Studying the issue of subtracting fractions with different denominators is found in the school subject Algebra in the eighth grade and it sometimes causes difficulties in understanding for children. To subtract fractions with different denominators, use the following formula:
The procedure for subtracting fractions is similar to addition, since it completely copies the principle of operation.
First, we calculate the smallest number that is a multiple of both the denominator.
Secondly, we multiply the numerator and denominator of each fraction by a certain number that will allow us to reduce the denominator to a given minimum common denominator.
Thirdly, the subtraction procedure itself occurs, when in the end the denominator is duplicated, and the numerator of the second fraction is subtracted from the first.
Example: 8/3 2/4 = 8/3 1/2 = 16/6 3/6 = 13/6 = 2 whole 1/6
First you need to bring them to the same denominator, and then subtract. For example, 1/2 - 1/4 = 2/4 - 1/4 = 1/4. Or, more difficult, 1/3 - 1/5 = 5/15 - 3/15 = 2/15. Do you need to explain how fractions are reduced to a common denominator?
When performing operations such as adding or subtracting ordinary fractions with different denominators, a simple rule applies - the denominators of these fractions are reduced to one number, and the operation itself is performed with the numbers in the numerator. That is, the fractions receive a common denominator and seem to be combined into one. Finding a common denominator for arbitrary fractions usually comes down to simply multiplying each fraction by the denominator of the other fraction. But in simpler cases, you can immediately find factors that will bring the denominators of the fractions to the same number.
Example of subtracting fractions: 2/3 - 1/7 = 2*7/3*7 - 1*3/7*3 = 14/21 - 3/21 = (14-3)/21 = 11/21
Many adults have already forgotten how to subtract fractions with different denominators, but this action relates to elementary mathematics.
To subtract fractions with different denominators, you need to bring them to a common denominator, that is, find the least common multiple of the denominators, then multiply the numerators by additional factors equal to the ratio of the least common multiple and the denominator.
Fraction signs are preserved. Once the fractions have the same denominators, you can subtract, and then, if possible, reduce the fraction.
Elena, have you decided to repeat your school mathematics course?)))
To subtract fractions with different denominators, they must first be reduced to the same denominator and then subtracted. The simplest option: Multiply the numerator and denominator of the first fraction by the denominator of the second fraction, and multiply the numerator and denominator of the second fraction by the denominator of the first fraction. We get two fractions with the same denominators. Now we subtract the numerator of the second fraction from the numerator of the first fraction, and they have the same denominator.
For example, three-fifths subtracting two sevenths is equal to twenty-one thirty-fifths subtracting ten thirty-fifths and this is equal to eleven thirty-fifths.
If the denominators are large numbers, then you can find their least common multiple, i.e. a number that will be divisible by one and the other denominator. And bring both fractions to a common denominator (least common multiple)
How to subtract fractions with different denominators is a very simple task - we bring the fractions to a common denominator and then do the subtraction in the numerator.
Many people encounter difficulties when there are integers next to these fractions, so I wanted to show how to do this with the following example:
subtracting fractions with whole parts and different denominators
first we subtract the whole parts 8-5 = 3 (the three remains near the first fraction);
we bring the fractions to a common denominator 6 (if the numerator of the first fraction is greater than the second, we do the subtraction and write it next to the whole part, in our case we move on);
we decompose the whole part 3 into 2 and 1;
We write 1 as a fraction 6/6;
We write 6/6+3/6-4/6 under the common denominator 6 and do the operations in the numerator;
write down the result found 2 5/6.
It is important to remember that fractions are subtracted if they have the same denominator. Therefore, when we have fractions with different denominators in difference, they simply need to be brought to a common denominator, which is not difficult to do. We simply have to factor the numerator of each fraction and calculate the least common multiple, which must not equal zero. Don’t forget to also multiply the numerators by the resulting additional factors, but here is an example for convenience:
If you want to subtract fractions with unlike denominators, you will first have to find the common denominator for the two fractions. And then subtract the second from the numerator of the first fraction. A new fraction is obtained, with a new meaning.
As far as I remember from the 3rd grade mathematics course, to subtract fractions with different denominators, you first need to calculate the common denominator and reduce it to it, and then simply subtract the numerators from each other and the denominator remains the same.
To subtract fractions with unlike denominators, we first have to find the lowest common denominator of those fractions.
Let's look at an example:
Divide the larger number 25 by the smaller 20. It is not divisible. This means we multiply the denominator 25 by such a number, the resulting sum can be divided by 20. This number will be 4. 25x4=100. 100:20=5. Thus we found the lowest common denominator - 100.
Now we need to find the additional factor for each fraction. To do this, divide the new denominator by the old one.
Multiply 9 by 4 = 36. Multiply 7 by 5 = 35.
Having a common denominator, we carry out the subtraction as shown in the example and get the result.
This lesson will cover adding and subtracting algebraic fractions with different denominators. We already know how to add and subtract common fractions with different denominators. To do this, the fractions must be reduced to a common denominator. It turns out that algebraic fractions follow the same rules. At the same time, we already know how to reduce algebraic fractions to a common denominator. Adding and subtracting fractions with different denominators is one of the most important and difficult topics in the 8th grade course. Moreover, this topic will appear in many topics in the algebra course that you will study in the future. As part of the lesson, we will study the rules for adding and subtracting algebraic fractions with different denominators, and also analyze a number of typical examples.
Let's look at the simplest example for ordinary fractions.
Example 1. Add fractions: .
Solution:
Let's remember the rule for adding fractions. To begin, fractions must be reduced to a common denominator. The common denominator for ordinary fractions is least common multiple(LCM) of the original denominators.
Definition
The smallest natural number that is divisible by both numbers and .
To find the LCM, you need to factor the denominators into prime factors, and then select all the prime factors that are included in the expansion of both denominators.
; . Then the LCM of numbers must include two twos and two threes: .
After finding the common denominator, you need to find an additional factor for each fraction (in fact, divide the common denominator by the denominator of the corresponding fraction).
Each fraction is then multiplied by the resulting additional factor. We get fractions with the same denominators, which we learned to add and subtract in previous lessons.
We get: 
.
Answer:.
Let us now consider the addition of algebraic fractions with different denominators. First, let's look at fractions whose denominators are numbers.
Example 2. Add fractions: .
Solution:
The solution algorithm is absolutely similar to the previous example. It is easy to find the common denominator of these fractions: and additional factors for each of them.
.
Answer:.
So, let's formulate algorithm for adding and subtracting algebraic fractions with different denominators:
1. Find the lowest common denominator of fractions.
2. Find additional factors for each of the fractions (by dividing the common denominator by the denominator of the given fraction).
3. Multiply the numerators by the corresponding additional factors.
4. Add or subtract fractions using the rules for adding and subtracting fractions with like denominators.
Let us now consider an example with fractions whose denominator contains letter expressions.
Example 3. Add fractions: .
Solution:
Since the letter expressions in both denominators are the same, you should find a common denominator for the numbers. The final common denominator will look like: . Thus, the solution to this example looks like:.
Answer:.
Example 4. Subtract fractions: .
Solution:
If you can’t “cheat” when choosing a common denominator (you can’t factor it or use abbreviated multiplication formulas), then you have to take the product of the denominators of both fractions as the common denominator.
Answer:.
In general, when solving such examples, the most difficult task is to find a common denominator.
Let's look at a more complex example.
Example 5. Simplify: .
Solution:
When finding a common denominator, you must first try to factor the denominators of the original fractions (to simplify the common denominator).
In this particular case:
Then it is easy to determine the common denominator: 
.
We determine additional factors and solve this example:
Answer:.
Now let's establish the rules for adding and subtracting fractions with different denominators.
Example 6. Simplify: .
Solution:
Answer:.
Example 7. Simplify: .
Solution:
.
Answer:.
Let us now consider an example in which not two, but three fractions are added (after all, the rules of addition and subtraction for a larger number of fractions remain the same).
Example 8. Simplify: .
Has your child brought homework from school and you don't know how to solve it? Then this mini lesson is for you!
How to add decimals
It is more convenient to add decimal fractions in a column. To add decimals, you need to follow one simple rule:
- The place must be under the place, the comma under the comma.
As you can see in the example, the whole units are located under each other, the tenths and hundredths digits are located under each other. Now we add the numbers, ignoring the comma. What to do with the comma? The comma is moved to the place where it stood in the integer category.
Adding fractions with equal denominators
To perform addition with a common denominator, you need to keep the denominator unchanged, find the sum of the numerators and get a fraction that will be the total sum.
Adding fractions with different denominators using the common multiple method
The first thing you need to pay attention to is the denominators. The denominators are different, whether one is divisible by the other, or whether they are prime numbers. First we need to bring it to one common denominator; there are several ways to do this:
- 1/3 + 3/4 = 13/12, to solve this example we need to find the least common multiple (LCM) that will be divisible by 2 denominators. To denote the smallest multiple of a and b – LCM (a;b). In this example LCM (3;4)=12. We check: 12:3=4; 12:4=3.
- We multiply the factors and add the resulting numbers, we get 13/12 - an improper fraction.
- In order to convert an improper fraction into a proper fraction, divide the numerator by the denominator, we get the integer 1, the remainder 1 is the numerator and 12 is the denominator.
Adding fractions using the cross-cross multiplication method
To add fractions with different denominators, there is another method using the “cross to cross” formula. This is a guaranteed way to equalize the denominators; to do this, you need to multiply the numerators with the denominator of one fraction and vice versa. If you are just at the initial stage of learning fractions, then this method is the simplest and most accurate way to get the correct result when adding fractions with different denominators.
Find the numerator and denominator. A fraction includes two numbers: the number that is located above the line is called the numerator, and the number that is located below the line is called the denominator. The denominator denotes the total number of parts into which a whole is divided, and the numerator is the number of such parts considered.
- For example, in the fraction ½ the numerator is 1 and the denominator is 2.
Determine the denominator. If two or more fractions have a common denominator, such fractions have the same number under the line, that is, in this case, a certain whole is divided into the same number of parts. Adding fractions with a common denominator is very simple, since the denominator of the summed fraction will be the same as the fractions being added. For example:
- The fractions 3/5 and 2/5 have a common denominator of 5.
- The fractions 3/8, 5/8, 17/8 have a common denominator of 8.
Determine the numerators. To add fractions with a common denominator, add their numerators and write the result above the denominator of the fractions being added.
- The fractions 3/5 and 2/5 have numerators 3 and 2.
- Fractions 3/8, 5/8, 17/8 have numerators 3, 5, 17.
Add up the numerators. In problem 3/5 + 2/5, add the numerators 3 + 2 = 5. In problem 3/8 + 5/8 + 17/8, add the numerators 3 + 5 + 17 = 25.
Write the total fraction. Remember that when adding fractions with a common denominator, it remains unchanged - only the numerators are added.
- 3/5 + 2/5 = 5/5
- 3/8 + 5/8 + 17/8 = 25/8
Convert the fraction if necessary. Sometimes a fraction can be written as a whole number rather than as a fraction or decimal. For example, the fraction 5/5 is easily converted to 1, since any fraction whose numerator is equal to its denominator is 1. Imagine a pie cut into three parts. If you eat all three parts, you will have eaten the whole (one) pie.
- Any fraction can be converted to a decimal; To do this, divide the numerator by the denominator. For example, the fraction 5/8 can be written as follows: 5 ÷ 8 = 0.625.
If possible, simplify the fraction. A simplified fraction is a fraction whose numerator and denominator do not have common factors.
- For example, consider the fraction 3/6. Here, both the numerator and the denominator have a common divisor equal to 3, that is, the numerator and denominator are completely divisible by 3. Therefore, the fraction 3/6 can be written as follows: 3 ÷ 3/6 ÷ 3 = ½.
If necessary, convert an improper fraction to a mixed fraction (mixed number). An improper fraction has a numerator greater than its denominator, for example, 25/8 (a proper fraction has a numerator less than its denominator). An improper fraction can be converted to a mixed fraction, which consists of an integer part (that is, a whole number) and a fraction part (that is, a proper fraction). To convert an improper fraction, such as 25/8, to a mixed number, follow these steps:
- Divide the numerator of an improper fraction by its denominator; write down the partial quotient (whole answer). In our example: 25 ÷ 8 = 3 plus some remainder. In this case, the whole answer is the whole part of the mixed number.
- Find the remainder. In our example: 8 x 3 = 24; subtract the resulting result from the original numerator: 25 - 24 = 1, that is, the remainder is 1. In this case, the remainder is the numerator of the fractional part of the mixed number.
- Write a mixed fraction. The denominator does not change (that is, it is equal to the denominator of the improper fraction), so 25/8 = 3 1/8.
Actions with fractions.
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
So, what are fractions, types of fractions, transformations - we remembered. Let's get to the main issue.
What can you do with fractions? Yes, everything is the same as with ordinary numbers. Add, subtract, multiply, divide.
All these actions with decimal working with fractions is no different from working with whole numbers. Actually, that’s what’s good about them, decimal ones. The only thing is that you need to put the comma correctly.
Mixed numbers, as I already said, are of little use for most actions. They still need to be converted to ordinary fractions.
But the actions with ordinary fractions they will be more cunning. And much more important! Let me remind you: all actions with fractional expressions with letters, sines, unknowns, and so on and so forth are no different from actions with ordinary fractions! Operations with ordinary fractions are the basis for all algebra. It is for this reason that we will analyze all this arithmetic in great detail here.
Adding and subtracting fractions.
Everyone can add (subtract) fractions with the same denominators (I really hope!). Well, let me remind those who are completely forgetful: when adding (subtracting), the denominator does not change. The numerators are added (subtracted) to give the numerator of the result. Type:
In short, in general terms:
What if the denominators are different? Then, using the basic property of a fraction (here it comes in handy again!), we make the denominators the same! For example:
Here we had to make the fraction 4/10 from the fraction 2/5. For the sole purpose of making the denominators the same. Let me note, just in case, that 2/5 and 4/10 are the same fraction! Only 2/5 are uncomfortable for us, and 4/10 are really okay.
By the way, this is the essence of solving any math problems. When we from uncomfortable we do expressions the same thing, but more convenient for solving.
Another example:
The situation is similar. Here we make 48 from 16. By simple multiplication by 3. This is all clear. But we came across something like:
How to be?! It's hard to make a nine out of a seven! But we are smart, we know the rules! Let's transform every fraction so that the denominators are the same. This is called “reduce to a common denominator”:
Wow! How did I know about 63? Very simple! 63 is a number that is divisible by 7 and 9 at the same time. Such a number can always be obtained by multiplying the denominators. If we multiply a number by 7, for example, then the result will certainly be divisible by 7!
If you need to add (subtract) several fractions, there is no need to do it in pairs, step by step. You just need to find the denominator common to all fractions and reduce each fraction to this same denominator. For example:
And what will be the common denominator? You can, of course, multiply 2, 4, 8, and 16. We get 1024. Nightmare. It’s easier to estimate that the number 16 is perfectly divisible by 2, 4, and 8. Therefore, from these numbers it’s easy to get 16. This number will be the common denominator. Let's turn 1/2 into 8/16, 3/4 into 12/16, and so on.
By the way, if you take 1024 as the common denominator, everything will work out, in the end everything will be reduced. But not everyone will get to this end, because of the calculations...
Complete the example yourself. Not some kind of logarithm... It should be 29/16.
So, the addition (subtraction) of fractions is clear, I hope? Of course, it is easier to work in a shortened version, with additional multipliers. But this pleasure is available to those who worked honestly in the lower grades... And did not forget anything.
And now we will do the same actions, but not with fractions, but with fractional expressions. New rake will be revealed here, yes...
So, we need to add two fractional expressions:
We need to make the denominators the same. And only with the help multiplication! This is what the main property of a fraction dictates. Therefore, I cannot add one to X in the first fraction in the denominator. (that would be nice!). But if you multiply the denominators, you see, everything grows together! So we write down the line of the fraction, leave an empty space at the top, then add it, and write the product of the denominators below, so as not to forget:
And, of course, we don’t multiply anything on the right side, we don’t open the parentheses! And now, looking at the common denominator on the right side, we realize: in order to get the denominator x(x+1) in the first fraction, you need to multiply the numerator and denominator of this fraction by (x+1). And in the second fraction - to x. This is what you get:
Note! Here are the parentheses! This is the rake that many people step on. Not parentheses, of course, but their absence. The parentheses appear because we are multiplying all numerator and all denominator! And not their individual pieces...
In the numerator of the right side we write the sum of the numerators, everything is as in numerical fractions, then we open the brackets in the numerator of the right side, i.e. We multiply everything and give similar ones. There is no need to open the parentheses in the denominators or multiply anything! In general, in denominators (any) the product is always more pleasant! We get:
So we got the answer. The process seems long and difficult, but it depends on practice. Once you solve the examples, get used to it, everything will become simple. Those who have mastered fractions in due time do all these operations with one left hand, automatically!
And one more note. Many smartly deal with fractions, but get stuck on examples with whole numbers. Like: 2 + 1/2 + 3/4= ? Where to fasten the two-piece? You don’t need to fasten it anywhere, you need to make a fraction out of two. It's not easy, but very simple! 2=2/1. Like this. Any whole number can be written as a fraction. The numerator is the number itself, the denominator is one. 7 is 7/1, 3 is 3/1 and so on. It's the same with letters. (a+b) = (a+b)/1, x=x/1, etc. And then we work with these fractions according to all the rules.
Well, the knowledge of addition and subtraction of fractions was refreshed. Converting fractions from one type to another was repeated. You can also get checked. Shall we settle it a little?)
Calculate:
Answers (in disarray):
71/20; 3/5; 17/12; -5/4; 11/6
Multiplication/division of fractions - in the next lesson. There are also tasks for all operations with fractions.
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