| (111) | a | and | c | are the same as | ad | and | bc |
| b | d | bd | bd |
| (112) | a | + | b | = | a + b |
| c | c | c | |||
| a | - | b | = | a - b | |
| c | c | c | |||
| (113) | a | + | c | = | ad + bc |
| b | d | bd | |||
| a | - | c | = | ad - bc | |
| b | d | bd | |||
| (118) | a | × | c | = | ac |
| b | d | bd | |||
| (121) | a | divᵈ. by | c | or | a/b | = | ad |
| b | d | c/d | bc |
125. These results are true even when the letters themselves represent fractions. For example, take the fraction
- a/b
- c/d
whose numerator and denominator are fractional, and multiply its numerator and denominator by the fraction
| e | , which gives | ae/bf |
| f | ce/df | |
| which (121) is | aedf | |
| bfce | ||
which, dividing the numerator and denominator by ef (108), is
- ad
- bc
But the original fraction itself is