Adding and subtracting ordinary fractions. Subtracting ordinary fractions: rules, examples, solutions
Fractions are ordinary numbers and can also be added and subtracted. But due to the fact that they contain a denominator, more complex rules than for integers.
Let's consider the simplest case, when there are two fractions with the same denominators. Then:
To add fractions with the same denominators, you need to add their numerators and leave the denominator unchanged.
To subtract fractions with the same denominators, you need to subtract the numerator of the second from the numerator of the first fraction, and again leave the denominator unchanged.
Within each expression, the denominators of the fractions are equal. By definition of adding and subtracting fractions we get:
As you can see, it’s nothing complicated: we just add or subtract the numerators and that’s it.
But even in such simple actions, people manage to make mistakes. What is most often forgotten is that the denominator does not change. For example, when adding them, they also begin to add up, and this is fundamentally wrong.
Get rid of bad habit Adding the denominators is quite simple. Try the same thing when subtracting. As a result, the denominator will be zero, and the fraction will (suddenly!) lose its meaning.
Therefore, remember once and for all: when adding and subtracting, the denominator does not change!
Many people also make mistakes when adding several negative fractions. There is confusion with the signs: where to put a minus and where to put a plus.
This problem is also very easy to solve. It is enough to remember that the minus before the sign of a fraction can always be transferred to the numerator - and vice versa. And of course, don’t forget two simple rules:
- Plus by minus gives minus;
- Two negatives make an affirmative.
Let's look at all this with specific examples:
Task. Find the meaning of the expression:
In the first case, everything is simple, but in the second, let’s add minuses to the numerators of the fractions:
What to do if the denominators are different
Directly adding fractions with different denominators it is forbidden. At least, this method is unknown to me. However, the original fractions can always be rewritten so that the denominators become the same.
There are many ways to convert fractions. Three of them are discussed in the lesson “Reducing fractions to a common denominator”, so we will not dwell on them here. Let's look at some examples:
Task. Find the meaning of the expression:
In the first case, we reduce the fractions to a common denominator using the “criss-cross” method. In the second we will look for the NOC. Note that 6 = 2 · 3; 9 = 3 · 3. The last factors in these expansions are equal, and the first ones are relatively prime. Therefore, LCM(6, 9) = 2 3 3 = 18.
What to do if a fraction has an integer part
I can please you: different denominators in fractions are not the biggest evil. Much more errors occurs when an integer part is isolated in the fraction terms.
Of course, there are own addition and subtraction algorithms for such fractions, but they are quite complex and require a long study. Better use simple diagram, given below:
- Convert all fractions containing an integer part to improper ones. We obtain normal terms (even with different denominators), which are calculated according to the rules discussed above;
- Actually, calculate the sum or difference of the resulting fractions. As a result, we will practically find the answer;
- If this is all that was required in the problem, we perform the inverse transformation, i.e. getting rid of improper fraction, highlighting a whole part in it.
The rules for moving to improper fractions and highlighting the whole part are described in detail in the lesson “What is a numerical fraction”. If you don’t remember, be sure to repeat it. Examples:
Task. Find the meaning of the expression:
Everything is simple here. The denominators inside each expression are equal, so all that remains is to convert all fractions to improper ones and count. We have:
To simplify the calculations, I have skipped some obvious steps in the last examples.
A small note on the last two examples, where fractions with the highlighted ones are subtracted whole part. The minus before the second fraction means that the entire fraction is subtracted, and not just its whole part.
Re-read this sentence again, look at the examples - and think about it. This is where beginners make a huge number of mistakes. They love to give such tasks to tests. You will also encounter them several times in the tests for this lesson, which will be published shortly.
Summary: general calculation scheme
In conclusion, I will give a general algorithm that will help you find the sum or difference of two or more fractions:
- If one or more fractions have an integer part, convert these fractions to improper ones;
- Bring all the fractions to a common denominator in any way convenient for you (unless, of course, the writers of the problems did this);
- Add or subtract the resulting numbers according to the rules for adding and subtracting fractions with like denominators;
- If possible, shorten the result. If the fraction is incorrect, select the whole part.
Remember that it is better to highlight the whole part at the very end of the task, immediately before writing down the answer.
Examples with fractions are one of the basic elements of mathematics. There are many different types equations with fractions. Below is detailed instructions for solving examples of this type.
How to solve examples with fractions - general rules
To solve examples with fractions of any type, be it addition, subtraction, multiplication or division, you need to know the basic rules:
- To add fractional expressions with same denominator(the denominator is the number located at the bottom of the fraction, the numerator is at the top), you need to add their numerators, and leave the denominator the same.
- In order to subtract a second fractional expression (with the same denominator) from one fraction, you need to subtract their numerators and leave the denominator the same.
- To add or subtract fractions with different denominators, you need to find the lowest common denominator.
- In order to find a fractional product, you need to multiply the numerators and denominators, and, if possible, reduce.
- To divide a fraction by a fraction, you multiply the first fraction by the second fraction reversed.
How to solve examples with fractions - practice
Rule 1, example 1:
Calculate 3/4 +1/4.
According to Rule 1, if two (or more) fractions have the same denominator, you simply add their numerators. We get: 3/4 + 1/4 = 4/4. If a fraction has the same numerator and denominator, the fraction will equal 1.
Answer: 3/4 + 1/4 = 4/4 = 1.
Rule 2, example 1:
Calculate: 3/4 – 1/4
Using rule number 2, to solve this equation you need to subtract 1 from 3 and leave the denominator the same. We get 2/4. Since two 2 and 4 can be reduced, we reduce and get 1/2.
Answer: 3/4 – 1/4 = 2/4 = 1/2.
Rule 3, Example 1
Calculate: 3/4 + 1/6
Solution: Using the 3rd rule, we find the lowest common denominator. The least common denominator is the number that is divisible by the denominators of all fractional expressions in the example. Thus, we need to find the minimum number that will be divisible by both 4 and 6. This number is 12. We write 12 as the denominator. Divide 12 by the denominator of the first fraction, we get 3, multiply by 3, write 3 in the numerator *3 and + sign. Divide 12 by the denominator of the second fraction, we get 2, multiply 2 by 1, write 2*1 in the numerator. So, we get a new fraction with a denominator equal to 12 and a numerator equal to 3*3+2*1=11. 11/12.
Answer: 11/12
Rule 3, Example 2:
Calculate 3/4 – 1/6. This example is very similar to the previous one. We do all the same steps, but in the numerator instead of the + sign, we write a minus sign. We get: 3*3-2*1/12 = 9-2/12 = 7/12.
Answer: 7/12
Rule 4, Example 1:
Calculate: 3/4 * 1/4
Using the fourth rule, we multiply the denominator of the first fraction by the denominator of the second and the numerator of the first fraction by the numerator of the second. 3*1/4*4 = 3/16.
Answer: 3/16
Rule 4, Example 2:
Calculate 2/5 * 10/4.
This fraction can be reduced. In the case of a product, the numerator of the first fraction and the denominator of the second and the numerator of the second fraction and the denominator of the first are canceled.
2 cancels from 4. 10 cancels from 5. We get 1 * 2/2 = 1*1 = 1.
Answer: 2/5 * 10/4 = 1
Rule 5, Example 1:
Calculate: 3/4: 5/6
Using the 5th rule, we get: 3/4: 5/6 = 3/4 * 6/5. We reduce the fraction according to the principle of the previous example and get 9/10.
Answer: 9/10.
How to solve examples with fractions - fractional equations
Fractional equations are examples where the denominator contains an unknown. In order to solve such an equation, you need to use certain rules.
Let's look at an example:
Solve the equation 15/3x+5 = 3
Let us remember that you cannot divide by zero, i.e. the denominator value must not be zero. When solving such examples, this must be indicated. For this purpose, there is an OA (permissible value range).
So 3x+5 ≠ 0.
Hence: 3x ≠ 5.
x ≠ 5/3
At x = 5/3 the equation simply has no solution.
Having indicated the ODZ, in the best possible way decide given equation will get rid of fractions. To do this, we first represent all non-fractional values in the form of a fraction, in in this case number 3. We get: 15/(3x+5) = 3/1. To get rid of fractions you need to multiply each of them by the lowest common denominator. In this case it will be (3x+5)*1. Sequencing:
- Multiply 15/(3x+5) by (3x+5)*1 = 15*(3x+5).
- Open the brackets: 15*(3x+5) = 45x + 75.
- We do the same with right side equations: 3*(3x+5) = 9x + 15.
- Equate the left and right sides: 45x + 75 = 9x +15
- Move the X's to the left, numbers to the right: 36x = – 50
- Find x: x = -50/36.
- We reduce: -50/36 = -25/18
Answer: ODZ x ≠ 5/3. x = -25/18.
How to solve examples with fractions - fractional inequalities
Fractional inequalities of the type (3x-5)/(2-x)≥0 are solved using the number axis. Let's look at this example.
Sequencing:
- We equate the numerator and denominator to zero: 1. 3x-5=0 => 3x=5 => x=5/3
2. 2-x=0 => x=2 - We draw a number axis, writing the resulting values on it.
- Draw a circle under the value. There are two types of circles - filled and empty. A filled circle means that given value is included in the range of solutions. An empty circle indicates that this value is not included in the solution range.
- Since the denominator cannot be equal to zero, there will be an empty circle under the 2nd.
- To determine the signs, we substitute any number greater than two into the equation, for example 3. (3*3-5)/(2-3)= -4. the value is negative, which means we write a minus above the area after the two. Then substitute for X any value of the interval from 5/3 to 2, for example 1. The value is again negative. We write a minus. We repeat the same with the area located up to 5/3. We substitute any number less than 5/3, for example 1. Again, minus.
- Since we are interested in the values of x at which the expression will be greater than or equal to 0, and there are no such values (there are minuses everywhere), this inequality has no solution, that is, x = Ø (an empty set).
Answer: x = Ø
Fraction calculator designed for quickly calculating operations with fractions, it will help you easily add, multiply, divide or subtract fractions.
Modern schoolchildren begin studying fractions already in the 5th grade, and exercises with them become more complicated every year. Mathematical terms and quantities that we learn at school can rarely be useful to us in life. adult life. However, fractions, unlike logarithms and powers, are found quite often in everyday life (measuring distances, weighing goods, etc.). Our calculator is designed for quick operations with fractions.
First, let's define what fractions are and what they are. Fractions are the ratio of one number to another; it is a number consisting of an integer number of fractions of a unit.
Types of fractions:
- Ordinary
- Decimal
- Mixed
Example ordinary fractions:
The top value is the numerator, the bottom is the denominator. The dash shows us that the top number is divisible by the bottom number. Instead of this writing format, when the dash is horizontal, you can write differently. You can put an inclined line, for example:
1/2, 3/7, 19/5, 32/8, 10/100, 4/1
Decimals are the most popular type of fractions. They consist of an integer part and a fractional part, separated by a comma.
Example of decimal fractions:
0.2 or 6.71 or 0.125
Consist of a whole number and a fractional part. To find out the value of this fraction, you need to add the whole number and the fraction.
Example mixed fractions:
The fraction calculator on our website is able to quickly perform any mathematical operations with fractions online:
- Addition
- Subtraction
- Multiplication
- Division
To carry out the calculation, you need to enter numbers in the fields and select an action. For fractions, you need to fill in the numerator and denominator; the whole number may not be written (if the fraction is ordinary). Don't forget to click on the "equal" button.
It’s convenient that the calculator immediately provides the process for solving an example with fractions, and not just a ready-made answer. It is thanks to the expanded solution that you can use this material when solving school tasks and for better mastery of the material covered.
You need to perform the example calculation:
After entering the indicators into the form fields, we get:
To make your own calculation, enter the data in the form.
Fraction calculator
Enter two fractions:| + - * : | |||||||
Related sections.
The next action that can be performed with ordinary fractions is subtraction. In this material, we will look at how to correctly calculate the difference between fractions with like and unlike denominators, how to subtract a fraction from a natural number and vice versa. All examples will be illustrated with problems. Let us clarify in advance that we will only examine cases where the difference of fractions results in a positive number.
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How to find the difference between fractions with like denominators
Let's start right away with clear example: Let's say we have an apple that has been divided into eight parts. Let's leave five parts on the plate and take two of them. This action can be written like this:
As a result, we have 3 eighths left, since 5 − 2 = 3. It turns out that 5 8 - 2 8 = 3 8.
Thereby simple example We saw exactly how the subtraction rule works for fractions whose denominators are the same. Let's formulate it.
Definition 1
To find the difference between fractions with like denominators, you need to subtract the numerator of the other from the numerator of one, and leave the denominator the same. This rule can be written as a b - c b = a - c b.
We will use this formula in the future.
Let's take specific examples.
Example 1
Subtract the common fraction 17 15 from the fraction 24 15.
Solution
We see that these fractions have the same denominators. So all we need to do is subtract 17 from 24. We get 7 and add the denominator to it, we get 7 15.
Our calculations can be written as follows: 24 15 - 17 15 = 24 - 17 15 = 7 15
If necessary, you can shorten a complex fraction or select an entire part from an improper fraction to make counting more convenient.
Example 2
Find the difference 37 12 - 15 12.
Solution
Let's use the formula described above and calculate: 37 12 - 15 12 = 37 - 15 12 = 22 12
It is easy to notice that the numerator and denominator can be divided by 2 (we already talked about this earlier when we examined the signs of divisibility). Shortening the answer, we get 11 6. This is an improper fraction, from which we will select the whole part: 11 6 = 1 5 6.
How to find the difference of fractions with different denominators
This mathematical operation can be reduced to what we have already described above. To do this, we simply reduce the necessary fractions to the same denominator. Let's formulate a definition:
Definition 2
To find the difference between fractions that have different denominators, you need to reduce them to the same denominator and find the difference between the numerators.
Let's look at an example of how this is done.
Example 3
Subtract the fraction 1 15 from 2 9.
Solution
The denominators are different, and you need to reduce them to the smallest overall value. In this case, the LCM is 45. The first fraction requires an additional factor of 5, and the second - 3.
Let's calculate: 2 9 = 2 5 9 5 = 10 45 1 15 = 1 3 15 3 = 3 45
We have two fractions with the same denominator, and now we can easily find their difference using the algorithm described earlier: 10 45 - 3 45 = 10 - 3 45 = 7 45
A short summary of the solution looks like this: 2 9 - 1 15 = 10 45 - 3 45 = 10 - 3 45 = 7 45.
Do not neglect reducing the result or separating an entire part from it, if necessary. IN in this example we don't need to do that.
Example 4
Find the difference 19 9 - 7 36.
Solution
Let's reduce the fractions indicated in the condition to the lowest common denominator 36 and get 76 9 and 7 36, respectively.
We calculate the answer: 76 36 - 7 36 = 76 - 7 36 = 69 36
The result can be reduced by 3 and get 23 12. The numerator is greater than the denominator, which means we can select the whole part. The final answer is 1 11 12.
A short summary of the entire solution is 19 9 - 7 36 = 1 11 12.
How to subtract a natural number from a common fraction
This action can also be easily reduced to simple subtraction of ordinary fractions. This can be done by representing a natural number as a fraction. Let's show it with an example.
Example 5
Find the difference 83 21 – 3 .
Solution
3 is the same as 3 1. Then you can calculate it like this: 83 21 - 3 = 20 21.
If the condition requires subtracting an integer from an improper fraction, it is more convenient to first separate the integer from it by writing it as a mixed number. Then the previous example can be solved differently.
From the fraction 83 21, when separating the whole part, you get 83 21 = 3 20 21.
Now let's just subtract 3 from it: 3 20 21 - 3 = 20 21.
How to subtract a fraction from a natural number
This action is done similarly to the previous one: we rewrite the natural number as a fraction, bring both to a single denominator and find the difference. Let's illustrate this with an example.
Example 6
Find the difference: 7 - 5 3 .
Solution
Let's make 7 a fraction 7 1. We do subtraction and conversion final result, isolating the whole part from it: 7 - 5 3 = 5 1 3.
There is another way to make calculations. It has some advantages that can be used in cases where the numerators and denominators of the fractions in the problem are large numbers.
Definition 3
If the fraction that needs to be subtracted is proper, then the natural number from which we are subtracting must be represented as the sum of two numbers, one of which is equal to 1. After this, you need to subtract the desired fraction from one and get the answer.
Example 7
Calculate the difference 1 065 - 13 62.
Solution
The fraction to be subtracted is proper because its numerator is less than its denominator. Therefore, we need to subtract one from 1065 and subtract the desired fraction from it: 1065 - 13 62 = (1064 + 1) - 13 62
Now we need to find the answer. Using the properties of subtraction, the resulting expression can be written as 1064 + 1 - 13 62. Let's calculate the difference in brackets. To do this, let's imagine unit as a fraction 1 1.
It turns out that 1 - 13 62 = 1 1 - 13 62 = 62 62 - 13 62 = 49 62.
Now let's remember about 1064 and formulate the answer: 1064 49 62.
We use old way to prove that it is less convenient. These are the calculations we would come up with:
1065 - 13 62 = 1065 1 - 13 62 = 1065 62 1 62 - 13 62 = 66030 62 - 13 62 = = 66030 - 13 62 = 66017 62 = 1064 4 6
The answer is the same, but the calculations are obviously more cumbersome.
We looked at the case where we need to subtract a proper fraction. If it is incorrect, we replace it with a mixed number and subtract according to familiar rules.
Example 8
Calculate the difference 644 - 73 5.
Solution
The second fraction is an improper fraction, and the whole part must be separated from it.
Now we calculate similarly to the previous example: 630 - 3 5 = (629 + 1) - 3 5 = 629 + 1 - 3 5 = 629 + 2 5 = 629 2 5
Properties of subtraction when working with fractions
The properties that subtraction has natural numbers, also apply to cases of subtracting ordinary fractions. Let's look at how to use them when solving examples.
Example 9
Find the difference 24 4 - 3 2 - 5 6.
Solution
We have already solved similar examples when we looked at subtracting a sum from a number, so we follow the already known algorithm. First, let's calculate the difference 25 4 - 3 2, and then subtract the last fraction from it:
25 4 - 3 2 = 24 4 - 6 4 = 19 4 19 4 - 5 6 = 57 12 - 10 12 = 47 12
Let's transform the answer by separating the whole part from it. Result - 3 11 12.
A short summary of the entire solution:
25 4 - 3 2 - 5 6 = 25 4 - 3 2 - 5 6 = 25 4 - 6 4 - 5 6 = = 19 4 - 5 6 = 57 12 - 10 12 = 47 12 = 3 11 12
If the expression contains both fractions and natural numbers, it is recommended to group them by type when calculating.
Example 10
Find the difference 98 + 17 20 - 5 + 3 5.
Solution
Knowing the basic properties of subtraction and addition, we can group numbers as follows: 98 + 17 20 - 5 + 3 5 = 98 + 17 20 - 5 - 3 5 = 98 - 5 + 17 20 - 3 5
Let's complete the calculations: 98 - 5 + 17 20 - 3 5 = 93 + 17 20 - 12 20 = 93 + 5 20 = 93 + 1 4 = 93 1 4
If you notice an error in the text, please highlight it and press Ctrl+Enter
Note! Before writing your final answer, see if you can shorten the fraction you received.
Subtracting fractions with like denominators, examples:
,
,
Subtracting a proper fraction from one.
If it is necessary to subtract a fraction from a unit that is proper, the unit is converted to the form of an improper fraction, its denominator is equal to the denominator of the subtracted fraction.
An example of subtracting a proper fraction from one:
Denominator of the fraction to be subtracted = 7 , i.e., we represent one as an improper fraction 7/7 and subtract it according to the rule for subtracting fractions with like denominators.
Subtracting a proper fraction from a whole number.
Rules for subtracting fractions - correct from a whole number (natural number):
- We convert given fractions that contain an integer part into improper ones. We obtain normal terms (it doesn’t matter if they have different denominators), which we calculate according to the rules given above;
- Next, we calculate the difference between the fractions that we received. As a result, we will almost find the answer;
- We perform the inverse transformation, that is, we get rid of the improper fraction - we select the whole part in the fraction.
Subtract a proper fraction from a whole number: represent the natural number as a mixed number. Those. We take one in a natural number and convert it to the form of an improper fraction, the denominator being the same as that of the subtracted fraction.
Example of subtracting fractions:
In the example, we replaced one with the improper fraction 7/7 and instead of 3 we wrote down a mixed number and subtracted a fraction from the fractional part.
Subtracting fractions with different denominators.
Or, to put it another way, subtracting different fractions.
Rule for subtracting fractions with different denominators. In order to subtract fractions with different denominators, it is necessary, first, to reduce these fractions to the lowest common denominator (LCD), and only after this, perform the subtraction as with fractions with the same denominators.
The common denominator of several fractions is LCM (least common multiple) natural numbers that are the denominators of these fractions.
Attention! If in the final fraction the numerator and denominator have common factors, then the fraction must be reduced. An improper fraction is best represented as a mixed fraction. Leaving the subtraction result without reducing the fraction where possible is an incomplete solution to the example!
Procedure for subtracting fractions with different denominators.
- find the LCM for all denominators;
- put additional factors for all fractions;
- multiply all numerators by an additional factor;
- We write the resulting products into the numerator, signing the common denominator under all fractions;
- subtract the numerators of fractions, signing the common denominator under the difference.
In the same way, addition and subtraction of fractions is carried out if there are letters in the numerator.
Subtracting fractions, examples:
Subtracting mixed fractions.
At subtracting mixed fractions (numbers) separately, the integer part is subtracted from the integer part, and the fractional part is subtracted from the fractional part.
The first option for subtracting mixed fractions.
If the fractional parts the same denominators and numerator of the fractional part of the minuend (we subtract it from it) ≥ numerator of the fractional part of the subtrahend (we subtract it).
For example:
The second option for subtracting mixed fractions.
When fractional parts different denominators. To begin with, we bring the fractional parts to a common denominator, and after that we subtract the whole part from the whole part, and the fractional part from the fractional part.
For example:
The third option for subtracting mixed fractions.
The fractional part of the minuend is less than the fractional part of the subtrahend.
Example:
Because Fractional parts have different denominators, which means, as in the second option, we first bring ordinary fractions to a common denominator.
The numerator of the fractional part of the minuend is less than the numerator of the fractional part of the subtrahend.3 < 14. This means we take a unit from the whole part and reduce this unit to the form of an improper fraction with the same denominator and numerator = 18.
In the numerator on the right side we write the sum of the numerators, then we open the brackets in the numerator on the right side, that is, we multiply everything and give similar ones. We do not open the parentheses in the denominator. It is customary to leave the product in the denominators. We get:
