• Adding and subtracting fractions with the same denominator
• Adding and subtracting fractions with different denominators
• Understanding the NOC
• Converting fractions to the same denominator
• How to add an integer and a fraction

1 Addition and subtraction of fractions with the same denominator

To add fractions with the same denominator, add their numerators, and leave the denominator the same, for example:

To subtract fractions with the same denominator, 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 need to add their whole parts separately, and then add their fractional parts, and write down the result with a mixed fraction,

Example 1:

Example 2:

If when adding fractional parts turned out improper fraction, select the whole part from it and add it to the whole part, for example:

2 Addition and subtraction of 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 multiple 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 (LCM) is the smallest natural number that is divisible by both 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:

1. Factor these numbers
2. Take the largest expansion, and write these numbers as a product
3. Select in other decompositions the numbers that do not occur in the largest decomposition (or occur in it a smaller number of times), and add them to the product.
4. Multiply all the numbers in the product, this will be the LCM.

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

4 Converting 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 reduce fractions to one indicator, you first need to find the LCM (that is, the smallest number that is divisible by both denominators) of the denominators of these fractions, then add additional factors to the numerators of the fractions. You can find them by dividing the common denominator (LCM) 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.

5 How to add an integer and a fraction

In order to add an integer and a fraction, you just need to add this number in front of the fraction, and you get a mixed fraction, for example:

If we add an integer and a mixed fraction, we add that number to the whole fraction, for example:

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Addition and subtraction of fractions with the same denominator.

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This test tests the ability to add fractions with the same denominator. In this case, two rules must be observed:

• If the result is an incorrect fraction, you need to convert it to a mixed number.
• If the fraction can be abbreviated, be sure to abbreviate it, otherwise the wrong answer will be counted.

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Initially, I wanted to include common denominator methods in the Adding and Subtracting Fractions paragraph. But there was so much information, and its importance is so great (after all, common denominators are not only for numeric fractions) that it is better to study this issue separately.

So, let's say we have two fractions with different denominators. And we want to make sure that the denominators become the same. The basic property of the fraction comes to the rescue, which, recall, sounds like this:

The fraction will not change if its numerator and denominator are multiplied by the same nonzero number.

Thus, if the factors are chosen correctly, the denominators of the fractions become equal - this process is called common denominator reduction. And the required numbers, "leveling" the denominators, are called additional factors.

Why do you even need to bring fractions to a common denominator? Here are just a few reasons:

1. Addition and subtraction of fractions with different denominators. There is no other way to perform this operation;
2. Comparison of fractions. Sometimes converting to a common denominator makes this task much easier;
3. Solving problems for shares and percentages. Percentages are, in fact, common expressions that contain fractions.

There are many ways to find numbers that, when multiplied by, make the denominators of fractions equal. We will consider only three of them - in order of increasing complexity and, in a sense, efficiency.

Cross-multiplication

The easiest and safest way to guarantee equalization of the denominators. We will go ahead: we multiply the first fraction by the denominator of the second fraction, and the second by the denominator of the first. As a result, the denominators of both fractions will become equal to the product of the original denominators. Take a look:

Consider the denominators of neighboring fractions as additional factors. We get:

Yes, it's that simple. If you are just starting to learn fractions, it is better to work with this particular method - this way you will insure yourself against many mistakes and are guaranteed to get the result.

The only drawback this method- you have to count a lot, because the denominators are multiplied "ahead of time", and as a result, very large numbers can be obtained. This is the price to pay for reliability.

Common divisors method

This technique helps to greatly reduce the calculations, but, unfortunately, it is rarely used. The method is as follows:

1. Before you go ahead (that is, the criss-cross method), take a look at the denominators. Perhaps one of them (the one that is larger) is divided by the other.
2. The number obtained as a result of such division will be an additional factor for the fraction with a lower denominator.
3. In this case, a fraction with a large denominator does not need to be multiplied by anything at all - this is the saving. At the same time, the probability of error is sharply reduced.

Task. Find the values ​​of the expressions:

Note that 84: 21 = 4; 72: 12 = 6. Since in both cases one denominator is divisible by the other without a remainder, we apply the method of common factors. We have:

Note that the second fraction was never multiplied by anything at all. In fact, we have cut the amount of computation in half!

By the way, I took the fractions in this example for a reason. If you're curious, try counting them crosswise. After the reduction, the answers will be the same, but there will be much more work.

This is the strength of the method of common divisors, but, I repeat, it can be applied only when one of the denominators is divisible by the other without a remainder. Which is rare enough.

Least Common Multiple Method

When we bring fractions to a common denominator, we are essentially trying to find a number that is divisible by each of the denominators. Then we bring the denominators of both fractions to this number.

There are a lot of such numbers, and the smallest of them will not necessarily be equal to the direct product of the denominators of the original fractions, as it is assumed in the "criss-cross" method.

For example, for the denominators 8 and 12, the number 24 is fine, since 24: 8 = 3; 24: 12 = 2. This number is much less than the product 8 12 = 96.

The smallest number that is divisible by each of the denominators is called their least common multiple (LCM).

Notation: the least common multiple of a and b is denoted by LCM (a; b). For example, LCM (16; 24) = 48; LCM (8; 12) = 24.

If you can find such a number, the total amount of computation will be minimal. Take a look at examples:

Task. Find the values ​​of the expressions:

Note that 234 = 117 · 2; 351 = 117 3. The factors 2 and 3 are relatively prime (they have no common divisors except 1), and the factor 117 is common. Therefore, the LCM (234; 351) = 117 2 3 = 702.

Similarly, 15 = 5 · 3; 20 = 5 4. The factors 3 and 4 are relatively prime, and the factor 5 is common. Therefore, LCM (15; 20) = 5 3 4 = 60.

Now we bring the fractions to common denominators:

Note how helpful factoring the original denominators was:

1. Having found the same factors, we immediately arrived at the least common multiple, which, generally speaking, is a nontrivial problem;
2. From the resulting expansion, you can find out which factors are "missing" for each of the fractions. For example, 234 3 = 702, therefore, for the first fraction, the additional factor is 3.

To estimate how colossal gains the least common multiple method gives, try calculating the same examples using the criss-cross method. Without a calculator, of course. I think after that comments will be superfluous.

Do not think that there will be no such complex fractions in the real examples. They meet all the time, and the above tasks are not the limit!

The only problem is how to find this very NOC. Sometimes everything is found in a few seconds, literally "by eye", but on the whole this is a complex computational task that requires separate consideration. We will not touch on this here.

Fractions have different or the same denominator. The same denominator or in another way is called common denominator have a fraction. Common denominator example:

\ (\ frac (17) (5), \ frac (1) (5) \)

An example of different denominators for fractions:

\ (\ frac (8) (3), \ frac (2) (13) \)

How to bring a fraction to a common denominator?

The first fraction has a denominator of 3, the second has 13. You need to find a number that is divisible by both 3 and 13. This number is 39.

The first fraction must be multiplied by additional factor 13. So that the fraction does not change, we must multiply both the numerator by 13 and the denominator.

\ (\ frac (8) (3) = \ frac (8 \ times \ color (red) (13)) (3 \ times \ color (red) (13)) = \ frac (104) (39) \)

The second fraction is multiplied by an additional factor of 3.

\ (\ frac (2) (13) = \ frac (2 \ times \ color (red) (3)) (13 \ times \ color (red) (3)) = \ frac (6) (39) \)

We have brought the fraction to a common denominator:

\ (\ frac (8) (3) = \ frac (104) (39), \ frac (2) (13) = \ frac (6) (39) \)

Lowest common denominator.

Let's consider another example:

Let us reduce the fractions \ (\ frac (5) (8) \) and \ (\ frac (7) (12) \) to a common denominator.

The common denominator for numbers 8 and 12 can be numbers 24, 48, 96, 120, ..., it is customary to choose lowest common denominator in our case, this number is 24.

Least common denominator Is the smallest number by which the denominator of the first and second fraction is divided.

How do you find the lowest common denominator?
By enumerating numbers, by which the denominator of the first and second fractions is divided and the smallest of them is chosen.

We need the fraction with denominator 8 to multiply by 3, and the fraction with denominator 12 to multiply by 2.

\ (\ begin (align) & \ frac (5) (8) = \ frac (5 \ times \ color (red) (3)) (8 \ times \ color (red) (3)) = \ frac (15 ) (24) \\\\ & \ frac (7) (12) = \ frac (7 \ times \ color (red) (2)) (12 \ times \ color (red) (2)) = \ frac ( 14) (24) \\\\ \ end (align) \)

If you do not immediately succeed in bringing the fractions to the lowest common denominator, there is nothing wrong with that, further solving the example you may have to get the answer

The common denominator can be found for any two fractions, it can be the product of the denominators of these fractions.

For example:
Reduce the fractions \ (\ frac (1) (4) \) and \ (\ frac (9) (16) \) to the lowest common denominator.

The easiest way to find a common denominator is the product of the denominators 4⋅16 = 64. 64 is not the lowest common denominator. According to the assignment, you need to find exactly the lowest common denominator. Therefore, we are looking further. We need a number that can be divisible by both 4 and 16, this is the number 16. Bring the fraction to a common denominator, multiply the fraction with the denominator of 4 by 4, and the fraction with the denominator of 16 by one. We get:

\ (\ begin (align) & \ frac (1) (4) = \ frac (1 \ times \ color (red) (4)) (4 \ times \ color (red) (4)) = \ frac (4 ) (16) \\\\ & \ frac (9) (16) = \ frac (9 \ times \ color (red) (1)) (16 \ times \ color (red) (1)) = \ frac ( 9) (16) \\\\ \ end (align) \)

It often turns out that actions with fractions do not cause difficulties for students. Finding a common denominator becomes the main problem. To deal with this issue, you need to remember the rule for reducing fractions to a common denominator and understand why this common denominator is needed at all.

What is a fraction?

In grade 5, students are told that a fraction is a whole divided into pieces. Moreover, the denominator denotes the number of parts into which an object was divided, and the numerator denotes the number of these parts, which was taken for calculation.

But in mathematics, there is another definition: a fraction is an incomplete division operation. This means that just like any fraction can be turned into division, so any division can be turned into a fraction. For example:

$$ (5 \ over (7)) = 5: 7 $$

$$ 7: 13 = (7 \ over (13)) $$

$$ 12: 9 = (12 \ over (9)) $$

You can endlessly give examples, but the meaning will not change from this: the slash of the fraction replaces the division sign.

Why find a common denominator?

In order to add or subtract two fractions, you need to turn two division operations into one. This is only possible if the divisor is the same. In the form of formulas, it looks like this:

a: b-c: e = (a * e) :( b * e) - (c * b) :( b * e) = ((a * e) - (c * b)) :( b * e )

That is, in order to add or subtract fractions, you need to bring them to a common denominator. Otherwise, it will simply not be possible to correctly solve the example.

To multiply and divide fractions, you do not need to bring fractions to a common denominator. There is a different theoretical basis for these operations, which suggests a different course of action.

How to find the common denominator of fractions

In order to find the common denominator of fractions, you need to find the greatest common multiple of the denominators. Let's give an example, let's solve a small expression:

$$ (3 \ over (5)) + (7 \ over (15)) $$

Find the LCM of the denominators. The number 15 is divisible by the number 5, which means

$$ (3 \ over (5)) + (7 \ over (15)) = ((3 * 3) \ over (15)) + (7 \ over (15)) = (9 \ over (15)) + (7 \ over (15)) = (16 \ over (15)) = 1 (1 \ over (15)) $$ - note that as the numerator increases, so does the denominator. At the end of the solution of the example with fractions, if possible, you should select the whole part of the expression.

It is possible to bring fractions to a common denominator only by using the basic property of a fraction. The formulation of this property is as follows: if the numerator and denominator of a fraction are multiplied by the same number, then the value of the fraction will not change. This means that when reducing the fraction to a common denominator, it is necessary to take into account the increase in the numerator.

The LCM can be found analytically, as we did in the example. But most often you have to resort to the factorization. In order to find the LCM of two numbers, you should:

• Factor these numbers
• Check which prime factors lacking in decomposition.
• The number with the smallest number of factors is taken and numbers are added to its expansion, which are in other expansions, but are mostly absent. The number of numbers is also taken into account. This means that if in the main decomposition there is one number 3, and in other decompositions there are two numbers 3, then you need to multiply the main decomposition by two triples.

What have we learned?

We talked about bringing fractions to a common denominator. They told why it is needed, and what operations with fractions can be performed without reducing to a common denominator. They gave an example and told how the numerator changes when the fractions are brought to a common denominator.

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Common denominator of fractions

Fractions AND have the same denominator. They say they have common denominator 25. Fractions and have different denominators, but they can be brought to a common denominator using the basic property of fractions. To do this, find a number that is divisible by 8 and 3, for example, 24. Let us bring the fractions to the denominator 24, for this we multiply the numerator and denominator of the fraction by additional factor 3. The additional factor is usually written on the left above the numerator:

Multiply the numerator and denominator of the fraction by an additional factor of 8:

Let us bring the fractions to a common denominator. Most often, fractions result in the lowest common denominator, which is the lowest common multiple of the fraction's denominator. Since the LCM (8, 12) = 24, the fractions can be reduced to the denominator 24. Find the additional factors of the fractions: 24: 8 = 3, 24:12 = 2. Then

Several fractions can be brought to a common denominator.

Example. Let us bring the fractions to a common denominator. Since 25 = 5 2, 10 = 2 5, 6 = 2 3, then LCM (25, 10, 6) = 2 3 5 2 = 150.

Let's find additional factors of fractions and bring them to the denominator 150:

Comparison of fractions

In fig. 4.7 shows a segment AB of length 1. It is divided by 7 equal parts... The segment AC has a length and the segment AD has a length.

The length of the segment AD is greater than the length of the segment AC, i.e. the fraction is greater than the fraction

Of the two fractions with a common denominator, the one with the larger numerator is larger, i.e.

For example, or

To compare any two fractions, they are brought to a common denominator, and then the rule for comparing fractions with a common denominator is applied.

Example. Compare fractions

Solution. LCM (8, 14) = 56. Then Since 21> 20, then

If the first fraction is less than the second, and the second is less than the third, then the first is less than the third.

Proof. Let three fractions be given. Let's bring them to a common denominator. Let after that they have the form Since the first fraction is less

second, then r< s. Так как вторая дробь меньше третьей, то s < t. Из полученных неравенств для natural numbers it follows that r< t, тогда первая дробь меньше третьей.

The fraction is called correct if its numerator is less than the denominator.

The fraction is called wrong if its numerator is greater than or equal to the denominator.

For example, fractions are correct and fractions are incorrect.

The correct fraction is less than 1 and the improper fraction is greater than or equal to 1.