Answer to Question #116658 in Algebra for Mannini Lebuso

Question

Use a standard algorithm to calculate 4 + 2/3 - 3/4

Answer 1

When adding fractions with different denominators, find a common denominator.

Then add the numerators

$${a \over b}+{c \over d}={ad+bc \over bd}$$

When adding fractions with different denominators, we don’t usually use this property. Instead we rewrite the fractions so that they have the smallest possible common denominator (often smaller than the product of the denominators). This denominator is the Least Common Denominator (LCD)

Then we use

$${k\over n}+{m \over n}={k+m \over n}$$

$$4+{2 \over 3}-{3 \over 4}$$

You can express an integer as a fraction by simply dividing by 1, or you can express any integer as a fraction by simply choosing a numerator and denominator so that the overall value is equal to the integer.

$$4={4 \over 1}$$

Factoring each denominator into prime factors gives

$$1=1, 3=3, 4=2^2$$

We find the least common denominator (LCD) by forming the product of all the factors that occur in these factorizations, using the highest power of each factor.

Tthe least common denominator is equal to the common denominator

$$4+{2 \over 3}-{3 \over 4}= {4 \over 1}+{2 \over 3}-{3 \over 4}={4\cdot3\cdot4+2\cdot4-3\cdot3 \over 3\cdot 4}=$$

$$={48+8-9 \over 12}={4 7\over 12}$$