PROPORTIONAL REASONING-1
KEY POINTS & FIGURE IT OUT
7.2 Ratios
In a ratio of the form a : b, we can say that for every 'a' units of the first quantity, there are 'b' units of the second quantity.
A more systematic way to compare whether the ratios are proportional is to reduce them to their simplest form and see if these simplest forms are the same.
7.3 Ratios in their Simplest Form
We can reduce ratios to their simplest form by dividing the terms by their HCF.
When two ratios are the same in their simplest forms, we say the ratios are proportional.
We use the symbol :: to show proportionality.
Example:
60 : 40 :: 30 : 20 and 60 : 40 :: 90 : 60.
7.4 Problem Solving with Proportional Reasoning
Example 1:
Are the ratios 3 : 4 and 72 : 96 proportional?
3 : 4 is already in its simplest form. To find the simplest form of 72 : 96, we need to divide both terms by their HCF.
The HCF of 72 and 96 is 24. Dividing both terms by 24, we get 3 : 4. Since both ratios in their simplest form are the same, they are proportional
Example 2:
Kesang wanted to make lemonade for a celebration. She made 6 glasses of lemonade in a vessel and added 10 spoons of sugar to the drink. Her father expected more people to join the celebration. So he asked her to make 18 more glasses of lemonade. To make the lemonade with the same sweetness, how many spoons of sugar should she add? 6 : 10 :: 18 : ?
The ratio of glasses of lemonade to spoons of sugar is 6 : 10.
6 : 10 :: 18 : x.
6 : 10 :: 18 : x
\[
\frac{6}{10} = \frac{18}{x} \implies x = \frac{18 \times 10}{6} = 30
\]
So, she should use 30 spoons of sugar to make 18 glasses of lemonade with the same sweetness as earlier
Example 3:
Nitin and Hari were constructing a compound wall around their house. Nitin was building the longer side, 60 ft in length, and Hari was building the shorter side, 40 ft in length. Nitin used 3 bags of cement but Hari used only 2 bags of cement. Nitin was worried that the wall Hari built would not be as strong as the wall he built because she used less cement. Is Nitin correct in his thinking?
The ratio in Nitin’s case is 60 : 3, That is , 20 : 1 (in its simplest form).
The ratio in Hari’s case is 40 : 2, That is , 20 : 1 (in its simplest form).
Since both ratios are proportional, the walls are equally strong. Nitin should not worry!
Example 4:
In my school, there are 5 teachers and 170 students. The ratio of teachers to students in my school is 5 : 170. Count the number of teachers and students in your school. What is the ratio of teachers to students in your school? Write it below. ______ : ______ Is the teacher-to-student ratio in your school proportional to the one in my school?
Example:
My school = 5:170
Your school = 6: 240
5:170::6:240 no not in proportional
Example 5:
Measure the width and height (to the nearest cm) of the blackboard in your classroom. What is the ratio of width to height of the blackboard? ______ : ______ Can you draw a rectangle in your notebook whose width and height are proportional to the ratio of the blackboard? Compare the rectangle you have drawn to those drawn by your classmates. Do they all look the same?
Black board Width: 120 cm Height: 90 cm HCF = 30
120:90 = 4:3
Draw rectangle with Width = 8 cm
Height = (8÷4)×3=6 cm
Yes, in terms of shape. All rectangles drawn with a 4 : 3 ratio will have the same shape. They will look like scaled versions of each other—some may be larger, some smaller, but the proportion of width to height will be identical. This is the essence of similarity. they look different in size, Precision, drawing accuracy.
Example 6:
When Neelima was 3 years old, her mother’s age was 10 times her age. What is the ratio of Neelima’s age to her mother’s age? What would be the ratio of their ages when Neelima is 12 years old? Would it remain the same?
The ratio of Neelima’s age to her mother’s age when Neelima is 3 years old is 3 : 30 (her mother’s age is 10 times Neelima’s age).
In the simplest form, it is 1 : 10.
When Neelima is 12 years old (, 9 years later), the ratio of their ages will be 12 : 39 (9 years later, her mother would be 39 years old).
In the simplest form, it is 4 : 13.
When we add (or subtract) the same number from the terms of a ratio, the ratio changes and is not necessarily proportional to the original ratio.
Example 7:
Fill in the missing numbers for the following ratios that are proportional to
14 : 21.
______ : 42
6 : ______
2 : ______What factor should we multiply 14 by to get 6? Can it be an integer? Or should it be a fraction?
14:21::28:42
14y = 6
π¦= 6/14 = 3/7
14 × 3/7 is 6 and 21 × 3/7 is 9.
So, the ratio is 6 : 9.
In the third ratio, the first term is 2.
14 : 21 divide by 7 (HCF of 14 and 21)
So, the ratio is 2 : 3
Factor = \( \frac{6}{14} = \frac{3}{7} \)
→ \( 21 \times \frac{3}{7} = 9 \)
Ratio = 6 : 9.
2. ? : 42
Factor = \( \frac{42}{21} = 2 \)
→ \( 14 \times 2 = 28 \)
Ratio = 28 : 42.
3. 2 : ?
HCF of 14, 21 = 7 → divide by 7 → 2 : 3.
7.5 Filter Coffee Example
Regular coffee: 15 ml decoction + 35 ml milk → ratio 15 : 35 → simplest 3 : 7.
Stronger: 20 ml decoction + 30 ml milk → ratio 20 : 30 → simplest 2 : 3.
Lighter: 10 ml decoction + 40 ml milk → ratio 10 : 40 → simplest 1 : 4.
Comparison:
- 2 : 3 (≈0.667) > 3 : 7 (≈0.429) → stronger.
- 1 : 4 (0.25) < 3 : 7 (≈0.429) → lighter.
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Figure it Out (Page 165-167)
### 1. Circle the true proportions:
(i) 4 : 7 :: 12 : 21 ✅
(ii) 8 : 3 :: 24 : 6 ❌
(iii) 7 : 12 :: 12 : 7 ❌
(iv) 21 : 6 :: 35 : 10 ✅
(v) 12 : 18 :: 28 : 12 ❌
(vi) 24 : 8 :: 9 : 3 ✅
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###
2. Give 3 ratios proportional to 4 : 9.
8 : 18, 12 : 27, 20 : 45
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###
3. Fill in the missing numbers for these ratios proportional to 18 : 24.
3 : 4, 12 : 16, 20 : \( \frac{80}{3} \), 27 : 36
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4. Which rectangles are similar?
Rectangles with same width : height ratio are similar.
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###
5. Draw smaller and bigger rectangles with same ratio (32 : 18 → 16 : 9).
Factor of change = 2:
Smaller: 16 mm × 9 mm
Larger: 64 mm × 36 mm
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###
6. Ratio of grey bricks to coloured bricks in pattern:
(a) Grey : Coloured = 9 : 6 → simplest = 3 : 2
(b) Grey : Coloured = 16 : 12 → simplest = 4 : 3
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7. Human figure proportions:
Example: head : torso = 25 : 60 → 5 : 12
Proportional drawing looks realistic because body parts maintain natural ratios.
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7.6 Rule of Three (Trairasika)
If a : b :: c : d, then
\[
d = \frac{b \times c}{a}
\]
Ancient Indian method:
- pramΔαΉa (a)
- phala (b)
- ichchhΔ (c)
- ichchhΔphala (d) = \( \frac{b \times c}{a} \)
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###
Example 8:
School: 120 students → 15 kg rice.
80 students → ? kg rice.
120 : 15 :: 80 : x
\[
x = \frac{15 \times 80}{120} = 10 \ \text{kg}
\]
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###
Example 9:
Car: 90 km in 150 min. Distance in 4 hours (240 min)?
150 : 90 :: 240 : x
\[
x = \frac{90 \times 240}{150} = 144 \ \text{km}
\]
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###
Example 10:
Himachal tea: 200 g → ₹200 → ratio 1 : 1.
Meghalaya tea: 1 kg (1000 g) → ₹800 → ratio 5 : 4.
Not proportional.
Cost per kg:
Himachal = ₹1000, Meghalaya = ₹800 → Himachal more expensive.
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###
Example 11:
Investment: Prashant ₹75,000, Bhuvan ₹25,000 → ratio 3 : 1.
Profit ₹4000 → Prashant = \( \frac{3}{4} \times 4000 = ₹3000 \), Bhuvan = \( \frac{1}{4} \times 4000 = ₹1000 \).
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Example 12:
Mixture: 40 kg, sand : cement = 3 : 1 → sand = 30 kg, cement = 10 kg.
New ratio = 5 : 2, sand unchanged (30 kg).
Cement needed = \( \frac{2}{5} \times 30 = 12 \) kg → add 2 kg.
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Figure it Out (Continued)
###
1. Divide ₹4500 in ratio 2 : 3:
First part = \( \frac{2}{5} \times 4500 = ₹1800 \)
Second part = \( \frac{3}{5} \times 4500 = ₹2700 \).
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2. Acid : water = 1 : 5, total 240 ml:
Acid = \( \frac{1}{6} \times 240 = 40 \) ml
Water = \( \frac{5}{6} \times 240 = 200 \) ml.
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###
3. Blue : yellow = 3 : 5, make 40 ml green:
Blue = \( \frac{3}{8} \times 40 = 15 \) ml
Yellow = \( \frac{5}{8} \times 40 = 25 \) ml.
Add 20 ml yellow → new ratio = 15 : 45 = 1 : 3.
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4. Rice : urad dal = 2 : 1, 6 cups mixture:
Rice = \( \frac{2}{3} \times 6 = 4 \) cups
Urad dal = \( \frac{1}{3} \times 6 = 2 \) cups.
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5. Red : yellow = 3 : 5 in one bucket. Add another bucket of yellow:
New ratio = 3 : 13.
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6. Orange juice : apple juice = 600 : 900 → simplest = 2 : 3.
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7. Buses: 162 people → 3 buses (54 each).
For 204 people → 4 buses needed (12 vacant seats).
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8. Delhi vs Mumbai crowding:
Mumbai more crowded because population density higher.
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9. Crane neck : body = 4 : 6.
If my height = 165 cm, neck height = \( \frac{4}{10} \times 165 = 66 \) cm.
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10. Ancient problem:
\( 2 \frac{1}{2} \) palas saffron → \( \frac{3}{7} \) niskas.
9 niskas → \( \frac{35}{6} \times 9 = 52.5 \) palas saffron.
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11. Harmain age 1, brother 5. When ratio 1 : 2?
After 3 years → ages 4 and 8 → ratio 1 : 2.
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12. Gold : water mass ratio = 37 : 2.
1 litre water = 1 kg → 1 litre gold = \( \frac{37}{2} = 18.5 \) kg.
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13. Manure for farming:
Plot = 200 ft × 500 ft = 100,000 sq ft.
Manure needed ≈ 22.96 tonnes.
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14. Tap fills 500 ml mug in 15 seconds.
10 L bucket → 300 seconds = 5 minutes.
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15. Land cost: 1 acre = 43,560 sq ft → ₹15,00,000.
2,400 sq ft → ₹82,664.63.
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16. Tractor vs oxen ploughing:
Oxen: 20 acres × 6 hours = 120 hours.
Tractor (4× faster) → 30 hours.
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17. ₹10 coin: copper : nickel = 3 : 1.
Mass = 7.74 g → copper = 5.805 g, nickel = 1.935 g.
Cost: copper = ₹5.26, nickel = ₹2.59.
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Unit Conversions (Reference)
Length:
1 m = 3.281 ft
Area:
1 m² = 10.764 ft²,
1 acre = 43,560 ft²,
1 hectare = 2.471 acres
Volume:
1 L = 1000 mL = 1000 cc
Temperature:
\[
°F = \frac{9}{5} \times °C + 32, \quad °C = \frac{5}{9} \times (°F - 32)
\]
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