Fractions: Formulas, Rules and Solved Questions
Fractions represent parts of a whole or the ratio of two numbers. This page explains proper, improper and mixed fractions, equivalent fractions, simplification, comparison, and operations such as addition, subtraction, multiplication and division. It also includes fraction formulas, shortcuts and solved examples for quantitative aptitude.
What Are Fractions?
Fractions with the same value are called equivalent fractions. Multiplying or dividing both numerator and denominator by the same non-zero number does not change the value. For example, 3/4 = 6/8 = 9/12. A fraction is in lowest terms when the numerator and denominator have no common factor other than 1.
Fractions Formula & Tricks
Important Formulas
Multiply each numerator by the other denominator, combine the results, and retain the product of the denominators. Reduce the answer if possible.
Multiply the numerators and denominators separately. Cancel common factors before multiplying when possible.
To divide by a fraction, multiply by its reciprocal.
Multiply the quantity by the fraction. For example, 3/5 of 40 = 3 × 40/5 = 24.
Multiply the whole number by the denominator, add the numerator, and place the result over the original denominator.
Quick Tricks
For addition or subtraction, take the least common multiple of the denominators as the common denominator. This usually gives smaller intermediate values than multiplying the denominators directly.
Cancel common factors between a numerator and a denominator before multiplying. The value remains unchanged and the calculation becomes shorter.
For positive fractions a/b and c/d, compare ad and bc. If ad > bc, then a/b > c/d; if ad < bc, then a/b < c/d.
Fractions Concepts
Types of Fractions
Examples are 3/8 for a proper fraction, 11/7 for an improper fraction, and 2 3/5 for a mixed fraction. An improper fraction greater than 1 can be converted into a mixed fraction by dividing its numerator by its denominator.
Equivalent Fractions and Simplification
To reduce a/b, divide a and b by GCD(a, b). For example, the GCD of 24 and 36 is 12, so 24/36 = 2/3. Multiplying both terms by the same non-zero number creates an equivalent fraction.
Addition and Subtraction of Fractions
For like denominators, a/c ± b/c = (a ± b)/c. For unlike denominators, use the LCM or the formula (ad ± bc)/bd, then reduce the result.
Multiplication and Division of Fractions
The reciprocal of a/b is b/a when a ≠ 0. In multiplication, common factors can be cancelled before multiplying. A denominator in the original fraction or divisor cannot be zero.
Comparing Fractions
For positive fractions a/b and c/d, compare ad and bc. If denominators are equal, the fraction with the larger numerator is greater. If numerators are equal and denominators are positive, the fraction with the smaller denominator is greater.
Fractions Video Lessons
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Fraction Operations: Add, Subtract, Multiply, Divide
Learn how to add, subtract, multiply, and divide fractions, including the core steps needed to simplify and solve fraction-based quantitative aptitude problems.
Practice Fractions Questions
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Fractions Revision Points
Use these rules for quick revision of fraction calculations.
- A fraction is a/b with b ≠ 0; a is the numerator and b is the denominator.
- For addition or subtraction, use a common denominator: a/b ± c/d = (ad ± bc)/bd.
- Multiply fractions directly: a/b × c/d = ac/bd.
- Divide by multiplying by the reciprocal: a/b ÷ c/d = ad/bc.
- Simplify a fraction by dividing its numerator and denominator by their GCD.
- Convert n a/b to an improper fraction using (nb + a)/b.
- For positive fractions, compare a/b and c/d by comparing ad and bc.
- Always check the sign, denominator, and whether the final fraction can be reduced.
Fractions FAQs
How do you add fractions with different denominators?
Take the LCM of the denominators, convert each fraction to that denominator, add the numerators, and simplify. For example, 1/6 + 3/8 = 4/24 + 9/24 = 13/24.
What is the reciprocal of a fraction?
The reciprocal of a/b is b/a, provided a ≠ 0. For example, the reciprocal of 5/9 is 9/5.
How do you convert an improper fraction into a mixed fraction?
Divide the numerator by the denominator. The quotient is the whole number and the remainder becomes the numerator over the original denominator. Thus, 23/6 = 3 5/6.
How do you find a fraction of a number?
Multiply the number by the fraction. For example, 7/12 of 48 = 7 × 48/12 = 28.
Which fraction is larger: 5/6 or 7/9?
Cross multiplication gives 5 × 9 = 45 and 7 × 6 = 42. Since 45 > 42, 5/6 is larger than 7/9.
Can numerator and denominator be divided separately while simplifying a fraction?
No. Both numerator and denominator must be divided by the same non-zero factor. For example, 20/32 becomes 5/8 after dividing both terms by 4.