Comparison of fractions: rules, examples, solutions. Number Comparison

When solving equations and inequalities, as well as problems with modules, it is required to locate the found roots on the real line.

As you know, the found roots can be different. They can be like this:, or they can be like this:,.

Accordingly, if the numbers are not rational but irrational (if you forgot what it is, look in the topic), or are complex mathematical expressions, then placing them on the number line is very problematic.

Moreover, calculators cannot be used in the exam, and an approximate calculation does not give 100% guarantees that one number is less than another (what if there is a difference between the compared numbers?).

Of course, you know that positive numbers are always greater than negative ones, and that if we represent a number axis, then when comparing, largest numbers will be located to the right than the smallest: ; ; etc.

But is it always so easy?

Where on the number line we mark .

How to compare them, for example, with a number? That's where the rub is...)

In this article, we will find a look at all the ways to compare numbers so that this is not a problem for you on the exam!

To begin with, let's talk in general terms about how and what to compare.

Important: it is desirable to make transformations in such a way that the inequality sign does not change! That is, in the course of transformations, it is undesirable to multiply by a negative number, and it is forbidden square if one of the parts is negative.

Fraction Comparison

So, we need to compare two fractions: and.

There are several options for how to do this.

Option 1. Bring fractions to a common denominator.

Let's write it as an ordinary fraction:

- (as you can see, I also reduced by the numerator and denominator).

Now we need to compare fractions:

Now we can continue to compare also in two ways. We can:

just reduce everything to a common denominator, presenting both fractions as improper (the numerator is greater than the denominator):
Which number is greater? That's right, the one whose numerator is greater, that is, the first.
“discard” (assume that we have subtracted one from each fraction, and the ratio of fractions to each other, respectively, has not changed) and we will compare the fractions:
We also bring them to a common denominator:
We got exactly the same result as in the previous case - the first number is greater than the second:
Let's also check whether we have correctly subtracted one? Let's calculate the difference in the numerator in the first calculation and the second:
1)
2)

So, we looked at how to compare fractions, bringing them to a common denominator. Let's move on to another method - comparing fractions by bringing them to a common ... numerator.

Option 2. Comparing fractions using reduction to a common numerator.

Yes Yes. This is not a typo. At school, this method is rarely taught to anyone, but very often it is very convenient. So that you quickly understand its essence, I will ask you only one question - “in what cases is the value of the fraction the largest?” Of course, you will say "when the numerator is as large as possible, and the denominator is as small as possible."

For example, you will definitely say that True? And if we need to compare such fractions: I think you, too, will immediately correctly put the sign, because in the first case they are divided into parts, and in the second into whole ones, which means that in the second case the pieces are very small, and accordingly:. As you can see, the denominators are different here, but the numerators are the same. However, in order to compare these two fractions, you do not need to find a common denominator. Although ... find it and see if the comparison sign is still wrong?

But the sign is the same.

Let's return to our original task - to compare and. We will compare and We bring these fractions not to a common denominator, but to a common numerator. For this it's simple numerator and denominator multiply the first fraction by. We get:

And. Which fraction is larger? That's right, the first one.

Option 3. Comparing fractions using subtraction.

How to compare fractions using subtraction? Yes, very simple. We subtract another from one fraction. If the result is positive, then the first fraction (reduced) is greater than the second (subtracted), and if negative, then vice versa.

In our case, let's try to subtract the first fraction from the second: .

As you already understood, we also translate into an ordinary fraction and get the same result -. Our expression becomes:

Further, we still have to resort to reduction to a common denominator. The question is how: in the first way, converting fractions into improper ones, or in the second, as if “removing” the unit? By the way, this action has a completely mathematical justification. Look:

I like the second option better, since multiplying in the numerator when reducing to a common denominator becomes many times easier.

We bring to a common denominator:

The main thing here is not to get confused about what number and where we subtracted from. Carefully look at the course of the solution and do not accidentally confuse the signs. We subtracted the first from the second number and got a negative answer, so? .. That's right, the first number is greater than the second.

Got it? Try comparing fractions:

Stop, stop. Do not rush to bring to a common denominator or subtract. Look: it can be easily converted to a decimal fraction. How much will it be? Right. What ends up being more?

This is another option - comparing fractions by reducing to a decimal.

Option 4. Comparing fractions using division.

Yes Yes. And so it is also possible. The logic is simple: when we divide a larger number by a smaller one, we get a number greater than one in the answer, and if we divide a smaller number by a larger one, then the answer falls on the interval from to.

To remember this rule, take for comparison any two prime numbers, for example, and. Do you know what's more? Now let's divide by. Our answer is . Accordingly, the theory is correct. If we divide by, what we get is less than one, which in turn confirms what is actually less.

Let's try to apply this rule on ordinary fractions. Compare:

Divide the first fraction by the second:

Let's shorten by and by.

The result is less, so the dividend is less than the divisor, that is:

We have analyzed all possible options for comparing fractions. As you can see there are 5 of them:

reduction to a common denominator;
reduction to a common numerator;
reduction to the form of a decimal fraction;
subtraction;
division.

Ready to workout? Compare fractions in the best way:

Let's compare the answers:

(- convert to decimal)
(divide one fraction by another and reduce by the numerator and denominator)
(select the whole part and compare fractions according to the principle of the same numerator)
(divide one fraction by another and reduce by the numerator and denominator).

2. Comparison of degrees

Now imagine that we need to compare not just numbers, but expressions where there is a degree ().

Of course, you can easily put a sign:

After all, if we replace the degree with multiplication, we get:

From this small and primitive example, the rule follows:

Now try to compare the following: . You can also easily put a sign:

Because if we replace exponentiation with multiplication...

In general, you understand everything, and it is not difficult at all.

Difficulties arise only when, when compared, the degrees have different bases and indicators. In this case, it is necessary to try to bring to a common basis. For example:

Of course, you know that this, accordingly, the expression takes the form:

Let's open the brackets and compare what happens:

A somewhat special case is when the base of the degree () is less than one.

If, then of two degrees or more, the one whose indicator is less.

Let's try to prove this rule. Let be.

We introduce some natural number as the difference between and.

Logical, isn't it?

Now let's pay attention to the condition - .

Respectively: . Hence, .

For example:

As you understand, we considered the case when the bases of the powers are equal. Now let's see when the base is in the range from to, but the exponents are equal. Everything is very simple here.

Let's remember how to compare this with an example:

Of course, you quickly calculated:

Therefore, when you come across similar problems for comparison, keep in mind some simple similar example that you can quickly calculate, and based on this example, put signs in a more complex one.

When performing transformations, remember that if you multiply, add, subtract or divide, then all actions must be done on both the left and right sides (if you multiply by, then you need to multiply both).

In addition, there are times when doing any manipulations is simply unprofitable. For example, you need to compare. In this case, it is not so difficult to raise to a power, and arrange the sign based on this:

Let's practice. Compare degrees:

Ready to compare answers? Here's what I got:

- the same as
- the same as
- the same as
- the same as

3. Comparison of numbers with a root

Let's start with what are roots? Do you remember this entry?

The root of a real number is a number for which equality holds.

Roots odd degree exist for negative and positive numbers, and even roots- Only for positive.

The value of the root is often an infinite decimal, which makes it difficult to accurately calculate it, so it is important to be able to compare roots.

If you forgot what it is and what it is eaten with -. If you remember everything, let's learn to compare the roots step by step.

Let's say we need to compare:

To compare these two roots, you do not need to do any calculations, just analyze the very concept of "root". Got what I'm talking about? Yes, about this: otherwise it can be written as the third power of some number, equal to the root expression.

What more? or? This, of course, you can compare without any difficulty. The larger the number we raise to a power, the larger the value will be.

So. Let's get the rule.

If the exponents of the roots are the same (in our case, this is), then it is necessary to compare the root expressions (and) - the larger the root number, the greater the value of the root with equal indicators.

Difficult to remember? Then just keep an example in mind and. That more?

The exponents of the roots are the same, since the root is square. The root expression of one number () is greater than another (), which means that the rule is really true.

But what if the radical expressions are the same, but the degrees of the roots are different? For example: .

It is also quite clear that when extracting a root of a greater degree, a smaller number will be obtained. Let's take for example:

Denote the value of the first root as, and the second - as, then:

You can easily see that there should be more in these equations, therefore:

If the root expressions are the same(in our case), and the exponents of the roots are different(in our case, this is and), then it is necessary to compare the exponents(And) - the larger the exponent, the smaller the given expression.

Try comparing the following roots:

Let's compare the results?

We have successfully dealt with this :). Another question arises: what if we are all different? And the degree, and the radical expression? Not everything is so difficult, we just need to ... "get rid" of the root. Yes Yes. Get rid of it.)

If we have different degrees and radical expressions, it is necessary to find the least common multiple (read the section about) for the root exponents and raise both expressions to a power equal to the least common multiple.

That we are all in words and in words. Here's an example:

We look at the indicators of the roots - and. Their least common multiple is .
Let's raise both expressions to a power:
Let's transform the expression and expand the brackets (more details in the chapter):
Let's consider what we have done, and put a sign:

4. Comparison of logarithms

So, slowly but surely, we approached the question of how to compare logarithms. If you don’t remember what kind of animal this is, I advise you to read the theory from the section first. Read? Then answer some important questions:

What is the argument of the logarithm and what is its base?
What determines whether a function is increasing or decreasing?

If you remember everything and learned it well - let's get started!

In order to compare logarithms with each other, you need to know only 3 tricks:

reduction to the same base;
casting to the same argument;
comparison with the third number.

First, pay attention to the base of the logarithm. You remember that if it is less, then the function decreases, and if it is greater, then it increases. This is what our judgments will be based on.

Consider comparing logarithms that have already been reduced to the same base or argument.

To begin with, let's simplify the problem: let in the compared logarithms equal grounds. Then:

The function, when increases on the interval from, means, by definition, then (“direct comparison”).
Example:- the bases are the same, respectively, we compare the arguments: , therefore:
The function, at, decreases on the interval from, which means, by definition, then (“reverse comparison”). - the bases are the same, respectively, we compare the arguments: , however, the sign of the logarithms will be “reverse”, since the function decreases: .

Now consider the cases where the bases are different, but the arguments are the same.

The base is bigger.
- . In this case, we use "reverse comparison". For example: - the arguments are the same, and. We compare the bases: however, the sign of the logarithms will be “reverse”:
Base a is in between.
- . In this case, we use "direct comparison". For example:
- . In this case, we use "reverse comparison". For example:

Let's write everything in a general tabular form:

, wherein	, wherein

Accordingly, as you already understood, when comparing logarithms, we need to bring to the same base, or argument, We come to the same base using the formula for moving from one base to another.

You can also compare logarithms with a third number and, based on this, infer what is less and what is more. For example, think about how to compare these two logarithms?

A little hint - for comparison, the logarithm will help you a lot, the argument of which will be equal.

Thought? Let's decide together.

We can easily compare these two logarithms with you:

Don't know how? See above. We just took it apart. What sign will be there? Right:

Agree?

Let's compare with each other:

You should get the following:

Now combine all our conclusions into one. Happened?

5. Comparison of trigonometric expressions.

What is sine, cosine, tangent, cotangent? What is the unit circle for and how to find the value of trigonometric functions on it? If you do not know the answers to these questions, I highly recommend that you read the theory on this topic. And if you know, then comparing trigonometric expressions with each other is not difficult for you!

Let's refresh our memory a bit. Let's draw a unit trigonometric circle and a triangle inscribed in it. Did you manage? Now mark on which side we have the cosine, and on which sine, using the sides of the triangle. (Of course, you remember that the sine is the ratio of the opposite side to the hypotenuse, and the cosine of the adjacent one?). Did you draw? Great! The final touch - put down where we will have it, where and so on. Put down? Phew) Compare what happened with me and you.

Phew! Now let's start the comparison!

Suppose we need to compare and . Draw these angles using the hints in the boxes (where we have marked where), laying out the points on the unit circle. Did you manage? Here's what I got.

Now let's lower the perpendicular from the points we marked on the circle to the axis ... Which one? Which axis shows the value of the sines? Right, . Here is what you should get:

Looking at this figure, which is bigger: or? Of course, because the point is above the point.

Similarly, we compare the value of cosines. We only lower the perpendicular onto the axis ... Right, . Accordingly, we look at which point is to the right (well, or higher, as in the case of sines), then the value is greater.

You probably already know how to compare tangents, right? All you need to know is what is tangent. So what is tangent?) That's right, the ratio of sine to cosine.

To compare the tangents, we also draw an angle, as in the previous case. Let's say we need to compare:

Did you draw? Now we also mark the values of the sine on the coordinate axis. Noted? And now indicate the values of the cosine on the coordinate line. Happened? Let's compare:

Now analyze what you have written. - we divide a large segment into a small one. The answer will be a value that is exactly greater than one. Right?

And when we divide the small one by the big one. The answer will be a number that is exactly less than one.

So the value of which trigonometric expression is greater?

Right:

As you now understand, the comparison of cotangents is the same, only in reverse: we look at how the segments that define cosine and sine relate to each other.

Try to compare the following trigonometric expressions yourself:

Examples.

Answers.

COMPARISON OF NUMBERS. AVERAGE LEVEL.

Which of the numbers is greater: or? The answer is obvious. And now: or? Not so obvious anymore, right? And so: or?

Often you need to know which of the numeric expressions is greater. For example, when solving an inequality, put points on the axis in the correct order.

Now I will teach you to compare such numbers.

If you need to compare numbers and, we put a sign between them (derived from the Latin word Versus or abbreviated vs. - against):. This sign replaces the unknown inequality sign (). Further, we will perform identical transformations until it becomes clear which sign should be put between the numbers.

The essence of comparing numbers is as follows: we treat the sign as if it were some kind of inequality sign. And with the expression, we can do everything we usually do with inequalities:

add any number to both parts (and subtract, of course, we can also)
“move everything in one direction”, that is, subtract one of the compared expressions from both parts. In place of the subtracted expression will remain: .
multiply or divide by the same number. If this number is negative, the inequality sign is reversed: .
Raise both sides to the same power. If this power is even, you must make sure that both parts have the same sign; if both parts are positive, the sign does not change when raised to a power, and if they are negative, then it changes to the opposite.
take the root of the same degree from both parts. If we extract the root of an even degree, you must first make sure that both expressions are non-negative.
any other equivalent transformations.

Important: it is desirable to make transformations in such a way that the inequality sign does not change! That is, in the course of transformations, it is undesirable to multiply by a negative number, and it is impossible to square if one of the parts is negative.

Let's look at a few typical situations.

1. Exponentiation.

Example.

Which is more: or?

Solution.

Since both sides of the inequality are positive, we can square to get rid of the root:

Example.

Which is more: or?

Solution.

Here, too, we can square, but this will only help us get rid of the square root. Here it is necessary to raise to such a degree that both roots disappear. This means that the exponent of this degree must be divisible by both (the degree of the first root) and by. This number is, so we raise it to the th power:

2. Multiplication by the conjugate.

Example.

Which is more: or?

Solution.

Multiply and divide each difference by the conjugate sum:

Obviously, the denominator on the right side is greater than the denominator on the left. Therefore, the right fraction is less than the left:

3. Subtraction

Let's remember that.

Example.

Which is more: or?

Solution.

Of course, we could square everything, regroup, and square again. But you can do something smarter:

It can be seen that each term on the left side is less than each term on the right side.

Accordingly, the sum of all terms on the left side is less than the sum of all terms on the right side.

But be careful! We were asked more...

The right side is larger.

Example.

Compare numbers and.

Solution.

Remember the trigonometry formulas:

Let us check in which quarters the points and lie on the trigonometric circle.

4. Division.

Here we also use a simple rule: .

With or, that is.

When the sign changes: .

Example.

Make a comparison: .

Solution.

5. Compare the numbers with the third number

If and, then (law of transitivity).

Example.

Compare.

Solution.

Let's compare the numbers not with each other, but with the number.

It's obvious that.

On the other side, .

Example.

Which is more: or?

Solution.

Both numbers are larger but smaller. Choose a number such that it is greater than one but less than the other. For example, . Let's check:

6. What to do with logarithms?

Nothing special. How to get rid of logarithms is described in detail in the topic. The basic rules are:

\[(\log _a)x \vee b(\rm( )) \Leftrightarrow (\rm( ))\left[ (\begin(array)(*(20)(l))(x \vee (a^ b)\;(\rm(at))\;a > 1)\\(x \wedge (a^b)\;(\rm(at))\;0< a < 1}\end{array}} \right.\] или \[{\log _a}x \vee {\log _a}y{\rm{ }} \Leftrightarrow {\rm{ }}\left[ {\begin{array}{*{20}{l}}{x \vee y\;{\rm{при}}\;a >1)\\(x \wedge y\;(\rm(at))\;0< a < 1}\end{array}} \right.\]

We can also add a rule about logarithms with different bases and the same argument:

It can be explained as follows: the larger the base, the less it will have to be raised in order to get the same one. If the base is smaller, then the opposite is true, since the corresponding function is monotonically decreasing.

Example.

Compare numbers: i.

Solution.

According to the above rules:

And now the advanced formula.

The rule for comparing logarithms can also be written shorter:

Example.

Which is more: or?

Solution.

Example.

Compare which of the numbers is greater: .

Solution.

COMPARISON OF NUMBERS. BRIEFLY ABOUT THE MAIN

1. Exponentiation

If both sides of the inequality are positive, they can be squared to get rid of the root

2. Multiplication by the conjugate

A conjugate is a multiplier that complements the expression to the formula for the difference of squares: - conjugate for and vice versa, because .

3. Subtraction

4. Division

At or that is

When the sign changes:

5. Comparison with the third number

If and then

6. Comparison of logarithms

Basic Rules:

Logarithms with different bases and the same argument:

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The rules for comparing ordinary fractions depend on the type of fraction (proper, improper, mixed fraction) and on the significant (same or different) of the compared fractions.

This section discusses options for comparing fractions that have the same numerator or denominator.

Rule. To compare two fractions with same denominators, it is necessary to compare their numerators. More (less) is the fraction whose numerator is greater (less).

For example, compare fractions:

Rule. To compare proper fractions with the same numerators, you need to compare their denominators. More (less) is the fraction whose denominator is less (greater).

For example, compare fractions:

Comparison of proper, improper and mixed fractions with each other

Rule. Improper and mixed fractions are always greater than any proper fraction.

A proper fraction is, by definition, less than 1, so improper and mixed fractions (having a number equal to or greater than 1) are greater than a proper fraction.

Rule. Of two mixed fractions, the larger (less) one is the one in which the integer part of the fraction is larger (less). When the integer parts of mixed fractions are equal, the fraction with the larger (less) fractional part is greater (less).

Not only prime numbers can be compared, but fractions too. After all, a fraction is the same number as, for example, natural numbers. You only need to know the rules by which fractions are compared.

Comparing fractions with the same denominators.

If two fractions have the same denominators, then it is easy to compare such fractions.

To compare fractions with the same denominators, you need to compare their numerators. The larger fraction has the larger numerator.

Consider an example:

Compare the fractions \(\frac(7)(26)\) and \(\frac(13)(26)\).

The denominators of both fractions are the same, equal to 26, so we compare the numerators. The number 13 is greater than 7. We get:

\(\frac(7)(26)< \frac{13}{26}\)

Comparison of fractions with equal numerators.

If a fraction has the same numerator, then the larger fraction is the one with the smaller denominator.

You can understand this rule if you give an example from life. We have cake. 5 or 11 guests can come to visit us. If 5 guests come, then we will cut the cake into 5 equal pieces, and if 11 guests come, we will divide it into 11 equal pieces. Now think about in which case one guest will have a larger piece of cake? Of course, when 5 guests come, the piece of cake will be bigger.

Or another example. We have 20 candies. We can evenly distribute candies to 4 friends or evenly divide candies between 10 friends. In which case will each friend have more candies? Of course, when we only divide by 4 friends, the number of candies each friend will have more. Let's check this problem mathematically.

\(\frac(20)(4) > \frac(20)(10)\)

If we do solve these fractions, then we get the numbers \(\frac(20)(4) = 5\) and \(\frac(20)(10) = 2\). We get that 5 > 2

This is the rule for comparing fractions with the same numerators.

Let's consider another example.

Compare fractions with the same numerator \(\frac(1)(17)\) and \(\frac(1)(15)\) .

Since the numerators are the same, the greater is the fraction where the denominator is less.

\(\frac(1)(17)< \frac{1}{15}\)

Comparison of fractions with different denominators and numerators.

To compare fractions with different denominators, you need to reduce the fractions to and then compare the numerators.

Compare the fractions \(\frac(2)(3)\) and \(\frac(5)(7)\).

First, find the common denominator of the fractions. It will be equal to the number 21.

\(\begin(align)&\frac(2)(3) = \frac(2 \times 7)(3 \times 7) = \frac(14)(21)\\\\&\frac(5) (7) = \frac(5 \times 3)(7 \times 3) = \frac(15)(21)\\\\ \end(align)\)

Then we move on to comparing numerators. Rule for comparing fractions with the same denominators.

\(\begin(align)&\frac(14)(21)< \frac{15}{21}\\\\&\frac{2}{3} < \frac{5}{7}\\\\ \end{align}\)

Comparison.

An improper fraction is always greater than a proper one. Because an improper fraction is greater than 1 and a proper fraction is less than 1.

Example:
Compare the fractions \(\frac(11)(13)\) and \(\frac(8)(7)\).

The fraction \(\frac(8)(7)\) is not correct and is greater than 1.

\(1 < \frac{8}{7}\)

The fraction \(\frac(11)(13)\) is correct and less than 1. Compare:

\(1 > \frac(11)(13)\)

We get, \(\frac(11)(13)< \frac{8}{7}\)

Related questions:
How do you compare fractions with different denominators?
Answer: it is necessary to bring the fractions to a common denominator and then compare their numerators.

How to compare fractions?
Answer: first you need to decide which category the fractions belong to: they have a common denominator, they have a common numerator, they do not have a common denominator and numerator, or you have a proper and improper fraction. After classifying fractions, apply the appropriate comparison rule.

What is the comparison of fractions with the same numerators?
Answer: If fractions have the same numerators, the larger fraction is the one with the smaller denominator.

Example #1:
Compare the fractions \(\frac(11)(12)\) and \(\frac(13)(16)\).

Solution:
Since there are no identical numerators or denominators, we apply the comparison rule with different denominators. We need to find a common denominator. The common denominator will be equal to 96. Let's bring the fractions to a common denominator. Multiply the first fraction \(\frac(11)(12)\) by an additional factor of 8, and multiply the second fraction \(\frac(13)(16)\) by 6.

\(\begin(align)&\frac(11)(12) = \frac(11 \times 8)(12 \times 8) = \frac(88)(96)\\\\&\frac(13) (16) = \frac(13 \times 6)(16 \times 6) = \frac(78)(96)\\\\ \end(align)\)

We compare fractions by numerators, that fraction is greater in which the numerator is greater.

\(\begin(align)&\frac(88)(96) > \frac(78)(96)\\\\&\frac(11)(12) > \frac(13)(16)\\\ \ \end(align)\)

Example #2:
Compare a proper fraction with a unit?

Solution:
Any proper fraction is always less than 1.

Task #1:
Father and son played football. The son of 10 approaches hit the gate 5 times. And dad hit the gate 3 times out of 5 approaches. Whose result is better?

Solution:
The son hit out of 10 possible approaches 5 times. We write as a fraction \(\frac(5)(10) \).
Dad hit out of 5 possible approaches 3 times. We write as a fraction \(\frac(3)(5) \).

Compare fractions. We have different numerators and denominators, let's bring it to the same denominator. The common denominator will be 10.

\(\begin(align)&\frac(3)(5) = \frac(3 \times 2)(5 \times 2) = \frac(6)(10)\\\\&\frac(5) (10)< \frac{6}{10}\\\\&\frac{5}{10} < \frac{3}{5}\\\\ \end{align}\)

Answer: Dad's result is better.

Lesson objectives:

Tutorials: learn how to compare fractions various kinds using various methods;
Developing: development of basic methods of mental activity, generalizations of comparison, highlighting the main thing; development of memory, speech.
Educational: learn to listen to each other, foster mutual assistance, a culture of communication and behavior.

Lesson steps:

1. Organizational.

Let's start the lesson with the words of the French writer A. France: "Learning can be fun .... To digest knowledge, you need to absorb it with appetite."

Let's follow this advice, try to be attentive, let's absorb knowledge with great desire, because. they will be useful to us in the future.

2. Actualization of students' knowledge.

1.) Frontal oral work of students.

Purpose: to repeat the material covered, which is required when learning a new one:

A) regular and improper fractions;
B) bringing fractions to a new denominator;
C) finding the lowest common denominator;

(Files are being worked on. Students have them available at every lesson. Answers are written on them with a marker, and then unnecessary information is erased.)

Tasks for oral work.

1. Name an extra fraction among the chain:

A) 5/6; 1/3; 7/10; 11/3; 4/7.
B) 2/6; 6/18; 1/3; 4/5; 4/12.

2. Bring fractions to a new denominator 30:

1/2; 2/3; 4/5; 5/6; 1/10.

Find the smallest common denominator of fractions:

1/5 and 2/7; 3/4 and 1/6; 2/9 and 1/2.

2.) Game situation.

Guys, our familiar clown (the students met him at the beginning of the school year) asked me to help him solve the problem. But I think you guys can help our friend without me. And the next task.

“Compare fractions:

a) 1/2 and 1/6;
b) 3/5 and 1/3;
c) 5/6 and 1/6;
d) 12/7 and 4/7;
e) 3 1/7 and 3 1/5;
f) 7 5/6 and 3 1/2;
g) 1/10 and 1;
h) 10/3 and 1;
i) 7/7 and 1.”

Guys, to help the clown, what should we learn?

The purpose of the lesson, tasks (students formulate independently).

The teacher helps them by asking questions:

a) which of the pairs of fractions can we already compare?

b) what tool do we need to compare fractions?

3. Guys in groups (in permanent multilevel).

Each group is given a task and instructions for its implementation.

First group : Compare mixed fractions:

a) 1 1/2 and 2 5/6;
b) 3 1/2 and 3 4/5

and derive a rule for equating mixed fractions with the same and different integer parts.

Instruction: Comparing mixed fractions (using a number beam)

compare the whole parts of fractions and draw a conclusion;
compare fractional parts (do not display the rule for comparing fractional parts);
make a rule - algorithm:

Second group: Compare fractions with different denominators and different numerators. (use number beam)

a) 6/7 and 9/14;
b) 5/11 and 1/22

Instruction

Compare denominators
Think about whether it is possible to reduce fractions to a common denominator
Start the rule with the words: “To compare fractions with different denominators, you need to ...”

Third group: Comparison of fractions with one.

a) 2/3 and 1;
b) 8/7 and 1;
c) 10/10 and 1 and formulate a rule.

Instruction

Consider all cases: (use number ray)

a) If the numerator of a fraction is equal to the denominator, ………;
b) If the numerator of a fraction is less than the denominator,………;
c) If the numerator of a fraction is greater than the denominator,………. .

Formulate a rule.

Fourth group: Compare fractions:

a) 5/8 and 3/8;
b) 1/7 and 4/7 and formulate a rule for comparing fractions with the same denominator.

Instruction

Use the number beam.

Compare the numerators and draw a conclusion, starting with the words: “From two fractions with the same denominators……”.

Fifth group: Compare fractions:

a) 1/6 and 1/3;
b) 4/9 and 4/3 using the number line:

0__.__.__1/6__.__.__1/3__.__.4/9__.__.__.__.__.__.__.__.__.__1__.__.__.__.__.__4/3__.__

Formulate a rule for comparing fractions with the same numerators.

Instruction

Compare the denominators and draw a conclusion, starting with the words:

“From two fractions with the same numerators………..”.

Sixth group: Compare fractions:

a) 4/3 and 5/6; b) 7/2 and 1/2 using number line

0__.__.__1/2__.__5/6__1__.__4/3__.__.__.__.__.__.__.__.__.__.__.__.__7/2__.__

Formulate a rule for comparing proper and improper fractions.

Instruction.

Think about which fraction is always greater, right or wrong.

4. Discussion of the conclusions made in groups.

Word to each group. Formulation of the rules of students and their comparison with the standards of the corresponding rules. Next, printouts of the rules for comparing various types of ordinary fractions are given to each student.

5. We return to the task set at the beginning of the lesson. (We solve the clown problem together).

6. Work in notebooks. Using the rules for comparing fractions, students, under the guidance of a teacher, compare fractions:

a) 8/13 and 8/25;
b) 11/42 and 3/42;
c) 7/5 and 1/5;
d) 18/21 and 7/3;
e) 2 1/2 and 3 1/5;
f) 5 1/2 and 5 4/3;

(it is possible to invite a student to the board).

7. Students are invited to perform a test comparing fractions for two options.

1 option.

1) compare fractions: 1/8 and 1/12

a) 1/8 > 1/12;
b) 1/8<1/12;
c) 1/8=1/12

2) Which is bigger: 5/13 or 7/13?

a) 5/13;
b) 7/13;
c) are equal

3) Which is smaller: 2/3 or 4/6?

a) 2/3;
b) 4/6;
c) are equal

4) Which of the fractions is less than 1: 3/5; 17/9; 7/7?

a) 3/5;
b) 17/9;
c) 7/7

5) Which of the fractions is greater than 1: ?; 7/8; 4/3?

a) 1/2;
b) 7/8;
c) 4/3

6) Compare fractions: 2 1/5 and 1 7/9

a) 2 1/5<1 7/9;
b) 2 1/5 = 1 7/9;
c) 2 1/5 >1 7/9

Option 2.

1) compare fractions: 3/5 and 3/10

a) 3/5 > 3/10;
b) 3/5<3/10;
c) 3/5=3/10

2) Which is bigger: 10/12 or 1/12?

a) are equal;
b) 10/12;
c) 1/12

3) Which is smaller: 3/5 or 1/10?

a) 3/5;
b) 1/10;
c) are equal

4) Which of the fractions is less than 1: 4/3; 1/15; 16/16?

a) 4/3;
b) 1/15;
c) 16/16

5) Which of the fractions is greater than 1: 2/5; 9/8; 11/12?

a) 2/5;
b) 9/8;
c) 11/12

6) Compare fractions: 3 1/4 and 3 2/3

a) 3 1/4 = 3 2/3;
b) 3 1/4 > 3 2/3;
c) 3 1/4< 3 2/3

Answers to the test:

Option 1: 1a, 2b, 3c, 4a, 5b, 6a

Option 2: 2a, 2b, 3b, 4b, 5b, 6c

8. Once again we return to the purpose of the lesson.

We check the comparison rules and give a differentiated homework:

1,2,3 groups - come up with two examples for each rule and solve them.

4,5,6 groups - No. 83 a, b, c, No. 84 a, b, c (from the textbook).

In this lesson we will learn how to compare fractions with each other. This is a very useful skill that is needed to solve a whole class of more complex problems.

First, let me remind you of the definition of the equality of fractions:

Fractions a /b and c /d are called equal if ad = bc.

5/8 = 15/24 because 5 24 = 8 15 = 120;
3/2 = 27/18 because 3 18 = 2 27 = 54.

In all other cases, the fractions are unequal, and one of the following statements is true for them:

The fraction a /b is greater than the fraction c /d ;
The fraction a /b is less than the fraction c /d .

The fraction a /b is called greater than the fraction c /d if a /b − c /d > 0.

A fraction x /y is called less than a fraction s /t if x /y − s /t< 0.

Designation:

Thus, the comparison of fractions is reduced to their subtraction. Question: how not to get confused with the notation "greater than" (>) and "less than" (<)? Для ответа просто приглядитесь к тому, как выглядят эти знаки:

The expanding part of the check is always directed towards the larger number;
The sharp nose of a jackdaw always indicates a lower number.

Often in tasks where you want to compare numbers, they put the sign "∨" between them. This is a jackdaw with its nose down, which, as it were, hints: the larger of the numbers has not yet been determined.

Task. Compare numbers:

Following the definition, we subtract the fractions from each other:

In each comparison, we needed to bring fractions to a common denominator. In particular, using the criss-cross method and finding the least common multiple. I deliberately did not focus on these points, but if something is not clear, take a look at the lesson " Addition and subtraction of fractions"- it is very easy.

Decimal Comparison

In the case of decimal fractions, everything is much simpler. There is no need to subtract anything here - just compare the digits. It will not be superfluous to remember what a significant part of a number is. For those who have forgotten, I suggest repeating the lesson “ Multiplication and division of decimal fractions"- it will also take just a couple of minutes.

A positive decimal X is greater than a positive decimal Y if it contains a decimal place such that:

The digit in this digit in the fraction X is greater than the corresponding digit in the fraction Y;

All digits older than given in fractions X and Y are the same.

12.25 > 12.16. The first two digits are the same (12 = 12), and the third is greater (2 > 1);
0,00697 < 0,01. Первые два разряда опять совпадают (00 = 00), а третий - меньше (0 < 1).

In other words, we are sequentially looking at the decimal places and looking for the difference. In this case, a larger number corresponds to a larger fraction.

However, this definition requires clarification. For example, how to write and compare digits up to the decimal point? Remember: any number written in decimal form can be assigned any number of zeros on the left. Here are a couple more examples:

0,12 < 951, т.к. 0,12 = 000,12 - приписали два нуля слева. Очевидно, 0 < 9 (речь идет о старшем разряде).
2300.5 > 0.0025, because 0.0025 = 0000.0025 - added three zeros on the left. Now you can see that the difference starts in the first bit: 2 > 0.

Of course, in the given examples with zeros there was an explicit enumeration, but the meaning is exactly this: fill in the missing digits on the left, and then compare.

Task. Compare fractions:

0,029 ∨ 0,007;

14,045 ∨ 15,5;

0,00003 ∨ 0,0000099;

1700,1 ∨ 0,99501.

By definition we have:

0.029 > 0.007. The first two digits are the same (00 = 00), then the difference begins (2 > 0);
14,045 < 15,5. Различие - во втором разряде: 4 < 5;
0.00003 > 0.0000099. Here you need to carefully count the zeros. The first 5 digits in both fractions are zero, but further in the first fraction is 3, and in the second - 0. Obviously, 3 > 0;
1700.1 > 0.99501. Let's rewrite the second fraction as 0000.99501, adding 3 zeros to the left. Now everything is obvious: 1 > 0 - the difference is found in the first digit.

Unfortunately, the above scheme for comparing decimal fractions is not universal. This method can only compare positive numbers. In the general case, the algorithm of work is as follows:

A positive fraction is always greater than a negative one;
Two positive fractions are compared according to the above algorithm;
Two negative fractions are compared in the same way, but at the end the inequality sign is reversed.

Well, isn't it weak? Now let's look at specific examples - and everything will become clear.

Task. Compare fractions:

0,0027 ∨ 0,0072;

−0,192 ∨ −0,39;

0,15 ∨ −11,3;

19,032 ∨ 0,0919295;

−750 ∨ −1,45.

0,0027 < 0,0072. Здесь все стандартно: две положительные дроби, различие начинается на 4 разряде (2 < 7);
-0.192 > -0.39. Fractions are negative, 2 digits are different. 1< 3, но в силу отрицательности знак неравенства меняется на противоположный;
0.15 > -11.3. A positive number is always greater than a negative one;
19.032 > 0.091. It is enough to rewrite the second fraction in the form of 00.091 to see that the difference occurs already in 1 digit;
−750 < −1,45. Если сравнить числа 750 и 1,45 (без минусов), легко видеть, что 750 >001.45. The difference is in the first category.