem 1:

If 15 workers can build a wall in 48 hours, how many workers will be required to do the same work in 30 hours?

Solution:

Let the required number of workers be xx. Since the number of workers and the time taken are in inverse proportion (more workers take less time), we use the formula: x1×y1=x2×y2x_1 \times y_1 = x_2 \times y_2 Here, x1=15x_1 = 15 workers, y1=48y_1 = 48 hours, y2=30y_2 = 30 hours, and we need to find x2x_2. 15×48=x2×3015 \times 48 = x_2 \times 30 720=30x2720 = 30x_2 x2=72030x_2 = \frac{720}{30} x2=24x_2 = 24 Therefore, 2424 workers are required.

Explanation:

This is an inverse proportion problem because as the time allowed decreases, the number of workers must increase to finish the same job.

Problem 2:

A can do a piece of work in 10 days and B can do the same work in 15 days. How many days will they take to complete the work if they work together?

Solution:

Work done by A in 11 day = 110\frac{1}{10} Work done by B in 11 day = 115\frac{1}{15} Work done by (A + B) in 11 day = 110+115\frac{1}{10} + \frac{1}{15} To add the fractions, find the L.C.M. of 1010 and 1515, which is 3030: (A+B)′s 1 day work=3+230=530=16(\text{A+B})'s\text{ 1 day work} = \frac{3 + 2}{30} = \frac{5}{30} = \frac{1}{6} Total time taken by A and B together = 11 day’s work=11/6=6\frac{1}{\text{1 day's work}} = \frac{1}{1/6} = 6 days.

Explanation:

To find the combined time, we first determine how much work each person does in a single day, sum those rates, and then take the reciprocal to find the total days.