110. When the terms of the fraction given are already in factors,[15] any one factor in the numerator may be divided by a number, provided some one factor in the denominator is divided by the same. This follows from (88) and (108). In the following examples the figures altered by division are accented. [15] A factor of a number is a number which divides it without remainder: thus, 4, 6, 8, are factors of 24, and 6 × 4, 8 × 3, 2 × 2 × 2 × 3, are several ways of decomposing 24 into factors. 12 × 11 × 10 3′ × 11 × 10 1′ × 11 × 5′ ------------ = ------------- = ------------- = 55. 2 × 3 × 4 2 × 3 × 1′ 1′ × 1′ × 1′ 18 × 15 × 13 2′ × 3′ × 1′ 1′ × 1′ × 1′ ------------ = ------------- = ------------- = ¹/₁₆. 20 × 54 × 52 4′ × 6′ × 4′ 2′ × 2′ × 4′ 27 × 28 3′ × 4′ 3′ × 2′ ------- = --------- = -------- = ⁶/₅. 9 × 70 1′ × 10′ 1′ × 5′ 111. As we can, by (108), multiply the numerator and denominator of a fraction by any number, without altering its value, we can now readily reduce two fractions to two others, which shall have the same value as the first two, and which shall have the same denominator. Take, for example, ⅔ and ⁴/₇; multiply both terms of ⅔ by 7, and both terms of ⁴/₇ by 3. It then appears that 2 × 7 ⅔ is ------- or ¹⁴/₂₁ 3 × 7 4 × 3 ⁴/₇ is ------- or ¹²/₂₁. 7 × 3 Here are then two fractions ¹⁴/₂₁ and ¹²/₂₁, equal to ⅔ and ⁴/₇, and having the same denominator, 21; in this case, ⅔ and ⁴/₇ are said to be _reduced to a common denominator_. It is required to reduce ⅒, ⅚, and ⁷/₉ to a common denominator. Multiply both terms of the first by the product of 6 and 9; of the second by the product of 10 and 9; and of the third by the product of 10 and 6. Then it appears (108) that 1 × 6 × 9 ⅒ is ----------- or ⁵⁴/₅₄₀ 10 × 6 × 9 5 × 10 × 9 ⅚ is ------------ or ⁴⁵⁰/₅₄₀ 6 × 10 × 9 7 × 10 × 6 ⁷/₉ is ------------ or ⁴²⁰/₅₄₀. 9 × 10 × 6