A city has two main roads which cross each other at the centre of the city. These two roads are along the North–South (N–S) direction and East–West (E–W) direction. All the other streets of the city run parallel to these roads and are 200 m apart. There are 10 streets in each direction.
- (i) Using 1 cm = 200 m, draw a model of the city in your notebook. Represent the roads/streets by single lines.
- (ii) (a) There are street intersections in the model. Each street intersection is formed by two streets — one running in the N–S direction and another in the E–W direction. Each street intersection is referred to in the following manner: If the second street running in the N–S direction and 5th street in the E–W direction meet at some crossing, then we call this street intersection (2, 5). Using this convention, find: (a) how many street intersections can be referred to as (4, 3).
- (ii) (b) how many street intersections can be referred to as (3, 4).
Step-by-step solution
Idea: 200 m on the ground becomes 1 cm in the model. Numbering the N–S streets from west to east and the E–W streets from south to north turns every crossing into an ordered pair, exactly like coordinates: the first number is always the N–S street.
(i) Using 1 cm = 200 m, draw a model of the city in your notebook. Represent the roads/streets by single lines.
- Streets are 200 m apart, and 200 m is shown as 1 cm, so draw the lines 1 cm apart.½ mark
- 10 streets in a direction have 9 gaps between them, so each set of lines spans 9 × 1 cm = 9 cm (9 × 200 m = 1.8 km on the ground).½ mark
- Draw 10 vertical lines (the N–S streets) and 10 horizontal lines (the E–W streets), 1 cm apart, making a 9 cm × 9 cm grid. Number the N–S streets 1 to 10 from west to east and the E–W streets 1 to 10 from south to north (see the figure). With 10 lines each way the centre of the city lies between the 5th and 6th streets. (If the 10 streets are counted besides the main roads, draw 11 lines each way, 10 cm × 10 cm, with the main roads as the 6th lines; the answers to (ii) do not change.)1 mark
(ii) (a) There are street intersections in the model. Each street intersection is formed by two streets — one running in the N–S direction and another in the E–W direction. Each street intersection is referred to in the following manner: If the second street running in the N–S direction and 5th street in the E–W direction meet at some crossing, then we call this street intersection (2, 5). Using this convention, find: (a) how many street intersections can be referred to as (4, 3).
- (4, 3) means the 4th N–S street and the 3rd E–W street. A vertical line and a horizontal line cross at exactly one point.½ mark
- So only one street intersection can be referred to as (4, 3) (red in the figure).½ mark
(ii) (b) how many street intersections can be referred to as (3, 4).
- (3, 4) means the 3rd N–S street and the 4th E–W street. Again these two lines meet at exactly one point.½ mark
- So only one intersection is (3, 4), and it is not the same as (4, 3) (blue in the figure): the order of the numbers matters, just as (4, 3) and (3, 4) are different points on a graph.½ mark
Check: Total crossings = 10 × 10 = 100 (or 11 × 11 = 121 if the main roads are drawn as extra lines), and each has its own pair (N–S number, E–W number), so no pair can name two crossings ✓.
Answer to write in the exam
(i)
Scale: 200 m = 1 cm ⇒ lines 1 cm apart
10 streets ⇒ 9 gaps ⇒ 9 cm (= 1.8 km)
Model: 10 vertical (N–S) and 10 horizontal (E–W) lines, 1 cm apart, forming a 9 cm × 9 cm grid; the city centre lies between the 5th and 6th lines.
(If the main roads are extra to the 10 streets: 11 lines each way, 10 cm × 10 cm, main roads as the 6th lines.)
(ii) (a)
(4, 3): 4th N–S street meets 3rd E–W street
Two such streets cross at exactly one point.
∴ Only one street intersection is (4, 3).
(ii) (b)
(3, 4): 3rd N–S street meets 4th E–W street
These cross at exactly one point, different from (4, 3).
∴ Only one street intersection is (3, 4).
Common mistakes that cost marks
- Spacing the lines 2 cm apart because the streets are 200 m apart. The scale is 200 m = 1 cm, so the lines are 1 cm apart.
- Saying (4, 3) and (3, 4) are the same crossing. The first number is always the N–S street, so they are different.
- Counting the streets from different starting sides in different parts of the answer. Fix the numbering (west to east, south to north) once and keep it.
How this can come in the exam
On a model drawn to the scale 1 cm = 200 m, two parallel streets 1.2 km apart are drawn
- 6 cm apart
- 12 cm apart
- 0.6 cm apart
- 2.4 cm apart
Show answer
(A) 6 cm apart
1.2 km = 1200 m, and 1200 ÷ 200 = 6, so 6 cm.
Assertion (A): In the city model, the street intersections (4, 3) and (3, 4) are different.
Reason (R): In the pair (a, b), a always names the N–S street and b the E–W street, so the order matters.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
R is exactly why the two pairs name different crossings.
Try one yourself
A town has 6 streets in each direction, 250 m apart. Using 1 cm = 250 m, how big is the model, and how many street intersections are there?
Show answer
6 streets have 5 gaps: the model is 5 cm × 5 cm. Intersections = 6 × 6 = 36.
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