Chapter 18 - Mastering Fractions

Of all the specialized branches of mathematics, fractions seem to be greeted with more general panic than all the others put together.

It does not have to be so. Fractions are really not much more complicated than multiplying or dividing. Perhaps the reason for their general unpopularity is that they are taught, to an even greater extent than is true for the other processes, almost entirely by rote. The rote itself simply has to have a few more steps and rules than do whole numbers.

You can add any two whole numbers together without doing anything to them first. But not fractions. The reason why this is so has apparently escaped the normal teaching methods. Many people have trouble understanding why you can multiply two fractions together and get an answer smaller than either of them. If you multiply two numbers together, isn't the answer larger than either? Again—not with fractions.

Both peculiarities, along with the other peculiarities, are inherent in the true nature of fractions. Let us approach their nature with some general observations.

A fraction is, in essence, a number that cannot be expressed normally in our decimal system of digits running from 1 to 9 and then starting over. It is usually smaller than 1, and our counting system has no way of expressing such a quantity other than the apparently awkward form of the fraction (other than a decimal, which is merely a fraction written in another way).

A fraction, even if we have no other way to indicate it than a fraction, is however a very real number or quantity. The form in which we show it is really a fabulously ingenious and useful method of expressing any conceivable quantity from any conceivable counting base in terms of the number system we know.

Imagine, if you will, that our base quantity "1" is a loaf of bread. We have built up a complete arithmetic based on loaves of bread; we have units of ten loaves; we have learned by heart how to add 3 loaves to 6 loaves, to start with 8 loaves and take away 4 loaves, to imagine that one group of 2 loaves has been doubled, or multiplied by 2. But then, suddenly, one of our loaves breaks into pieces and we must account for the pieces.
This is a fraction. The loaf may have broken into "3" pieces, but we have no arithmetic with which to handle it. The only units we know are in terms of loaves of bread. Yet this "1"—this loaf—is no longer 1. It is less than 1.

How do we express the quantity represented by each of these pieces? Some genius or geniuses, centuries ago, suggested that we represent it by "1"—because it once was 1 loaf— divided by "3"—as if each of the pieces were now a loaf. The 3 came from 1, so the essential quantity is the one expressed by a division of 3 into 1.

We write it ⅓.

This is the basic fact about all fractions. They are real quantities, but quantities that cannot be expressed in our regular number system, so we express them in terms of divisions.

A fraction is, then, merely a division problem.

When we write the quantity 2/5, we really intend to convey the idea of a quantity that is outside our number system, and can best be expressed by dividing 2 by 5, or 2 ÷ 5, or 5 pi. Because we wish to show it as a quantity more than a problem, we write it 2/5.

Thinking of a fraction as really a problem in division, which also expresses a specific quantity, may help you to gain an emotional grasp of the entire system.

Why 2 x 2 Is "Less" Than 2

One of the most baffling habits of fractions is that when you multiply two of them together, your answer is less than either of them alone. We are so accustomed to thinking of multiplication as an increasing process that this jars our basic number sense.

If you think of multiplying as counting a number a certain number of times—which is precisely what it is—the concept becomes clearer. If you count a number more than once, then the result is obviously larger than the number was. But if you count the number less than once, as you do when you count it only ⅓ times, for instance, then the answer must be smaller than the number was when you started. If the number you counted was less than 1 to start with, such as ¼, then the answer will obviously also be smaller than the number of times you counted it—because to get an answer as large as your "counting" number you would have to count another number at least as large as 1.

This is why multiplying by two fractions smaller than 1 gives you an answer smaller than either of the fractions. You count a number that is smaller than 1 to begin with, and you don't even count it one whole time. When you multiply ¼ x ½, you are saying in effect "count ½ exactly ¼ times."

This fact leads us into the first natural rule for handling fractions with understanding as well as memorized rules: to multiply fractions, multiply the top numbers together for the top of the answer, and multiply the bottom numbers together for the bottom of the answer.

Note that we define this rule in terms of top numbers and bottom numbers. Arithmetic has become topheavy with special names such as "numerator" and "denominator" that confuse things more than they clarify them for most of us. If you agree, "top" and "bottom" is instantly and unmistakably clear.

Following this rule, then, count 3/5 exactly ¼ times—or, if it sounds clearer, ¼ of one time:

The top of our answer is 3, which is what we get when we multiply 3X1. The bottom is 20, which is produced by multiplying 5x4. This is what is produced when you start with a quantity expressed by dividing 5 into 3 (3/5) and count it not even once, but a number of times expressed by the division of 4 into 1 (¼).

In order to refresh your memory, try it yourself:

This is simple, naturally, but if you are at all rusty it would help to cover up the answer with your pad and write down the answer.

Multiplying the top numbers, we get 6. Multiplying the bottom numbers, we get 28. The answer is 6/28.

This is true, but 6/28 is a fairly complex fraction. Is there a simpler expression of the same quantity? 6 and 28 are both evenly divisible by 2. If we divide both the top and bottom by 2, our fraction becomes 3/14.

Think about this fact for a bit. Your memory of the rules undoubtedly tells you that it is the same, but visualize the two expressions and see if you can feel their identity.

This leads us to a general rule for all fractions:

If you multiply or divide both the top and bottom numbers of a fraction by the same number, the quantity remains unchanged.
By this rule, 6/8 is the same quantity as ¾. Is it? You know by training that it is. But can you feel it? As a good exercise in number sense, try expressing this quantity by 6 dots above a line with 8 dots below it. Thoughtfully connect each two adjacent dots so they become 1 line, in pairs, and note that you now have 3 lines over 4 lines. You have not changed the relationship of the quantities above and below the line, but you have changed the numbers.

Try a few multiplication exercises on your pad:

Work these out. They are elementary, but important.

The raw answers are, of course, 6/72, 5/60, ⅜, and ⅛. We say "raw" because some of these can be reduced to simpler terms. 6/72, for instance, is reduceable at sight to 3/36, and this in turn is clearly 1/12. Check your other answers for reduction possibilities.

Short-Cut Multiplying

If any fraction whose top and bottom numbers can be evenly divided by the same number can be reduced to a simpler form by dividing, then two fractions to be multiplied can also go through the same process even before they are multiplied.

This means that often you can do part of the reducing before you multiply, rather than after.

The secret that makes this possible is that it does not make a bit of difference in what order you multiply or divide numbers: the result will be the same. 4 x 8 x 6 is the same as 8 x 6 x 4 is the same as 6 x 4 x 8—as well as 8 x 4 x 6 and 6 x 8 x 4.

If this fact is not instinctive with you, work out each of the above multiplications and make it instinctive.
 
When we start out with a problem such as the first one above, we note that more than one top and bottom can be divided by the same number:

The top 3 and bottom 9 are both divisible by 3. They become, respectively, 1 and 3. So now we have:

But the top 2 and bottom 8 are also divisible by the same number. Dividing both by 2, we have:

Now our answer comes out as 1/12. This is the same as our answer the first time we tried it, after reduction.

This process is traditionally called "cancellation." It might more sensibly be called "reduction," because that is what you really do. You do not cancel anything; you reduce numbers where you can.

Try the short-cut method of reduction on these:

Use your pad to play with these before checking the reduced forms below.

4/7 x 5/12 can be reduced by dividing the top 4 and bottom 12 by 4. The reduced form is 1/7 x 5/3, giving the answer 5/21

3/5 x 5/9 has two reductions. The two 5's can both be divided by 5, which gives us 3/1 x 1/9. The top 3 and bottom 9 can both be divided by 3, giving 1/1 x ⅓, The answer must be ⅓.

2/3 x 6/7 offers only the bottom 3 and top 6, both to be divided by 3. Now the problem is 2/1 x 2/7, which gives 4/7.

5/6 x 3/20 offers two reductions. 5 goes into the top 5 and bottom 20, reducing the problem to 1/6 x ¾. 3 goes evenly into the bottom 6 and top 3, further reducing the problem to ½ x ¼. Answer, ⅛.

There is an important reason why I refuse to call this process "cancellation." The technique is usually taught as an "X" process, from the top of one fraction to the bottom of another. It is definitely not necessarily so; any top and bottom (never a top and top or bottom and bottom, of course) will do, in the same fraction or in any of the fractions to be multiplied.

You can reduce 6/8 x 2/12 the same way you would 6/12 x 2/8, using any top and bottom that can be divided evenly by the same number. The only difference is that usually fractions are presented in arithmetic books already reduced for such problems. In our real-life figure work, they are not always so reduced for us. Look for reducing possibilities everywhere.
 
Dividing Fractions

Just as it may seem peculiar to multiply two quantities (if they are fractions) and get an answer smaller than either of them, so may it appear outrageous to divide one quantity into another and (if they are fractions) get an answer larger than either.

Keep firmly in mind that division is merely the reverse of multiplication, and review in your mind the reasons for the strange results you get in multiplication. In effect, the fraction divided is the answer to an imaginary multiplication, and the purpose of the division is to find the missing partner in the multiplication.

Let us start into the division of fractions with a simple example:

If we multiply by multiplying the respective tops and bottoms, then we might expect to divide by dividing them. In a problem this simple, we can indeed: 1 into 3 gives 3, and 2 into 4 gives 2: 3/2 is the answer.

Do not worry about that 3/2 yet. We will get into so-called improper fractions later.

The technique of simple division will theoretically work with any problem, but since every number does not "go into" every other number evenly we sometimes would end up with awkward decimal remainders and create some really difficult-to-handle answers.

This is why the standard trick of "inversion" has been developed. The trick has this rule:

To divide by a fraction, turn it upside down and multiply by it.

If this seems at all odd, reinforce your grasp of the reason why, as well as the rule, by considering that all division is merely an inversion of multiplication. When you multiply by 4, you count the other number 4 times. When you divide by 4, you count the other number ¼ times.
 
Another way of saying "divide by 27" is to say "multiply by ½7."
So another way to say "divide by ¾" is to invert the fraction and say "multiply by 4/3."

The single greatest source of confusion to many people is remembering which fraction to invert. If you fully understand the why, you cannot ever again become confused. To make extra sure, run through the comparison once more.
 
In order to divide 28 by 14, would you set it up as

So in order to divide ½ by ¼, would you set it up as

It is the fraction by which you divide that you invert —always.

Pull out your pad and do these examples with inversion:

Inverting the divider of the first problem gives us ¾ x 2/1. The answer is 6/4, which reduces to 3/2. If you used short-cut reduction, you would have converted the problem to 3/2 x 1/1 before multiplying.

The second problem becomes ⅞ x 5/2, which gives an answer of 35/16. This fraction cannot be reduced.

The third, when you invert the divider, becomes 5/6 x 3/2. This can be reduced to 5/2 x ½, with the answer 5/4.

Short-Cut Division

If you are sure of your technique, there is no need to rewrite such division problems with the divider inverted. You can do the inversion in your head by following this rule:
 
To divide, multiply the top of the fraction divided by the bottom of the divider, and put it on top. Multiply the bottom of the fraction divided by the top of the divider, and put it on the bottom.

In other words, you simply multiply each top by the other bottom. Keep your answer straight by using the fraction divided as your guide for the answer: the product of this top and the other bottom becomes the top of the answer. The entire process automatically inverts the divider without rewriting.

Here is an example:

 Top of fraction divided (2) times the other bottom (4) is 8. Since you used the top of the fraction divided, this 8 goes on top of the answer. Bottom of fraction divided (3) times the other top (3) is 9. This goes on the bottom of the answer. The answer is 8/9.

Try one yourself:

Keep your top and bottom straight by matching to the fraction divided rather than to the divider, and you can read the answer at sight. It is 5/8.

Beware of one possible misunderstanding here. When you divide in this fashion, you cannot reduce in the normal fashion by dividing tops and bottoms simultaneously. This is because you would invert the divider if you rewrote it before multiplying, so in essence the top of the divider becomes its bottom and vice versa. You can, if you take care to keep track of the proper tops and bottoms, reduce by dividing both tops or both bottoms by any number that will go into them evenly, because you invert the divider in multiplying anyway.

Adding Fractions

Adding and subtracting fractions is, surprisingly, more work than multiplying or dividing them. The reason is simple, and is based on the fact that it makes no difference in what order you multiply a series of numbers—but it makes a big difference in what order you multiply and add. Consider this quantity:

Does it make any difference whether you treat this as 2X3, plus A—or as 2 x the sum of 3 + 4? Try it and see. One handling gives you 10. The other gives you 14.

It is critically important to add and multiply in the proper portions. 2x (3 + 4) is not the same as (2 x 3) +4. Examine the two expressions carefully and you will discover the cause for the difference. In the first handling, the 4 gets multiplied by the 2 after it has been added to the 3. In the second, the 4 never gets multiplied by the 2 at all. So the end result is quite different.

Another way of approaching the special rules for adding and subtracting fractions is to remember that each fraction is, depending on its bottom number, in its own special number system—one not accounted for in our regular digits and expressible in our digit system only as a division problem. ½, %, and ¾ are all quantities based on one-third of 1. But ½, ¾, ¾, and % are quantities based on one-fourth of 1. Thirds and fourths are not in the same number system at all, and trying to add or subtract combinations of the two is like adding gallons and litres.

The first job in adding or subtracting fractions, then, is to get them all into the same number system, Fortunately, it is not hard at all.

There is a very simple way of converting different fractions into the same system. We just multiply the bottoms and adjust the tops. We can even forget that forbidding schoolroom phrase "lowest common denominator," because we do not need it. All we have to do is multiply.

Here is how it works:

First, in order to determine the number system in which we can express both quantities, we multiply the bottoms. 4X3 is 12. This 12 will be the bottom of the answer, because it is a number system that can express both fourths and thirds accurately.

Before we can add, however, we must convert each fraction to this new system. ¾ is ¾, but it is not 3/12. How many twelfths is it? The simplest way to convert is to multiply each top by the other bottom, because this is the number by which we multiplied the bottom and, as we know, multiplying top and bottom by the same number does not change the value of the fraction.

3 x 3 is 9, so ¾ is 9/12. We do not worry about that in working, however. All we care about at the moment is the 9. For the second fraction, we multiply 2x4 and get 8. Now we add the two products, and this becomes the top of the answer. 9 + 8 is 17. The answer is 17/12.

Once again, look at these four expressions and try to feel their identity:

This answer is a fraction larger than 1. We will get to the handling of such fractions soon. First, let us finish addition and subtraction.

Try the simplified rule on the following addition. The rule, in one sentence, reads: To add fractions, multiply the bottoms for the new bottom; multiply each top by the other bottom and add these products for the new top.

You do not have to go through the entire step-by-step visualization above each time you do it. Just multiply the bottoms for the bottom of the answer. Multiply each top by the other bottom and add the products for the top of the answer.

For the problem above, our bottom is 2 x 3 or 6. 1 x 3 is 3, plus 2 x 2 is 4, gives 7 as the top. The answer is 7/6.
 
Try a few more with this technique. It is really simpler and faster than worrying about common denominators:

Work out and reduce where possible the answers to these on your pad.

The answers, in order, are 17/20, 16/12, 10/21 and 23/20. The second answer—16/20—can be reduced to 4/3.

Adding More Than Two

The rule becomes just a little more complicated when you add three or more fractions. You have to reduce all of them to the same number system.

The rule is not very much more complicated, however. Let us take it in two steps:

1. Multiply all the bottoms together. This will be the bottom of the answer.

2. Multiply each top by all the bottoms except its own, and add all the products. This will be the top of the answer.

This rule is precisely the same as the rule for adding two fractions, generalized to handle any number of fractions. Here is an example:

The first step is to multiply all the bottoms together. 2 x 3 is 6, X5 is 30. The bottom of the answer is 30.

The second step is to multiply each top by all the bottoms except its own. 1 x 3 is 3, x 5 is 15. 2 x 2 is 4, x 5 is 20. 3 x 2 is 6, x 3 is 18. Add 15 and 20 and 18 to get the top of the answer: 53. The answer is 53/30.

Examine carefully the steps in this addition, and you will see that in each case we are really multiplying each fraction's top and bottom by the same number: the products of the bottoms of all the other fractions. This translates all the fractions into the same number system and adjusts all the tops at the same time, without changing the quantity of each fraction. Do one on your own with this method:

First, find the bottom of the answer. 4 x 3 is 12, x 5 is 60.

Now for the top. 3 x 3 is 9, x 5 is 45. 2 x 4 is 8, x 5 is 40. 2 x 4 is 8, x 3 is 24. Adding 45, 40, and 24, we get 109 as the top of the answer . . . 109/60.

Special Cases

There is a further short cut in adding a series of fractions in which some of them are already in the same terms. The usual method is to hunt through all the bottoms for the "lowest common denominator," which takes a bit of figuring and then adjustment of each top.

It is far easier simply to add all like fractions first; then add the resulting unlike fractions in the method just described.

In order to add like fractions (all thirds, say, or all fifths), you simply add the tops. Do nothing to the bottoms. The sum 1/5 and 2/5 is the sum of the tops—3—over the same bottom: 3/5.

Here is how to handle a typical situation:

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 2/3. 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 x 8 is 32, for the bottom of the answer; 3 x 8 is 24, and 5 x 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; ¾ 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 x 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 3x4 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 x 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 x 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 l.

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.

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 l½. 5 + l½ 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 ⅔ + 5/7 gives is 29/21, which translates into l8/21. Add this to 13 for the final answer, 148/21.

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 l7/20.
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. 1⅜ becomes 11/8—because 3 + 8 total 11. So 6⅜ becomes 511/8 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 112/16, and the fractional part quickly reduces to ¾. Final answer, l¾.

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 164/3. 8 from 16 leaves 8. 7/8 from 4/3 leaves 11/24. Answer,

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 146/7 x 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 7⅜ 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 7⅜.

Follow this multiplication:

Both numbers must first be turned into improper fractions. 7⅔ becomes 23/3 (7 x 3, plus 2). 3¾ becomes 15/4 (3 x 4, plus 3).

Note that one top and one bottom are both divisible by 3. Divide both by 3 before going on:
 
Pause for a moment to see if the problem can be reduced in any way before continuing:

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 28 ¾.

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

The answer is 143¼8. You got it by, first, translating 4% into 3%. Then you translated 31/6 to 19/16. This multiplication shows no reduction possibilities, so you multiply top by top and bottom by bottom to get 703/48. Translate this back to a mixed number by dividing 703 by 48, and produce the final proper answer of 1431/48.

Dividing by mixed numbers is just the reverse of multiplying. Translate each to an improper fraction, then invert the divider and multiply as always.

Let's do the last example as a division:

The two mixed numbers translate to the same improper fractions: 37/8 and 19/6. Since this is division, however, we turn the divider upside down and handle it as a multiplication:

Pause to look for reduction possibilities. The 6 and the 8 are both divisible by 2, so we can simplify the problem a bit to read:

Multiplying top by top and bottom by bottom, we get the answer 111/76- This is an improper fraction, but the top is not twice the bottom so we do not divide. We put down a 1 for the whole-number part, and subtract the bottom from the top to find the top of the fractional part. The final answer is 135/76.

Most of the fractions and examples in this chapter have been more complex than the ones you normally run into in your number work. This has been entirely on purpose. Learn to handle those in this chapter well, and simpler ones should be easy.

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