Fractions – Formula & Equation Sheet
Essential formulas and equations from Ganita Prakash, tailored for Class 6 in Mathematics.
This one-pager compiles key formulas and equations from the Fractions chapter of Ganita Prakash. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formulas
Fraction = Part/Whole
A fraction expresses a part of a whole. 'Part' is represented by the numerator, while 'Whole' is the denominator. For example, in 1/2, 1 is the part and 2 is the whole.
Unit Fraction = 1/n
Unit fractions have a numerator of 1. For example, 1/5 represents one part of five equal parts. This is useful for dividing items equally.
Comparing Fractions: a/b vs c/d
To compare fractions, cross-multiply: a*d vs b*c. The larger result indicates the greater fraction. For example, to compare 1/4 and 1/3, calculate 1*3 vs 4*1.
Adding Fractions: a/b + c/d = (ad + bc) / bd
To add fractions with different denominators, convert them to a common denominator. For example, to add 1/3 and 1/4: (1*4) + (3*1) / (3*4) = 4/12 + 3/12 = 7/12.
Subtracting Fractions: a/b - c/d = (ad - bc) / bd
Similar to addition, subtract by ensuring common denominators. For example, 3/4 - 1/2 = (3*2 - 1*4) / (4*2) = 6/8 - 4/8 = 2/8 = 1/4.
Multiplying Fractions: (a/b) x (c/d) = (ac)/(bd)
Multiply numerators and denominators. For example, (1/2) x (3/4) = 3/8. This is useful in calculating portions or shares.
Dividing Fractions: (a/b) ÷ (c/d) = (a/b) x (d/c)
Change division to multiplication by the reciprocal. For example, (1/2) ÷ (3/4) = (1/2) x (4/3) = 4/6 = 2/3.
Simplifying Fractions: a/b = (a ÷ gcd(a, b)) / (b ÷ gcd(a, b))
Reduce fractions to their simplest form by dividing both numerator and denominator by their greatest common divisor (gcd). For example, 4/8 simplifies to 1/2.
Equivalent Fractions: a/b = (ka)/(kb)
Fractions that represent the same value. For example, 1/2 = 2/4, achieved by multiplying both numerator and denominator by k=2.
Fraction of a Number: (a/b) of n = (a*n)/b
To find a fraction of a quantity, multiply the quantity by the numerator, then divide by the denominator. For example, 1/3 of 9 = (1*9)/3 = 3.
Equations
1/4 + 1/4 = 2/4 = 1/2
Adding two equal fractions results in a sum that can often be simplified. It demonstrates how fractions can combine to form a larger whole.
3/5 - 1/5 = 2/5
Subtracting fractions with the same denominator keeps the denominator constant while subtracting numerators. This model is useful in practical scenarios, like sharing resources.
1/3 * 3/4 = 3/12 = 1/4
This multiplication example illustrates how a fraction of a fraction can yield a smaller portion, relevant for understanding ratios in recipes.
1/2 ÷ 1/4 = 1/2 * 4/1 = 2
Division of fractions shows how many times one fraction fits into another, highlighting relationships in measurement conversions.
2/3 = x/6
This equation can be solved by cross multiplication to find x. Such problems are common in finding equivalent fractions.
5/6 + x/6 = 1
To find x in this equation, subtract 5/6 from 1. It showcases how to work with fractional equations to find unknown quantities.
x/8 = 1/4
To solve, cross-multiply: x = 8/4 = 2. This kind of equation highlights solving for fractions in algebraic contexts.
a/b + b/a = (a² + b²) / ab
This equation shows that two fractions can be added by converting to a common denominator. It is useful in algebraic manipulations.
3/7 > 2/7
This inequality shows comparison of fractions based on their numerators while having the same denominator, useful in ranking quantities.
x/10 = 3/5
Cross multiplication here leads to x = (3*10)/5 = 6. Such equations help practice understanding ratios and proportion.
