Proportional Reasoning - I

Chapter at a Glance

This chapter introduces the mathematical foundations of ratio and proportion, exploring how quantities scale and relate to one another. Starting with visual similarity in digital image scaling, it transitions into ratios in simplest form and the cross-multiplication test for proportions. The chapter details historical methods like Aryabhata’s Rule of Three (Trairāśika), the algebra of non-equal division sharing, and practical unit conversions (covering length, area, volume, and temperature). It emphasizes distinguishing between directly proportional change, non-proportional additive change, and inversely proportional quantities.

Key Definitions & Terminology

• Ratio: A comparative relationship between two quantities showing how many times one value contains another, written as $a : b$.
• Terms: The individual numbers composing a ratio ($a$ and $b$ are the terms of $a : b$).
• Proportion: An equation stating that two ratios are equivalent, denoted by $a : b :: c : d$ or $\frac{a}{b} = \frac{c}{d}$.
• Simplest Form: A ratio whose terms are reduced by dividing both by their Highest Common Factor (HCF), leaving them coprime.
• Rule of Three (Trairāśika): An ancient Indian mathematical method for finding an unknown fourth proportional value when three related values are known.
• Pramāṇa (Measure): The reference quantity ($a$) in the Rule of Three.
• Phala (Fruit): The result corresponding to the reference quantity ($b$) in the Rule of Three.
• Ichchhā (Requisition): The desired new quantity ($c$) for which the yield is sought in the Rule of Three.
• Ichchhāphala (Yield): The unknown fourth quantity ($d$) to be calculated.
• Cross-Multiplication: The method of checking proportion by multiplying the outer terms (extremes) and inner terms (means) to see if they are equal ($ad = bc$).

Formulas, Rules & Properties

• Proportionality Scaling Condition:
• Two ratios $a : b$ and $c : d$ are proportional if there exists a scaling factor $f$ such that:
  $$c = fa \quad \text{and} \quad d = fb \quad \rightarrow \quad \frac{c}{a} = \frac{d}{b} = f$$
• Cross Multiplication Rule:
• If $a : b :: c : d$, then:
  $$ad = bc \quad \rightarrow \quad d = \frac{bc}{a}$$
• Trairāśika (Rule of Three) Formula:
• $$\text{ichchhāphala} = \frac{\text{phala} \times \text{ichchhā}}{\text{pramāṇa}}$$
• Ratio Partitioning (Sharing) Formula:
• If a total quantity $x$ is shared between two parts in the ratio $m : n$, the shares are:
  $$\text{First Part} = m \times \frac{x}{m + n} \quad \text{and} \quad \text{Second Part} = n \times \frac{x}{m + n}$$
• Unit Conversions:
• Length: $1\text{ metre} = 3.281\text{ feet}$
• Area: $1\text{ sq. metre} = 10.764\text{ sq. feet}$; $1\text{ acre} = 43,560\text{ sq. feet}$; $1\text{ hectare} = 10,000\text{ sq. metres} = 2.471\text{ acres}$
• Volume: $1\text{ millilitre (mL)} = 1\text{ cubic centimetre (cc)}$; $1\text{ litre} = 1,000\text{ mL} = 1,000\text{ cc}$
• Temperature:
  $$F = \frac{9}{5}C + 32 \quad \text{and} \quad C = \frac{5}{9}(F - 32)$$

Core Concepts & Topics

• Direct vs. Non-Proportional Additive Change:
• Direct scaling is multiplicative. Adding or subtracting the same constant to the terms of a ratio does not preserve proportion. For example, if a child is $3$ and her mother is $30$, the ratio is $1:10$. In $9$ years, their ages are $12$ and $39$, altering the simplest form ratio to $4:13$.
• Limits of the Rule of Three:
• Situations where a variable decreases as another increases (inverse relationships like speed vs. travel time) cannot be solved using direct Trairāśika. If speed increases, travel time decreases, meaning $50\text{ km/h} : 2\text{ hours} \neq 75\text{ km/h} : x\text{ hours}$.
• Image Scaling Proportionality:
• An image maintains its aspect ratio and shape without distortion if both its width and height are scaled by the exact same multiplier.

Worked Examples

• Lemonade Sweetness Scaling (Page 222 Example 2):
• Problem: Kesang added $10$ spoons of sugar to make $6$ glasses of lemonade. To make $18$ more glasses of lemonade with the same sweetness, how many spoons of sugar must she add?
• Solution: Set up the proportion:
  $$6 : 10 :: 18 : x \rightarrow \frac{18}{6} = 3 \text{ (scaling factor)}$$
  $$x = 10 \times 3 = 30 \text{ spoons}$$
• Earth Distance Travel (Figure it Out Q1, Page 229):
• Problem: The Earth travels $940$ million km around the Sun in a year ($52$ weeks). How far does it travel in one week?
• Solution:
  • Let $x$ be the distance traveled in $1$ week.
  • Set up the proportion:
    $$52\text{ weeks} : 940,000,000\text{ km} :: 1\text{ week} : x\text{ km}$$
    $$x = \frac{940,000,000 \times 1}{52} \approx 18,076,923\text{ km}$$
• Mason Brick Calculation (Figure it Out Q2, Page 229):
• Problem: A mason requires $1450$ bricks to build a $10$-foot wall section. How many bricks does he need to build a house layout with walls of total length $108\text{ ft}$? (Assume wall height and thickness are constant).
• Solution: Set up the proportion:
  $$10 : 1450 :: 108 : x \rightarrow x = \frac{1450 \times 108}{10} = 15,660\text{ bricks}$$
• Sharing Investments in Ratio (Example 11, Page 233):
• Problem: Prashanti invested ₹$75,000$ and Bhuvan invested ₹$25,000$. They make a monthly profit of ₹$4,000$ and share it in the ratio of their investments. Find their shares.
• Solution:
  • Ratio of investment $= 75000 : 25000 = 3 : 1$.
  • Sum of parts $= 3 + 1 = 4$.
  • Size of each part $= 4000 \div 4 = 1000$.
  • Prashanti's share $= 3 \times 1000 = $ ₹$3,000$.
  • Bhuvan's share $= 1 \times 1000 = $ ₹$1,000$.
• Līlāvatī Saffron Problem (Figure it Out Q5, Page 235):
• Problem: If $2\frac{1}{2}$ palas of saffron cost $\frac{3}{7}$ niskas, what quantity of saffron can be bought for $9$ niskas?
• Solution:
  • Let saffron quantity bought for $9$ niskas be $x$ palas.
  • Set up Trairāśika:
    $$\frac{3}{7} : 2.5 :: 9 : x \rightarrow \frac{3}{7}x = 2.5 \times 9 = 22.5$$
    $$x = 22.5 \times \frac{7}{3} = 7.5 \times 7 = 52.5\text{ palas}$$
• Cow Manure Field Calculation (Figure it Out Q8, Page 236):
• Problem: Good farming practice requires applying $10$ tonnes of cow manure per $1$ acre. A farmer has a tomato plot of size $200\text{ ft}$ by $500\text{ ft}$. How much manure (in kg) does he need?
• Solution:
  • Area of plot $= 200 \times 500 = 100,000\text{ sq. ft}$.
  • $1\text{ acre} = 43,560\text{ sq. ft}$.
  • $10\text{ tonnes} = 10,000\text{ kg}$.
  • Set up proportion:
    $$43,560 : 10,000 :: 100,000 : x \rightarrow x = \frac{100,000 \times 10,000}{43,560} \approx 22,956.8\text{ kg}$$
• Cupro-Nickel Coin Alloy Cost (Figure it Out Q12, Page 236):
• Problem: A ₹$10$ coin of mass $7.74\text{ g}$ is a cupro-nickel alloy of copper and nickel in a $3 : 1$ ratio. If copper costs ₹$906/\text{kg}$ and nickel costs ₹$1,341/\text{kg}$, what is the cost of the metals in the coin?
• Solution:
  • Divide mass in ratio $3 : 1$:
  • Mass of copper $= \frac{3}{4} \times 7.74 = 5.805\text{ g} = 0.005805\text{ kg}$.
  • Mass of nickel $= \frac{1}{4} \times 7.74 = 1.935\text{ g} = 0.001935\text{ kg}$.
  • Calculate costs:
  • Cost of copper $= 0.005805 \times 906 = $ ₹$5.26$.
  • Cost of nickel $= 0.001935 \times 1341 = $ ₹$2.59$.
  • Total cost of metals $= 5.26 + 2.59 = $ ₹$7.85$.

Practical Activities & Experiments

• lemonade Taste Test: Prepare three cups of lemonade: (1) mixing $10\text{ mL}$ lemon juice with $20\text{ mL}$ water ($1:2$), (2) mixing $15\text{ mL}$ lemon juice with $30\text{ mL}$ water ($1:2$), and (3) mixing $10\text{ mL}$ lemon juice with $30\text{ mL}$ water ($1:3$). Have volunteers taste them to confirm that cups (1) and (2) taste identical (proportional sweetness), while cup (3) tastes significantly lighter (non-proportional).
• Classroom Map Scale Drawing: Measure the width and height of the classroom whiteboard. Scale down these dimensions using a factor of $\frac{1}{10}$ (dividing both by $10$) and draw the rectangle in your notebook. Measure and verify that the ratio of the notebook drawing's dimensions matches the whiteboard's original ratio.