Elementary arithmetic, with brief notices of its history... by Robert Potts.

RESULTS, HINTS, ETC., FOR THE EXERCISES.
ABSTRACT FRACTIONS.
I.
1. Art. 1. 2. Arts. 1, 9. 3. The improper fractions are given. 4.
12+9+6 12X9X6 1
12+9- 6=4+3+2 =9 = =, 1 and 9= 1 of 216. 5. Art.
3 3 24:
11 and note. 6. Art. 12. 7. Art. 13, note. 8. Art. 15 and note. 9. Art. 10
and note. 10. Arts. 15, 16, and note. 11. Art. 16.
12. The numbers which are aliquot parts of 12 are 1, 2, 3, 4, 6. Thus I is -I of
12, 2 is I of 12, 3 is ~ of 12, 4 is I of 12, 6 is 2 of 12. The other numbers less than
12, namely, 5, 7, 8, 9, 10, 11, can be divided into two or more aliquot parts.
5 4 1 1 1 1 f 9 6 3 1 1 1
Thus, =- + =-+ of. l+ =+ of
12 12 12 3 4 3 12 12 12 2+2 2
7 6 1 1 1 1 10 6 4 1 1
12-12 12 2 6 2 12 12 -12 2+3
8 6 2 1 1 1 11 6 3 2 1 1 1
12 12+12=-23 of 1212 12 12 246
It may also be noted that the aliquot parts of 20 and of 100 are much employed,
in calculations connected with mercantile transactions.
II.
The following are the fractions in the lowest terms found by dividing the
numerator and denominator of each fraction by their greatest common measure
respectively:11 55 4 55 1309 60 2993 23 19 131 276 8 7 13 61 10996,
20 6' 6 5' 8' 36' 77' 7 32' 33 ' 3 9' ' 1' 397' 9' 9' 14' 211' 34661
11761
122387
III.
The following are the fractions as required:1. 5 and 3: 8 and 9 35 20 an 12 168 240 an 175
15 15 12 12 105 105 105 420 420 420
4680 2457 2912 ad 5544 2376 2475 1760 d 2880
65:an and
6552' 6552' 6552 5525' 3960' 3960' 3960 3960
2. and4 8 12 7 and 20 18 3 and 5: 40 75 35 and 51
2. and I -and - and -- - and9 9 18' 18 18 36' ' 36 36 100' 100' 100 100
160 375 175 16 3
1000' 1000' 1000' 1000 1000
3 15 d28 30 24 25 45 42 44 105 98 24 and 56
3. __ a _and and: and: and
100 100 600 600 600 72' 72 72 168' 168' 168 168
120 100 32 210 and 75
240' 240' 240, 240 240
IV.
1.. 2. 2.. 3. 7. 4. 557. 5. 1. 2 6. 7,. 7. 8. 5-. 9.
1. 10. 465T. 11. 1-.. 12. 43-. 13. 305m1. 14. 3536. 15. 33i-l.
7 OVO 42