GSEB Class 8 Maths Solutions Chapter 13 Direct and Inverse Proportions InText Questions

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Question 1. Observe the following tables and find if x and y are directly proportional?
(i)

(ii)
(iii)
Answer:
(i) To check for direct proportionality, we calculate the ratio \( \frac{x}{y} \) for each pair of values in the table. \( \frac{20}{40} = \frac{1}{2} \)
\( \frac{17}{34} = \frac{1}{2} \)
\( \frac{14}{28} = \frac{1}{2} \)
\( \frac{11}{22} = \frac{1}{2} \)
\( \frac{8}{16} = \frac{1}{2} \)
\( \frac{5}{10} = \frac{1}{2} \)
\( \frac{2}{4} = \frac{1}{2} \)
Each ratio is the same, which means \( x \) and \( y \) are directly proportional.
(ii) To check for direct proportionality, we calculate the ratio \( \frac{x}{y} \) for each pair of values in the table. \( \frac{6}{4} = \frac{3}{2} \)
\( \frac{10}{8} = \frac{5}{4} \)
\( \frac{14}{12} = \frac{7}{6} \)
\( \frac{18}{16} = \frac{9}{8} \)
\( \frac{22}{20} = \frac{11}{10} \)
\( \frac{26}{24} = \frac{13}{12} \)
\( \frac{30}{28} = \frac{15}{14} \)
All the ratios are not the same, so \( x \) and \( y \) are not directly proportional.
(iii) To check for direct proportionality, we calculate the ratio \( \frac{x}{y} \) for each pair of values in the table. \( \frac{5}{15} = \frac{1}{3} \)
\( \frac{8}{24} = \frac{1}{3} \)
\( \frac{12}{36} = \frac{1}{3} \)
\( \frac{15}{60} = \frac{1}{4} \)
\( \frac{18}{72} = \frac{1}{4} \)
\( \frac{20}{100} = \frac{1}{5} \)
All the ratios are not the same, so \( x \) and \( y \) are not directly proportional.
In simple words: To see if two things are directly proportional, you divide the numbers from one row by the numbers in the other row. If all the answers are the same, then they are directly proportional. If the answers are different, they are not.

Exam Tip: For direct proportionality, the ratio \( \frac{x}{y} \) (or \( \frac{y}{x} \)) must remain constant for all corresponding pairs of values. Make sure to simplify each ratio to its lowest terms for easy comparison.

 

Question 2. Principal = Rs. 1000, Rate = 8% per annum. Fill in the following table and find which type of interest (simple or compound) change is in direct proportion with time period?
Answer: We need to fill the table for simple interest and compound interest and then check for direct proportionality with the time period.

Case of simple interest [P = Rs. 1000, r = 8% p.a.]

In each case, the ratio \( \frac{SI}{T} \) is the same.
\( \implies \) The simple interest changes in direct proportion with the time period.

Case of Compound Interest [P = Rs. 1000, r = 8% p.a.]
The ratio \( \frac{CI}{T} \) is not the same in each case.
\( \implies \) The compound interest does not change in direct proportion with the time period.
In simple words: Simple interest is directly proportional to time because the interest earned per year stays the same. But compound interest is not directly proportional because the amount of interest changes each year as it's calculated on a growing principal.

Exam Tip: Remember that simple interest grows linearly with time, always maintaining the same interest amount per unit of time. Compound interest, however, grows exponentially, meaning the interest amount increases over time, making it not directly proportional to the period.

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Question 1. On a squared paper, draw five squares of different sides. Write the following information in a tabular form? Find whether the length of a side is in direct proportion to:
(a) The perimeter of the square
(b) The area of the square
Answer: We draw the following squares I, II, III, IV, and V, having side lengths of 1 cm, 2 cm, 3 cm, 4 cm, and 5 cm, respectively. We then calculate their perimeters and areas to complete the table.

(a) The length of a side (L) and the perimeter of the square (P) are in direct proportion because the ratio \( \frac{L}{P} \) is constant (\( \frac{1}{4} \)).
(b) However, the length of a square (L) and the area of the square (A) are not in direct proportion because the ratio \( \frac{L}{A} \) is not constant.
In simple words: When you make a square bigger, its outside edge (perimeter) grows at the same speed as its side length. But its inside space (area) grows much faster, not at the same speed. So, side length and perimeter are directly linked, but side length and area are not.

Exam Tip: Remember that for direct proportionality, the ratio of the two quantities must always be constant. The perimeter of a square is \( 4 \times L \), so \( P/L = 4 \) (constant). The area is \( L^2 \), so \( A/L = L \), which is not constant, thus not directly proportional.

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Question 1. Observe the following tables and find which pair of variables (here x and y) are in inverse proportion?
(i)

(ii)
(iii)
Answer:
(i) To check for inverse proportionality, we calculate the product \( x \times y \) for each pair of values in the table.
\( x_1y_1 = 50 \times 5 = 250 \)
\( x_2y_2 = 40 \times 6 = 240 \)
\( x_3y_3 = 30 \times 7 = 210 \)
\( x_4y_4 = 20 \times 8 = 160 \)
Since \( x_1y_1 \neq x_2y_2 \neq x_3y_3 \neq x_4y_4 \), the values of \( x \) and \( y \) are not in inverse proportion.

(ii) To check for inverse proportionality, we calculate the product \( x \times y \) for each pair of values in the table.
\( x_1y_1 = 100 \times 60 = 6000 \)
\( x_2y_2 = 200 \times 30 = 6000 \)
\( x_3y_3 = 300 \times 20 = 6000 \)
\( x_4y_4 = 400 \times 15 = 6000 \)
Since \( x_1y_1 = x_2y_2 = x_3y_3 = x_4y_4 \), the values of \( x \) and \( y \) are in inverse proportion.

(iii) To check for inverse proportionality, we calculate the product \( x \times y \) for each pair of values in the table.
\( x_1y_1 = 90 \times 10 = 900 \)
\( x_2y_2 = 60 \times 15 = 900 \)
\( x_3y_3 = 45 \times 20 = 900 \)
\( x_4y_4 = 30 \times 25 = 750 \)
\( x_5y_5 = 20 \times 30 = 600 \)
\( x_6y_6 = 5 \times 35 = 175 \)
Since \( x_1y_1 = x_2y_2 = x_3y_3 \neq x_4y_4 \neq x_5y_5 \neq x_6y_6 \), the values of \( x \) and \( y \) are not in inverse proportion.
In simple words: To see if two things are inversely proportional, you multiply the numbers from one row by the numbers in the other row. If all the answers are the same, then they are inversely proportional. If the answers are different, they are not.

Exam Tip: For inverse proportionality, the product \( x \times y \) must be constant for all corresponding pairs of values. This means as one quantity increases, the other decreases in a way that their product stays the same.

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Question 1. Take a sheet of paper. Fold it as shown in the figure. Count the number of parts and the area of a part in each case? Tabulate your observations and discuss with your friends. Is it a case inverse proportion? Why?
Answer: We can observe the pattern of folding the paper and how it affects the number of parts and the area of each part.

Here, as the number of parts increases, the area of each part decreases. Specifically, if you double the number of parts, the area of each part halves. Thus, it is a case of "inverse proportion."
In simple words: When you fold a paper more times, you get more pieces. But each piece gets smaller. This shows that the number of pieces and the size of each piece are connected in an opposite way—one goes up, the other goes down. This is called inverse proportion.

Exam Tip: Inverse proportion describes a relationship where an increase in one quantity leads to a proportional decrease in another, such that their product remains constant. Think of sharing a fixed quantity: more people mean a smaller share for each.

 

Question 2. Take a few containers of different sizes with circular bases. Fill the same amount of water in each container Note the diameter of each container and the respective height at which the water level stands. Tabulate your observations. Is it a case of inverse proportion?
Answer: We take containers with varying diameters \( d_1, d_2, d_3 \) and fill them with the same amount of water. We then observe the corresponding water levels \( h_1, h_2, h_3 \).

Here, we observe that as the diameter of the container gets smaller, the level of water in the container rises. Conversely, a larger diameter results in a lower water level for the same volume of water. Therefore, the diameter and the height of the water level are in inverse proportion.
In simple words: If you pour the same amount of water into different sized containers, a wide container will have a low water level, and a narrow container will have a high water level. This shows that the width of the container and the water's height are inversely related.

Exam Tip: For a fixed volume of liquid, the area of the base and the height of the liquid are inversely proportional. A wider base (larger diameter) means a lower height, and a narrower base (smaller diameter) means a higher height.