Speed Mathematics Simplified (58 page)

The simplified way is first to add the like fractions. 1/5 and 2/5 are in the same terms, so they total 3/5. 2/7 and 4/7 are in the same terms, so they total 6/7. Now merely add 3/5 and 6/7 as you have done before, and get 51/35.

Try it yourself:

The first and last fractions are both in ninths, so simply add the tops: 6/9. The second and third fractions are both in terms of fifths, so add the tops and get 3/5. The sum of 6/9 and 3/5 is 57/45, which reduces to 19/15.

Reducing As You Go

You can save work by reducing fractions as you go. The 6/9 in the last example can be reduced at sight to 2/5. This gives the final answer, 19/15, directly. It is obviously easier to reduce 6/9 to 2/3 than to note that 57/45 is also divisible, top and bottom, by 3.

It is good practice, then, to reduce each fraction in your problem, or any of the intermediate working figures, to its simplest form before continuing.

Another form of reduction-as-you-go is to avoid multiplying all the bottoms, when you can. Suppose, for instance, you start out to add this problem:

You will get the right answer if you follow the general rule: 4 × 8 is 32, for the bottom of the answer; 3 × 8 is 24, and 5 × 4 is 20, totaling 44 for the top: 44/32. This reduces to 11/8.

Note, however, as you look at the bottoms, that 4 goes into 8 exactly twice. If we simply double ¾ to 6/8, the fraction is in the same terms (eighths) as the other. We can then add the tops and find the answer, 11/8. This is easier.

The intermediate step here, 6/8, is not the simplest expression of the quantity; 3/4 is. Yet because it puts the quantity into the same numerical system as the other, 6/8 is the simplest expression for this problem.

The same lesson applies when you add a series of fractions. The short cut is first to add all like fractions, then add the results. If inspection shows you as you start multiplying the bottoms that your product so far is identical with (or divisible by) one or more of the other bottoms, stop right there and add the fractions so far before continuing.

Here is an example:

Using the general rule, you start to find the bottom of the answer by multiplying the bottoms. 2 × 3 is 6, times…the next number is identical.

This means that the first two fractions can be expressed in sixths. The last fraction is already in sixths. So—this is non-standard, but a definite short cut—first add the two fractions on the left, then add their total to the last fraction. Work it out both ways, if you wish, and see that your final answer is 12/6 either way. This is a very “improper” fraction indeed, but we will get through subtraction before taking up that subject.

Rather than seeking lowest common denominators, then, start the addition of any series of fractions by multiplying the
smallest
bottoms first. Often, your running product will be identical with larger bottoms when you get to them, or evenly divisible into them or by them. In this case, add up the fractions so far and then add this sum to the others. It is an easier approach.

Your number sense is the best guide to partial addition before completing a problem. If you start to add three fractions with bottoms of 12, 3, and 4, you will note that 3 × 4 is 12. So first adjust and add these two fractions, then add the sum to the other…which is already in twelfths.

Subtracting Fractions

If you are completely and confidently at home with adding fractions, subtraction poses no problems. The rules are all identical in reverse. Instead of adding the adjusted tops, you subtract the top of the smaller fraction from the top of the larger fraction. (Larger and smaller applies not to the individual tops or bottoms, of course, but describes which fraction is subtracted from which. It is easier than “minuend and subtrahend.”)

One example should make this clear:

Start just the way you would in adding. Multiply the two bottoms to find the bottom of the answer. 4 × 9 is 36.

Now, however, you clearly separate one top from the other top, because it makes a great deal of difference in subtraction although none in addition. The top of the larger fraction is 3. Multiplying this by the bottom of the other, we have 27. The top of the smaller fraction is 2. 2 × 4 is 8. 8 from 27 is (complement and slash) 19. The answer is 19/36.

Do one yourself:

Use your pad to finish this before going on.

The answer, of course, is 1/20.

Improper Fractions

In many of the examples, we have produced answers such as 9/4 or 53/40. If the top of a fraction is larger than its bottom, then the quantity expressed by the fraction is larger than 1. A fraction expressing a quantity larger than 1 is called improper because the quantity is really a whole number plus a fraction.

Nevertheless, we often deal with “improper” fractions, because these are frequently the most convenient ways of expressing the quantities we are handling.

The method for translating an improper fraction into proper form is simple. You merely divide the top by the bottom. A fraction, you recall, is merely a special way of writing a division problem anyway.

In most cases, the answer to the division of an improper fraction will be a number and a remainder. This remainder will be in the same terms as the improper fraction you started with, so it merely becomes the top of the new fraction. The answer to the division becomes the whole number.

Let us try translating the two improper fractions mentioned above into mixed numbers. The two fractions are 9/4 and 53/40:

Most improper fractions turn out to be 1 plus a fraction when translated to mixed numbers. If you see by inspection that the top of the improper fraction is less that
twice
the bottom, you do not even have to divide to translate it. Merely put down a 1 for the whole number, and subtract the bottom from the top to produce the top of your fractional part. 17/12, by this short cut, is 1 plus 5/12—the 5 being produced by subtracting 12 from 17. The reason for this is obvious, since 12/12 is equal to 1.

Translate these improper fractions to proper mixed numbers:

Cover the answers with your pad, please.

The proper equivalents for the above fractions are

Mixed Numbers

The “proper” form of many quantities equal to more than 1, but not any even whole number, is expressed in the number-plus-fraction form you just created from improper fractions. Often, you must calculate with such mixed numbers.