When adding and subtracting algebraic fractions with different denominators, the fractions first lead to common denominator. This means that they find such a single denominator, which is divided by the original denominator of each algebraic fraction that is part of this expression.

As you know, if the numerator and denominator of a fraction are multiplied (or divided) by the same number other than zero, then the value of the fraction will not change. This is the main property of a fraction. Therefore, when fractions lead to a common denominator, in fact, the original denominator of each fraction is multiplied by the missing factor to a common denominator. In this case, it is necessary to multiply by this factor and the numerator of the fraction (it is different for each fraction).

For example, given the following sum of algebraic fractions:

It is required to simplify the expression, i.e., add two algebraic fractions. To do this, first of all, it is necessary to reduce the terms-fractions to a common denominator. The first step is to find a monomial that is divisible by both 3x and 2y. In this case, it is desirable that it be the smallest, i.e., find the least common multiple (LCM) for 3x and 2y.

For numerical coefficients and variables, the LCM is searched separately. LCM(3, 2) = 6 and LCM(x, y) = xy. Further, the found values ​​are multiplied: 6xy.

Now we need to determine by what factor we need to multiply 3x to get 6xy:
6xy ÷ 3x = 2y

This means that when reducing the first algebraic fraction to a common denominator, its numerator must be multiplied by 2y (the denominator has already been multiplied when reduced to a common denominator). The factor for the numerator of the second fraction is similarly searched for. It will be equal to 3x.

Thus, we get:

Further, it is already possible to act as with fractions with the same denominators: the numerators are added, and one common is written in the denominator:

After transformations, a simplified expression is obtained, which is one algebraic fraction, which is the sum of two original ones:

Algebraic fractions in the original expression may contain denominators that are polynomials rather than monomials (as in the above example). In this case, before finding a common denominator, factor the denominators (if possible). Further, the common denominator is collected from different factors. If the factor is in several initial denominators, then it is taken once. If the factor has different degrees in the original denominators, then it is taken with a larger one. For example:

Here the polynomial a 2 - b 2 can be represented as a product (a - b)(a + b). The factor 2a – 2b is expanded as 2(a – b). Thus, the common denominator will be equal to 2(a - b)(a + b).

I originally 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, not only numerical fractions have common denominators), 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 main property of a fraction comes to the rescue, which, let me remind you, sounds like this:

A fraction does not change if its numerator and denominator are multiplied by the same non-zero number.

Thus, if you choose the right factors, the denominators of the fractions will be equal - this process is called reduction to a common denominator. And the desired numbers, "leveling" the denominators, are called additional factors.

Why do you 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. Fraction comparison. Sometimes reduction to a common denominator greatly simplifies this task;
3. Solving problems on shares and percentages. Percentages are, in fact, ordinary expressions that contain fractions.

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

Multiplication "criss-cross"

The simplest and most reliable way, which is guaranteed to equalize the denominators. We will act "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:

As additional factors, consider the denominators of neighboring fractions. We get:

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

The only drawback of this method is that you have to count a lot, because the denominators are multiplied "ahead", and as a result, very large numbers can be obtained. That's the price of reliability.

Common divisor method

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

1. Look at the denominators before you go "thru" (i.e., "criss-cross"). Perhaps one of them (the one that is larger) is divisible by the other.
2. The number resulting from such a division will be an additional factor for a fraction with a smaller denominator.
3. At the same time, a fraction with a large denominator does not need to be multiplied by anything at all - this is the savings. At the same time, the probability of error is sharply reduced.

Task. Find expression values:

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

Note that the second fraction was not multiplied by anything at all. In fact, we have cut the amount of calculations in half!

By the way, I took the fractions in this example for a reason. If you're interested, try counting them using the criss-cross method. 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, again, it can only be applied when one of the denominators is divided by the other without a remainder. Which happens quite rarely.

Least common multiple method

When we reduce 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 is assumed in the "crosswise" method.

For example, for denominators 8 and 12, the number 24 is quite suitable, 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 manage to find such a number, the total amount of calculations will be minimal. Look at the examples:

Task. Find expression values:

Note that 234 = 117 2; 351 = 117 3 . Factors 2 and 3 are coprime (have no common divisors except 1), and factor 117 is common. Therefore LCM(234; 351) = 117 2 3 = 702.

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

Now let's bring the fractions to common denominators:

Note how useful the factorization of the original denominators turned out to be:

1. Having found the same factors, we immediately reached the least common multiple, which, generally speaking, is a non-trivial problem;
2. From the resulting expansion, you can find out which factors are “missing” for each of the fractions. For example, 234 3 \u003d 702, therefore, for the first fraction, the additional factor is 3.

To see how much of a win the least common multiple method gives, try calculating the same examples using the criss-cross method. Of course, without a calculator. I think after that comments will be redundant.

Do not think that such complex fractions will not be in real examples. They meet all the time, and the above tasks are not the limit!

The only problem is how to find this NOC. Sometimes everything is found in a few seconds, literally “by eye”, but in general this is a complex computational problem that requires separate consideration. Here we will not touch on this.

To solve examples with fractions, you need to be able to find the smallest common denominator. Below is a detailed instruction.

How to find the lowest common denominator - concept

The least common denominator (LCD) in simple words is the minimum number that is divisible by the denominators of all the fractions of a given example. In other words, it is called the Least Common Multiple (LCM). NOZ is used only if the denominators of the fractions are different.

How to find the lowest common denominator - examples

Let's consider examples of finding NOZ.

Calculate: 3/5 + 2/15.

Solution (Sequence of actions):

• We look at the denominators of fractions, make sure that they are different and the expressions are reduced as much as possible.
• We find the smallest number that is divisible by both 5 and 15. This number will be 15. Thus, 3/5 + 2/15 = ?/15.
• We figured out the denominator. What will be in the numerator? An additional multiplier will help us figure this out. An additional factor is the number obtained by dividing the NOZ by the denominator of a particular fraction. For 3/5, the additional factor is 3, since 15/5 = 3. For the second fraction, the additional factor is 1, since 15/15 = 1.
• Having found out the additional factor, we multiply it by the numerators of the fractions and add the resulting values. 3/5 + 2/15 = (3*3+2*1)/15 = (9+2)/15 = 11/15.

Answer: 3/5 + 2/15 = 11/15.

If in the example not 2, but 3 or more fractions are added or subtracted, then the NOZ must be searched for as many fractions as given.

Calculate: 1/2 - 5/12 + 3/6

Solution (sequence of actions):

• Finding the lowest common denominator. The minimum number divisible by 2, 12 and 6 is 12.
• We get: 1/2 - 5/12 + 3/6 = ?/12.
• We are looking for additional multipliers. For 1/2 - 6; for 5/12 - 1; for 3/6 - 2.
• We multiply by the numerators and assign the corresponding signs: 1/2 - 5/12 + 3/6 = (1 * 6 - 5 * 1 + 2 * 3) / 12 = 7/12.

Answer: 1/2 - 5/12 + 3/6 = 7/12.