One of the most important sciences, the application of which can be seen in disciplines such as chemistry, physics and even biology, is mathematics. Studying this science allows you to develop some mental qualities and improve your ability to concentrate. One of the topics that deserve special attention in the Mathematics course is adding and subtracting fractions. Many students find it difficult to study. Perhaps our article will help you better understand this topic.
How to subtract fractions whose denominators are the same
Fractions are the same numbers with which you can perform various operations. Their difference from whole numbers lies in the presence of a denominator. That is why, when performing operations with fractions, you need to study some of their features and rules. Most simple case is subtraction ordinary fractions, whose denominators are represented as the same number. Performing this action will not be difficult if you know a simple rule:
- In order to subtract a second from one fraction, it is necessary to subtract the numerator of the subtracted fraction from the numerator of the fraction being reduced. We write this number into the numerator of the difference, and leave the denominator the same: k/m - b/m = (k-b)/m.
Examples of subtracting fractions whose denominators are the same
7/19 - 3/19 = (7 - 3)/19 = 4/19.
From the numerator of the fraction “7” we subtract the numerator of the fraction “3” to be subtracted, we get “4”. We write this number in the numerator of the answer, and in the denominator we put the same number that was in the denominators of the first and second fractions - “19”.
The picture below shows several more similar examples.
Let's consider a more complex example where fractions with like denominators are subtracted:
29/47 - 3/47 - 8/47 - 2/47 - 7/47 = (29 - 3 - 8 - 2 - 7)/47 = 9/47.
From the numerator of the fraction “29” being reduced by subtracting in turn the numerators of all subsequent fractions - “3”, “8”, “2”, “7”. As a result, we get the result “9”, which we write down in the numerator of the answer, and in the denominator we write down the number that is in the denominators of all these fractions - “47”.
Adding fractions that have the same denominator
Adding and subtracting ordinary fractions follows the same principle.
- In order to add fractions whose denominators are the same, you need to add the numerators. The resulting number is the numerator of the sum, and the denominator will remain the same: k/m + b/m = (k + b)/m.
Let's see what this looks like using an example:
1/4 + 2/4 = 3/4.
To the numerator of the first term of the fraction - “1” - add the numerator of the second term of the fraction - “2”. The result - “3” - is written into the numerator of the sum, and the denominator is left the same as that present in the fractions - “4”.
Fractions with different denominators and their subtraction
Action with fractions that have same denominator, we have already considered. As we see, knowing simple rules, solving such examples is quite easy. But what to do if you need to perform an operation with fractions that have different denominators? Many secondary school students are confused by such examples. But even here, if you know the principle of the solution, the examples will no longer be difficult for you. There is also a rule here, without which the solution similar fractions It's simply impossible.
- 2/3 - one three and one two are missing in the denominator:
2/3 = (2 x 3 x 2)/(3 x 3 x 2) = 12/18. - 7/9 or 7/(3 x 3) - the denominator is missing a two:
7/9 = (7 x 2)/(9 x 2) = 14/18. - 5/6 or 5/(2 x 3) - the denominator is missing a three:
5/6 = (5 x 3)/(6 x 3) = 15/18. - The number 18 is made up of 3 x 2 x 3.
- The number 15 is made up of 5 x 3.
- The common multiple will be the following factors: 5 x 3 x 3 x 2 = 90.
- 90 divided by 15. The resulting number “6” will be a multiplier for 3/15.
- 90 divided by 18. The resulting number “5” will be a multiplier for 4/18.
- Convert all fractions that have an integer part to improper ones. Speaking in simple words, remove the whole part. To do this, multiply the number of the integer part by the denominator of the fraction, and add the resulting product to the numerator. The number that comes out after these actions is the numerator improper fraction. The denominator remains unchanged.
- If fractions have different denominators, they should be reduced to the same denominator.
- Perform addition or subtraction with the same denominators.
- When receiving an improper fraction, select the whole part.
To subtract fractions from different denominators, it is necessary to reduce them to the same lowest denominator.
We will talk in more detail about how to do this.
Property of a fraction
In order to bring several fractions to the same denominator, you need to use the main property of a fraction in the solution: after dividing or multiplying the numerator and denominator by same number you get a fraction equal to the given one.
So, for example, the fraction 2/3 can have denominators such as “6”, “9”, “12”, etc., that is, it can have the form of any number that is a multiple of “3”. After we multiply the numerator and denominator by “2”, we get the fraction 4/6. After we multiply the numerator and denominator of the original fraction by “3”, we get 6/9, and if we perform a similar operation with the number “4”, we get 8/12. One equality can be written as follows:
2/3 = 4/6 = 6/9 = 8/12…
How to convert multiple fractions to the same denominator
Let's look at how to reduce multiple fractions to the same denominator. For example, let's take the fractions shown in the picture below. First you need to determine which number can become the denominator for all of them. To make things easier, let's factorize the existing denominators.
The denominator of the fraction 1/2 and the fraction 2/3 cannot be factorized. The denominator 7/9 has two factors 7/9 = 7/(3 x 3), the denominator of the fraction 5/6 = 5/(2 x 3). Now we need to determine which factors will be the smallest for all these four fractions. Since the first fraction has the number “2” in the denominator, it means that it must be present in all denominators; in the fraction 7/9 there are two triplets, which means that both of them must also be present in the denominator. Taking into account the above, we determine that the denominator consists of three factors: 3, 2, 3 and is equal to 3 x 2 x 3 = 18.
Let's consider the first fraction - 1/2. There is a “2” in its denominator, but there is not a single “3” digit, but there should be two. To do this, we multiply the denominator by two triples, but, according to the property of a fraction, we must multiply the numerator by two triples:
1/2 = (1 x 3 x 3)/(2 x 3 x 3) = 9/18.
We perform the same operations with the remaining fractions.
All together it looks like this:
How to subtract and add fractions that have different denominators
As mentioned above, in order to add or subtract fractions that have different denominators, they must be reduced to the same denominator, and then use the rules for subtracting fractions that have the same denominator, which have already been discussed.
Let's look at this as an example: 4/18 - 3/15.
Finding the multiple of numbers 18 and 15:
After the denominator has been found, it is necessary to calculate the factor that will be different for each fraction, that is, the number by which it will be necessary to multiply not only the denominator, but also the numerator. To do this, divide the number that we found (the common multiple) by the denominator of the fraction for which additional factors need to be determined.
The next stage of our solution is to reduce each fraction to the denominator “90”.
We have already talked about how this is done. Let's see how this is written in an example:
(4 x 5)/(18 x 5) - (3 x 6)/(15 x 6) = 20/90 - 18/90 = 2/90 = 1/45.
If fractions with small numbers, then you can common denominator determine as in the example shown in the picture below.
The same is true for those with different denominators.
Subtraction and having integer parts
We have already discussed in detail the subtraction of fractions and their addition. But how to subtract if the fraction has whole part? Again, let's use a few rules:
There is another way in which you can add and subtract fractions with whole parts. To do this, actions are performed separately with whole parts, and actions with fractions separately, and the results are recorded together.
The example given consists of fractions that have the same denominator. In the case when the denominators are different, they must be brought to the same value, and then perform the actions as shown in the example.
Subtracting fractions from whole numbers
Another type of action with fractions is the case when the fraction must be subtracted from At first glance similar example seems difficult to solve. However, everything is quite simple here. To solve it, you need to convert the integer into a fraction, and with the same denominator that is in the subtracted fraction. Next, we perform a subtraction similar to subtraction with identical denominators. In an example it looks like this:
7 - 4/9 = (7 x 9)/9 - 4/9 = 53/9 - 4/9 = 49/9.
The subtraction of fractions (grade 6) presented in this article is the basis for solving more complex examples that are covered in subsequent grades. Knowledge of this topic is subsequently used to solve functions, derivatives, and so on. Therefore, it is very important to understand and understand the operations with fractions discussed above.
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 that is 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 view:
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 out of 16. By simple multiplication at 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 have worked honestly in junior classes... And I didn’t 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 the fraction line at the top empty place Let’s leave it, 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 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.
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
Actions with fractions. In this article we will look at examples, everything in detail with explanations. We will consider ordinary fractions. We'll look at decimals later. I recommend watching the whole thing and studying it sequentially.
1. Sum of fractions, difference of fractions.
Rule: when adding fractions with equal denominators, as a result we get a fraction - the denominator of which remains the same, and its numerator will be equal to the sum numerators of fractions.
Rule: when calculating the difference between fractions with the same denominators, we obtain a fraction - the denominator remains the same, and the numerator of the second is subtracted from the numerator of the first fraction.
Formal notation for the sum and difference of fractions with equal denominators:
Examples (1):
It is clear that when ordinary fractions are given, then everything is simple, but what if they are mixed? Nothing complicated...
Option 1– you can convert them into ordinary ones and then calculate them.
Option 2– you can “work” separately with the integer and fractional parts.
Examples (2):
More:
And if the difference of two is given mixed fractions and the numerator of the first fraction will be less than the numerator of the second? You can also act in two ways.
Examples (3):
*Converted to ordinary fractions, calculated the difference, converted the resulting improper fraction to a mixed fraction.
*We broke it down into integer and fractional parts, got a three, then presented 3 as the sum of 2 and 1, with one represented as 11/11, then found the difference between 11/11 and 7/11 and calculated the result. The meaning of the above transformations is to take (select) a unit and present it in the form of a fraction with the denominator we need, then we can subtract another from this fraction.
Another example:
Conclusion: there is a universal approach - in order to calculate the sum (difference) of mixed fractions with equal denominators, they can always be converted to improper ones, then perform the necessary action. After this, if the result is an improper fraction, we convert it to a mixed fraction.
Above we looked at examples with fractions that have equal denominators. What if the denominators are different? In this case, the fractions are reduced to the same denominator and the specified action is performed. To change (transform) a fraction, the basic property of the fraction is used.
Let's look at simple examples:
In these examples, we immediately see how one of the fractions can be transformed to get equal denominators.
If we designate ways to reduce fractions to the same denominator, then we will call this one METHOD ONE.
That is, immediately when “evaluating” a fraction, you need to figure out whether this approach will work - we check whether the larger denominator is divisible by the smaller one. And if it is divisible, then we perform a transformation - we multiply the numerator and denominator so that the denominators of both fractions become equal.
Now look at these examples:
This approach is not applicable to them. There are also ways to reduce fractions to a common denominator; let’s consider them.
Method TWO.
We multiply the numerator and denominator of the first fraction by the denominator of the second, and the numerator and denominator of the second fraction by the denominator of the first:
*In fact, we reduce fractions to form when the denominators become equal. Next, we use the rule for adding fractions with equal denominators.
Example:
*This method can be called universal, and it always works. The only downside is that after the calculations you may end up with a fraction that will need to be further reduced.
Let's look at an example:
It can be seen that the numerator and denominator are divisible by 5:
Method THREE.
You need to find the least common multiple (LCM) of the denominators. This will be the common denominator. What kind of number is this? This is the least natural number, which is divisible by each of the numbers.
Look, here are two numbers: 3 and 4, there are many numbers that are divisible by them - these are 12, 24, 36, ... The smallest of them is 12. Or 6 and 15, they are divisible by 30, 60, 90 .... The least is 30. The question is - how to determine this least common multiple?
There is a clear algorithm, but often this can be done immediately without calculations. For example, according to the above examples (3 and 4, 6 and 15) no algorithm is needed, we took large numbers (4 and 15), doubled them and saw that they are divisible by the second number, but pairs of numbers can be others, for example 51 and 119.
Algorithm. In order to determine the least common multiple of several numbers, you must:
- decompose each number into SIMPLE factors
— write down the decomposition of the BIGGER of them
- multiply it by the MISSING factors of other numbers
Let's look at examples:
50 and 60 => 50 = 2∙5∙5 60 = 2∙2∙3∙5
in decomposition more one five is missing
=> LCM(50,60) = 2∙2∙3∙5∙5 = 300
48 and 72 => 48 = 2∙2∙2∙2∙3 72 = 2∙2∙2∙3∙3
in the expansion of a larger number two and three are missing
=> LCM(48.72) = 2∙2∙2∙2∙3∙3 = 144
* Least common multiple of two prime numbers equal to their product
Question! Why is finding the least common multiple useful, since you can use the second method and simply reduce the resulting fraction? Yes, it is possible, but it is not always convenient. Look at the denominator for the numbers 48 and 72 if you simply multiply them 48∙72 = 3456. You will agree that it is more pleasant to work with smaller numbers.
Let's look at examples:
*51 = 3∙17 119 = 7∙17
the expansion of a larger number is missing a triple
=> NOC(51,119) = 3∙7∙17
Now let's use the first method:
*Look at the difference in the calculations, in the first case there are a minimum of them, but in the second you need to work separately on a piece of paper, and even the fraction you received needs to be reduced. Finding the LOC simplifies the work significantly.
More examples:
*In the second example it is clear that smallest number which is divisible by 40 and 60 is equal to 120.
RESULT! GENERAL COMPUTING ALGORITHM!
— we reduce fractions to ordinary ones if there is an integer part.
- we bring fractions to a common denominator (first we look at whether one denominator is divisible by another; if it is divisible, then we multiply the numerator and denominator of this other fraction; if it is not divisible, we act using the other methods indicated above).
- Having received fractions with equal denominators, we perform operations (addition, subtraction).
- if necessary, we reduce the result.
- if necessary, then select the whole part.
2. Product of fractions.
The rule is simple. When multiplying fractions, their numerators and denominators are multiplied:
Examples:
Task. 13 tons of vegetables were brought to the base. Potatoes make up ¾ of all imported vegetables. How many kilograms of potatoes were brought to the base?
Let's finish with the piece.
*I previously promised to give you a formal explanation of the main property of a fraction through a product, please:
3. Division of fractions.
Dividing fractions comes down to multiplying them. It is important to remember here that the fraction that is the divisor (the one that is divided by) is turned over and the action changes to multiplication:
This action can be written in the form of a so-called four-story fraction, because the division “:” itself can also be written as a fraction:
Examples:
That's all! Good luck to you!
Sincerely, Alexander Krutitskikh.
Instructions
Reduction to a common denominator.
Let the fractions a/b and c/d be given.
The numerator and denominator of the first fraction are multiplied by LCM/b
The numerator and denominator of the second fraction are multiplied by LCM/d
An example is shown in the figure.
To compare fractions, you need to add them to a common denominator, then compare the numerators. For example, 3/4< 4/5, см. .
Adding and subtracting fractions.
To find the sum of two ordinary fractions, they need to be brought to a common denominator, then add the numerators, leaving the denominator unchanged. An example of adding fractions 1/2 and 1/3 is shown in the figure.
The difference of fractions is found in a similar way; after finding the common denominator, the numerators of the fractions are subtracted, see the figure.
When multiplying ordinary fractions, the numerators and denominators are multiplied together.
In order to divide two fractions, a fraction of the second fraction is necessary, i.e. change its numerator and denominator, and then multiply the resulting fractions.
Video on the topic
Sources:
- fractions grade 5 using an example
- Basic fraction problems
Module represents absolute value expressions. Straight brackets are used to denote a module. The values contained in them are considered modulo. The solution to the module is to expand the parentheses according to certain rules and finding the set of expression values. In most cases, the module is expanded in such a way that the submodular expression receives a series of positive and negative values including zero value. Based on these properties of the module, further equations and inequalities of the original expression are compiled and solved.
Instructions
Write the original equation with . To do this, open the module. Consider each submodular expression. Determine at what value of the unknown quantities included in it the expression in modular brackets becomes zero.
To do this, equate the submodular expression to zero and find the resulting equation. Write down the values you find. In the same way, determine the values of the unknown variable for each module in given equation.
Draw a number line and plot the resulting values on it. The values of the variable in the zero module will serve as constraints when solving the modular equation.
In the original equation, you need to expand the modular ones, changing the sign so that the values of the variable correspond to those displayed on the number line. Solve the resulting equation. Check the found value of the variable against the constraint specified by the module. If the solution satisfies the condition, it is true. Roots that do not satisfy the restrictions must be discarded.
Similarly, expand the modules of the original expression, taking into account the sign, and calculate the roots of the resulting equation. Write down all the resulting roots that satisfy the constraint inequalities.
Fractional numbers can be expressed in in different forms exact value quantities. You can do the same with fractions mathematical operations, as with whole numbers: subtraction, addition, multiplication and division. To learn to decide fractions, we must remember some of their features. They depend on the type fractions, the presence of an integer part, a common denominator. Some arithmetic operations after execution they require reduction of the fractional part of the result.
You will need
- - calculator
Instructions
Look closely at the numbers. If among the fractions there are decimals and irregular ones, sometimes it is more convenient to first perform operations with decimals, and then convert them to the irregular form. Can you translate fractions in this form initially, writing the value after the decimal point in the numerator and putting 10 in the denominator. If necessary, reduce the fraction by dividing the numbers above and below by one divisor. Fractions in which the whole part is isolated must be converted to the wrong form by multiplying it by the denominator and adding the numerator to the result. This value will become the new numerator fractions. To select a whole part from an initially incorrect one fractions, you need to divide the numerator by the denominator. Write the whole result from fractions. And the remainder of the division will become the new numerator, denominator fractions it does not change. For fractions with an integer part, it is possible to perform actions separately, first for the integer and then for the fractional parts. For example, the sum of 1 2/3 and 2 ¾ can be calculated:
- Converting fractions to the wrong form:
- 1 2/3 + 2 ¾ = 5/3 + 11/4 = 20/12 + 33/12 = 53/12 = 4 5/12;
- Summation of separately integer and fractional parts of terms:
- 1 2/3 + 2 ¾ = (1+2) + (2/3 + ¾) = 3 +(8/12 + 9/12) = 3 + 12/17 = 3 + 1 5/12 = 4 5 /12.
For with values below the line, find the common denominator. For example, for 5/9 and 7/12 the common denominator will be 36. For this, the numerator and denominator of the first fractions you need to multiply by 4 (you get 28/36), and the second one - by 3 (you get 15/36). Now you can perform the calculations.
If you are going to calculate the sum or difference of fractions, first write the found common denominator under the line. Execute necessary actions between the numerators, and write the result above the new line fractions. Thus, the new numerator will be the difference or sum of the numerators of the original fractions.
To calculate the product of fractions, multiply the numerators of the fractions and write the result in place of the numerator of the final fractions. Do the same for the denominators. When dividing one fractions write down one fraction on the other, and then multiply its numerator by the denominator of the second. In this case, the denominator of the first fractions multiplied accordingly by the second numerator. In this case, a kind of revolution occurs fractions(divisor). The final fraction will be the result of multiplying the numerators and denominators of both fractions. It's not hard to learn fractions, written in the condition in the form of “four-story” fractions. If it separates two fractions, rewrite them using the “:” separator and continue regular division.
For getting final result Reduce the resulting fraction by dividing the numerator and denominator by one whole number, the largest possible in in this case. In this case, there must be integers above and below the line.
note
Do not perform arithmetic with fractions whose denominators are different. Choose a number such that when you multiply the numerator and denominator of each fraction by it, the result is that the denominators of both fractions are equal.
When recording fractional numbers The dividend is written above the line. This quantity is designated as the numerator of the fraction. The divisor, or denominator, of the fraction is written below the line. For example, one and a half kilograms of rice will be written as a fraction in the following way: 1 ½ kg rice. If the denominator of a fraction is 10, the fraction is called a decimal. In this case, the numerator (dividend) is written to the right of the whole part, separated by a comma: 1.5 kg of rice. For ease of calculation, such a fraction can always be written in in the wrong form: 1 2/10 kg potatoes. To simplify, you can reduce the numerator and denominator values by dividing them by one integer. IN in this example may be divided by 2. The result will be 1 1/5 kg of potatoes. Make sure that the numbers you are going to perform arithmetic with are presented in the same form.
Instructions
Click once on the “Insert” menu item, then select “Symbol”. This is one of the most simple ways inserts fractions into the text. It consists in the following. The set of ready-made symbols includes fractions. Their number, as a rule, is small, but if you need to write ½ in the text rather than 1/2, then this option will be the most optimal for you. In addition, the number of fraction characters may depend on the font. For example, for the Times New Roman font there are slightly fewer fractions than for the same Arial. Vary fonts to find the best option when it comes to simple expressions.
Click on the “Insert” menu item and select the “Object” sub-item. A window will appear in front of you with a list of possible objects to insert. Choose among them Microsoft Equation 3.0. This app will help you type fractions. And not only fractions, but also complex mathematical expressions, containing various trigonometric functions and other elements. Double-click on this object with the left mouse button. A window will appear in front of you containing many symbols.
To print a fraction, select the symbol representing a fraction with an empty numerator and denominator. Click on it once with the left mouse button. An additional menu will appear, clarifying the scheme itself. fractions. There may be several options. Select the one that suits you best and click on it once with the left mouse button.
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:
- In order to add fractional expressions with the same denominator (the denominator is the number at the bottom of the fraction, the numerator 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 lowest common denominator is a number that is divisible by the denominators of all fractional expressions 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, it turned out 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 present all non-fractional values as a fraction, in this case the 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 the right side of the equation: 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 the given value is within the solution range. An empty circle indicates that this value is not included in the solution range.
- Since the denominator cannot be equal to zero, under the 2nd there will be an empty circle.
- 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