How to compare mixed fractions with different denominators. Comparison of mixed fractions. Comparison of a mixed number and a fraction

This article will focus on comparison of mixed numbers... First, we'll figure out which mixed numbers are called equal and which are unequal. Next, we will give a rule for comparing unequal mixed numbers, which allows you to find out which number is greater and which is less, and consider examples. Finally, we will focus on comparing mixed numbers with natural numbers and fractions.

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Equal and unequal mixed numbers

First you need to know which mixed numbers are called equal and which are unequal. Let us give the corresponding definitions.

Definition.

Equal mixed numbers - these are mixed numbers, which have equal and whole parts, and fractional parts.

In other words, two mixed numbers are said to be equal if their records completely coincide. If the records of mixed numbers are different, then such mixed numbers are called unequal.

Definition.

Unequal mixed numbers Are mixed numbers with different notations.

The sounded definitions allow you to determine at a glance whether the given mixed numbers are equal or not. For example, mixed numbers and equal, since their entries completely coincide. These numbers have equal whole parts and equal fractional parts. Mixed numbers and are unequal, since they have unequal whole parts. Other examples of unequal mixed numbers are and as well as and.

Sometimes it becomes necessary to find out which of two unequal mixed numbers is greater than the other, and which is less. We will consider how this is done in the next paragraph.

Comparison of mixed numbers

Comparing mixed numbers can be reduced to comparing ordinary fractions. To do this, it is enough to convert the mixed numbers to improper fractions.

For example, let's compare a mixed number and a mixed number by representing them as improper fractions. We have and. So the comparison of the original mixed numbers is reduced to the comparison of fractions with different denominators and. Since, then.

Comparing mixed numbers by comparing equal fractions is not the best solution. It is much more convenient to use the following mixed number comparison rule: greater is the mixed number, the integer part of which is greater, if the whole parts are equal, then the greater is the mixed number, the fractional part of which is greater.

Let's consider how the mixed numbers are compared according to the sounded rule. To do this, we will analyze the solutions of examples.

Example.

Which is the mixed number or more?

Decision.

The integer parts of the compared mixed numbers are equal, so the comparison is reduced to comparing the fractional parts and. Since then ... Thus, the mixed number is greater than the mixed number.

Answer:

Comparison of a mixed number and a natural number

Let's figure out how to compare a mixed number and a natural number.

It's fair a rule for comparing a mixed number with a natural number: if the integer part of the mixed number is less than this natural number, then the mixed number is less than this natural number, and if the whole part of the mixed number is greater than or equal to this mixed number, then the mixed number is greater than this natural number.

Let's look at examples of comparing a mixed number and a natural number.

Example.

Compare the numbers 6 and.

Decision.

The integer part of the mixed number is 9. Since it is greater than the natural number 6, then.

Answer:

Example.

Given a mixed number and a natural number 34, which of the numbers is less?

Decision.

The whole part of the mixed number is less than 34 (11<34 ), поэтому .

Answer:

The mixed number is less than 34.

Example.

Compare the number 5 and the mixed number.

Decision.

The integer part of this mixed number is equal to the natural number 5, therefore, this mixed number is greater than 5.

Answer:

In conclusion of this paragraph, we note that any mixed number is greater than one. This statement follows from the rule for comparing a mixed number and a natural number, and also from the fact that the integer part of any mixed number is either greater than 1 or equal to 1.

Comparison of a mixed number and a fraction

First let's say about comparison of a mixed number and a regular fraction... Any regular fraction is less than one (see right and wrong fractions), therefore, any regular fraction is less than any mixed number (since any mixed number is greater than 1).

This article looks at comparing fractions. Here we will find out which of the fractions is greater or less, apply the rule, analyze examples of solutions. Let's compare fractions with both the same and different denominators. Let's compare an ordinary fraction with a natural number.

Comparing fractions with the same denominator

When fractions with the same denominators are compared, we work only with the numerator, which means we are comparing the fractions of a number. If there is a fraction 3 7, then it has 3 parts 1 7, then the fraction 8 7 has 8 such parts. In other words, if the denominator is the same, the numerators of these fractions are compared, that is, 3 7 and 8 7, the numbers 3 and 8 are compared.

Hence the rule for comparing fractions with the same denominators follows: of the available fractions with the same indicators, the fraction with the larger numerator is considered larger and vice versa.

This suggests that you should pay attention to the numerators. To do this, consider an example.

Example 1

Compare the given fractions 65 126 and 87 126.

Decision

Since the denominators of the fractions are the same, go to the numerators. From the numbers 87 and 65, it is obvious that 65 is less. Based on the rule for comparing fractions with the same denominators, we have that 87 126 is more than 65 126.

Answer: 87 126 > 65 126 .

Comparison of fractions with different denominators

Comparing such fractions can be compared to comparing fractions with the same indicators, but there is a difference. Now it is necessary to bring the fractions to a common denominator.

If there are fractions with different denominators, to compare them you need:

find a common denominator;
compare fractions.

Let's consider these actions by example.

Example 2

Compare the fractions 5 12 and 9 16.

Decision

First of all, it is necessary to bring the fractions to a common denominator. This is done in this way: the LCM is found, that is, the least common divisor, 12 and 16. This number is 48. It is necessary to inscribe additional factors to the first fraction 5 12, this number is found from the quotient 48: 12 \u003d 4, for the second fraction 9 16 - 48: 16 \u003d 3. Let's write down the result in this way: 5 12 \u003d 5 4 12 4 \u003d 20 48 and 9 16 \u003d 9 3 16 3 \u003d 27 48.

After comparing the fractions, we find that 20 48< 27 48 . Значит, 5 12 меньше 9 16 .

Answer: 5 12 < 9 16 .

There is another way to compare fractions with different denominators. It runs without converting to a common denominator. Let's look at an example. To compare the fractions a b and c d, we bring to a common denominator, then b d, that is, the product of these denominators. Then the additional factors for fractions will be the denominators of the adjacent fraction. It will be written as a d b d and c b d b. Using the rule with the same denominators, we have that the comparison of fractions has been reduced to comparisons of the products a d and c b. From this we get the rule for comparing fractions with different denominators: if a d\u003e b c, then a b\u003e c d, but if a d< b · c , тогда a b < c d . Рассмотрим сравнение с разными знаменателями.

Example 3

Compare fractions 5 18 and 23 86.

Decision

This example has a \u003d 5, b \u003d 18, c \u003d 23, and d \u003d 86. Then it is necessary to calculate a · d and b · c. This implies that a d \u003d 5 86 \u003d 430 and b c \u003d 18 23 \u003d 414. But 430\u003e 414, then the given fraction 5 18 is greater than 23 86.

Answer: 5 18 > 23 86 .

Comparing fractions with the same numerators

If the fractions have the same numerators and different denominators, then you can perform the comparison according to the previous paragraph. The comparison result is possible when comparing their denominators.

There is a rule for comparing fractions with the same numerators : of two fractions with the same numerators, the larger the fraction with the lower denominator, and vice versa.

Let's look at an example.

Example 4

Compare fractions 54 19 and 54 31.

Decision

We have that the numerators are the same, which means that the fraction with the denominator 19 is greater than the fraction with the denominator 31. This is understandable based on the rule.

Answer: 54 19 > 54 31 .

Otherwise, you can consider an example. There are two plates on which 1 2 cakes, Anna the other 1 16. If you eat 1 2 cakes, then you will fill up faster than just 1 16. Hence the conclusion that the largest denominator with the same numerators is the smallest when comparing fractions.

Comparison of fraction with natural number

Comparing an ordinary fraction with a natural number is the same as comparing two fractions with the denominators written in the form 1. Below is an example for detailed consideration.

Example 4

Comparison of 63 8 and 9 is required.

Decision

It is necessary to represent the number 9 as a fraction 9 1. Then we have to compare the fractions 63 8 and 9 1. This is followed by reduction to a common denominator by finding additional factors. After that, we see that we need to compare fractions with the same denominators 63 8 and 72 8. Based on the comparison rule, 63< 72 , тогда получаем 63 8 < 72 8 . Значит, заданная дробь меньше целого числа 9 , то есть имеем 63 8 < 9 .

Answer: 63 8 < 9 .

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To compare mixed fractions, there is a two-step sequence of actions:

Step 1. Compare whole parts of mixed
numbers (fractions).
Of two fractions with different integer parts, more
the one whose whole part is larger.
Step 2. Compare the fractional part of mixed
numbers (fractions).
For two fractions with the same integer part
the larger is the one whose fractional part is larger.

Comment:

Any mixed fraction (mixed
number) is greater than its integer part and less
natural number following it.
For instance,
2 < 2½ < 3;
1 < 1¼ < 2;
5 < 5¾ < 6.

Examples.

Further in the form of pictures will be given
examples of mixed numbers (fractions).
Try to compare them logically first,
and then using the rule.

Which buttons are more: blue or orange?

1) 3¾
3½

Which buttons are more: blue or orange?

3¾\u003e
3½

Which buttons are more: blue or orange?

3¾\u003e
3½
Why did we come to this conclusion?
Amount of both orange and blue
buttons can be expressed as fractions as shown above. It is obvious that these
mixed fractions (numbers) have the same whole parts, but different fractions.
As a rule, in such cases it is the fractional parts that need to be compared. Consider them
separately.

Which buttons are more: blue or orange?

¾
>
½
Even just looking at these images, we can say that
the orange button piece is larger than the blue one.
And if we compare the fractions themselves, we get that\u003e ½.

10. Which buttons are more: blue or orange?

3¾\u003e
3½
Answer: More orange buttons

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.

Comparison of fractions with the same denominator.

If two fractions have the same denominator, then such fractions are easy to compare.

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

Let's consider an example:

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

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

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

Comparison of fractions with equal numerators.

If a fraction has the same numerators, then the fraction with the lower denominator is larger.

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

Or another example. We have 20 chocolates. We can distribute sweets equally to 4 friends or equally share sweets among 10 friends. When will each friend have more sweets? Of course, when we divide by only 4 friends, each friend will have more candies. Let's check this problem mathematically.

\\ (\\ frac (20) (4)\u003e \\ frac (20) (10) \\)

If we solve these fractions before we get the numbers \\ (\\ frac (20) (4) \u003d 5 \\) and \\ (\\ frac (20) (10) \u003d 2 \\). We get that 5\u003e 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 larger is the fraction where the denominator is smaller.

\\ (\\ 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) \u003d \\ frac (2 \\ times 7) (3 \\ times 7) \u003d \\ frac (14) (21) \\\\\\\\ & \\ frac (5) (7) \u003d \\ frac (5 \\ times 3) (7 \\ times 3) \u003d \\ frac (15) (21) \\\\\\\\ \\ end (align) \\)

Then we move on to comparing the numerators. The rule for comparing fractions with the same denominator.

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

Comparison.

An incorrect fraction is always more correct.Because the improper fraction is greater than 1 and the proper fraction is less than 1.

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

Fraction \\ (\\ frac (8) (7) \\) is incorrect and it is greater than 1.

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

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

\\ (1\u003e \\ frac (11) (13) \\)

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

Questions on the topic:
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 do you 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 right and wrong fraction. After classification of fractions, apply the appropriate comparison rule.

What is comparing fractions with the same numerators?
Answer: if the fractions have the same numerators, the larger fraction has the lower denominator.

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

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

\\ (\\ begin (align) & \\ frac (11) (12) \u003d \\ frac (11 \\ times 8) (12 \\ times 8) \u003d \\ frac (88) (96) \\\\\\\\ & \\ frac (13) (16) \u003d \\ frac (13 \\ times 6) (16 \\ times 6) \u003d \\ frac (78) (96) \\\\\\\\ \\ end (align) \\)

Compare fractions with numerators, the larger fraction which has a larger numerator.

\\ (\\ begin (align) & \\ frac (88) (96)\u003e \\ frac (78) (96) \\\\\\\\ & \\ frac (11) (12)\u003e \\ frac (13) (16) \\\\\\ Example # 2:

Compare a correct fraction with one?
Any regular fraction is always less than 1.

Decision:
Task number 1:

The son and father played football. The son hit the goal 5 times out of 10 approaches. And dad hit the goal 3 times out of 5 approaches. Whose result is better?
{!LANG-6ed6c9022494d08a6a1750b4ae120782!}

Decision:
The son hit 5 times out of 10 possible approaches. Let us write it as a fraction \\ (\\ frac (5) (10) \\).
Dad hit out of 5 possible approaches 3 times. Let us write it as a fraction \\ (\\ frac (3) (5) \\).

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

\\ (\\ begin (align) & \\ frac (3) (5) \u003d \\ frac (3 \\ times 2) (5 \\ times 2) \u003d \\ frac (6) (10) \\\\\\\\ & \\ frac (5) (ten)< \frac{6}{10}\\\\&\frac{5}{10} < \frac{3}{5}\\\\ \end{align}\)

Answer: dad has a better result.

The purpose of the lesson:develop skills for comparing mixed numbers.

Lesson Objectives:

Learn to compare mixed numbers.
Develop thinking, attention.
Cultivate accuracy when drawing rectangles.

Equipment:table "Common fractions", a set of circles "Fractions and fractions"

During the classes

I. Organizational moment.

Writing the date in a notebook.

What is the date today? What month? what year? What's the month? What's the lesson?

II. Oral work

1. Work on the plate:

347

999

200

127

Read the numbers.
Name the largest, smallest number.
Name the numbers in descending, ascending order.
Name the neighbors of each number.
Comparison of 1 and 2 numbers.
Compare numbers 2 and 3.
How much 3 is the number less than 4.
Decompose the last number into the sum of the digit terms, name: how many units are there in this number, how many tens, how many hundreds.

2. What numbers are we studying now? (Fractional.)