What does phraseologism mean to bring to a common denominator. Come to a common denominator

From the royal cause to the common ... The five-episode historical film "The Romanovs. The Tsar's affair" (producer and scriptwriter E. Chavchavadze, director G. Ogurnaya) at first glance examines the history of Russia from the point of view of the nobility, whose role in the formation of the state

I originally wanted to include the 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 factors correctly, 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 learn 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 without a remainder by the other, 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 equal the direct product of the denominators of the original fractions, as is assumed in the "cross-wise" 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 appreciate 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.

Nothing prevents you from seeing more than your own point of view.
V. Kabrut

I can say about someone that he is a scoundrel or mediocrity. Talk to him. And this is natural, because I have accumulated a lot of anger towards people. I do not want to meet them halfway, try to understand, show sympathy. Yes, I agree that by doing this I do not bring people closer to me, but only push them away. But basically, I don't care.
It's all about me. The only obstacle to contact is myself.
I blame myself for not giving him what he expects from me, not giving him pleasure, not filling his desires.
Let's say he loves to listen and read about high, spiritual things. He strives to feel this spirituality in himself and wants the high to be expressed in the appropriate high style. He's all ablaze with anticipatory pleasure!
And suddenly I come and speak about such an important subject to him in extremely low, ugly words. I'm deprecating the topic. This is completely against his expectations. And it makes you suffer! Suffer from the fact that I do not express myself in the way he would like to see, hear. Not the way it appears to him. He can't stand it and criticizes me... He doesn't want to suffer, pushes away suffering... No! Get someone else to tell me what I want to hear!
Why are we not ready to understand and accept another? Is it really better to consider only one's worldview as true, and oneself, if not perfect, then still a special personality?
If I could find something in common between me and another, on which we both would agree, then how much we would advance in understanding each other, we would contact each other ...
We would find that despite all the differences between us, there is still room for unity. Moreover, unification is impossible without the existence of disagreements.
Because it is only out of mutual misunderstanding that the opportunity to yield is born. Without mutual concessions, there is no connection.
In the meantime, my opinion, as it were, draws a line: everything, I think so! But there is also a general opinion. This thing is special. By absorbing information from others, we develop each other and come to a common denominator. Each of us is enriched from all the others.
As a result, while remaining ourselves, we acquire knowledge of others, add sensations and representations of them to ourselves. And then the individual strength of each increases at times ...
And then it’s already a stone’s throw to admiration for each other ...
By linking together, we will discover a new reality. Let's expand the boundaries of the body, and our common desires will show the way there, to a new world, a huge magnificent one. He will open for us, open us. He will take it to himself, and we will continue to live in it.
And so, trying together to tighten the knot of communication between us, we merge into one cell and another, and another ... Seven billion knots create a network on which the whole world rests!
I understood all this when I decided to accept the other as he is ... I threw out my own assessments, judgments ...
Opened up my heart... Let it do whatever it wants. The main thing is not to be afraid, the heart is not a brain, it does not know how to make mistakes.
I just crossed a certain threshold, beyond which my cruel nature remained, which created my character in me, gave me the appropriate upbringing.
Broke my stone heart with love...
We are at a crossroads. We need to come to a new way of thinking. To the idea that everything in the world is interconnected and interdependent. That love should rule the world...
In fact, since such thoughts are increasingly manifested in people, it means that we are already experiencing such a transition, we are already in it.
By acting on love, we will create a universal human, inclusive of all and included in all. This is the only thing we can do...
And we will be happy.