Proportional Reasoning-2 – Formula & Equation Sheet
Essential formulas and equations from Ganita Prakash Part II, tailored for Class 8 in Mathematics.
This one-pager compiles key formulas and equations from the Proportional Reasoning-2 chapter of Ganita Prakash Part II. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formulas
a : b = c : d
This notation represents two ratios a to b and c to d. They are said to be proportional if a × d = b × c. Useful for establishing equivalence in ratios.
Cross-Multiplication: a/b = c/d
This equation shows the cross-multiplication method to check the proportion of two ratios. Useful in verifying equivalence quickly.
k = a/b
k is the constant of proportionality. It indicates that quantities a and b maintain the ratio represented by k. Important in direct proportionality problems.
x/y = z/t
This equation implies that x is to y as z is to t. It is used in solving problems involving direct proportions.
Percentage Formula: P = (part/whole) × 100
P is the percentage, 'part' refers to the portion being compared, and 'whole' is the total amount. Essential for finding percentages in various scenarios.
Unit Conversion: 1 km = 1000 m
This conversion is crucial when dealing with distances in different units. Understanding conversions is key in solving measurement problems.
Proportional Increase: New Value = Original Value × (1 + r)
Where r is the rate of increase (in decimal). This formula helps calculate the new value after a proportional increase.
Ratio of Areas: A1/A2 = (L1/L2)²
A1 and A2 are areas of two similar shapes, while L1 and L2 are corresponding lengths. This relationship is vital in geometry.
Ratio of Volumes: V1/V2 = (L1/L2)³
V1 and V2 represent volumes of similar 3D shapes, where L1 and L2 are corresponding lengths. Important in volume comparison.
Simple Interest: SI = (P × R × T)/100
SI is the simple interest, P is the principal amount, R is the rate of interest, and T is the time in years. Used in financial mathematics.
Equations
a/b = c/d
This equation defines the relationship between two ratios a : b and c : d, asserting their proportionality. Useful for resolving ratio problems.
E = mc²
In this context, it represents the conversion of mass to energy. Although primarily used in physics, understanding conversion principles aids in proportional reasoning.
Proportionality Statement: y = kx
y is directly proportional to x, with k as the constant of proportionality. This statement is fundamental in understanding linear relationships.
a + b = c + d
This equation can represent balance in proportional scenarios, where combined quantities remain equal. Helpful in problem-solving involving mixtures.
P1/P2 = Q1/Q2
This represents the proportionality of two sets of parameters (P and Q). Essential for comparative analysis in experiments.
Average = (sum of values) / (number of values)
The formula calculates the average of a data set, which relates to proportional reasoning when considering ratios.
x = (y × d)/b
Solves for x when the ratios depend on y and b. This equation is critical for manipulating variables in proportional relationships.
C = πd
C is the circumference of a circle, d is the diameter, and π is approximately 3.14. This formula shows how circumference scales with size.
v = d/t
This formula defines speed (v) as distance (d) over time (t). It showcases basic proportionality in rates of motion.
R = (f × d) / v
This equation finds the relationship of radius (R), frequency (f), distance (d), and velocity (v). Useful in physics and circular motion.
