Learning Objective: Explore map scales where 1 cm represents a large geographical distance (e.g., 1:60,00,000).
Learning Objective: Scale multi-term ratios like a:b:c :: p:q:r proportionally.
Learning Objective: Verify that for inverse proportions, the product xy = k remains constant.
Two ratios a:b and c:d are proportional if a × d = b × c, or a / c = b / d.
Ratios can have multiple terms if each quantity changes by the same factor. For instance, a spice mix of coriander, red chillies, toor dal, and fenugreek in ratio 8:4:2:1 maintains its taste when scaled proportionally. When two multi-term ratios are proportional: a:b:c:d :: p:q:r:s implies a / p = b / q = c / r = d / s.
To divide a quantity x in a multi-term ratio a:b:c:..., each part is found by multiplying x by the term's fractional share over the sum of all terms: Part = x × a / (a + b + c + ...).
1. Cricket practice session time allocation in ratio 3:4:3:5 (warm-up/cool-down : batting : bowling : fielding) with a total duration of 150 minutes:
Step 1: Find total parts = 3 + 4 + 3 + 5 = 15 parts.
Step 2: Value of one part = 150 / 15 = 10 minutes.
Step 3: Multiply each ratio term by 10:
- Warm-up/cool-down = 3 × 10 = 30 minutes.
- Batting = 4 × 10 = 40 minutes.
- Bowling = 3 × 10 = 30 minutes.
- Fielding = 5 × 10 = 50 minutes.
2. School library books ratio of Odiya : Hindi : English is 3:2:1, with 288 Odiya books:
Step 1: 3 parts correspond to 288 books.
Step 2: Value of one part = 288 / 3 = 96 books.
Step 3: Calculate other books:
- Hindi books = 2 × 96 = 192 books.
- English books = 1 × 96 = 96 books.
3. Total of 100 coins in ratio Rs 10 : Rs 5 : Rs 2 : Rs 1 :: 4:3:2:1:
Step 1: Total parts = 4 + 3 + 2 + 1 = 10 parts.
Step 2: Value of one part = 100 / 10 = 10 coins.
Step 3: Count and value of each coin type:
- Rs 10 coins = 4 parts = 40 coins (Value = 400).
- Rs 5 coins = 3 parts = 30 coins (Value = 150).
- Rs 2 coins = 2 parts = 20 coins (Value = 40).
- Rs 1 coins = 1 part = 10 coins (Value = 10).
Step 4: Total money = 400 + 150 + 40 + 10 = 600.
4. Construct a triangle with sidelengths in ratio 3:4:5. Will all triangles drawn be congruent?
Step 1: 3:4:5 represents right-angled triangles.
Step 2: Conclusion: No, they will not be congruent; they will be similar triangles because scaling the ratio creates triangles of different sizes with identical angles.
5. Can you construct a triangle with sidelengths in the ratio 1:3:5?
Step 1: Check triangle inequality theorem (sum of any two sides must be greater than the third side).
Step 2: 1 + 3 = 4, which is less than 5.
Step 3: Conclusion: No, such a triangle cannot exist.
Pie charts illustrate proportions of a whole using sectors of a circle where angles are proportional to data values. Since the total angle in a circle is 360 degrees, data categories are divided proportionally across 360 degrees.
1. A group of 360 people voted for their favorite season: 90 summer, 120 rainy, rest winter. Draw a pie chart:
Step 1: Winter votes = 360 - (90 + 120) = 150 people.
Step 2: Sector angles (out of 360 degrees):
- Summer angle = (90 / 360) × 360 = 90 degrees.
- Rainy angle = (120 / 360) × 360 = 120 degrees.
- Winter angle = (150 / 360) × 360 = 150 degrees.
2. Favorite TV channels: Entertainment 50%, Sports 25%, News 15%, Information 10%:
Step 1: Convert percentages into angles out of 360 degrees:
- Entertainment = 50% of 360 = 180 degrees.
- Sports = 25% of 360 = 90 degrees.
- News = 15% of 360 = 54 degrees.
- Information = 10% of 360 = 36 degrees.
Two quantities x and y vary in inverse proportion if their product remains constant: xy = k. When one quantity increases by a factor n, the other changes by the inverse factor 1/n.
1. Which pairs of quantities are in inverse proportion? (Page 67)
- (i) Taps filling tank and time taken: Inverse proportion.
- (ii) Painters and days needed: Inverse proportion.
- (iii) Distance traveled and petrol amount: Direct proportion.
- (iv) Speed of cyclist and time taken: Inverse proportion.
- (v) Length of cloth and price paid: Direct proportion.
- (vi) Pages in a book and reading time: Direct proportion.
2. If 24 pencils cost 120, how much will 20 such pencils cost?
Cost of 1 pencil = 120 / 24 = 5. Cost of 20 pencils = 20 × 5 = 100.
3. Tank supplies 20 families for 6 days. If 10 more families join (total 30), how long will water last?
Total family-days = 20 × 6 = 120. Days for 30 families = 120 / 30 = 4 days.
5. Pie chart survey on transport modes (Walk 90 degrees, Bus 120 degrees, Cycle 60 degrees, Two-wheeler 60 degrees, Car 30 degrees):
(i) Most common: Bus.
(ii) Fraction by car: 30 / 360 = 1/12.
(iii) If 18 children travel by car, total participants = 18 × 12 = 216 children.
(iv) Equal numbers: Cycle and Two-wheeler (both 60 degrees).
6. Three workers paint fence in 4 days. If one more worker joins (total 4), how many days?
Worker-days = 3 × 4 = 12. Days for 4 workers = 12 / 4 = 3 days.
7. Takes 6 hours to fill 2 tanks with 1 pump. How long to fill 5 tanks with same pump?
Time per tank = 6 / 2 = 3 hours. Time for 5 tanks = 5 × 3 = 15 hours.
8. Chairs in 25 rows with 12 chairs each. Rearranged with 20 chairs per row, how many rows?
Total chairs = 25 × 12 = 300. Rows = 300 / 20 = 15 rows.
9. School has 8 periods of 45 mins. If 9 periods a day, how long is each period?
Total daily minutes = 8 × 45 = 360 minutes. Length of 9 periods = 360 / 9 = 40 minutes.
10. Small pump fills tank in 3 hours (rate 1/3), large pump in 2 hours (rate 1/2). Together time?
Combined rate = 1/3 + 1/2 = 5/6 tank per hour. Time = 6/5 hours = 1.2 hours (1 hour 12 mins).
11. Factory requires 42 machines for 63 days. How many machines for 54 days?
Total machine-days = 42 × 63 = 2646. Machines needed = 2646 / 54 = 49 machines.
12. Car takes 2 hours at 60 km/h (Distance = 120 km). Time at 80 km/h?
Time = 120 / 80 = 1.5 hours.