Addition of fractions. Addition and subtraction of algebraic fractions with different denominators (basic rules, simplest cases)

Actions with fractions. In this article, we will analyze examples, everything is detailed with explanations. We will consider ordinary fractions. In the future, we will analyze decimals. I recommend to watch the whole and study sequentially.

1. Sum of fractions, difference of fractions.

Rule: when adding fractions with equal denominators, the result is a fraction - the denominator of which remains the same, and its numerator will be equal to the sum of the numerators of the fractions.

Rule: when calculating the difference of fractions with the same denominators, we get a fraction - the denominator remains the same, and the numerator of the second is subtracted from the numerator of the first fraction.

Formal notation of the sum and difference of fractions with equal denominators:

Examples (1):

It is clear that when ordinary fractions are given, then everything is simple, but if they are mixed? Nothing complicated...

Option 1- you can convert them into ordinary ones and then calculate them.

Option 2- you can separately "work" with the integer and fractional parts.

Examples (2):

Yet:

And if the difference of two mixed fractions and the numerator of the first fraction will be less than the numerator of the second? It can also be done in two ways.

Examples (3):

* Converted to ordinary fractions, calculated the difference, translated the resulting not proper fraction into a mixed one.

* Divided into integer and fractional parts, got three, then presented 3 as the sum of 2 and 1, with the unit presented as 11/11, then found the difference between 11/11 and 7/11 and calculated the result. The meaning of the above transformations is to take (select) the unit and present it as a fraction with the denominator we need, then from this fraction we can already subtract another.

Another example:

Conclusion: there is a universal approach - in order to calculate the sum (difference) of mixed fractions with equal denominators, they can always be converted into improper ones, then perform the necessary action. After that, if as a result we get an improper fraction, we translate it into a mixed one.

Above, we looked at examples with fractions that have equal denominators. What if the denominators differ? In this case, the fractions are reduced to the same denominator and the specified action is performed. To change (transform) a fraction, the main property of the fraction is used.

Consider simple examples:

In these examples, we immediately see how one of the fractions can be converted to get equal denominators.

If we designate ways to reduce fractions to one denominator, then this one will be called METHOD ONE.

That is, immediately when “evaluating” the fraction, you need to figure out whether such an approach will work - we check whether the larger denominator is divisible by the smaller one. And if it is divided, then we perform the transformation - we multiply the numerator and denominator so that the denominators of both fractions become equal.

Now look at these examples:

This approach does not apply to them. There are other ways to reduce fractions to a common denominator, consider them.

Method SECOND.

Multiply the numerator and denominator of the first fraction by the denominator of the second, and the numerator and denominator of the second fraction by the denominator of the first:

*In fact, we bring fractions to the form when the denominators become equal. Next, we use the rule of adding timid with equal denominators.

Example:

*This method can be called universal, and it always works. The only negative is that after the calculations, a fraction may turn out that will need to be further reduced.

Consider an example:

It can be seen that the numerator and denominator are divisible by 5:

Method THIRD.

Find the least common multiple (LCM) of the denominators. This will be the common denominator. What is this number? It's the smallest natural number, which is divisible by each of the numbers.

Look, here are two numbers: 3 and 4, there are many numbers that are divisible by them - these are 12, 24, 36, ... The smallest of them is 12. Or 6 and 15, 30, 60, 90 are divisible by them .... Least 30. Question - how to determine this least common multiple?

There is a clear algorithm, but often this can be done immediately without calculations. For example, according to the above examples (3 and 4, 6 and 15), no algorithm is needed, we took large numbers (4 and 15), doubled them and saw that they are divisible by the second number, but pairs of numbers can be others, such as 51 and 119.

Algorithm. In order to determine the least common multiple of several numbers, you must:

- decompose each of the numbers into SIMPLE factors

- write out the decomposition of the BIGGER of them

- multiply it by the MISSING factors of other numbers

Consider examples:

50 and 60 50 = 2∙5∙5 60 = 2∙2∙3∙5

in the expansion of a larger number, one five is missing

=> LCM(50,60) = 2∙2∙3∙5∙5 = 300

48 and 72 48 = 2∙2∙2∙2∙3 72 = 2∙2∙2∙3∙3

in the expansion of a larger number, two and three are missing

=> LCM(48,72) = 2∙2∙2∙2∙3∙3 = 144

* Least common multiple of two prime numbers equal to their product

Question! And why is it useful to find the least common multiple, because you can use the second method and simply reduce the resulting fraction? Yes, you can, but it's not always convenient. See what the denominator will be for the numbers 48 and 72 if you simply multiply them 48∙72 = 3456. Agree that it is more pleasant to work with smaller numbers.

Consider examples:

*51 = 3∙17 119 = 7∙17

in the expansion of a larger number, a triple is missing

=> LCM(51,119) = 3∙7∙17

And now we apply the first method:

* Look at the difference in the calculations, in the first case there is a minimum of them, and in the second you need to work separately on a piece of paper, and even the fraction that you got needs to be reduced. Finding the LCM simplifies the work considerably.

More examples:

* In the second example, it is clear that smallest number, which is divided by 40 and 60 is equal to 120.

TOTAL! GENERAL CALCULATION ALGORITHM!

- we bring fractions to ordinary ones, if there is an integer part.

- we bring fractions to a common denominator (first we look to see if one denominator is divisible by another, if it is divisible, then we multiply the numerator and denominator of this other fraction; if it is not divisible, we act through the other methods indicated above).

- having received fractions with equal denominators, we perform actions (addition, subtraction).

- if necessary, we reduce the result.

- if necessary, select the whole part.

2. Product of fractions.

The rule is simple. When multiplying fractions, their numerators and denominators are multiplied:

Examples:

The next action that can be performed with ordinary fractions is subtraction. As part of this material, we will consider how to correctly calculate the difference between fractions with the same and different denominators how to subtract a fraction from a natural number and vice versa. All examples will be illustrated with tasks. Let us clarify in advance that we will analyze only cases where the difference of fractions results in a positive number.

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How to find the difference between fractions with the same denominator

Let's start right away with an illustrative example: let's say we have an apple that has been divided into eight parts. Let's leave five parts on the plate and take two of them. This action can be written like this:

We end up with 3 eighths because 5 − 2 = 3 . It turns out that 5 8 - 2 8 = 3 8 .

With this simple example, we have seen exactly how the subtraction rule works for fractions with the same denominators. Let's formulate it.

Definition 1

To find the difference between fractions with the same denominators, you need to subtract the numerator of one from the numerator of the other, and leave the denominator the same. This rule can be written as a b - c b = a - c b .

We will use this formula in what follows.

Let's take concrete examples.

Example 1

Subtract from the fraction 24 15 the common fraction 17 15 .

Solution

We see that these fractions have the same denominators. So all we have to do is subtract 17 from 24. We get 7 and add a denominator to it, we get 7 15 .

Our calculations can be written like this: 24 15 - 17 15 \u003d 24 - 17 15 \u003d 7 15

If necessary, you can shorten compound fraction or select the whole part from the wrong one to make it more convenient to count.

Example 2

Find the difference 37 12 - 15 12 .

Solution

Let's use the formula described above and calculate: 37 12 - 15 12 = 37 - 15 12 = 22 12

It is easy to see that the numerator and denominator can be divided by 2 (we already talked about this earlier when we analyzed the signs of divisibility). Reducing the answer, we get 11 6 . This is an improper fraction, from which we will select the whole part: 11 6 \u003d 1 5 6.

How to find the difference between fractions with different denominators

Such a mathematical operation can be reduced to what we have already described above. To do this, simply bring the desired fractions to the same denominator. Let's formulate the definition:

Definition 2

To find the difference between fractions that have different denominators, you need to bring them to the same denominator and find the difference between the numerators.

Let's look at an example of how this is done.

Example 3

Subtract 1 15 from 2 9 .

Solution

The denominators are different, and you need to reduce them to the smallest common sense. In this case, the LCM is 45. For the first fraction, an additional factor of 5 is required, and for the second - 3.

Let's calculate: 2 9 = 2 5 9 5 = 10 45 1 15 = 1 3 15 3 = 3 45

We got two fractions with the same denominator, and now we can easily find their difference using the algorithm described earlier: 10 45 - 3 45 = 10 - 3 45 = 7 45

A brief record of the solution looks like this: 2 9 - 1 15 \u003d 10 45 - 3 45 \u003d 10 - 3 45 \u003d 7 45.

Do not neglect the reduction of the result or the selection of a whole part from it, if necessary. In this example, we do not need to do this.

Example 4

Find the difference 19 9 - 7 36 .

Solution

We bring the fractions indicated in the condition to the lowest common denominator 36 and obtain 76 9 and 7 36 respectively.

We consider the answer: 76 36 - 7 36 \u003d 76 - 7 36 \u003d 69 36

The result can be reduced by 3 to get 23 12 . The numerator is greater than the denominator, which means we can extract the whole part. The final answer is 1 11 12 .

The summary of the whole solution is 19 9 - 7 36 = 1 11 12 .

How to subtract a natural number from a common fraction

This action can also be easily reduced to a simple subtraction ordinary fractions. This can be done by representing a natural number as a fraction. Let's show an example.

Example 5

Find the difference 83 21 - 3 .

Solution

3 is the same as 3 1 . Then you can calculate like this: 83 21 - 3 \u003d 20 21.

If in the condition it is necessary to subtract an integer from improper fraction, it is more convenient to first extract an integer from it, writing it as a mixed number. Then the previous example can be solved differently.

From the fraction 83 21, when you select the integer part, you get 83 21 \u003d 3 20 21.

Now just subtract 3 from it: 3 20 21 - 3 = 20 21 .

How to subtract a fraction from a natural number

This action is done similarly to the previous one: we rewrite a natural number as a fraction, bring both to a common denominator and find the difference. Let's illustrate this with an example.

Example 6

Find the difference: 7 - 5 3 .

Solution

Let's make 7 a fraction 7 1 . We do the subtraction and transform the final result, extracting the integer part from it: 7 - 5 3 = 5 1 3 .

There is another way to make calculations. It has some advantages that can be used in cases where the numerators and denominators of the fractions in the problem are large numbers.

Definition 3

If the fraction to be subtracted is correct, then the natural number from which we are subtracting must be represented as the sum of two numbers, one of which is equal to 1. After that, you need to subtract the desired fraction from unity and get the answer.

Example 7

Calculate the difference 1 065 - 13 62 .

Solution

The fraction to be subtracted is correct, because its numerator is less than the denominator. Therefore, we need to subtract one from 1065 and subtract the desired fraction from it: 1065 - 13 62 \u003d (1064 + 1) - 13 62

Now we need to find the answer. Using the properties of subtraction, the resulting expression can be written as 1064 + 1 - 13 62 . Let's calculate the difference in brackets. To do this, we represent the unit as a fraction 1 1 .

It turns out that 1 - 13 62 \u003d 1 1 - 13 62 \u003d 62 62 - 13 62 \u003d 49 62.

Now let's remember about 1064 and formulate the answer: 1064 49 62 .

We use old way to prove that it is less convenient. Here are the calculations we would get:

1065 - 13 62 = 1065 1 - 13 62 = 1065 62 1 62 - 13 62 = 66030 62 - 13 62 = = 66030 - 13 62 = 66017 62 = 1064 4 6

The answer is the same, but the calculations are obviously more cumbersome.

We considered the case when you need to subtract the correct fraction. If it's wrong, we replace it with a mixed number and subtract according to the familiar rules.

Example 8

Calculate the difference 644 - 73 5 .

Solution

The second fraction is improper, and the whole part must be separated from it.

Now we calculate similarly to the previous example: 630 - 3 5 = (629 + 1) - 3 5 = 629 + 1 - 3 5 = 629 + 2 5 = 629 2 5

Subtraction properties when working with fractions

The properties that the subtraction of natural numbers possesses also apply to the cases of subtracting ordinary fractions. Let's see how to use them when solving examples.

Example 9

Find the difference 24 4 - 3 2 - 5 6 .

Solution

We have already solved similar examples when we analyzed the subtraction of a sum from a number, so we act on already known algorithm. First, we calculate the difference 25 4 - 3 2, and then subtract the last fraction from it:

25 4 - 3 2 = 24 4 - 6 4 = 19 4 19 4 - 5 6 = 57 12 - 10 12 = 47 12

Let's transform the answer by extracting the integer part from it. The result is 3 11 12.

Brief summary of the whole solution:

25 4 - 3 2 - 5 6 = 25 4 - 3 2 - 5 6 = 25 4 - 6 4 - 5 6 = = 19 4 - 5 6 = 57 12 - 10 12 = 47 12 = 3 11 12

If the expression contains both fractions and natural numbers, it is recommended to group them by types when calculating.

Example 10

Find the difference 98 + 17 20 - 5 + 3 5 .

Solution

Knowing the basic properties of subtraction and addition, we can group numbers as follows: 98 + 17 20 - 5 + 3 5 = 98 + 17 20 - 5 - 3 5 = 98 - 5 + 17 20 - 3 5

Let's complete the calculations: 98 - 5 + 17 20 - 3 5 = 93 + 17 20 - 12 20 = 93 + 5 20 = 93 + 1 4 = 93 1 4

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

Actions with fractions.

Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")

So, what are fractions, types of fractions, transformations - we remembered. Let's tackle the main question.

What can you do with fractions? Yes, all that with ordinary numbers. Add, subtract, multiply, divide.

All these actions with decimal operations with fractions are no different from operations with integers. Actually, this is what they are good for, decimal. The only thing is that you need to put the comma correctly.

mixed numbers , as I said, are of little use for most actions. They still need to be converted to ordinary fractions.

And here are the actions with ordinary fractions will be smarter. And much more important! Let me remind you: all actions with fractional expressions with letters, sines, unknowns, and so on and so forth are no different from actions with ordinary fractions! Operations with ordinary fractions are the basis for all algebra. It is for this reason that we will analyze all this arithmetic in great detail here.

Addition and subtraction of fractions.

Everyone can add (subtract) fractions with the same denominators (I really hope!). Well, let me remind you that I’m completely forgetful: when adding (subtracting), the denominator does not change. The numerators are added (subtracted) to give the numerator of the result. Type:

In short, in general view:

What if the denominators are different? Then, using the main property of the fraction (here it came in handy again!), We make the denominators the same! For example:

Here we had to make the fraction 4/10 from the fraction 2/5. Solely for the purpose of making the denominators the same. I note, just in case, that 2/5 and 4/10 are the same fraction! Only 2/5 is uncomfortable for us, and 4/10 is even nothing.

By the way, this is the essence of solving any tasks in mathematics. When we're out uncomfortable expressions do the same, but more convenient to solve.

Another example:

The situation is similar. Here we make 48 out of 16. By simple multiplication by 3. This is all clear. But here we come across something like:

How to be?! It's hard to make a nine out of a seven! But we are smart, we know the rules! Let's transform every fraction so that the denominators are the same. This is called "reduce to a common denominator":

How! How did I know about 63? Very simple! 63 is a number that is evenly divisible by 7 and 9 at the same time. Such a number can always be obtained by multiplying the denominators. If we multiply some number by 7, for example, then the result will certainly be divided by 7!

If you need to add (subtract) several fractions, there is no need to do it in pairs, step by step. You just need to find the denominator that is common to all fractions, and bring each fraction to this same denominator. For example:

And what will be the common denominator? You can, of course, multiply 2, 4, 8, and 16. We get 1024. Nightmare. It is easier to estimate that the number 16 is perfectly divisible by 2, 4, and 8. Therefore, it is easy to get 16 from these numbers. This number will be the common denominator. Let's turn 1/2 into 8/16, 3/4 into 12/16, and so on.

By the way, if we take 1024 as a common denominator, everything will work out too, in the end everything will be reduced. Only not everyone will get to this end, because of the calculations ...

Solve the example yourself. Not a logarithm... It should be 29/16.

So, with the addition (subtraction) of fractions is clear, I hope? Of course, it is easier to work in a shortened version, with additional multipliers. But this pleasure is available to those who honestly worked in the lower grades ... And did not forget anything.

And now we will do the same actions, but not with fractions, but with fractional expressions. New rakes will be found here, yes ...

So, we need to add two fractional expressions:

We need to make the denominators the same. And only with the help multiplication! So the main property of the fraction says. Therefore, I cannot add one to x in the first fraction in the denominator. (But that would be nice!). But if you multiply the denominators, you see, everything will grow together! So we write down, the line of the fraction, leave an empty space on top, then add it, and write the product of the denominators below, so as not to forget:

And, of course, we don’t multiply anything on the right side, we don’t open brackets! And now, looking at the common denominator of the right side, we think: in order to get the denominator x (x + 1) in the first fraction, we need to multiply the numerator and denominator of this fraction by (x + 1). And in the second fraction - x. You get this:

Note! Parentheses are here! This is the rake that many step on. Not brackets, of course, but their absence. Parentheses appear because we multiply the whole numerator and the whole denominator! And not their individual pieces ...

In the numerator of the right side, we write the sum of the numerators, everything is as in numerical fractions, then we open the brackets in the numerator of the right side, i.e. multiply everything and give like. You don't need to open the brackets in the denominators, you don't need to multiply something! In general, in denominators (any) the product is always more pleasant! We get:

Here we got the answer. The process seems long and difficult, but it depends on practice. Solve examples, get used to it, everything will become simple. Those who have mastered the fractions in the allotted time, do all these operations with one hand, on the machine!

And one more note. Many famously deal with fractions, but hang on examples with whole numbers. Type: 2 + 1/2 + 3/4= ? Where to fasten a deuce? No need to fasten anywhere, you need to make a fraction out of a deuce. It's not easy, it's very simple! 2=2/1. Like this. Any whole number can be written as a fraction. The numerator is the number itself, the denominator is one. 7 is 7/1, 3 is 3/1 and so on. It's the same with letters. (a + b) \u003d (a + b) / 1, x \u003d x / 1, etc. And then we work with these fractions according to all the rules.

Well, on addition - subtraction of fractions, knowledge was refreshed. Transformations of fractions from one type to another - repeated. You can also check. Shall we settle a little?)

Calculate:

Answers (in disarray):

71/20; 3/5; 17/12; -5/4; 11/6

Multiplication / division of fractions - in the next lesson. There are also tasks for all actions with fractions.

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