What is more than 9 5 or 7 6. Comparison of fractions: rules, examples, solutions

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.

1. 5/8 = 15/24 because 5 24 = 8 15 = 120;
2. 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:

1. The fraction a /b is greater than the fraction c /d ;
2. 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:

1. The expanding part of the check is always directed towards the larger number;
2. 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 intentionally 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 has a decimal place such that:

1. The digit in this digit in the fraction X is greater than the corresponding digit in the fraction Y;
2. All digits older than given in fractions X and Y are the same.

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

In other words, we sequentially look through 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:

1. 0,12 < 951, т.к. 0,12 = 000,12 - приписали два нуля слева. Очевидно, 0 < 9 (we are talking about senior level).
2. 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:

1. 0,029 ∨ 0,007;
2. 14,045 ∨ 15,5;
3. 0,00003 ∨ 0,0000099;
4. 1700,1 ∨ 0,99501.

By definition we have:

1. 0.029 > 0.007. The first two digits are the same (00 = 00), then the difference begins (2 > 0);
2. 14,045 < 15,5. Различие - во втором разряде: 4 < 5;
3. 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;
4. 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 comparison scheme decimal fractions not universal. This method can only compare positive numbers. In the general case, the algorithm of work is as follows:

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

Well, isn't it weak? Now consider concrete examples- and everything will become clear.

Task. Compare fractions:

1. 0,0027 ∨ 0,0072;
2. −0,192 ∨ −0,39;
3. 0,15 ∨ −11,3;
4. 19,032 ∨ 0,0919295;
5. −750 ∨ −1,45.

1. 0,0027 < 0,0072. Здесь все стандартно: две положительные дроби, различие начинается на 4 разряде (2 < 7);
2. -0.192 > -0.39. Fractions are negative, 2 digits are different. one< 3, но в силу отрицательности знак неравенства меняется на противоположный;
3. 0,15 > −11,3. positive number always more negative;
4. 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;
5. −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.

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

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

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

Mathematicians have already taken care of ready-made rules that allow you 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.

Lesson content

Comparing fractions with the same denominators

The fractions to be compared come across different. The most successful case is when fractions have the same denominators, but different numerators. In this case, the following rule applies:

From two fractions same denominators The greater is the fraction with the greater numerator. And accordingly, the smaller fraction will be, in which the numerator is smaller.

For example, let's compare fractions and and answer which of these fractions is larger. Here the denominators are the same, but the numerators are different. A fraction has a larger numerator than a fraction. So the fraction is greater than . So we answer. Reply using the more icon (>)

This example can be easily understood if we think about pizzas that are divided into four parts. more pizzas than pizzas:

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

Comparing fractions with the same numerator

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 numerator, the fraction with the smaller denominator is larger. The fraction with the larger denominator is therefore smaller.

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

This example can be easily understood if we think about pizzas that are divided into three and four parts. more pizzas than pizzas:

Everyone agrees 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 it will be easy to determine which fraction is greater or less.

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

Now we find additional factors for each fraction. 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 over 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 over the second fraction:

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 denominators, the larger fraction is the one with the larger numerator:

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

After selecting the integer part in the fraction, we get the following expression:

Now you can easily understand why more than . Let's draw these fractions in the form of pizzas:

2 whole pizzas and pizzas, more than pizzas.

Subtraction of mixed numbers. Difficult cases.

When subtracting mixed numbers, sometimes you find that things don't go 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 response be received.

For example, 10−8=2

10 - reduced

8 - subtracted

2 - difference

The minus 10 is greater than the subtracted 8, so we got the normal answer 2.

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

5 - reduced

7 - subtracted

−2 is the difference

In this case, we go beyond the numbers we are used 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 an appropriate mathematical background, which we have not yet received.

If, when solving examples for subtraction, 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 it will be possible to get a normal answer. And in order to understand whether the reduced fraction is greater than the subtracted one, you need to be able to compare these fractions.

For example, let's solve an example.

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

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

Now let's solve this example

Check if the reduced fraction is greater than the subtracted one. We find that it is less:

In this case, it is more reasonable to stop and not continue further calculation. We will return to this example when we study negative numbers.

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

First, check whether the reduced mixed number is greater than the subtracted one. To do this, we translate mixed numbers into improper fractions:

We got 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're having trouble, be sure to repeat.

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

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

A fraction has a larger numerator than a fraction. So the fraction is greater than the fraction.

This means that the minuend is greater than the subtrahend.

So we can go back to our example and boldly solve it:

Example 3 Find the value of an expression

Check if the minuend is greater than the subtrahend.

Convert mixed numbers to improper fractions:

We got fractions with different numerators and different denominators. We bring these fractions to the same (common) denominator.

Two unequal fractions are subject to further comparison to find out which fraction is larger and which fraction is smaller. To compare two fractions, there is a rule for comparing fractions, which we will formulate below, and we will also analyze examples of the application of this rule when comparing fractions with the same and different denominators. In conclusion, we will show how to compare fractions with the same numerators without reducing them to a common denominator, and also consider how to compare an ordinary fraction with a natural number.

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Comparing fractions with the same denominators

Comparing fractions with the same denominators is essentially a comparison of the number of equal shares. For example, the common fraction 3/7 determines 3 parts 1/7, and the fraction 8/7 corresponds to 8 parts 1/7, so comparing fractions with the same denominators 3/7 and 8/7 comes down to comparing the numbers 3 and 8, that is , to comparing numerators.

From these considerations it follows rule for comparing fractions with the same denominator: Of two fractions with the same denominator, the larger fraction is the one whose numerator is larger, and the smaller is the fraction whose numerator is smaller.

The stated rule explains how to compare fractions with the same denominators. Consider an example of applying the rule for comparing fractions with the same denominators.

Example.

Which fraction is larger: 65/126 or 87/126?

Decision.

The denominators of the compared ordinary fractions are equal, and the numerator 87 of the fraction 87/126 is greater than the numerator 65 of the fraction 65/126 (if necessary, see the comparison of natural numbers). Therefore, according to the rule for comparing fractions with the same denominators, the fraction 87/126 is greater than the fraction 65/126.

Answer:

Comparing fractions with different denominators

Comparing fractions with different denominators can be reduced to comparing fractions with the same denominators. To do this, you only need to compare common fractions lead to a common denominator.

So, to compare two fractions with different denominators, you need

• bring fractions to a common denominator;
• compare the resulting fractions with the same denominators.

Let's take a look at an example solution.

Example.

Compare the fraction 5/12 with the fraction 9/16.

Decision.

First, we bring these fractions with different denominators to a common denominator (see the rule and examples of reducing fractions to a common denominator). As a common denominator, take the lowest common denominator equal to LCM(12, 16)=48 . Then the additional factor of the fraction 5/12 will be the number 48:12=4 , and the additional factor of the fraction 9/16 will be the number 48:16=3 . We get and .

Comparing the resulting fractions, we have . Therefore, the fraction 5/12 is smaller than the fraction 9/16. This completes the comparison of fractions with different denominators.

Answer:

Let's get another way to compare fractions with different denominators, which will allow you to compare fractions without reducing them to a common denominator and all the difficulties associated with this process.

To compare fractions a / b and c / d, they can be reduced to a common denominator b d, equal to the product denominators of compared fractions. In this case, the additional factors of the fractions a/b and c/d are the numbers d and b, respectively, and the original fractions are reduced to fractions and with a common denominator b d . Recalling the rule for comparing fractions with the same denominators, we conclude that the comparison of the original fractions a/b and c/d has been reduced to comparing the products of a d and c b .

From this follows the following rule for comparing fractions with different denominators: if a d>b c , then , and if a d

Consider comparing fractions with different denominators in this way.

Example.

Compare the common fractions 5/18 and 23/86.

Decision.

In this example, a=5 , b=18 , c=23 and d=86 . Let's calculate the products a d and b c . We have a d=5 86=430 and b c=18 23=414 . Since 430>414 , the fraction 5/18 is greater than the fraction 23/86 .

Answer:

Comparing fractions with the same numerator

Fractions with the same numerators and different denominators can certainly be compared using the rules discussed in the previous paragraph. However, the result of comparing such fractions is easy to obtain by comparing the denominators of these fractions.

There is such rule for comparing fractions with the same numerator: Of two fractions with the same numerator, the one with the smaller denominator is the larger, and the one with the larger denominator is the smaller.

Let's consider an example solution.

Example.

Compare the fractions 54/19 and 54/31.

Decision.

Since the numerators of the compared fractions are equal, and the denominator 19 of the fraction is 54/19 less than the denominator 31 fractions 54/31, then 54/19 is greater than 54/31.

Of two fractions with the same denominator, the one with the larger numerator is the larger, and the one with the smaller numerator is the smaller.. In fact, after all, 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 each whole circle was divided by the same number 5 , but they took different amount parts: they took more - a large fraction and it turned out.

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

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

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

Examples. Compare ordinary fractions:

Let's bring these fractions to the smallest common denominator. NOZ(4 ; 6)=12. We find additional factors for each of the fractions. For the 1st fraction, an additional multiplier 3 (12: 4=3 ). For the 2nd fraction, an additional multiplier 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.

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 solve these fractions up to, 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. He will is 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.

Not proper fraction always more correct. because improper fraction 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 \(\frac(13)(16)\).

Decision:
Since there are no identical numerators or denominators, we apply the comparison rule with different denominators. We need to find a common denominator. 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?

Decision:
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?

Decision:
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.

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

Answer: Dad's result is better.