NCERT Solutions Class 9 Mathematics Ganita Manjari 2 Chapter 13 Two Variables One Line Exercise 13.3

Exercise Set 13.3

Question 1. The following diagrams represent ski hills. Rank the hills in order of their steepness, from least to greatest.

Answer:
Steepness = slope = \( \frac{\text{rise}}{\text{run}} \).
Hill A: \( \frac{60}{70} \) = \( \frac{6}{7} \) ≈ 0.857.
Hill B: \( \frac{60}{110} \) = \( \frac{6}{11} \) ≈ 0.545.
Hill C: \( \frac{80}{100} \) = \( \frac{4}{5} \) = 0.8.
Since 0.545 < 0.8 < 0.857, i.e. \( \frac{6}{11} \) < \( \frac{4}{5} \) < \( \frac{6}{7} \):
Order from least to greatest steepness: B, C, A.

Teacher's Note:
1. Hills A and B have the same rise; the one with the shorter run is steeper.
2. Turning fractions into decimals is the easiest way to compare them.

Question 2. The ramp at a loading dock rises 2.5 metres over a run of 4 metres. Find the slope of the ramp.

Answer:
Slope = \( \frac{\text{rise}}{\text{run}} \) = \( \frac{2.5}{4} \) = \( \frac{25}{40} \) = \( \frac{5}{8} \).
The slope of the ramp is \( \frac{5}{8} \) (= 0.625): it climbs 5 m for every 8 m measured horizontally.

Question 3. Find the slope of the line ℓ in each of the following diagrams.

Answer:
Take one grid square as 1 unit.
(i) ℓ is horizontal, so y never changes and the rise is 0. Slope = \( \frac{0}{\text{run}} \) = 0.
(ii) Marked points P(−2, 2) and Q(2, −2). Slope = \( \frac{-2 - 2}{2 - (-2)} \) = \( \frac{-4}{4} \) = −1. The line drops 1 unit for every unit to the right.
(iii) Marked points P(−2, −1) and Q(1, 1). Slope = \( \frac{1 - (-1)}{1 - (-2)} \) = \( \frac{2}{3} \). The line climbs 2 units for every 3 units to the right.
(iv) ℓ is vertical, so x never changes and the run is 0. Slope = \( \frac{\text{rise}}{0} \), which is not defined.
The answers do not depend on the scale, because rise and run are measured in the same unit.

Teacher's Note:
1. Horizontal line: slope 0. Vertical line: slope undefined. Students often mix these up.
2. Always subtract the coordinates in the same order in the numerator and the denominator.

Question 4. An accessibility ramp for wheelchairs is to be made alongside the staircase. The guidelines given for the slope of the ramp is 1 cm vertical rise to 12 cm horizontal length. What should be the horizontal length of the ramp if the total height of the stairs is 18 cm?
Answer:
Required slope = \( \frac{1}{12} \) (1 cm up for every 12 cm across).
Let the horizontal length be L cm. The rise is the full height of the stairs, 18 cm.
\( \frac{18}{L} \) = \( \frac{1}{12} \) → L = 18 × 12 = 216.

The ramp should have a horizontal length of 216 cm (2.16 m).