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
Let's 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:
In this lesson we will learn how to compare fractions with each other. This is a very useful skill that is necessary for solving a whole class of more complex problems.
First, let me remind you of the definition of equality of fractions:
Fractions a /b and c /d are said to be equal if ad = bc.
- 5/8 = 15/24, since 5 24 = 8 15 = 120;
- 3/2 = 27/18, since 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 said to be greater than the fraction c /d if a /b − c /d > 0.
A fraction x /y is said to be smaller than a fraction s /t if x /y − s /t< 0.
Designation:
Thus, comparing fractions comes down to subtracting them. Question: how not to get confused with the notations “more than” (>) and “less than” (<)? Для ответа просто приглядитесь к тому, как выглядят эти знаки:
- The flared part of the jackdaw always points towards the larger number;
- The sharp nose of a jackdaw always points to a lower number.
Often in problems where you need to compare numbers, a “∨” sign is placed between them. This is a daw with its nose down, which seems to hint: the larger of the numbers has not yet been determined.
Task. Compare numbers:
Following the definition, subtract the fractions from each other:
In each comparison, we were required to reduce fractions to a common denominator. Specifically, 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 “Adding and subtracting fractions” - it is very easy.
Comparison of decimals
In the case of decimal fractions, everything is much simpler. There is no need to subtract anything here - just compare the digits. It’s a good idea to remember what the significant part of a number is. For those who have forgotten, I suggest repeating the lesson “Multiplying and dividing decimals” - this 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 place in the fraction X is greater than the corresponding digit in the fraction Y;
- All digits higher than this for the 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 go through the decimal places one by one and look 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 decimal places? Remember: any number written in decimal form can have any number of zeros added to 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 - three zeros were added to the left. Now you can see that the difference begins in the first digit: 2 > 0.
Of course, in the given examples with zeros there was an obvious overkill, but the point is exactly this: fill in the missing bits 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 coincide (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 then in the first fraction there 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 detected in the first digit.
Unfortunately, the given scheme for comparing decimal fractions is not universal. This method can only compare positive numbers. In the general case, the operating algorithm is as follows:
- A positive fraction is always greater than a negative fraction;
- Two positive fractions are compared using the above algorithm;
- Two negative fractions are compared in the same way, but at the end the inequality sign is reversed.
Well, not bad? 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, 2nd digit is 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 number;
- 19.032 > 0.091. It is enough to rewrite the second fraction in the form 00.091 to see that the difference arises already in the 1st 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.
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.
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.
Lesson contentComparing 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. Let's 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 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.
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