Proportional Reasoning-1 – Formula & Equation Sheet
Essential formulas and equations from Ganita Prakash Part I, tailored for Class 8 in Mathematics.
This one-pager compiles key formulas and equations from the Proportional Reasoning-1 chapter of Ganita Prakash Part I. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formulas
Proportion: a/b = c/d
Here, a, b, c, and d are quantities in a proportional relationship. This states that the ratio of a to b is equal to the ratio of c to d. It is used in solving problems involving equivalent ratios.
Cross Multiplication: ad = bc
If a/b = c/d, then cross multiplying gives ad = bc. This is useful for finding unknown values in proportional relationships.
Scale Factor: k = new value/original value
k represents the scale factor which indicates how much an image is enlarged or reduced. This is critical when resizing figures while maintaining similarity.
Percentage Change: Percentage Change = ((new value - old value)/old value) × 100
This formula calculates the percentage increase or decrease of a quantity, useful in assessing changes in values over time.
Direct Variation: y = kx
In this relationship, y varies directly with x, where k is the constant of variation. It represents scenarios where one quantity increases, and the other does too.
Inverse Variation: y = k/x
Here, y varies inversely with x, meaning as one increases, the other decreases. It’s essential in cases like speed and time.
Ratio Relationship: a:b = c:d
This expresses that quantities a and b relate similarly to quantities c and d. Used extensively in comparing dimensions.
Fixed Proportion: a/b = k
This indicates that the ratio of a to b is constant (k). It occurs in scenarios where two quantities always maintain the same ratio.
Graph of Proportionality: y = mx
In this equation, m represents the slope, and shows the relationship on a graph where y is proportional to x through a constant rate.
Ratio of Areas of Similar Figures: A1/A2 = (s1/s2)²
A1 and A2 are areas of similar figures, and s1 and s2 are corresponding sides. This formula helps in determining area relationships between similar shapes.
Equations
Height Ratio: H1/H2 = W1/W2
This equation states that the ratio of heights of two similar images equals the ratio of their respective widths, establishing proportionality in dimensions.
Width to Height Ratio: W/H = constant
This indicates that the ratio of width to height remains constant in similar figures, crucial for maintaining the shape during resizing.
Area of Rectangle: A = l × w
A is the area, l is the length, and w is the width. This is fundamental in calculating areas in proportional reasoning.
Volume of Prism: V = A_base × h
V is volume, A_base is the area of the base, and h is the height. It shows how to determine the volume of similar three-dimensional shapes.
Equivalence of Ratios: (a/c) = (b/d)
This states that two ratios a/c and b/d are equal, used in solving proportional equations or problems.
Rate of Change: (y2 - y1)/(x2 - x1)
This represents the rate of change between two points in a graph, helpful in analyzing linear relationships between variables.
Proportional Relationship: y = kx^n (n=1 for direct)
This expression shows how y is dependent on x when proportional relationships are involved, with 'k' being a constant. Applicable in various mathematical models.
Pythagorean Theorem: a² + b² = c²
In right triangles, where c is the hypotenuse, this theorem relates the squares of the sides to find unknown lengths. Useful in geometric applications.
Gradient Formula: m = (y2 - y1)/(x2 - x1)
This formula calculates slope (m) from two points, important for linear relationships in graphical representation.
Distance Formula: d = √((x2 - x1)² + (y2 - y1)²)
This finds the distance between two points in the coordinate plane, vital for problems involving spatial relationships.
