Solving fractions with the same denominators. Addition of fractions

Find the numerator and denominator. A fraction consists of two numbers: the number above the line is called the numerator, and the number below the line is called the denominator. The denominator indicates the total number of parts into which a whole is broken, and the numerator is the considered number of such parts.

For example, in the fraction ½, the numerator is 1 and the denominator is 2.

Determine the denominator. If two or more fractions have a common denominator, such fractions have the same number under the line, that is, in this case, some whole is divided into the same number of parts. Adding fractions with a common denominator is very easy, since the denominator of the total fraction will be the same as that of the fractions being added. For example:

The fractions 3/5 and 2/5 have a common denominator 5.
Fractions 3/8, 5/8, 17/8 have a common denominator 8.

Determine the numerators. To add fractions with a common denominator, add their numerators, and write the result above the denominator of the added fractions.

The fractions 3/5 and 2/5 have numerators 3 and 2.
Fractions 3/8, 5/8, 17/8 have numerators 3, 5, 17.

Add up the numerators. In problem 3/5 + 2/5 add the numerators 3 + 2 = 5. In problem 3/8 + 5/8 + 17/8 add the numerators 3 + 5 + 17 = 25.

Write down the total. Remember that when adding fractions with a common denominator, it remains unchanged - only the numerators are added.

3/5 + 2/5 = 5/5
3/8 + 5/8 + 17/8 = 25/8

Convert the fraction if necessary. Sometimes a fraction can be written as a whole number rather than as a common or decimal fraction. For example, the fraction 5/5 easily converts to 1, since any fraction whose numerator is equal to the denominator is 1. Imagine a pie cut into three parts. If you eat all three parts, then you will eat the whole (one) pie.

Any common fraction can be converted to a decimal; To do this, divide the numerator by the denominator. For example, the fraction 5/8 can be written like this: 5 ÷ 8 = 0.625.

Simplify the fraction if possible. A simplified fraction is a fraction whose numerator and denominator do not have a common divisor.

For example, consider the fraction 3/6. Here, both the numerator and the denominator have a common divisor equal to 3, that is, the numerator and denominator are completely divisible by 3. Therefore, the fraction 3/6 can be written as follows: 3 ÷ 3/6 ÷ 3 = ½.

If necessary, convert the improper fraction to a mixed fraction (mixed number). For an improper fraction, the numerator is greater than the denominator, for example, 25/8 (for a proper fraction, the numerator is less than the denominator). An improper fraction can be converted to a mixed fraction, which consists of an integer part (that is, a whole number) and a fractional part (that is, a proper fraction). To convert an improper fraction such as 25/8 to a mixed number, follow these steps:

Divide the numerator of the improper fraction by its denominator; write down the incomplete quotient (the whole answer). In our example: 25 ÷ 8 = 3 plus some remainder. In this case, the whole answer is the integer part of the mixed number.
Find the rest. In our example: 8 x 3 = 24; subtract the result from the original numerator: 25 - 24 \u003d 1, that is, the remainder is 1. In this case, the remainder is the numerator of the fractional part of the mixed number.
Write a mixed fraction. The denominator does not change (that is, it is equal to the denominator of the improper fraction), so 25/8 = 3 1/8.

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, everything is the same as 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, I’ll remind you 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 terms:

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 do not need to open brackets in the denominators, you do not need to multiply something! In general, in denominators (any) the work 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 crack down on 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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Adding and subtracting fractions with the same denominators
Adding and subtracting fractions with different denominators
The concept of the NOC
Bringing fractions to the same denominator
How to add a whole number and a fraction

1 Adding and subtracting fractions with the same denominators

To add fractions with the same denominators, you need to add their numerators, and leave the denominator the same, for example:

To subtract fractions with the same denominators, subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same, for example:

To add mixed fractions, you must separately add their whole parts, and then add their fractional parts, and write the result as a mixed fraction,

If, when adding the fractional parts, an improper fraction is obtained, we select the integer part from it and add it to the integer part, for example:

2 Adding and subtracting fractions with different denominators

In order to add or subtract fractions with different denominators, you must first bring them to the same denominator, and then proceed as indicated at the beginning of this article. The common denominator of several fractions is the LCM (least common multiple). For the numerator of each of the fractions, additional factors are found by dividing the LCM by the denominator of this fraction. We'll look at an example later, after we figure out what an LCM is.

3 Least common multiple (LCM)

The least common multiple of two numbers (LCM) is the smallest natural number that is divisible by both of these numbers without a remainder. Sometimes the LCM can be found orally, but more often, especially when working with large numbers, you have to find the LCM in writing, using the following algorithm:

In order to find the LCM of several numbers, you need:

Decompose these numbers into prime factors
Take the largest expansion, and write these numbers as a product
Select in other expansions the numbers that do not occur in the largest expansion (or occur in it a smaller number of times), and add them to the product.
Multiply all the numbers in the product, this will be the LCM.

For example, let's find the LCM of numbers 28 and 21:

4Reducing fractions to the same denominator

Let's go back to adding fractions with different denominators.

When we reduce fractions to the same denominator, equal to the LCM of both denominators, we must multiply the numerators of these fractions by additional multipliers. You can find them by dividing the LCM by the denominator of the corresponding fraction, for example:

Thus, in order to bring fractions to one indicator, you must first find the LCM (that is, the smallest number that is divisible by both denominators) of the denominators of these fractions, then put additional factors on the numerators of the fractions. You can find them by dividing the common denominator (LCD) by the denominator of the corresponding fraction. Then you need to multiply the numerator of each fraction by an additional factor, and put the LCM as the denominator.

5How to add a whole number and a fraction

In order to add a whole number and a fraction, you just need to add this number before the fraction, and you get a mixed fraction, for example.

Fractional expressions are difficult for a child to understand. Most people have difficulties with . When studying the topic "addition of fractions with integers", the child falls into a stupor, finding it difficult to solve the task. In many examples, a series of calculations must be performed before an action can be performed. For example, convert fractions or convert an improper fraction to a proper one.

Explain to the child clearly. Take three apples, two of which will be whole, and the third will be cut into 4 parts. Separate one slice from the cut apple, and put the remaining three next to two whole fruits. We get ¼ apples on one side and 2 ¾ on the other. If we combine them, we get three whole apples. Let's try to reduce 2 ¾ apples by ¼, that is, remove one more slice, we get 2 2/4 apples.

Let's take a closer look at actions with fractions, which include integers:

First, let's recall the calculation rule for fractional expressions with a common denominator:

At first glance, everything is easy and simple. But this applies only to expressions that do not require conversion.

How to find the value of an expression where the denominators are different

In some tasks, it is necessary to find the value of an expression where the denominators are different. Consider a specific case:
3 2/7+6 1/3

Find the value of this expression, for this we find a common denominator for two fractions.

For numbers 7 and 3, this is 21. We leave the integer parts the same, and reduce the fractional parts to 21, for this we multiply the first fraction by 3, the second by 7, we get:
6/21+7/21, do not forget that whole parts are not subject to conversion. As a result, we get two fractions with one denominator and calculate their sum:
3 6/21+6 7/21=9 15/21
What if the result of addition is an improper fraction that already has an integer part:
2 1/3+3 2/3
In this case, we add the integer parts and fractional parts, we get:
5 3/3, as you know, 3/3 is one, so 2 1/3+3 2/3=5 3/3=5+1=6

With finding the sum, everything is clear, let's analyze the subtraction:

From all that has been said, the rule of operations on mixed numbers follows, which sounds like this:

If it is necessary to subtract an integer from a fractional expression, it is not necessary to represent the second number as a fraction, it is enough to operate only on integer parts.

Let's try to calculate the value of expressions on our own:

Let's take a closer look at the example under the letter "m":

4 5/11-2 8/11, the numerator of the first fraction is less than the second. To do this, we take one integer from the first fraction, we get,
3 5/11+11/11=3 whole 16/11, subtract the second from the first fraction:
3 16/11-2 8/11=1 whole 8/11

Be careful when completing the task, do not forget to convert improper fractions to mixed ones, highlighting the whole part. To do this, it is necessary to divide the value of the numerator by the value of the denominator, what happened, takes the place of the integer part, the remainder will be the numerator, for example:

19/4=4 ¾, check: 4*4+3=19, in the denominator 4 remains unchanged.

Summarize:

Before proceeding with the task related to fractions, it is necessary to analyze what kind of expression it is, what transformations need to be performed on the fraction in order for the solution to be correct. Look for more rational solutions. Don't go the hard way. Plan all the actions, decide first in a draft version, then transfer to a school notebook.

To avoid confusion when solving fractional expressions, it is necessary to follow the sequence rule. Decide everything carefully, without rushing.

To add an integer to a fraction, it is enough to perform a series of actions, or rather, calculations.

For example, you have 7 - an integer, you need to add it to the fraction 1/2.

We act as follows:

We multiply 7 by the denominator (2), it turns out 14,
to 14 we add the upper part (1), it turns out 15,
and substitute the denominator.
the result is 15/2.

In this simple way, you can add whole numbers to fractional ones.

And to select a whole number from a fraction, you need to divide the numerator by the denominator, and the remainder will be a fraction.

The operation of adding an integer to a regular fraction is not difficult and sometimes it simply consists in the formation of a mixed fraction, in which the integer part is placed to the left of the fractional part, for example, such a fraction will be mixed:

However, more often, when you add an integer to a fraction, you get an improper fraction, in which the numerator is greater than the denominator. This operation is performed as follows: an integer is represented as an improper fraction with the same denominator as the fraction being added, and then the numerators of both fractions are simply added. For example, it will look like this:

5+1/8 = 5*8/8+1/8 = 40/8+1/8 = 41/8

I think it's very simple.

For example, we have a fraction 1/4 (this is the same as 0.25, that is, a quarter of a whole number).

And to this quarter you can add any integer, for example 3. It turns out three and a quarter:

3.25. Or in a fraction it is expressed like this: 3 1/4

Here, following the example of this example, you can add any fractions with any integers.

You need to raise an integer to a fraction with a denominator of 10 (6/10). Next, bring the existing fraction to a common denominator 10 (35=610). Well, perform the operation as with ordinary fractions 610+610=1210 total 12.

You can do this in two ways.

one). A fraction can be converted to a whole number and added. For example, 1/2 is 0.5; 1/4 equals 0.25; 2/5 is 0.4 and so on.

We take the integer 5, to which we need to add the fraction 4/5. Let's convert the fraction: 4/5 is 4 divided by 5 and we get 0.8. Add 0.8 to 5 and get 5.8 or 5 4/5.

2). Second way: 5 + 4/5 = 29/5 = 5 4/5.

Adding fractions is a simple mathematical operation, for example, you need to add the integer 3 and the fraction 1/7. To add these two numbers you must have the same denominator so you have to multiply three times seven and divide by that number, then you get 21/7+1/7, denominator one, add 21 and 1, you get the answer 22/7 .

Just take and add an integer to this fraction. Let's say 6+1/2=6 1/2. Well, if this is a decimal fraction, then for example, 6 + 1.2 = 7.2.

To add a fraction and an integer, you need to add a fractional number to an integer and write them down as a complex number, for example, when adding an ordinary fraction to an integer, we get: 1/2 +3 \u003d 3 1/2; when adding a decimal fraction: 0.5 +3 \u003d 3.5.

A fraction in itself is not an integer, because it does not reach it in quantity, and therefore there is no need to convert an integer into this fraction. Therefore, the integer remains an integer and fully demonstrates the full denomination, and the fraction is added to it, and demonstrates how much this integer lacks before the next full point is added.

academic example.

10 + 7/3 = 10 integers and 7/3.

If, of course, there are integers, then they are summed with integers.

12 + 5 7/9 = 17 and 7/9.

What is a whole number and what is a fraction.

If a both terms are positive, this fraction should be assigned to an integer. You get a mixed number. Moreover, there may be 2 cases.

Case 1

The fraction is correct, i.e. the numerator is less than the denominator. Then the mixed number obtained after attribution will be the answer.

4/9 + 10 = 10 4/9 (ten point four ninths).

Case 2

The fraction is incorrect, i.e. the numerator is greater than the denominator. Then a little transformation is required. An improper fraction should be turned into a mixed number, in other words, highlight the whole part. It is done like this:

After that, you need to add the integer part of the improper fraction to the integer and add its fractional part to the resulting amount. In the same way, a whole is added to a mixed number.

1) 11/4 + 5 = 2 3/4 + 5 = 7 3/4 (7 whole three fourths).

2) 5 1/2 + 6 = 11 1/2 (11 whole one second).

If one of the terms or both negative, then addition is performed according to the rules for adding numbers with different or identical signs. An integer is represented as the ratio of this number and 1, and then both the numerator and denominator are multiplied by a number equal to the denominator of the fraction to which the integer is added.

3) 1/5 + (-2)= 1/5 + -2/1 = 1/5 + -10/5 = -9/5 = -1 4/5 (minus 1 whole four fifths).

4) -13/3 + (-4) = -13/3 + -4/1 = -13/3 + -12/3 = -25/3 = -8 1/3 (minus 8 point one third).

Comment.

After getting acquainted with negative numbers, when studying actions with them, students in grade 6 should understand that adding a positive integer to a negative fraction is the same as subtracting a fraction from a natural number. This action, as you know, is performed like this:

In fact, in order to add a fraction and an integer, you simply need to simply reduce the existing integer to a fractional one, and doing this is as easy as shelling pears. You just need to take the denominator of the fraction (available in the example) and make it the denominator of an integer by multiplying it by this denominator and dividing, here is an example:

2+2/3 = 2*3/3+2/3 = 6/3+2/3 = 8/3