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 denotes total parts into which some whole is divided, 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 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. AT this case the whole answer is whole part 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.

Lesson content

Adding fractions with the same denominators

Adding fractions is of two types:

1. Adding fractions with the same denominators
2. Adding fractions with different denominators

Let's start with adding fractions with the same denominators. Everything is simple here. To add fractions with the same denominators, you need to add their numerators, and leave the denominator unchanged. For example, let's add the fractions and . We add the numerators, and leave the denominator unchanged:

This example can be easily understood if we think of a pizza that is divided into four parts. If you add pizza to pizza, you get pizza:

Example 2 Add fractions and .

The answer turned out not proper fraction. If the end of the task comes, then it is customary to get rid of improper fractions. To get rid of an improper fraction, you need to select the whole part in it. In our case, the integer part is allocated easily - two divided by two is equal to one:

This example can be easily understood if we think of a pizza that is divided into two parts. If you add more pizzas to the pizza, you get one whole pizza:

Example 3. Add fractions and .

Again, add the numerators, and leave the denominator unchanged:

This example can be easily understood if we think of a pizza that is divided into three parts. If you add more pizzas to pizza, you get pizzas:

Example 4 Find the value of an expression

This example is solved in exactly the same way as the previous ones. The numerators must be added and the denominator left unchanged:

Let's try to depict our solution using a picture. If you add pizzas to a pizza and add more pizzas, you get 1 whole pizza and more pizzas.

As you can see, adding fractions with the same denominators is not difficult. It is enough to understand the following rules:

1. To add fractions with the same denominator, you need to add their numerators, and leave the denominator unchanged;

Adding fractions with different denominators

Now we will learn how to add fractions with different denominators. When adding fractions, the denominators of those fractions must be the same. But they are not always the same.

For example, fractions can be added because they have the same denominators.

But fractions cannot be added at once, because these fractions have different denominators. In such cases, fractions must be reduced to the same (common) denominator.

There are several ways to reduce fractions to the same denominator. Today we will consider only one of them, since the rest of the methods may seem complicated for a beginner.

The essence of this method lies in the fact that first (LCM) of the denominators of both fractions is sought. Then the LCM is divided by the denominator of the first fraction and the first additional factor is obtained. They do the same with the second fraction - the LCM is divided by the denominator of the second fraction and the second additional factor is obtained.

Then the numerators and denominators of the fractions are multiplied by their additional factors. As a result of these actions, fractions that had different denominators turn into fractions that have the same denominators. And we already know how to add such fractions.

Example 1. Add fractions and

First of all, we find the least common multiple of the denominators of both fractions. The denominator of the first fraction is the number 3, and the denominator of the second fraction is the number 2. The least common multiple of these numbers is 6

LCM (2 and 3) = 6

Now back to fractions and . First, we divide the LCM by the denominator of the first fraction and get the first additional factor. LCM is the number 6, and the denominator of the first fraction is the number 3. Divide 6 by 3, we get 2.

The resulting number 2 is the first additional factor. We write it down to the first fraction. To do this, we make a small oblique line above the fraction and write down the found additional factor above it:

We do the same with the second fraction. We divide the LCM by the denominator of the second fraction and get the second additional factor. LCM is the number 6, and the denominator of the second fraction is the number 2. Divide 6 by 2, we get 3.

The resulting number 3 is the second additional factor. We write it to the second fraction. Again, we make a small oblique line above the second fraction and write the found additional factor above it:

Now we are all set to add. It remains to multiply the numerators and denominators of fractions by their additional factors:

Look closely at what we have come to. We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to add such fractions. Let's complete this example to the end:

Thus the example ends. To add it turns out.

Let's try to depict our solution using a picture. If you add pizzas to a pizza, you get one whole pizza and another sixth of a pizza:

Reduction of fractions to the same (common) denominator can also be depicted using a picture. Bringing the fractions and to a common denominator, we get the fractions and . These two fractions will be represented by the same slices of pizzas. The only difference will be that this time they will be divided into equal shares (reduced to the same denominator).

The first drawing shows a fraction (four pieces out of six) and the second picture shows a fraction (three pieces out of six). Putting these pieces together we get (seven pieces out of six). This fraction is incorrect, so we have highlighted the integer part in it. The result was (one whole pizza and another sixth pizza).

Note that we have painted this example in too much detail. AT educational institutions it is not customary to write in such a detailed manner. You need to be able to quickly find the LCM of both denominators and additional factors to them, as well as quickly multiply the additional factors found by your numerators and denominators. While at school, we would have to write this example as follows:

But there is also the other side of the coin. If detailed notes are not made at the first stages of studying mathematics, then questions of the kind “Where does that number come from?”, “Why do fractions suddenly turn into completely different fractions? «.

To make it easier to add fractions with different denominators, you can use the following step-by-step instructions:

1. Find the LCM of the denominators of fractions;
2. Divide the LCM by the denominator of each fraction and get an additional multiplier for each fraction;
3. Multiply the numerators and denominators of fractions by their additional factors;
4. Add fractions that have the same denominators;
5. If the answer turned out to be an improper fraction, then select its whole part;

Example 2 Find the value of an expression .

Let's use the instructions above.

Step 1. Find the LCM of the denominators of fractions

Find the LCM of the denominators of both fractions. The denominators of the fractions are the numbers 2, 3 and 4

Step 2. Divide the LCM by the denominator of each fraction and get an additional multiplier for each fraction

Divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 2. Divide 12 by 2, we get 6. We got the first additional factor 6. We write it over the first fraction:

Now we divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 3. We divide 12 by 3, we get 4. We got the second additional factor 4. We write it over the second fraction:

Now we divide the LCM by the denominator of the third fraction. LCM is the number 12, and the denominator of the third fraction is the number 4. Divide 12 by 4, we get 3. We got the third additional factor 3. We write it over the third fraction:

Step 3. Multiply the numerators and denominators of fractions by your additional factors

We multiply the numerators and denominators by our additional factors:

Step 4. Add fractions that have the same denominators

We came to the conclusion that fractions that had different denominators turned into fractions that have the same (common) denominators. It remains to add these fractions. Add up:

The addition didn't fit on one line, so we moved the remaining expression to the next line. This is allowed in mathematics. When an expression does not fit on one line, it is carried over to the next line, and it is necessary to put an equal sign (=) at the end of the first line and at the beginning of a new line. The equal sign on the second line indicates that this is a continuation of the expression that was on the first line.

Step 5. If the answer turned out to be an improper fraction, then select the whole part in it

Our answer is an improper fraction. We must single out the whole part of it. We highlight:

Got an answer

Subtraction of fractions with the same denominators

There are two types of fraction subtraction:

1. Subtraction of fractions with the same denominators
2. Subtraction of fractions with different denominators

First, let's learn how to subtract fractions with the same denominators. Everything is simple here. To subtract another from one fraction, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same.

For example, let's find the value of the expression . To solve this example, it is necessary to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged. Let's do this:

This example can be easily understood if we think of a pizza that is divided into four parts. If you cut pizzas from a pizza, you get pizzas:

Example 2 Find the value of the expression .

Again, from the numerator of the first fraction, subtract the numerator of the second fraction, and leave the denominator unchanged:

This example can be easily understood if we think of a pizza that is divided into three parts. If you cut pizzas from a pizza, you get pizzas:

Example 3 Find the value of an expression

This example is solved in exactly the same way as the previous ones. From the numerator of the first fraction, you need to subtract the numerators of the remaining fractions:

As you can see, there is nothing complicated in subtracting fractions with the same denominators. It is enough to understand the following rules:

1. To subtract another from one fraction, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged;
2. If the answer turned out to be an improper fraction, then you need to select the whole part in it.

Subtraction of fractions with different denominators

For example, a fraction can be subtracted from a fraction, since these fractions have the same denominators. But a fraction cannot be subtracted from a fraction, because these fractions have different denominators. In such cases, fractions must be reduced to the same (common) denominator.

The common denominator is found according to the same principle that we used when adding fractions with different denominators. First of all, find the LCM of the denominators of both fractions. Then the LCM is divided by the denominator of the first fraction and the first additional factor is obtained, which is written over the first fraction. Similarly, the LCM is divided by the denominator of the second fraction and a second additional factor is obtained, which is written over the second fraction.

The fractions are then multiplied by their additional factors. As a result of these operations, fractions that had different denominators turn into fractions that have the same denominators. And we already know how to subtract such fractions.

Example 1 Find the value of an expression:

These fractions have different denominators, so you need to bring them to the same (common) denominator.

First, we find the LCM of the denominators of both fractions. The denominator of the first fraction is the number 3, and the denominator of the second fraction is the number 4. The least common multiple of these numbers is 12

LCM (3 and 4) = 12

Now back to fractions and

Let's find an additional factor for the first fraction. To do this, we divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 3. Divide 12 by 3, we get 4. We write the four over the first fraction:

We do the same with the second fraction. We divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 4. Divide 12 by 4, we get 3. Write a triple over the second fraction:

Now we are all set for subtraction. It remains to multiply the fractions by their additional factors:

We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to subtract such fractions. Let's complete this example to the end:

Got an answer

Let's try to depict our solution using a picture. If you cut pizzas from a pizza, you get pizzas.

This is the detailed version of the solution. Being at school, we would have to solve this example in a shorter way. Such a solution would look like this:

Reduction of fractions and to a common denominator can also be depicted using a picture. Bringing these fractions to a common denominator, we get the fractions and . These fractions will be represented by the same pizza slices, but this time they will be divided into the same fractions (reduced to the same denominator):

The first drawing shows a fraction (eight pieces out of twelve), and the second picture shows a fraction (three pieces out of twelve). By cutting off three pieces from eight pieces, we get five pieces out of twelve. The fraction describes these five pieces.

Example 2 Find the value of an expression

These fractions have different denominators, so you first need to bring them to the same (common) denominator.

Find the LCM of the denominators of these fractions.

The denominators of the fractions are the numbers 10, 3 and 5. The least common multiple of these numbers is 30

LCM(10, 3, 5) = 30

Now we find additional factors for each fraction. To do this, we divide the LCM by the denominator of each fraction.

Let's find an additional factor for the first fraction. LCM is the number 30, and the denominator of the first fraction is the number 10. Divide 30 by 10, we get the first additional factor 3. We write it over the first fraction:

Now we find an additional factor for the second fraction. Divide the LCM by the denominator of the second fraction. LCM is the number 30, and the denominator of the second fraction is the number 3. Divide 30 by 3, we get the second additional factor 10. We write it over the second fraction:

Now we find an additional factor for the third fraction. Divide the LCM by the denominator of the third fraction. LCM is the number 30, and the denominator of the third fraction is the number 5. Divide 30 by 5, we get the third additional factor 6. We write it over the third fraction:

Now everything is ready for subtraction. It remains to multiply the fractions by their additional factors:

We came to the conclusion that fractions that had different denominators turned into fractions that have the same (common) denominators. And we already know how to subtract such fractions. Let's finish this example.

The continuation of the example will not fit on one line, so we move the continuation to the next line. Don't forget about the equal sign (=) on the new line:

The answer turned out to be a correct fraction, and everything seems to suit us, but it is too cumbersome and ugly. We should make it easier. What can be done? You can reduce this fraction.

To reduce a fraction, you need to divide its numerator and denominator by (gcd) the numbers 20 and 30.

So, we find the GCD of the numbers 20 and 30:

Now we return to our example and divide the numerator and denominator of the fraction by the found GCD, that is, by 10

Got an answer

Multiplying a fraction by a number

To multiply a fraction by a number, you need to multiply the numerator of the given fraction by this number, and leave the denominator the same.

Example 1. Multiply the fraction by the number 1.

Multiply the numerator of the fraction by the number 1

The entry can be understood as taking half 1 time. For example, if you take pizza 1 time, you get pizza

From the laws of multiplication, we know that if the multiplicand and the multiplier are interchanged, then the product will not change. If the expression is written as , then the product will still be equal to . Again, the rule for multiplying an integer and a fraction works:

This entry can be understood as taking half of the unit. For example, if there is 1 whole pizza and we take half of it, then we will have pizza:

Example 2. Find the value of an expression

Multiply the numerator of the fraction by 4

The answer is an improper fraction. Let's take a whole part of it:

The expression can be understood as taking two quarters 4 times. For example, if you take pizzas 4 times, you get two whole pizzas.

And if we swap the multiplicand and the multiplier in places, we get the expression. It will also be equal to 2. This expression can be understood as taking two pizzas from four whole pizzas:

Multiplication of fractions

To multiply fractions, you need to multiply their numerators and denominators. If the answer is an improper fraction, you need to select the whole part in it.

Example 1 Find the value of the expression .

Got an answer. It is desirable to reduce this fraction. The fraction can be reduced by 2. Then the final solution will take the following form:

The expression can be understood as taking a pizza from half a pizza. Let's say we have half a pizza:

How to take two-thirds from this half? First you need to divide this half into three equal parts:

And take two from these three pieces:

We'll get pizza. Remember what a pizza looks like divided into three parts:

One slice from this pizza and the two slices we took will have the same dimensions:

In other words, we are talking about the same pizza size. Therefore, the value of the expression is

Example 2. Find the value of an expression

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

The answer is an improper fraction. Let's take a whole part of it:

Example 3 Find the value of an expression

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

The answer turned out to be a correct fraction, but it will be good if it is reduced. To reduce this fraction, you need to divide the numerator and denominator of this fraction by the greatest common divisor (GCD) of the numbers 105 and 450.

So, let's find the GCD of the numbers 105 and 450:

Now we divide the numerator and denominator of our answer to the GCD that we have now found, that is, by 15

Representing an integer as a fraction

Any whole number can be represented as a fraction. For example, the number 5 can be represented as . From this, five will not change its meaning, since the expression means “the number five divided by one”, and this, as you know, is equal to five:

Reverse numbers

Now we will get acquainted with interesting topic in mathematics. It's called "reverse numbers".

Definition. Reverse to numbera is the number that, when multiplied bya gives a unit.

Let's substitute in this definition instead of a variable a number 5 and try to read the definition:

Reverse to number 5 is the number that, when multiplied by 5 gives a unit.

Is it possible to find a number that, when multiplied by 5, gives one? It turns out you can. Let's represent five as a fraction:

Then multiply this fraction by itself, just swap the numerator and denominator. In other words, let's multiply the fraction by itself, only inverted:

What will be the result of this? If we continue to solve this example, we get one:

This means that the inverse of the number 5 is the number, since when 5 is multiplied by one, one is obtained.

The reciprocal can also be found for any other integer.

You can also find the reciprocal for any other fraction. To do this, it is enough to turn it over.

Division of a fraction by a number

Let's say we have half a pizza:

Let's divide it equally between two. How many pizzas will each get?

It can be seen that after splitting half of the pizza, two equal slices were obtained, each of which makes up a pizza. So everyone gets a pizza.

Division of fractions is done using reciprocals. Reciprocals allow you to replace division with multiplication.

To divide a fraction by a number, you need to multiply this fraction by the reciprocal of the divisor.

Using this rule, we will write down the division of our half of the pizza into two parts.

So, you need to divide the fraction by the number 2. Here the dividend is a fraction and the divisor is 2.

To divide a fraction by the number 2, you need to multiply this fraction by the reciprocal of the divisor 2. The reciprocal of the divisor 2 is a fraction. So you need to multiply by

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 an integer from a fraction, you need to divide the numerator by the denominator, and the remainder will be a fraction.

The operation of adding to the correct common fraction whole number is not complicated 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 to know 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 from natural number fraction. 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

your child brought homework from school and you don't know how to solve it? Then this mini tutorial is for you!

How to add decimals

It is more convenient to add decimal fractions in a column. To add decimals, you need to follow one simple rule:

• The digit must be under the digit, comma under the comma.

As you can see in the example, whole units are under each other, tenths and hundredths are under each other. Now we add the numbers, ignoring the comma. What to do with a comma? The comma is transferred to the place where it stood in the discharge of integers.

Adding fractions with equal denominators

To perform addition with a common denominator, you need to keep the denominator unchanged, find the sum of the numerators and get a fraction, which will be the total amount.

Adding fractions with different denominators by finding a common multiple

The first thing to pay attention to is the denominators. The denominators are different, are they not divisible by one another, are they prime numbers. First you need to bring to one common denominator, there are several ways to do this:

• 1/3 + 3/4 = 13/12, to solve this example, we need to find the least common multiple (LCM) that will be divisible by 2 denominators. To denote the smallest multiple of a and b - LCM (a; b). In this example LCM (3;4)=12. Check: 12:3=4; 12:4=3.
• We multiply the factors and perform the addition of the resulting numbers, we get 13/12 - an improper fraction.

• In order to convert an improper fraction to a proper one, we divide the numerator by the denominator, we get the integer 1, the remainder 1 is the numerator and 12 is the denominator.

Adding fractions using cross multiplication

For adding fractions with different denominators, there is another way according to the “cross by cross” formula. This is a guaranteed way to equalize the denominators, for this you need to multiply the numerators with the denominator of one fraction and vice versa. If you are only on initial stage learning fractions, then this method is the easiest and most accurate, how to get the right result when adding fractions with different denominators.

The rules for adding fractions with different denominators are very simple.

Consider the rules for adding fractions with different denominators in steps:

1. Find the LCM (least common multiple) of the denominators. The resulting LCM will be the common denominator of the fractions;

2. Bring fractions to a common denominator;

3. Add fractions reduced to a common denominator.

On the simple example Learn how to add fractions with different denominators.

Example

An example of adding fractions with different denominators.

Add fractions with different denominators:

1	+	5
6		12

Let's decide step by step.

1. Find the LCM (least common multiple) of the denominators.

The number 12 is divisible by 6.

From this we conclude that 12 is the least common multiple of the numbers 6 and 12.

Answer: the nok of the numbers 6 and 12 is 12:

LCM(6, 12) = 12

The resulting NOC will be the common denominator of the two fractions 1/6 and 5/12.

2. Bring fractions to a common denominator.

In our example, only the first fraction needs to be reduced to a common denominator of 12, because the second fraction already has a denominator of 12.

Divide the common denominator of 12 by the denominator of the first fraction:

2 has an additional multiplier.

Multiply the numerator and denominator of the first fraction (1/6) by an additional factor of 2.