Comparison of fractions. How to compare fractions with different denominators? Comparison of mixed numbers: rules, examples, solutions How to compare mixed numbers

Rules of comparison of ordinary fractions depend on the type of fraction (correct, incorrect, mixed fraction) and from denominator (the same or different) in compared fractions. Rule. To compare two fractions with the same denominators, it is necessary to compare their numerals. More (less) is the fraction that has a numerator greater (less). for exampleCompare fractions:

Comparison of the right, incorrect and mixed frains among themselves.

Rule. Incorrect and mixed fraction always more than any correct fraction. The correct fraction by definition is less than 1, therefore, incorrect and mixed fraction (having a number in its composition, equal to or greater than 1) more correct fractions.

Rule. Of the two mixed fractions more (less), which has a whole piece of fraction (less). In the equality of the integers of mixed fractions more (less), the fraction in which more (less) fractional part.

for exampleCompare fractions:

Similarly, the comparison of natural numbers on the numerical axis is a large fraction worth the right to less fraction.

This article considers the comparison of fractions. Here we will find out which of the frains is more or less, we will apply the rule, we will analyze examples of the solution. Compare fractions with both the same and different denominators. Producing a comparison of ordinary fraction with natural number.

Comparison of fractions with the same denominators

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

From here, a rule of comparison of fractions with the same denominants: from the existing fractions with the same indicators is considered to be greater that fraction, which has a numerator more and vice versa.

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

Example 1.

Comparison of the specified fractions 65 126 and 87 126.

Decision

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

Answer: 87 126 > 65 126 .

Comparison of fractions with different denominators

Comparison of such fractions can be correlated with the comparison of fractions with the same indicators, but there is a difference. Now it is necessary to bring the fraction to bring common denominator.

If there are fractions with different denominators, it is necessary to compare them:

• find a common denominator;
• compare fractions.

Consider these actions on the example.

Example 2.

Comparison fractions 5 12 and 9 16.

Decision

First of all, it is necessary to bring the fraction for a common denominator. This is done in this way: there is a nok, that is, the smallest common divider, 12 and 16. This is the number 48. It is necessary to protest additional faults to the first fraction 5 12, this number is located from the private 48: 12 \u003d 4, for the second fraction 9 16 - 48: 16 \u003d 3. We write the resulting: 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 get 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 is performed without bringing to a common denominator. Consider on the example. To compare the fractions a B and C d, lead to a common denominator, then b · d, that is, the product of these denominators. Then additional faults for fractions will be denominators of the neighboring fraction. This will be written so a · d b · d and c · b d · b. Using the rule with the same denominants, we have that the comparison of the frains was reduced to the comparisons of the works A · D and C · b. From here we obtain the comparison rule with different denominants: 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.

Comparison 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. It follows 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 more than 23 86.

Answer: 5 18 > 23 86 .

Comparison of fractions with the same numerals

If the fractions have the same numerals and different denominators, then you can compare the previous item. The result of comparison may occur when comparing their denominators.

There is a rule comparison of fractions with the same numerals : of the two fractions with the same numerals, more than the fraction, which has a smaller denominator and vice versa.

Consider on the example.

Example 4.

Comparison fractions 54 19 and 54 31.

Decision

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

Answer: 54 19 > 54 31 .

Otherwise, you can consider on the example. There are two plates on which 1 2 pie, Anna Other 1 16. If you eat 1 2 pie, you will be fast, rather than 1 16. Hence the conclusion that the largest denominator with the same numerals is the smallest when comparing fractions.

Comparison of fractions with natural number

A comparison of an ordinary fraction with a natural number goes like a comparison of two fractions with a record of denominators in the form of 1. For detailed consideration below, we give an example.

Example 4.

You must compare 63 8 and 9.

Decision

It is necessary to represent the number 9 in the form of fractions 9 1. Then we have the need to compare fractions 63 8 and 9 1. Next, it is necessary to bring to a common denominator by finding additional factors. After that, we see that you need to compare the 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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The purpose of the lesson:to form the skills of comparing mixed numbers.

Tasks lesson:

1. Teach to compare mixed numbers.
2. Develop thinking, attention.
3. Educating accuracy during the drawing of rectangles.

Equipment:table " Ordinary fractions", Set" Fruit and Shares "circles

During the classes

I. Organizational moment.

Record dates in a notebook.

What number today? What month? What year? What is the month? What is the lesson?

II. Oral work

1. Work on the plate:

347

999

200

127

• Read numbers.
• Name the biggest, the smallest number.
• Name numbers in descending order, increasing.
• Name the neighbors of each number.
• Comparison 1 and 2 numbers.
• Compare 2 and 3 numbers.
• How much 3 is less than 4.
• Explore the last number on the amount of discharge terms, name: how many units are among this, how much dozen, how many hundreds.

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

• Name fractional numbers (1 each each).
• Name the mixed numbers (1 each each)

3. With the help of a set on the magnets "Shares and fractions" to show numbers and.

Today we will learn to compare such numbers. Writing in notebook theme lesson.

III. Studying the topic of lesson.

1. Compare with the help of circles:

and

2. Build rectangles and note the numbers and.

Conclusion: Of the two mixed numbers, the number that has more than the integer.

3. Work on the textbook: p. 83, Figure 12.

(Depicted whole apples and shares.)

We read the rule in the textbook (teacher, then 2-3 times children)

IV. Physical traffic.

Conducted by the teacher and students for muscles of the back and torso.

The purpose of the lesson:to form the skills of comparing mixed numbers.

Tasks lesson:

1. Teach to compare mixed numbers.
2. Develop thinking, attention.
3. Educating accuracy during the drawing of rectangles.

Equipment:table "Ordinary fractions", set of "Fruit and Shares" circles

During the classes

I. Organizational moment.

Record dates in a notebook.

What number today? What month? What year? What is the month? What is the lesson?

II. Oral work

1. Work on the plate:

347

999

200

127

• Read numbers.
• Name the biggest, the smallest number.
• Name numbers in descending order, increasing.
• Name the neighbors of each number.
• Comparison 1 and 2 numbers.
• Compare 2 and 3 numbers.
• How much 3 is less than 4.
• Explore the last number on the amount of discharge terms, name: how many units are among this, how much dozen, how many hundreds.

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

• Name fractional numbers (1 each each).
• Name the mixed numbers (1 each each)

3. With the help of a set on the magnets "Shares and fractions" to show numbers and.

Today we will learn to compare such numbers. Writing in notebook theme lesson.

III. Studying the topic of lesson.

1. Compare with the help of circles:

and

2. Build rectangles and note the numbers and.

Conclusion: Of the two mixed numbers, the number that has more than the integer.

3. Work on the textbook: p. 83, Figure 12.

(Depicted whole apples and shares.)

We read the rule in the textbook (teacher, then 2-3 times children)

IV. Physical traffic.

Conducted by the teacher and students for muscles of the back and torso.

V. Fixing material.

1. Repetition on the "Ordinary Fruit" table.

(Numbers, when whole parts are identical, are considered in the next lesson.)

2. Compare.

Vi. Homework According to individual cards, learn the rule on page 83 of the textbook.

VII. Individual work on cards.

VIII. The outcome of the lesson.

Estimation.

To compare mixed fractions there is a sequence of actions from two steps:

Step 1. Compare entire parts of mixed
numbers (fractions).
Of two fractions with a different whole part more
That whose whole part is more.
Step 2. Compare fraction of mixed
numbers (fractions).
For two fractions with the same whole part
More that whose fractional part is more.

Comment:

Any mixed fraction (mixed
number) more than its whole part and less
Natural number following him.
For example,
2 < 2½ < 3;
1 < 1¼ < 2;
5 < 5¾ < 6.

Examples.

Next in the form of pictures will be given
Examples of mixed numbers (fractions).
Try to compare them first logically
And after - using the rule.

1)

What buttons are larger: blue or orange?

1) 3¾.
3½

What buttons are larger: blue or orange?

32\u003e
3½

What buttons are larger: blue or orange?

32\u003e
3½
Why did we make such a conclusion?
Number and orange and blue
Buttons can be expressed in the form of fractions, as shown above. Obviously, these
Mixed fractions (numbers) have the same integers, but different fractional.
According to the rule, in such cases you need to compare fractional parts. Consider them
separately.

What buttons are larger: blue or orange?

¾
>
½
Even just looking at these images we can say that
Orange slice button more than blue.
Yes, and if you compare the fractions yourself, we get that ¾\u003e ½.

10. What buttons are larger: blue or orange?

32\u003e
3½
Answer: More orange buttons