How to solve ordinary fractions with like denominators. Addition and subtraction of algebraic fractions with different denominators (basic rules, simplest cases)
Note! Before writing your final answer, see if you can shorten the fraction you received.
Subtracting fractions from same 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 the unit in the form improper fraction 7/7 and subtract 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 get normal terms (it doesn’t matter if they are with 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 reverse 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: imagine natural number as a mixed number. Those. We take a unit 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 under all fractions common denominator;
- 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:
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 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 improper fraction to 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 the 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.
Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ... discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Aporia of Zeno"]. Everyone understands that they are being fooled, but no one understands what the deception consists of.
From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.
If we turn our usual logic around, everything falls into place. Achilles runs with constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But it is not complete solution Problems. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells about a flying arrow:
A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (of course, you still need additional data for calculations, trigonometry will help you). What I want to draw special attention to is that two points in time and two points in space are different things that should not be confused, because they provide different possibilities for research.
Wednesday, July 4, 2018
The differences between set and multiset are described very well on Wikipedia. Let's see.
As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.
Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase “mind me, I’m in the house,” or rather, “mathematics studies abstract concepts,” there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.
First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: on different coins there is different quantities dirt, crystal structure and atomic arrangement of each coin is unique...
And now I have the most interesting question: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.
Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.
Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.
2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic symbols into numbers. This is not a mathematical operation.
4. Add the resulting numbers. Now that's mathematics.
The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.
From a mathematical point of view, it does not matter in which number system we write a number. So, in different systems In calculus, the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. WITH a large number 12345 I don’t want to fool my head, let’s look at the number 26 from the article about . Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.
As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of a rectangle in meters and centimeters, you would get completely different results.
Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.
Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?
Female... The halo on top and the arrow down are male.
If such a work of design art flashes before your eyes several times a day,
Then it’s not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: a minus sign, the number four, a designation of degrees). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.
1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.
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 in school can rarely be useful to us in 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. 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 of 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 detailed solution that you can use this material to solve school problems and to better master 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.
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 whole line typical examples.
Let's consider 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 literal 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 deciding similar examples, the most difficult task is finding 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 more fractions remain the same).
Example 8. Simplify: .
