A rational number (or fraction) is represented as a ratio between two integers, a and b, and has the form \({a \over b}\) where a is the numerator and b is the denominator. An improper fraction (\({5 \over 3} \)) has a numerator with a greater absolute value than the denominator and can be converted into a mixed number (\(1 {2 \over 3} \)) which has a whole number part and a fractional part.

The absolute value is the positive magnitude of a particular number or variable and is indicated by two vertical lines: \(\left|-5\right| = 5\). In the case of a variable absolute value (\(\left|a\right| = 5\)) the value of a can be either positive or negative (a = -5 or a = 5).

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80] and the first few multiples of 10 are [10, 20, 30, 40, 50, 60, 70, 80, 90]. The first few multiples they share are [40, 80] making 40 the smallest multiple 8 and 10 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{2 x 5}{8 x 5} \) + \( \frac{7 x 4}{10 x 4} \)

\( \frac{10}{40} \) + \( \frac{28}{40} \)

Now, because the fractions share a common denominator, you can add them:

\( \frac{10 + 28}{40} \) = \( \frac{38}{40} \) = \(\frac{19}{20}\)

The least common multiple (LCM) is the smallest positive integer that is a multiple of two or more integers.

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{1}{5} \) x \( \frac{1}{7} \) = \( \frac{1 x 1}{5 x 7} \) = \( \frac{1}{35} \) = \(\frac{1}{35}\)