Proportional reasoning helps us understand how quantities change in relation to each other. This chapter studies ratios, proportions, and how to use them for solving real-world problems involving scaling, mixing, and sharing.
Observing Similarity in Change
When we resize digital images, some look similar to the original while others appear distorted. Understanding why this happens teaches us about proportional changes.
Ratios
Ratios are mathematical way to express proportional relationships between two quantities.
Understanding Ratios
A ratio like 60:40 means for every 60 units of first quantity, there are 40 units of second quantity.
For Image A:
- Ratio of width to height = 60:40
- Numbers 60 and 40 are called terms of the ratio
General Form:
In ratio a:b, we say for every ‘a’ units of first quantity, there are ‘b’ units of second quantity.
Comparing Ratios
To check if ratios are proportional, we can:
- Multiply both terms by same factor
- See if we get the other ratio
Example:
- Image A ratio: 60:40
- Multiply both terms by 1/2: (60 × 1/2):(40 × 1/2) = 30:20
- This is ratio of Image C
- So ratios are proportional
Ratios in their Simplest Form
We can reduce ratios to simplest form by dividing terms by their HCF (Highest Common Factor).
For Image A (60:40):
- HCF of 60 and 40 is 20
- Dividing both terms by 20: 60÷20 : 40÷20 = 3:2
- Simplest form is 3:2
For Image D (90:60):
- HCF of 90 and 60 is 30
- Dividing both: 90÷30 : 60÷30 = 3:2
- Simplest form is also 3:2
For Image B (40:20):
- HCF is 20
- Simplest form: 2:1 (different from 3:2)
For Image E (60:60):
- HCF is 60
- Simplest form: 1:1 (also different from 3:2)
Notation for Proportion
When two ratios are same in simplest form, we say they are in proportion. We use :: symbol.
Written as: a:b :: c:d means ratios a:b and c:d are proportional
Examples:
- 60:40 :: 30:20 (both simplify to 3:2)
- 60:40 :: 90:60 (both simplify to 3:2)
Problem Solving with Proportional Reasoning
Example 1: Checking if Ratios are Proportional
Question: Are ratios 3:4 and 72:96 proportional?
Solution:
- 3:4 is already in simplest form
- For 72:96, find HCF of 72 and 96 which is 24
- Divide both terms: 72÷24 : 96÷24 = 3:4
- Both ratios have same simplest form
- Therefore, they are proportional
Example 2: Lemonade Problem
Question: Kesang made 6 glasses of lemonade with 10 spoons of sugar. She needs to make 18 more glasses. How many spoons of sugar needed?
Solution:
- For same sweetness, ratio must remain proportional
- Original ratio: 6:10
- New ratio: 18:?
- Setting up proportion: 6:10 :: 18:?
- First term changed from 6 to 18, factor is 18÷6 = 3
- Second term must also change by factor 3
- 10 × 3 = 30
- Answer: She needs 30 spoons of sugar
Example 3: Wall Construction
Question: Nitin built 60 ft wall with 3 cement bags. Hari built 40 ft wall with 2 cement bags. Is Hari’s wall less strong?
Solution:
- Nitin’s ratio (length:cement) = 60:3 = 20:1 in simplest form
- Hari’s ratio = 40:2 = 20:1 in simplest form
- Both ratios are proportional
- Walls are equally strong
- Nitin should not worry
Example 4: Age Ratio Problem
Question: When Neelima was 3 years old, her mother was 10 times her age. What ratio when Neelima is 12?
Solution:
- At age 3, Neelima’s mother was 30 years old
- Ratio: 3:30 = 1:10 in simplest form
- After 9 years, Neelima is 12 and mother is 39
- New ratio: 12:39 = 4:13 in simplest form
- Ratios are not same
Imp Point: When we add (or subtract) same number from ratio terms, the ratio changes and may not be proportional to original.
Example 5: Finding Missing Terms
Question: Fill missing numbers for ratios proportional to 14:21
- _:42
- 6:_
- 2:_
Solution:
For first ratio (_:42):
- Second term is 42, which is 2 times 21
- First term must also be 2 times 14
- 14 × 2 = 28
- Answer: 28:42
For second ratio (6:_):
- First term is 6
- 14 needs to be multiplied by what factor to get 6?
- 14 × y = 6, so y = 6/14 = 3/7
- Second term: 21 × 3/7 = 9
- Answer: 6:9
For third ratio (2:_):
- HCF of 14 and 21 is 7
- Dividing both by 7: 14÷7 = 2 and 21÷7 = 3
- Answer: 2:3
Filter Coffee Problem
Manjunath mixes coffee decoction with milk in different ratios.
Regular coffee: 15 mL decoction + 35 mL milk = 15:35 ratio
Stronger coffee: 20 mL decoction + 30 mL milk = 20:30 ratio
- This is stronger because ratio of coffee to milk is higher
- Simplifying: 20:30 = 2:3 compared to 15:35 = 3:7
- More coffee relative to milk makes it stronger
Lighter coffee: 10 mL decoction + 40 mL milk = 10:40 ratio
- This is lighter because less coffee relative to milk
- Simplifying: 10:40 = 1:4
- Less coffee relative to milk makes it lighter
Trairasika—The Rule of Three
This is ancient Indian method for solving proportion problems where three quantities are known and fourth needs to be found.
Historical Background
Āryabhaṭa (499 CE) and other ancient Indian mathematicians called these “Rule of Three” problems. The three known quantities were:
- Pramāṇa (measure) – represented as ‘a’
- Phala (fruit/result) – represented as ‘b’
- Ichchhā (requisition/desire) – represented as ‘c’
To find ichchhāphala (yield) – represented as ‘d’:
Āryabhaṭa’s Method:
“Multiply phala by ichchhā and divide by pramāṇa”
Formula: ichchhāphala = (phala × ichchhā) / pramāṇa
Cross Multiplication Method
For proportion a:b :: c:d, we have:
- ad = bc (cross multiplication)
- This can be rearranged to find unknown term
To find d: d = (bc)/a
Example 6: Mid-day Meal Problem
Question: For 120 students, cook makes 15 kg rice. On rainy day, only 80 students came. How much rice should cook make?
Solution:
- Setting up proportion: 120:15 :: 80:?
- Factor of change in students: 80/120 = 2/3
- Rice should also change by same factor
- 15 × 2/3 = 10
- Answer: Cook should make 10 kg rice
Example 7: Car Travel Problem
Question: Car travels 90 km in 150 minutes. How far will it go in 4 hours at same speed?
Solution:
- Convert 4 hours to same unit: 4 hours = 240 minutes
- Setting up proportion: 150:90 :: 240:x
- Cross multiplication: 150 × x = 240 × 90
- x = (240 × 90)/150 = 144
- Answer: Car will travel 144 km
Imp Note: Always use same units on both sides of proportion.
Example 8: Tea Price Comparison
Question: Himachal farmer sells 200g tea for ₹200. Meghalaya estate sells 1kg tea for ₹800. Which is more expensive?
Solution:
- Convert to same units: 1 kg = 1000 g
- Himachal ratio: 200:200 = 1:1 in simplest form
- Meghalaya ratio: 1000:800 = 5:4 in simplest form
- Ratios are not proportional
To compare prices:
- Meghalaya: 1 kg costs ₹800
- Himachal: 200g costs ₹200, so 1kg costs ₹1000
- Himachal tea is more expensive
Sharing, but Not Equally
Sometimes we need to divide quantity in specific ratio rather than equally.
Understanding Unequal Division
Example with 12 counters:
If divided equally between 2 people:
- Each gets 6
- Ratio is 6:6 = 1:1
If divided in ratio 3:1:
- Total parts = 3 + 1 = 4
- Size of each part = 12 ÷ 4 = 3
- First person gets: 3 × 3 = 9
- Second person gets: 1 × 3 = 3
General Method for Division in Ratio m:n
To divide quantity x in ratio m:n:
Step 1: Find total parts = m + n
Step 2: Find size of each part = x/(m+n)
Step 3: First part gets = m × [x/(m+n)]
Step 4: Second part gets = n × [x/(m+n)]
Example 9: Profit Sharing
Question: Prashanti invested ₹75,000 and Bhuvan invested ₹25,000 in business. Profit is ₹4,000. How to share profit in ratio of investment?
Solution:
- Investment ratio: 75000:25000 = 3:1 in simplest form
- Total parts = 3 + 1 = 4
- Size of each part = 4000 ÷ 4 = 1000
- Prashanti’s share = 3 × 1000 = ₹3,000
- Bhuvan’s share = 1 × 1000 = ₹1,000
Example 10: Mixture Problem
Question: 40 kg mixture has sand and cement in ratio 3:1. How much cement to add to make ratio 5:2?
Solution:
- Original mixture: total 40 kg in ratio 3:1
- Sand = 3/(3+1) × 40 = 30 kg
- Cement = 1/(3+1) × 40 = 10 kg
- Sand remains 30 kg in new mixture
- New ratio should be 5:2
- If ratio is 5:2, second term is 2/5 of first term
- Cement needed = 2/5 × 30 = 12 kg
- Already have 10 kg, need to add 2 kg more
Unit Conversions
For solving proportion problems, often need to convert units.
Important Conversions
Length:
- 1 metre = 3.281 feet
Area:
- 1 square metre = 10.764 square feet
- 1 acre = 43,560 square feet
- 1 hectare = 10,000 square metres
- 1 hectare = 2.471 acres
Volume:
- 1 millilitre (mL) = 1 cubic centimetre (cc)
- 1 litre = 1,000 mL or 1,000 cc
Temperature:
- Fahrenheit = (9/5) × Celsius + 32
- Celsius = (5/9) × (Fahrenheit – 32)
- Example: 25°C = 77°F
Questions and Answers
Which images look similar and which look different?
- Images A, C, and D look similar because their width and height changed by the same multiplicative factor, maintaining the same ratio in simplest form (3:2)
- Images B and E look different from A, C, and D because their dimensions changed by different factors – Image B has ratio 2:1 and Image E has ratio 1:1, neither matching the 3:2 ratio of similar images
Why does Image B look different even though it’s a rectangle like A?
- Even though Image B is rectangular and both width and height decreased by same amount (20 mm), they didn’t decrease by same factor
- Height of B is half of A’s height, but width of B is not half of A’s width
- Proportional change requires same multiplicative factor, not same additive difference
- This unequal scaling makes Image B appear elongated compared to Image A
What makes images A, C, and D appear similar?
- Their width and height changed by same factor through multiplication
- All three have width:height ratio that simplifies to 3:2
- When dimensions scale proportionally, shape remains same even if size changes
- This is why they look similar despite having different actual measurements
By what factor should we multiply ratio 60:40 to get 90:60?
- Comparing first terms: 90 = 60 × 1.5, so factor is 3/2
- Checking second terms: 60 = 40 × 1.5 = 40 × 3/2
- Both terms multiply by factor 3/2
- This confirms ratios are proportional
Are ratios 3:4 and 72:96 proportional?
- Reduce 72:96 to simplest form by finding HCF which is 24
- Dividing both terms by 24 gives 3:4
- Since both ratios have same simplest form (3:4), they are proportional
- This can also be verified by cross multiplication: 3 × 96 = 288 and 4 × 72 = 288
How can we find factor of change in ratio?
- Divide new value by original value for any one term
- For example, if first term changes from 6 to 18, factor is 18/6 = 3
- Apply same factor to other term to maintain proportion
- This works because proportional ratios have all terms changing by same factor
When does adding same number to both terms not keep ratio proportional?
- Adding or subtracting same number changes the ratio and usually breaks proportionality
- Example: ratio 1:10 becomes 12:39 when we add 11 to each term, simplifying to 4:13 which is different from original 1:10
- Only multiplication or division by same non-zero number maintains proportionality
- This is why age ratios change over time even though same years are added to both ages
Why is Himachal tea more expensive than Meghalaya tea?
- Himachal: 200g costs ₹200, so 1kg (1000g) would cost ₹1000
- Meghalaya: 1kg costs ₹800
- Comparing same quantity (1kg) shows Himachal tea costs ₹200 more
- Reasons could include quality, production methods, transportation, or local market conditions
How to divide 42 counters in ratio 4:3?
- Total parts = 4 + 3 = 7
- Size of each part = 42 ÷ 7 = 6
- First person gets 4 parts = 4 × 6 = 24 counters
- Second person gets 3 parts = 3 × 6 = 18 counters
- Verify: 24 + 18 = 42 and 24:18 = 4:3
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