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Fractions in Mathematics

In mathematics, a fraction represents a part of a whole. It consists of a numerator (the number of equal parts being considered) and a denominator (the total number of parts in the whole). For example, in 3/8, 3 is the numerator and 8 is the denominator.

Visualize a pie with 8 slices. Eating 3 slices represents the fraction 3/8, leaving 5/8 of the pie. Note that a fraction cannot have a denominator of 0, as this makes it undefined.

Adding Fractions

To add fractions, they must share a common denominator. One method is multiplying each fraction by the product of the denominators:

a/b + c/d = (a×d)/(b×d) + (c×b)/(d×b) = (ad + bc)/(bd)

Example:

3/4 + 1/6 = (3×6)/(4×6) + (1×4)/(6×4) = 18/24 + 4/24 = 22/24 = 11/12

Another method is using the least common multiple (LCM) of denominators to simplify the process:

1/4 + 1/6 + 1/2 = 3/12 + 2/12 + 6/12 = 11/12

Subtracting Fractions

Subtraction is similar to addition. Ensure a common denominator first:

a/b - c/d = (a×d)/(b×d) - (c×b)/(d×b) = (ad - bc)/(bd)

Example:

3/4 - 1/6 = (3×6)/(4×6) - (1×4)/(6×4) = 18/24 - 4/24 = 14/24 = 7/12

Multiplying Fractions

To multiply fractions, multiply numerators together and denominators together:

a/b × c/d = (a×c)/(b×d)

Example:

3/4 × 1/6 = 3/24 = 1/8

Dividing Fractions

Divide fractions by multiplying the first fraction by the reciprocal of the second:

a/b ÷ c/d = a/b × d/c = (a×d)/(b×c)

Example:

3/4 ÷ 1/6 = 3/4 × 6/1 = 18/4 = 9/2

Simplifying Fractions

Fractions are easier to work with in simplified form. Divide numerator and denominator by their greatest common factor:

220/440 = 1/2

Converting Fractions and Decimals

To convert a decimal to a fraction, identify the place value of the last digit. Example: 0.1234 = 1234/10000, which simplifies to 617/5000.

To convert fractions to decimals, divide the numerator by the denominator. Example: 1/2 = 0.5, 5/100 = 0.05.

Conclusion

Understanding fractions is fundamental in mathematics. With proper knowledge of addition, subtraction, multiplication, division, simplification, and conversion, fractions can be accurately represented and used in real-world applications.