The next topic is how to subtract fractions. Adding fractions with different denominators by finding a common multiple. Rule for adding and subtracting algebraic fractions with the same denominators
In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". Here's how it sounds:Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about 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 universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.
From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.
If we turn the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of its 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 infinitely quickly overtake the tortoise."
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise crawls 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 one hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet 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 of a flying arrow:
A flying arrow is motionless, since at each 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 the flying arrow rests at different points in space, which, in fact, is movement. There is another point 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 the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points space at one point in time, but it is impossible to determine the fact of movement from them (naturally, additional data for calculations are still needed, trigonometry will help you). What do I want to focus on Special attention, is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for exploration.
Wednesday, July 4, 2018
Very well the differences between set and multiset are described in Wikipedia. We look.
As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Reasonable beings will never understand such logic of absurdity. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.
Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of 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. Applicable mathematical theory sets to the mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash desk, paying salaries. Here a mathematician comes to us for his money. We count the entire amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.
First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will begin to convulsively recall physics: on different coins there is different amount dirt, crystal structure and atomic arrangement of each coin is unique...
And now I have the most interest Ask: where is the boundary beyond which 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 here is not even close.
Look here. We select football stadiums with the same field area. The area of the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller pulls out of his sleeve Trump ace and starts telling 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 will 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 they are shamans for that, 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 "Sum of Digits of a Number" page. She doesn't exist. There is no formula in mathematics by which you can find the sum of the digits of any number. After all, numbers are graphic symbols, with the help of 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 elementarily.
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's say we 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 to a number graphic symbol. This is not a mathematical operation.
2. We cut one received picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic characters to numbers. This is not a mathematical operation.
4. Add up 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" from shamans used by mathematicians. But that's not all.
From the point of view of mathematics, it does not matter in which number system we write the number. So, in different number systems, 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 of 12345, I don’t want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We will not consider each step under a microscope, we have 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 like finding the area of a rectangle in meters and centimeters would give you completely different results.
Zero in all number systems looks the same and has no sum of digits. This is another argument in favor of the fact that . A question for mathematicians: how is it denoted in mathematics that which is not a number? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists, no. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement of numbers. Because we can't compare numbers with different units measurements. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then it has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical action does not depend on the value of the number, the unit of measure used, and on who performs this action.
Ouch! Isn't this the women's restroom?
- Young woman! This is a laboratory for studying the indefinite holiness of souls upon ascension to heaven! Nimbus on top and arrow up. What other toilet?
Female... A halo on top and an arrow down is male.
If you have such a work of design art flashing before your eyes several times a day,
Then it is not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort on myself to see minus four degrees in a pooping person (one picture) (composition of several pictures: minus sign, number four, degrees designation). And I do not consider this girl a fool who does not know physics. She just has an arc stereotype of perception of graphic images. And mathematicians teach us this all the time. Here is an example.
1A is not "minus four degrees" or "one a". This is "pooping man" or the number "twenty-six" in the hexadecimal number system. Those people who constantly work in this number system automatically perceive the number and letter as one graphic symbol.
On the this lesson addition and subtraction will be considered 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 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 be found in many topics of the algebra course, which 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, as well as analyze a number of typical examples.
Consider the simplest example for ordinary fractions.
Example 1 Add fractions: .
Solution:
Remember the rule for adding fractions. To begin with, 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, it is necessary to expand the denominators into prime factors, and then choose all the prime factors that are included in the expansion of both denominators.
; . Then the LCM of numbers must include two 2s and two 3s: .
After finding the common denominator, it is necessary for each of the fractions to find an additional factor (in fact, divide the common denominator by the denominator of the corresponding fraction).
Then each fraction is multiplied by the resulting additional factor. Fractions are obtained from same denominators, add and subtract which we learned in previous lessons.
We get: 
.
Answer:.
Consider now the addition of algebraic fractions with different denominators. First consider 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 a common denominator for 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 smallest common denominator of fractions.
2. Find additional factors for each of the fractions (by dividing the common denominator by the denominator of this fraction).
3. Multiply the numerators by the appropriate additional factors.
4. Add or subtract fractions using the rules for adding and subtracting fractions with the same denominators.
Consider now an example with fractions in the denominator of which there are literal expressions.
Example 3 Add fractions: .
Solution:
Since the literal expressions in both denominators are the same, you should find a common denominator for numbers. The final common denominator will look like: . So the solution to this example is:
Answer:.
Example 4 Subtract fractions: .
Solution:
If you can’t “cheat” when choosing a common denominator (you can’t factor it or use the abbreviated multiplication formulas), then you have to take the product of the denominators of both fractions as a 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 factorize 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 we will fix the rules for adding and subtracting fractions with different denominators.
Example 6 Simplify: .
Solution:
Answer:.
Example 7 Simplify: .
Solution:
.
Answer:.
Consider now an example in which not two, but three fractions are added (after all, the rules for addition and subtraction for more fractions remain the same).
Example 8 Simplify: .
The next action that can be performed with ordinary fractions is subtraction. As part of this material, we will consider how to correctly calculate the difference between fractions with the same and different denominators, how to subtract a fraction from a natural number and vice versa. All examples will be illustrated with tasks. Let us clarify in advance that we will analyze only cases where the difference of fractions results in a positive number.
Yandex.RTB R-A-339285-1
How to find the difference between fractions with the same denominator
Let's start right away with an illustrative 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:
We end up with 3 eighths because 5 − 2 = 3 . It turns out that 5 8 - 2 8 = 3 8 .
Thereby a simple example we have seen 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 the same denominators, you need to subtract the numerator of one from the numerator of the other, 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 what follows.
Let's take specific examples.
Example 1
Subtract from the fraction 24 15 the common fraction 17 15 .
Solution
We see that these fractions have the same denominators. So all we have to do is subtract 17 from 24. We get 7 and add a denominator to it, we get 7 15 .
Our calculations can be written like this: 24 15 - 17 15 \u003d 24 - 17 15 \u003d 7 15
If necessary, you can shorten compound fraction or select the whole part from the wrong one to make it more convenient to count.
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 see that the numerator and denominator can be divided by 2 (we already talked about this earlier when we analyzed the signs of divisibility). Reducing the answer, we get 11 6 . This is an improper fraction, from which we will select the whole part: 11 6 \u003d 1 5 6.
How to find the difference between fractions with different denominators
Such a mathematical operation can be reduced to what we have already described above. To do this, simply bring the desired fractions to the same denominator. Let's formulate the definition:
Definition 2
To find the difference between fractions that have different denominators, you need to bring 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 1 15 from 2 9 .
Solution
The denominators are different, and you need to reduce them to the smallest common sense. In this case, the LCM is 45. For the first fraction, an additional factor of 5 is required, and for the second - 3.
Let's calculate: 2 9 = 2 5 9 5 = 10 45 1 15 = 1 3 15 3 = 3 45
We got 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 brief record of the solution looks like this: 2 9 - 1 15 \u003d 10 45 - 3 45 \u003d 10 - 3 45 \u003d 7 45.
Do not neglect the reduction of the result or the selection of a whole part from it, if necessary. AT this example we don't have to do that.
Example 4
Find the difference 19 9 - 7 36 .
Solution
We bring the fractions indicated in the condition to the lowest common denominator 36 and obtain 76 9 and 7 36 respectively.
We consider the answer: 76 36 - 7 36 \u003d 76 - 7 36 \u003d 69 36
The result can be reduced by 3 to get 23 12 . The numerator is greater than the denominator, which means we can extract the whole part. The final answer is 1 11 12 .
The summary of the whole solution is 19 9 - 7 36 = 1 11 12 .
How to subtract a natural number from a common fraction
Such an action can also be easily reduced to a simple subtraction of ordinary fractions. This can be done by representing a natural number as a fraction. Let's show an example.
Example 5
Find the difference 83 21 - 3 .
Solution
3 is the same as 3 1 . Then you can calculate like this: 83 21 - 3 \u003d 20 21.
If in the condition it is necessary to subtract an integer from an improper fraction, it is more convenient to first extract the integer from it, writing it as a mixed number. Then the previous example can be solved differently.
From the fraction 83 21, when you select the integer part, you get 83 21 \u003d 3 20 21.
Now 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 a natural number as a fraction, bring both to a common 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 the subtraction and transform the final result, extracting the integer 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 to be subtracted is correct, 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 that, you need to subtract the desired fraction from unity and get the answer.
Example 7
Calculate the difference 1 065 - 13 62 .
Solution
The fraction to be subtracted is correct, because its numerator is less than the denominator. Therefore, we need to subtract one from 1065 and subtract the desired fraction from it: 1065 - 13 62 \u003d (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, we represent the unit as a fraction 1 1 .
It turns out that 1 - 13 62 \u003d 1 1 - 13 62 \u003d 62 62 - 13 62 \u003d 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. Here are the calculations we would get:
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 have considered the case when we need to subtract proper fraction. If it's wrong, we replace it with a mixed number and subtract according to the familiar rules.
Example 8
Calculate the difference 644 - 73 5 .
Solution
The second fraction is improper, 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
Subtraction properties when working with fractions
The properties that subtraction has natural numbers, extend to the cases of subtraction of ordinary fractions. Let's see 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 analyzed the subtraction of a sum from a number, so we act on already known algorithm. First, we 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 extracting the integer part from it. The result is 3 11 12.
Brief summary of the whole 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 types 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 a mistake in the text, please highlight it and press Ctrl+Enter
Note! Before writing a final answer, see if you can reduce the fraction you received.
Subtraction of fractions with the same denominators examples:
,
,
Subtracting a proper fraction from one.
If it is necessary to subtract from the unit a fraction that is correct, 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:
The denominator of the fraction to be subtracted = 7 , i.e., we represent the unit as an improper fraction 7/7 and subtract according to the rule for subtracting fractions with the same denominators.
Subtracting a proper fraction from a whole number.
Rules for subtracting fractions - correct from integer (natural number):
- We translate the given fractions, which contain an integer part, into improper ones. We get normal terms (it does not matter if they have different denominators), which we consider according to the rules given above;
- Next, we calculate the difference of 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 integer part in the fraction.
Subtract a proper fraction from a whole number: we represent a natural number as a mixed number. Those. we take a unit in a natural number and translate it into the form of an improper fraction, the denominator is the same as that of the subtracted fraction.
Fraction subtraction example:
In the example, we replaced the unit with an improper fraction 7/7 and instead of 3 we wrote mixed number and the fraction was taken away from the fractional part.
Subtraction of fractions with different denominators.
Or, to put it another way, subtraction of different fractions.
Rule for subtracting fractions with different denominators. In order to subtract fractions with different denominators, it is necessary, first, to bring these fractions to the lowest common denominator (LCD), and only after that to subtract 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 the given fractions.
Attention! If in 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 result of the subtraction without reducing the fraction where possible is an unfinished solution to the example!
Procedure for subtracting fractions with different denominators.
- find the LCM for all denominators;
- put additional multipliers for all fractions;
- multiply all numerators by an additional factor;
- we write the resulting products in the numerator, signing a 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 in the presence of letters in the numerator.
Subtraction of fractions, examples:
Subtraction of mixed fractions.
At subtraction 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 is to subtract mixed fractions.
If the fractional parts the same denominators and numerator of the fractional part of the minuend (we subtract from it) ≥ the numerator of the fractional part of the subtrahend (we subtract it).
For example:
The second option is to subtract mixed fractions.
When the fractional parts various denominators. To begin with, we reduce the fractional parts to a common denominator, and then we subtract the integer part from the integer, and the fractional from the fractional.
For example:
The third option is to subtract 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. So, we take a unit from the integer part and bring this unit to the form of an improper fraction with the same denominator and numerator = 18.
In the numerator from the right side we write the sum of the numerators, then we open the brackets in the numerator from the right side, that is, we multiply everything and give similar ones. We do not open brackets in the denominator. It is customary to leave the product in the denominators. We get:
Fractions are ordinary numbers, they can also be added and subtracted. But due to the fact that they have a denominator, more complex rules are required here than for integers.
Consider the simplest case, when there are two fractions with the same denominators. Then:
To add fractions with the same denominators, add their numerators and leave the denominator unchanged.
To subtract fractions with the same denominators, it is necessary 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 addition and subtraction of fractions, we get:
As you can see, nothing complicated: just add or subtract the numerators - and that's it.
But even in such simple actions people manage to make mistakes. Most often they forget 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 easy enough. Try to do the same when subtracting. As a result, the denominator will be zero, and the fraction (suddenly!) will lose its meaning.
Therefore, remember once and for all: when adding and subtracting, the denominator does not change!
Also, many people make mistakes when adding several negative fractions. There is confusion with the signs: where to put a minus, and where - a plus.
This problem is also very easy to solve. It is enough to remember that the minus before the fraction sign can always be transferred to the numerator - and vice versa. And of course, do not forget two simple rules:
- Plus times minus gives minus;
- Two negatives make an affirmative.
Let's analyze all this with specific examples:
A task. Find the value of the expression:
In the first case, everything is simple, and in the second, we will add minuses to the numerators of fractions:
What if the denominators are different
You cannot directly add fractions with different denominators. 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 " Bringing fractions to a common denominator", so we will not dwell on them here. Let's take a look at some examples:
A task. Find the value of the expression:
In the first case, we bring the fractions to a common denominator using the "cross-wise" method. In the second, we will look for the LCM. Note that 6 = 2 3; 9 = 3 · 3. The last factors in these expansions are equal, and the first ones are coprime. Therefore, LCM(6; 9) = 2 3 3 = 18.
What if the fraction has an integer part
I can please you: different denominators of fractions are not the greatest evil. Much more errors occur when the whole part is highlighted in the fractional terms.
Of course, for such fractions there are own addition and subtraction algorithms, but they are rather complicated and require a long study. Better use a simple circuit below:
- Convert all fractions containing an integer part to improper. We get normal terms (even if 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 task, we perform the inverse transformation, i.e. we get rid of the improper fraction, highlighting the integer part in it.
Transition rules to improper fractions and selection of the integer part are described in detail in the lesson "What is a fraction". If you don't remember, be sure to repeat. Examples:
A task. Find the value of the expression:
Everything is simple here. The denominators inside each expression are equal, so it remains to convert all fractions to improper ones and count. We have:
To simplify the calculations, I skipped some obvious steps in the last examples.
A small note to the last two examples, where fractions with highlighted whole part. The minus before the second fraction means that it is the whole fraction that is subtracted, and not just its whole part.
Reread this sentence again, look at the examples, and think about it. This is where beginners allow great amount errors. They love to give such tasks to control work. You will also meet them repeatedly in the tests for this lesson, which will be published shortly.
Summary: General Scheme of Computing
In conclusion, I will give a general algorithm that will help you find the sum or difference of two or more fractions:
- If an integer part is highlighted in one or more fractions, convert these fractions to improper ones;
- Bring all the fractions to a common denominator in any way convenient for you (unless, of course, the compilers of the problems did this);
- Add or subtract the resulting numbers according to the rules for adding and subtracting fractions with the same denominators;
- Reduce the result if possible. If the fraction turned out to be incorrect, select the whole part.
Remember that it is better to highlight the whole part at the very end of the task, just before writing the answer.
