Ratio and Proportion Lecture Notes

This flashcard set covers the fundamental principles, unique terminology, and proprietary problem-solving heuristics for Ratio and Proportion as outlined in the lecture.

Ratio

A mathematical expression of comparison between two quantities that allows for describing relationships without revealing exact values.

Mathematical Equivalence of Ratios

The principle that the ratio $a:b$ is functionally equivalent to the fraction $\frac{a}{b}$; both act as tools for comparison.

Simplification

The process of reducing a ratio to its final form by dividing both terms by their greatest common divisor (e.g., $45:95$ becomes $9:19$).

Rule of Unit Fidelity

The requirement that ratios only be established between quantities of the same unit; disparate units result in a "Cannot be Determined" (CND) status.

Proportion

The comparison between two pairs of ratios that result in the same simplified value (e.g., $2:3$ and $40:60$).

Unique Number (Variable $x$)

A constant used to transition from a simplified ratio back to real-world values, converting the ratio into algebraic terms like $4x$ and $5x$.

Internal Gap Concept

The logic that ratios can only be subtracted directly when the difference (gap) between the terms of each ratio is identical.

Value Distribution Procedure

A methodology where you sum ratio units, divide the total sum by those units to find the value of "one ratio," and then multiply by individual components.

Bahurupiya (Shape-shifter)

A technique for combining ratios like $a:b$ and $b:c$ by identifying the common term ($b$) and equalizing its value in both.

Ulta N (Inverted N)

A visual multiplication path used to calculate $a:b:c$ following the sequence: $a \times b_1 \rightarrow b_1 \times b_2 \rightarrow b_2 \times c$.

Ladhkan (Hanging)

A method for solving equations like $2a = 3b = 4c$ by finding the LCM of coefficients and using the resulting denominators as the ratio.

Hide & Seek (Luka-Chupi)

A proprietary heuristic for equations like $2a = 3b = 4c$ where one variable's value is found by hiding its coefficient and multiplying the remaining coefficients.

Land Grab (Zameen Kabza)

A technique for calculating multi-variable ratios (e.g., $a:b:c:d$) by writing ratios in rows and filling empty spots with the neighboring number.

Sandwich Approach

A method for calculating $a:b:c:d$ that utilizes vertical multiplication for outer terms and specific cross-multiplication for inner terms.

Mahabharata

An optimized approach for finding only the first and last parts of a long ratio chain (e.g., $a:d$) through vertical multiplication of left ("Kaurava") and right ("Pandava") teams.

Jiski Lathi, Uski Bhains

The governing principle for coin problems stating the ratio type must match the given total (use Rupee ratio for total Rupees; use Coin ratio for a total count of coins).

Panda Approach

A universal cross-multiplication method used in income-expenditure problems to find the value of one ratio unit when savings are identical or different.

Succession/Laddering Formula

A method to bridge values from a previous year to the current year using: $(\frac{\text{Previous Value}}{\text{Old Ratio Term}}) \times \text{New Ratio Term}$.

Fractional Ratio Rule

Ratios should never be left in fractional form; they must be multiplied by the LCM of the denominators to convert them to whole numbers.

Inverse Relationship

The principle seen in problems like $2a = 3b$, where the values of the variables are inversely related to their coefficients (i.e., $a:b = 3:2$).