Compare negative fractions. Comparing fractions

In everyday life, we often have to compare fractional quantities. Most often this does not cause any difficulties. Indeed, everyone understands that half an apple is larger than a quarter. But when it comes to writing it down as a mathematical expression, it can get confusing. By applying the following mathematical rules, you can easily solve this problem.

How to compare fractions with the same denominators

Such fractions are most convenient to compare. In this case, use the rule:

Of two fractions with the same denominators but different numerators, the larger is the one whose numerator is larger, and the smaller is the one whose numerator is smaller.

For example, compare the fractions 3/8 and 5/8. The denominators in this example are equal, so we apply this rule. 3<5 и 3/8 меньше, чем 5/8.

Indeed, if you cut two pizzas into 8 slices, then 3/8 of a slice is always less than 5/8.

Comparing fractions with like numerators and unlike denominators

In this case, the sizes of the denominator shares are compared. The rule to be applied is:

If two fractions have equal numerators, then the fraction whose denominator is smaller is greater.

For example, compare the fractions 3/4 and 3/8. In this example, the numerators are equal, which means we use the second rule. The fraction 3/4 has a smaller denominator than the fraction 3/8. Therefore 3/4>3/8

Indeed, if you eat 3 slices of pizza divided into 4 parts, you will be more full than if you ate 3 slices of pizza divided into 8 parts.

Comparing fractions with different numerators and denominators

We apply the third rule:

Comparing fractions with different denominators should lead to comparing fractions with the same denominators. To do this, you need to reduce the fractions to a common denominator and use the first rule.

For example, you need to compare fractions and . To determine the larger fraction, we reduce these two fractions to a common denominator:

• Now let's find the second additional factor: 6:3=2. We write it above the second fraction:

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

Yandex.RTB R-A-339285-1

Comparing fractions with the same denominators

When comparing fractions with the same denominators, we work only with the numerator, which means we compare the fractions of the 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 are compared to the numbers 3 and 8.

This follows the rule for comparing fractions with the same denominators: of the existing fractions with the same exponents, 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, let's look at an example.

Example 1

Compare the given fractions 65 126 and 87 126.

Solution

Since the denominators of the fractions are the same, we move on 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 greater than 65,126.

Answer: 87 126 > 65 126 .

Comparing fractions with different denominators

Comparison of such fractions can be correlated with comparison of fractions with the same exponents, but there is a difference. Now you need to reduce the fractions to a common denominator.

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

• find a common denominator;
• compare fractions.

Let's look at these actions using an example.

Example 2

Compare the fractions 5 12 and 9 16.

Solution

First of all, it is necessary to reduce the fractions to a common denominator. This is done in this way: find the LCM, that is, the least common divisor, 12 and 16. This number is 48. It is necessary to add additional factors to the first fraction 5 12, this number is found from the quotient 48: 12 = 4, for the second fraction 9 16 – 48: 16 = 3. Let's write the result this way: 5 12 = 5 4 12 4 = 20 48 and 9 16 = 9 3 16 3 = 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 reduction to a common denominator. Let's look at an example. To compare fractions a b and c d, we reduce them to a common denominator, then b · d, that is, the product of these denominators. Then additional factors for fractions will be the denominators of the neighboring fraction. This will be written as a · d b · d and c · b d · b . Using the rule with identical denominators, we have that the comparison of fractions has been reduced to comparisons of the products a · d and c · b. From here we get the rule for comparing fractions with different denominators: if a · d > b · c, then a b > c d, but if a · d< b · c , тогда a b < c d . Рассмотрим сравнение с разными знаменателями.

Example 3

Compare the fractions 5 18 and 23 86.

Solution

This example has a = 5, b = 18, c = 23 and d = 86. Then it is necessary to calculate a·d and b·c. It follows that a · d = 5 · 86 = 430 and b · c = 18 · 23 = 414. But 430 > 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 the comparison can be made according to the previous point. The result of the comparison is possible by comparing their denominators.

There is a rule for comparing fractions with the same numerators : Of two fractions with the same numerators, the fraction that has the smaller denominator is greater and vice versa.

Let's look at an example.

Example 4

Compare the fractions 54 19 and 54 31.

Solution

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

Answer: 54 19 > 54 31 .

Otherwise, we can look at an example. There are two plates on which there are 1 2 pies, and another 1 16 anna. If you eat 1 2 pies, you will be full faster than just 1 16. Hence the conclusion is that the largest denominator with equal numerators is the smallest when comparing fractions.

Comparing a fraction with a 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. For a detailed look, we give an example below.

Example 4

A comparison needs to be made between 63 8 and 9 .

Solution

It is necessary to represent the number 9 as a fraction 9 1. Then we need to compare the fractions 63 8 and 9 1. This is followed by reduction to a common denominator by finding additional factors. After this 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 .

If you notice an error in the text, please highlight it and press Ctrl+Enter

Not only can prime numbers 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 fraction that has a larger numerator is larger.

Let's look at an example:

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

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

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

Comparing fractions with equal numerators.

If a fraction has the same numerators, then the fraction with the smaller denominator is greater.

This rule can be understood by giving 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, then we will divide it into 11 equal pieces. Now think about in what case would there be a larger piece of cake per guest? Of course, when 5 guests arrive, the piece of cake will be larger.

Or another example. We have 20 candies. We can give the candy equally to 4 friends or divide the candy equally among 10 friends. In what case will each friend have more candies? Of course, when we divide among only 4 friends, the number of candies for each friend will be greater. Let's check this problem mathematically.

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

If we solve these fractions before, 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 look at another example.

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

Since the numerators are the same, the fraction with the smaller denominator is larger.

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

Comparing 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, let's 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 larger than a proper fraction. 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 improper and is greater than 1.

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

The fraction \(\frac(11)(13)\) is correct and it is less than 1. Let’s compare:

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

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

Related questions:
How to compare fractions with different denominators?
Answer: you need 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 comparing fractions with the same numerators?
Answer: If fractions have the same numerators, the fraction with the smaller denominator is larger.

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

Solution:
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's reduce 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 with numerators, the fraction with the larger numerator is larger.

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

Example #2:
Compare a proper fraction to one?

Solution:
Any proper fraction is always less than 1.

Task #1:
The son and father were playing 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?

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

Let's compare fractions. We have different numerators and denominators, let's reduce them to one 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 has a better result.

Let's continue to study fractions. Today we will talk about their comparison. The topic is interesting and useful. It will allow a beginner to feel like a scientist in a white coat.

The essence of comparing fractions is to find out which of two fractions is greater or less.

To answer the question which of two fractions is greater or less, use such as more (>) or less (<).

Mathematicians have already taken care of ready-made rules that allow them to immediately answer the question of which fraction is larger and which is smaller. These rules can be safely applied.

We will look at all these rules and try to figure out why this happens.

Comparing fractions with the same denominators

The fractions that need to be compared are different. The best case is when the fractions have the same denominators, but different numerators. In this case, the following rule applies:

Of two fractions with the same denominator, the fraction with the larger numerator is greater. And accordingly, the fraction with the smaller numerator will be smaller.

For example, let's compare fractions and answer which of these fractions is larger. Here the denominators are the same, but the numerators are different. The fraction has a greater numerator than the fraction. This means the fraction is greater than . That's how we answer. You must answer using the more icon (>)

This example can be easily understood if we remember about pizzas, which are divided into four parts. There are more pizzas than pizzas:

Everyone will agree that the first pizza is bigger than the second.

Comparing fractions with the same numerators

The next case we can get into is when the numerators of the fractions are the same, but the denominators are different. For such cases, the following rule is provided:

Of two fractions with the same numerators, the fraction with the smaller denominator is greater. And accordingly, the fraction whose denominator is larger is smaller.

For example, let's compare the fractions and . These fractions have the same numerators. A fraction has a smaller denominator than a fraction. This means that the fraction is greater than the fraction. So we answer:

This example can be easily understood if we remember about pizzas, which are divided into three and four parts. There are more pizzas than pizzas:

Everyone will agree that the first pizza is bigger than the second.

Comparing fractions with different numerators and different denominators

It often happens that you have to compare fractions with different numerators and different denominators.

For example, compare fractions and . To answer the question which of these fractions is greater or less, you need to bring them to the same (common) denominator. Then you can easily determine which fraction is greater or less.

Let's bring the fractions to the same (common) denominator. Let's find the LCM of the denominators of both fractions. LCM of the denominators of the fractions and this is the number 6.

Now we find additional factors for each fraction. Let's divide the LCM by the denominator of the first fraction. LCM is the number 6, and the denominator of the first fraction is the number 2. Divide 6 by 2, we get an additional factor of 3. We write it above the first fraction:

Now let's find the second additional factor. Divide the LCM by the denominator of the second fraction. LCM is the number 6, and the denominator of the second fraction is the number 3. Divide 6 by 3, we get an additional factor of 2. We write it above the second fraction:

Let's multiply the fractions by their additional factors:

We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to compare such fractions. Of two fractions with the same denominator, the fraction with the larger numerator is greater:

The rule is the rule, and we will try to figure out why it is more than . To do this, select the whole part in the fraction. There is no need to highlight anything in the fraction, since the fraction is already proper.

After isolating the integer part in the fraction, we obtain the following expression:

Now you can easily understand why more than . Let's draw these fractions as pizzas:

2 whole pizzas and pizzas, more than pizzas.

Subtraction of mixed numbers. Difficult cases.

When subtracting mixed numbers, you can sometimes find that things aren't going as smoothly as you'd like. It often happens that when solving an example, the answer is not what it should be.

When subtracting numbers, the minuend must be greater than the subtrahend. Only in this case will a normal answer be received.

For example, 10−8=2

10 - decrementable

8 - subtrahend

2 - difference

The minuend 10 is greater than the subtrahend 8, so we get the normal answer 2.

Now let's see what happens if the minuend is less than the subtrahend. Example 5−7=−2

5—decreasable

7 - subtrahend

−2 — difference

In this case, we go beyond the limits of the numbers we are accustomed to and find ourselves in the world of negative numbers, where it is too early for us to walk, and even dangerous. To work with negative numbers, we need appropriate mathematical training, which we have not yet received.

If, when solving subtraction examples, you find that the minuend is less than the subtrahend, then you can skip such an example for now. It is permissible to work with negative numbers only after studying them.

The situation is the same with fractions. The minuend must be greater than the subtrahend. Only in this case will it be possible to get a normal answer. And in order to understand whether the fraction being reduced is greater than the fraction being subtracted, you need to be able to compare these fractions.

For example, let's solve the example.

This is an example of subtraction. To solve it, you need to check whether the fraction being reduced is greater than the fraction being subtracted. more than

so we can safely return to the example and solve it:

Now let's solve this example

We check whether the fraction being reduced is greater than the fraction being subtracted. We find that it is less:

In this case, it is wiser to stop and not continue further calculation. Let's return to this example when we study negative numbers.

It is also advisable to check mixed numbers before subtraction. For example, let's find the value of the expression .

First, let's check whether the mixed number being reduced is greater than the mixed number being subtracted. To do this, we convert mixed numbers to improper fractions:

We received fractions with different numerators and different denominators. To compare such fractions, you need to bring them to the same (common) denominator. We will not describe in detail how to do this. If you have difficulty, be sure to repeat.

After reducing the fractions to the same denominator, we obtain the following expression:

Now you need to compare the fractions and . These are fractions with the same denominators. Of two fractions with the same denominator, the fraction with the larger numerator is greater.

The fraction has a greater numerator than the fraction. This means that the fraction is greater than the fraction.

This means that the minuend is greater than the subtrahend

This means we can return to our example and safely solve it:

Example 3. Find the value of an expression

Let's check whether the minuend is greater than the subtrahend.

Let's convert mixed numbers to improper fractions:

We received fractions with different numerators and different denominators. Let us reduce these fractions to the same (common) denominator.

Of two fractions with the same denominators, the one with the larger numerator is greater, and the one with the smaller numerator is smaller.. In fact, the denominator shows how many parts one whole value was divided into, and the numerator shows how many such parts were taken.

It turns out that we divided each whole circle by the same number 5 , but they took different numbers of parts: the more they took, the larger the fraction you got.

Of two fractions with the same numerators, the one with the smaller denominator is greater, and the one with the larger denominator is smaller. Well, in fact, if we divide one circle into 8 parts, and the other on 5 parts and take one part from each of the circles. Which part will be larger?

Of course, from a circle divided by 5 parts! Now imagine that they were dividing not circles, but cakes. Which piece would you prefer, or rather, which share: a fifth or an eighth?

To compare fractions with different numerators and different denominators, you must reduce the fractions to their lowest common denominator and then compare fractions with the same denominators.

Examples. Compare common fractions:

Let's reduce these fractions to their lowest common denominator. NOZ(4 ; 6)=12. We find additional factors for each of the fractions. For the 1st fraction an additional factor 3 (12: 4=3 ). For the 2nd fraction an additional factor 2 (12: 6=2 ). Now we compare the numerators of the two resulting fractions with the same denominators. Since the numerator of the first fraction is less than the numerator of the second fraction ( 9<10) , then the first fraction itself is less than the second fraction.