Speed Mathematics Simplified (59 page)

In adding or subtracting, you simply handle the two parts of each number separately. If the fractional parts give you an improper fraction at the end, translate it. Then add the entire result to the whole part of the answer. Watch:

Handle this as two separate additions. 2 + 3 is 5. 5/8 + 7/8 is 12/8. First, reduce this to 3/2. Now translate it to 1½. 5 + 1½ is 6½—the final answer.

The principle does not change when unlike fractions are involved:

First we add 6 + 7 to get 13. Our technique for adding 2/3 + 5/7 gives is 29/21, which translates into 1 8/21. Add this to 13 for the final answer, 14 8/21.

Subtraction is handled in the same way:

First, subtract the whole numbers. 3 – 2 is 1. Now use the standard subtraction method on the fractions to get 7/20. The answer is 1 7/20.

Sometimes, however, we must subtract mixed numbers in which the fraction in the smaller number is larger than the fraction in the larger number. In this case, we make an improper fraction by “borrowing” 1 from the whole part of the larger number. This is just the reverse of translating an improper fraction to a mixed number.

The technique is very simple. After you “borrow” 1, reducing the value of the whole number by 1, you make an improper fraction by merely adding the
bottom and top
of the fraction to make the new top. I have never seen it described like this, but it works like magic. 1 3/8 becomes 11/8—because 3 + 8 total 11. So 6 3/8 becomes 5 11/8 after you “borrow” 1 from the 6.

Here is a case in which this technique is required:

You cannot subtract the fractions, because you cannot subtract 7 from 3.
Nor
can you use a complement, because the base here is 16, not 10. The solution is to “borrow” 1 from the 4 and translate 3/16 to 19/16 by adding top and bottom for the top of the improper fraction. Now the problem looks like this:

The answer is natural now. It is 1 12/16, and the fractional part quickly reduces to ¾. Final answer, 1¾.

The principle does not change when the fractional part of the problem is in unlike fractions. You still raise the fraction in the larger number by borrowing, if you need to, and then subtract, using the general technique for subtracting:

First borrow 1 from 17 and raise the fraction so you have 16 4/3. 8 from 16 leaves 8. 7/8 from 4/3 leaves 11/24. Answer, 8 11/24.

Multiplying and Dividing

When you come to multiplying and dividing mixed numbers, however, the situation is quite different. This is because multiplying or dividing affects
every
part of every number. If we multiply 14 6/7 × ¾, for instance, we must “count”
both
the 14 and the 6/7 exactly ¾ of one time.

The easiest general rule is to turn every mixed number into an improper fraction when you must multiply or divide. Since you are usually “borrowing” far more than 1 from the whole number—you “borrow” the entire number—you do not just add the top and bottom of the fraction for the new fraction. The rule, however, is not much more complicated:

To turn any mixed number into an improper fraction, multiply the whole number by the bottom of the fraction, add the top of the fraction, and put the result over the bottom.

Turn 7 3/8 into an improper fraction by this rule. First, multiply the (whole) 7 by the (bottom) 8: 56. Second, add the (top) 3: 59. Put this result over the bottom: 59/8.

If you try translating 59/8 back into “proper” form, you will find that it does come out to 7 3/8.

Follow this multiplication:

Both numbers must first be turned into improper fractions. 7 2/3 becomes 23/3 (7 × 3, plus 2). 3¾ becomes 15/4 (3 × 4, plus 3).

Pause for a moment to see if the problem can be reduced in any way before continuing:

Note that one top and one bottom are both divisible by 3. Divide both by 3 before going on:

Now multiply the top by the top, and the bottom by the bottom, as you always do in multiplying fractions. The result is:

Remember the general rule for translating improper fractions into mixed numbers: divide the top by the bottom. The answer is the whole number, and the remainder is the top of the fractional part. This answer translates to 28¾.

Cover the answer below with your pad while you exercise the technique on this problem: