1.3.9 a × a/b Trick

The following is when you are multiplying an integer times a fraction in the form a×aba \times \frac{a}{b}. The derivation of the trick is not of importance, only the result is:

a×ab=[a+(a−b)]+(a−b)2ba \times \frac{a}{b} = [a + (a - b)] + \frac{(a - b)^2}{b}

Let’s look at a couple of examples:

11×1113=11+(11−13)+(11−13)21311 \times \frac{11}{13} = 11 + (11 - 13) + \frac{(11 - 13)^2}{13} =11−2+413=9413= 11 - 2 + \frac{4}{13} = 9 \frac{4}{13}

It also works for multiplying by fractions larger than 1:

13×1312=13+(13−12)+(13−12)21213 \times \frac{13}{12} = 13 + (13 - 12) + \frac{(13 - 12)^2}{12} =13+1+112=14112= 13 + 1 + \frac{1}{12} = 14 \frac{1}{12}

As you can see, when you are multiplying by a fraction less than 1 you will be subtracting the difference between the numerator and denominator while when you are multiplying by a fraction greater than 1 you will be adding the difference.

It should be noted that there are exceptions (usually on the fourth column) where applying this trick is relatively difficult and it is much easier to just convert to improper fractions then subtract. An example of this is:

7×715−7=(7−8)+8215−7=−8+64157 \times \frac{7}{15} - 7 = (7 - 8) + \frac{8^2}{15} - 7 = -8 + \frac{64}{15} =−8+4+415=−31115= -8 + 4 + \frac{4}{15} = -3 \frac{11}{15}

The above expression was relatively difficult to compute, however if we convert to improper fractions:

7×715−7=7⋅715−7⋅15157 \times \frac{7}{15} - 7 = \frac{7 \cdot 7}{15} - \frac{7 \cdot 15}{15} =7⋅(7−15)15=−5615=−31115= \frac{7 \cdot (7 - 15)}{15} = \frac{-56}{15} = -3 \frac{11}{15}

This method is a lot less cumbersome and gets the answer relatively swiftly. However, it should be noted that the majority of times the trick is applicable and should definitely be used.

Problem Set 1.3.9

11×111411 \times \frac{11}{14}
22×222522 \times \frac{22}{25}
19×192319 \times \frac{19}{23}
27×273227 \times \frac{27}{32}
16×161916 \times \frac{16}{19}
29×293429 \times \frac{29}{34}
31×313431 \times \frac{31}{34}
14×1417−314 \times \frac{14}{17} - 3
11×1114+311 \times \frac{11}{14} + 3
13×1316+1313 \times \frac{13}{16} + 13
13×1317+413 \times \frac{13}{17} + 4
13×1314−1313 \times \frac{13}{14} - 13
17×1718−1717 \times \frac{17}{18} - 17
22×2225−2222 \times \frac{22}{25} - 22
14×1417−1414 \times \frac{14}{17} - 14
17×1172117 \times 1 \frac{17}{21}
13×1316−1313 \times \frac{13}{16} - 13
11×1112−1111 \times \frac{11}{12} - 11
7×715−77 \times \frac{7}{15} - 7
14×1417−1414 \times \frac{14}{17} - 14
15×1517−1515 \times \frac{15}{17} - 15
35×1353835 \times 1 \frac{35}{38}
13×1315−1313 \times \frac{13}{15} - 13