To add fractions with the same denominators. Subtraction of ordinary fractions: rules, examples, solutions

Note! Before writing a final answer, see if you can reduce the fraction you received.

Subtraction of fractions with the same denominators examples:

,

,

Subtracting a proper fraction from one.

If it is necessary to subtract from the unit a fraction that is correct, the unit is converted to the form of an improper fraction, its denominator is equal to the denominator of the subtracted fraction.

An example of subtracting a proper fraction from one:

The denominator of the fraction to be subtracted = 7 , i.e., we represent the unit as an improper fraction 7/7 and subtract according to the rule for subtracting fractions with the same denominators.

Subtracting a proper fraction from a whole number.

Rules for subtracting fractions - correct from integer (natural number):

• We translate the given fractions, which contain an integer part, into improper ones. We get the normal terms (it doesn't matter if they are different denominators), which we consider according to the rules given above;
• Next, we calculate the difference of the fractions that we received. As a result, we will almost find the answer;
• We perform the inverse transformation, that is, we get rid of the improper fraction - we select the integer part in the fraction.

Let us subtract a proper fraction from a whole number: we present natural number as a mixed number. Those. we take a unit in a natural number and translate it into the form of an improper fraction, the denominator is the same as that of the subtracted fraction.

Fraction subtraction example:

In the example, we replaced the unit with an improper fraction 7/7 and instead of 3 we wrote down a mixed number and subtracted a fraction from the fractional part.

Subtraction of fractions with different denominators.

Or, to put it another way, subtraction of different fractions.

Rule for subtracting fractions with different denominators. In order to subtract fractions with different denominators, it is necessary, first, to bring these fractions to the lowest common denominator (LCD), and only after that to subtract as with fractions with the same denominators.

The common denominator of several fractions is LCM (least common multiple) natural numbers that are the denominators of the given fractions.

Attention! If the numerator and denominator have common factors in the final fraction, then the fraction must be reduced. An improper fraction is best represented as a mixed fraction. Leaving the result of the subtraction without reducing the fraction where possible is an unfinished solution to the example!

Procedure for subtracting fractions with different denominators.

• find the LCM for all denominators;
• put additional multipliers for all fractions;
• multiply all numerators by an additional factor;
• we write the resulting products in the numerator, signing a common denominator under all fractions;
• subtract the numerators of fractions, signing the common denominator under the difference.

In the same way, addition and subtraction of fractions is carried out in the presence of letters in the numerator.

Subtraction of fractions, examples:

Subtraction of mixed fractions.

At subtraction of mixed fractions (numbers) separately, the integer part is subtracted from the integer part, and the fractional part is subtracted from the fractional part.

The first option is to subtract mixed fractions.

If the fractional parts the same denominators and numerator of the fractional part of the minuend (we subtract from it) ≥ the numerator of the fractional part of the subtrahend (we subtract it).

For example:

The second option is to subtract mixed fractions.

When the fractional parts various denominators. To begin with, we reduce the fractional parts to a common denominator, and then we subtract the integer part from the integer, and the fractional from the fractional.

For example:

The third option is to subtract mixed fractions.

The fractional part of the minuend is less than the fractional part of the subtrahend.

Example:

Because fractional parts have different denominators, which means, as in the second option, we first bring ordinary fractions to a common denominator.

The numerator of the fractional part of the minuend is less than the numerator of the fractional part of the subtrahend.3 < 14. So, we take a unit from the integer part and bring this unit to the form of an improper fraction with the same denominator and numerator = 18.

In the numerator from the right side we write the sum of the numerators, then we open the brackets in the numerator from the right side, that is, we multiply everything and give similar ones. We do not open brackets in the denominator. It is customary to leave the product in the denominators. We get:

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, 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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You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

In this lesson, we will consider the addition and subtraction of algebraic fractions with different denominators. We already know how to add and subtract common fractions with different denominators. To do this, the fractions must be reduced to a common denominator. It turns out that algebraic fractions follow the same rules. At the same time, we already know how to reduce algebraic fractions to a common denominator. Adding and subtracting fractions with different denominators is one of the most important and difficult topics in the 8th grade course. Moreover, this topic will be found in many topics of the algebra course, which you will study in the future. As part of the lesson, we will study the rules for adding and subtracting algebraic fractions with different denominators, and also analyze whole line typical examples.

Consider the simplest example for ordinary fractions.

Example 1 Add fractions: .

Solution:

Remember the rule for adding fractions. To begin with, fractions must be reduced to a common denominator. The common denominator for ordinary fractions is least common multiple(LCM) of the original denominators.

Definition

The smallest natural number that is divisible by both numbers and .

To find the LCM, it is necessary to decompose the denominators into prime factors, and then select all the prime factors that are included in the expansion of both denominators.

; . Then the LCM of numbers must include two 2s and two 3s: .

After finding the common denominator, it is necessary to find an additional factor for each of the fractions (in fact, divide the common denominator by the denominator of the corresponding fraction).

Then each fraction is multiplied by the resulting additional factor. We get fractions with the same denominators, which we learned to add and subtract in previous lessons.

We get: .

Answer:.

Consider now the addition of algebraic fractions with different denominators. First consider fractions whose denominators are numbers.

Example 2 Add fractions: .

Solution:

The solution algorithm is absolutely similar to the previous example. It is easy to find a common denominator for these fractions: and additional factors for each of them.

.

Answer:.

So let's formulate algorithm for adding and subtracting algebraic fractions with different denominators:

1. Find the smallest common denominator of fractions.

2. Find additional factors for each of the fractions (by dividing the common denominator by the denominator of this fraction).

3. Multiply the numerators by the appropriate additional factors.

4. Add or subtract fractions using the rules for adding and subtracting fractions with the same denominators.

Consider now an example with fractions in the denominator of which there are literal expressions.

Example 3 Add fractions: .

Solution:

Since the literal expressions in both denominators are the same, you should find a common denominator for numbers. The final common denominator will look like: . So the solution to this example is:

Answer:.

Example 4 Subtract fractions: .

Solution:

If you can’t “cheat” when choosing a common denominator (you can’t factor it or use the abbreviated multiplication formulas), then you have to take the product of the denominators of both fractions as a common denominator.

Answer:.

In general, when solving such examples, the most difficult task is to find a common denominator.

Let's look at a more complex example.

Example 5 Simplify: .

Solution:

When finding a common denominator, you must first try to factorize the denominators of the original fractions (to simplify the common denominator).

In this particular case:

Then it is easy to determine the common denominator: .

We determine additional factors and solve this example:

Answer:.

Now we will fix the rules for adding and subtracting fractions with different denominators.

Example 6 Simplify: .

Solution:

Answer:.

Example 7 Simplify: .

Solution:

.

Answer:.

Consider now an example in which not two, but three fractions are added (after all, the rules for addition and subtraction for more fractions remain the same).

Example 8 Simplify: .

Fractions are ordinary numbers, they can also be added and subtracted. But due to the fact that they have a denominator, more complex rules are required here than for integers.

Consider the simplest case, when there are two fractions with the same denominators. Then:

To add fractions with the same denominators, add their numerators and leave the denominator unchanged.

To subtract fractions with the same denominators, it is necessary to subtract the numerator of the second from the numerator of the first fraction, and again leave the denominator unchanged.

Within each expression, the denominators of the fractions are equal. By definition of addition and subtraction of fractions, we get:

As you can see, nothing complicated: just add or subtract the numerators - and that's it.

But even in such simple actions, people manage to make mistakes. Most often they forget that the denominator does not change. For example, when adding them, they also begin to add up, and this is fundamentally wrong.

Get rid of bad habit Adding the denominators is easy enough. Try to do the same when subtracting. As a result, the denominator will be zero, and the fraction (suddenly!) will lose its meaning.

Therefore, remember once and for all: when adding and subtracting, the denominator does not change!

Also, many people make mistakes when adding several negative fractions. There is confusion with the signs: where to put a minus, and where - a plus.

This problem is also very easy to solve. It is enough to remember that the minus before the fraction sign can always be transferred to the numerator - and vice versa. And of course, do not forget two simple rules:

1. Plus times minus gives minus;
2. Two negatives make an affirmative.

Let's analyze all this with specific examples:

A task. Find the value of the expression:

In the first case, everything is simple, and in the second, we will add minuses to the numerators of fractions:

What if the denominators are different

You cannot directly add fractions with different denominators. At least, this method is unknown to me. However, the original fractions can always be rewritten so that the denominators become the same.

There are many ways to convert fractions. Three of them are discussed in the lesson " Bringing fractions to a common denominator", so we will not dwell on them here. Let's take a look at some examples:

A task. Find the value of the expression:

In the first case, we bring the fractions to a common denominator using the "cross-wise" method. In the second, we will look for the LCM. Note that 6 = 2 3; 9 = 3 · 3. The last factors in these expansions are equal, and the first ones are coprime. Therefore, LCM(6; 9) = 2 3 3 = 18.

What if the fraction has an integer part

I can please you: different denominators of fractions are not the greatest evil. Much more errors occur when the whole part is highlighted in the fractional terms.

Of course, for such fractions there are own addition and subtraction algorithms, but they are rather complicated and require a long study. Better use a simple circuit below:

1. Convert all fractions containing an integer part to improper. We get normal terms (even if with different denominators), which are calculated according to the rules discussed above;
2. Actually, calculate the sum or difference of the resulting fractions. As a result, we will practically find the answer;
3. If this is all that was required in the task, we perform the inverse transformation, i.e. we get rid of the improper fraction, highlighting the integer part in it.

The rules for switching to improper fractions and highlighting the integer part are described in detail in the lesson "What is a numerical fraction". If you don't remember, be sure to repeat. Examples:

A task. Find the value of the expression:

Everything is simple here. The denominators inside each expression are equal, so it remains to convert all fractions to improper ones and count. We have:

To simplify the calculations, I skipped some obvious steps in the last examples.

A small note to the last two examples, where fractions with a highlighted integer part are subtracted. The minus before the second fraction means that it is the whole fraction that is subtracted, and not just its whole part.

Reread this sentence again, look at the examples, and think about it. This is where beginners make a lot of mistakes. They love to give such tasks to control work. You will also meet them repeatedly in the tests for this lesson, which will be published shortly.

Summary: General Scheme of Computing

In conclusion, I will give a general algorithm that will help you find the sum or difference of two or more fractions:

1. If an integer part is highlighted in one or more fractions, convert these fractions to improper ones;
2. Bring all the fractions to a common denominator in any way convenient for you (unless, of course, the compilers of the problems did this);
3. Add or subtract the resulting numbers according to the rules for adding and subtracting fractions with the same denominators;
4. Reduce the result if possible. If the fraction turned out to be incorrect, select the whole part.

Remember that it is better to highlight the whole part at the very end of the task, just before writing the answer.

Mixed fractions can be subtracted just like simple fractions. To subtract mixed numbers of fractions, you need to know a few subtraction rules. Let's study these rules with examples.

Subtraction of mixed fractions with the same denominators.

Consider an example with the condition that the integer and fractional part to be reduced are greater than the integer and fractional parts to be subtracted, respectively. Under such conditions, the subtraction occurs separately. The integer part is subtracted from the integer part, and the fractional part from the fractional.

Consider an example:

Subtract mixed fractions \(5\frac(3)(7)\) and \(1\frac(1)(7)\).

\(5\frac(3)(7)-1\frac(1)(7) = (5-1) + (\frac(3)(7)-\frac(1)(7)) = 4\ frac(2)(7)\)

The correctness of the subtraction is checked by addition. Let's check the subtraction:

\(4\frac(2)(7)+1\frac(1)(7) = (4 + 1) + (\frac(2)(7) + \frac(1)(7)) = 5\ frac(3)(7)\)

Consider an example with the condition that the fractional part of the minuend is less than the fractional part of the subtrahend, respectively. In this case, we borrow one from the integer in the minuend.

Consider an example:

Subtract mixed fractions \(6\frac(1)(4)\) and \(3\frac(3)(4)\).

The reduced \(6\frac(1)(4)\) has a smaller fractional part than the fractional part of the subtracted \(3\frac(3)(4)\). That is, \(\frac(1)(4)< \frac{1}{3}\), поэтому сразу отнять мы не сможем. Займем у целой части у 6 единицу, а потом выполним вычитание. Единицу мы запишем как \(\frac{4}{4} = 1\)

\(\begin(align)&6\frac(1)(4)-3\frac(3)(4) = (6 + \frac(1)(4))-3\frac(3)(4) = (5 + \color(red) (1) + \frac(1)(4))-3\frac(3)(4) = (5 + \color(red) (\frac(4)(4)) + \frac(1)(4))-3\frac(3)(4) = (5 + \frac(5)(4))-3\frac(3)(4) = \\\\ &= 5\frac(5)(4)-3\frac(3)(4) = 2\frac(2)(4) = 2\frac(1)(4)\\\\ \end(align)\)

Next example:

\(7\frac(8)(19)-3 = 4\frac(8)(19)\)

Subtracting a mixed fraction from a whole number.

Example: \(3-1\frac(2)(5)\)

The reduced 3 does not have a fractional part, so we cannot immediately subtract. Let's take the integer part of y 3 unit, and then perform the subtraction. We write the unit as \(3 = 2 + 1 = 2 + \frac(5)(5) = 2\frac(5)(5)\)

\(3-1\frac(2)(5)= (2 + \color(red) (1))-1\frac(2)(5) = (2 + \color(red) (\frac(5 )(5)))-1\frac(2)(5) = 2\frac(5)(5)-1\frac(2)(5) = 1\frac(3)(5)\)

Subtraction of mixed fractions with different denominators.

Consider an example with the condition if the fractional parts of the minuend and the subtrahend have different denominators. It is necessary to reduce to a common denominator, and then perform a subtraction.

Subtract two mixed fractions with different denominators \(2\frac(2)(3)\) and \(1\frac(1)(4)\).

The common denominator is 12.

\(2\frac(2)(3)-1\frac(1)(4) = 2\frac(2 \times \color(red) (4))(3 \times \color(red) (4) )-1\frac(1 \times \color(red) (3))(4 \times \color(red) (3)) = 2\frac(8)(12)-1\frac(3)(12 ) = 1\frac(5)(12)\)

Related questions:
How to subtract mixed fractions? How to solve mixed fractions?
Answer: you need to decide what type the expression belongs to and apply the solution algorithm according to the type of expression. Subtract the integer from the integer part, subtract the fractional part from the fractional part.

How to subtract a fraction from a whole number? How to subtract a fraction from a whole number?
Answer: you need to take a unit from an integer and write this unit as a fraction

\(4 = 3 + 1 = 3 + \frac(7)(7) = 3\frac(7)(7)\),

and then subtract the whole from the whole, subtract the fractional part from the fractional part. Example:

\(4-2\frac(3)(7) = (3 + \color(red) (1))-2\frac(3)(7) = (3 + \color(red) (\frac(7 )(7)))-2\frac(3)(7) = 3\frac(7)(7)-2\frac(3)(7) = 1\frac(4)(7)\)

Example #1:
Subtract a proper fraction from one: a) \(1-\frac(8)(33)\) b) \(1-\frac(6)(7)\)

Solution:
a) Let's represent the unit as a fraction with a denominator of 33. We get \(1 = \frac(33)(33)\)

\(1-\frac(8)(33) = \frac(33)(33)-\frac(8)(33) = \frac(25)(33)\)

b) Let's represent the unit as a fraction with a denominator of 7. We get \(1 = \frac(7)(7)\)

\(1-\frac(6)(7) = \frac(7)(7)-\frac(6)(7) = \frac(7-6)(7) = \frac(1)(7) \)

Example #2:
Subtract a mixed fraction from an integer: a) \(21-10\frac(4)(5)\) b) \(2-1\frac(1)(3)\)

Solution:
a) Let's take 21 units from an integer and write it like this \(21 = 20 + 1 = 20 + \frac(5)(5) = 20\frac(5)(5)\)

\(21-10\frac(4)(5) = (20 + 1)-10\frac(4)(5) = (20 + \frac(5)(5))-10\frac(4)( 5) = 20\frac(5)(5)-10\frac(4)(5) = 10\frac(1)(5)\\\\\)

b) Let's take 1 from the integer 2 and write it like this \(2 = 1 + 1 = 1 + \frac(3)(3) = 1\frac(3)(3)\)

\(2-1\frac(1)(3) = (1 + 1)-1\frac(1)(3) = (1 + \frac(3)(3))-1\frac(1)( 3) = 1\frac(3)(3)-1\frac(1)(3) = \frac(2)(3)\\\\\)

Example #3:
Subtract an integer from a mixed fraction: a) \(15\frac(6)(17)-4\) b) \(23\frac(1)(2)-12\)

a) \(15\frac(6)(17)-4 = 11\frac(6)(17)\)

b) \(23\frac(1)(2)-12 = 11\frac(1)(2)\)

Example #4:
Subtract a proper fraction from a mixed fraction: a) \(1\frac(4)(5)-\frac(4)(5)\)

\(1\frac(4)(5)-\frac(4)(5) = 1\\\\\)

Example #5:
Compute \(5\frac(5)(16)-3\frac(3)(8)\)

\(\begin(align)&5\frac(5)(16)-3\frac(3)(8) = 5\frac(5)(16)-3\frac(3 \times \color(red) ( 2))(8 \times \color(red) (2)) = 5\frac(5)(16)-3\frac(6)(16) = (5 + \frac(5)(16))- 3\frac(6)(16) = (4 + \color(red) (1) + \frac(5)(16))-3\frac(6)(16) = \\\\ &= (4 + \color(red) (\frac(16)(16)) + \frac(5)(16))-3\frac(6)(16) = (4 + \color(red) (\frac(21 )(16)))-3\frac(3)(8) = 4\frac(21)(16)-3\frac(6)(16) = 1\frac(15)(16)\\\\ \end(align)\)