Fractions can be ordinary and decimal. When a student learns of the existence of the latter, he begins, at every opportunity, to translate everything that is possible into decimal form, even if this is not required.
Oddly enough, the preferences of high school students and students change, because it is easier to perform many arithmetic operations with ordinary fractions. And the values that graduates deal with can sometimes be simply impossible to convert to decimal form without loss. As a result, both types of fractions are, one way or another, adapted to the case and have their own advantages and disadvantages. Let's see how to work with them.
Definition
Fractions are the same fractions. If there are ten slices in an orange, and you were given one, then you have 1/10 of the fruit in your hand. With such a record, as in the previous sentence, the fraction will be called ordinary. If you write the same as 0.1 - decimal. Both options are equal, but they have their own advantages. The first option is more convenient for multiplication and division, the second for addition, subtraction, and in a number of other cases.
How to convert a fraction to another form
Suppose you have a fraction and you want to make a decimal out of it. What do I need to do?
By the way, you need to decide in advance that not every number can be written in decimal form without problems. Sometimes you have to round off the result, losing a certain number of decimal places, and in many areas - for example, in the exact sciences - this is a completely impermissible luxury. At the same time, actions with decimal and ordinary fractions in grade 5 allow such a transfer from one type to another without interference, at least as a training.
If from the denominator, by multiplying or dividing by an integer, you can get a value that is a multiple of 10, the translation will take place without any difficulties: ¾ turns into 0.75, 13/20 - into 0.65.
The reverse procedure is even easier, since you can always get an ordinary one from a decimal fraction without loss of accuracy. For example, 0.2 becomes 1/5 and 0.08 becomes 4/25.
Internal conversions
Before carrying out joint actions with ordinary fractions, you need to prepare the numbers for possible mathematical operations.
First of all, you need to bring all the fractions in the example to one general form. They must be either regular or decimal. Let's make a reservation right away that it is more convenient to perform multiplication and division with the former.
In preparing the numbers for further actions, you will be helped by the rule, which is well-known and used both in the first years of studying the subject, and in higher mathematics, which is studied at universities.
Fraction properties
Let's say you have some meaning. Let's say 2/3. What changes if you multiply the numerator and denominator by 3? It turns out 6/9. And if a million? 2,000,000 / 3,000,000. But wait, the number does not change qualitatively at all - 2/3 remain equal to 2,000,000/3,000,000. It is only the form that changes, not the content. The same will happen when dividing both parts by the same value. This is the main property of the fraction, which will repeatedly help you perform actions with decimal and ordinary fractions on tests and exams.
Multiplying the numerator and denominator by the same number is called fraction expansion, and division is called contraction. I must say that crossing out the same numbers at the top and bottom when multiplying and dividing fractions is a surprisingly pleasant procedure (within the framework of a math lesson, of course). One gets the impression that the answer is already close and the example is practically solved.
Incorrect fractions
An irregular fraction is one in which the numerator is greater than or equal to the denominator. In other words, if it is possible to select a whole part of it, it falls under this definition.
If such a number (greater than or equal to one) is presented as an ordinary fraction, it will be called incorrect. And if the numerator is less than the denominator, it is correct. Both types are equally convenient when performing possible actions with ordinary fractions. They can be freely multiplied and divided, added and subtracted.
If the whole part is selected at the same time and there is a remainder in the form of a fraction, the resulting number will be called mixed. In the future, you will come across various ways of combining such structures with variables, as well as solving equations where this knowledge is required.
Arithmetic operations
If everything is clear with the basic property of a fraction, then how to behave when multiplying fractions? Actions with ordinary fractions in grade 5 imply all types of arithmetic operations that are performed in two different ways.
Multiplication and division are very easy. In the first case, the numerators and denominators of two fractions are simply multiplied. In the second - the same thing, only crosswise. Thus, the numerator of the first fraction is multiplied by the denominator of the second, and vice versa.
To perform addition and subtraction, you need to perform an additional action - bring all the components of the expression to a common denominator. This means that the lower parts of the fractions must be changed to the same value - a multiple of both existing denominators. For example, for 2 and 5 it will be 10. For 3 and 6 - 6. But then what to do with the top? We cannot leave it as it was if we changed the lower one. According to the basic property of the fraction, we will multiply the numerator by the same number as the denominator. This operation must be performed with each of the numbers that we will add or subtract. However, such actions with ordinary fractions in the 6th grade are already performed "automatically", and difficulties arise only at the initial stage of studying the topic.
Comparison
If two fractions have the same denominator, then the one with the larger numerator will be larger. If the upper parts are the same, then the larger will be the one with the smaller denominator. It should be borne in mind that such successful situations for comparison are rare. Most likely, both the top and bottom parts of the expressions will not match. Then you need to remember about the possible actions with ordinary fractions and use the technique used in addition and subtraction. Also, remember that if we are talking about negative numbers, then the largest fraction will be smaller in absolute value.
Benefits of common fractions
It happens that teachers tell children one phrase, the content of which can be expressed as follows: the more information is given when formulating an assignment, the easier the solution will be. Sounds weird? But really: with a large number of known quantities, you can use almost any formulas, but if only a couple of numbers are provided, additional reflections may be required, you will have to remember and prove theorems, give arguments in favor of your innocence ...
Why are we doing this? And besides, ordinary fractions, for all their cumbersomeness, can greatly simplify the student's life, allowing for multiplication and division to reduce whole strings of values, and when calculating the sum and difference, take out common arguments and, again, reduce them.
When it is required to carry out joint actions with ordinary and decimal fractions, transformations are carried out in favor of the former: how do you convert 3/17 to decimal? Only with the loss of information, not otherwise. But 0.1 can be represented as 1/10, and then - as 17/170. And then the two resulting numbers can be added or subtracted: 30/170 + 17/170 = 47/170.
Why decimal fractions are useful
If it is more convenient to carry out actions with ordinary fractions, then writing everything down with their help is extremely inconvenient, decimal here have a significant advantage. Compare: 1748/10000 and 0.1748. It is the same meaning, presented in two different ways. Of course, the second way is easier!
In addition, decimal fractions are easier to represent, since all data have a common base that differs only by orders of magnitude. For example, we are easily aware of a 30% discount and even estimate it as significant. Do you immediately understand which is more - 30% or 137/379? Thus, decimal fractions provide a standardized calculation.
In high school, students solve quadratic equations. Performing actions with ordinary fractions here is already extremely problematic, since the formula for calculating the values of a variable contains the square root of the sum. In the presence of a fraction that cannot be reduced to a decimal, the solution becomes so complicated that it becomes almost impossible to calculate the exact answer without a calculator.
So, each way of representing fractions has its advantages in its respective context.
Recording forms
There are two ways to write actions with ordinary fractions: through a horizontal line, in two "tiers", and through a slash (aka "slash") - into a line. When a student writes in a notebook, the first option is usually more convenient and therefore more common. The distribution of a number of numbers in the cells contributes to the development of attentiveness when calculating and carrying out transformations. When writing to a string, you can inadvertently confuse the order of actions, lose any data - that is, make a mistake.
Quite often nowadays there is a need to print numbers on a computer. You can separate fractions with the traditional horizontal bar using a feature in Microsoft Word 2010 and later. The fact is that in these versions of the software there is an option called "formula". It displays a rectangular transformable field, within which you can combine any mathematical symbols, make up both two- and four-story fractions. In the denominator and numerator, you can use parentheses, operation signs. As a result, you will be able to write down any joint actions with ordinary and decimal fractions in the traditional form, that is, the way they are taught to do it in school.
If you use the standard Notepad text editor, then all fractional expressions will need to be written with a slash. Unfortunately, there is no other way here.
Conclusion
So we examined all the basic actions with ordinary fractions, of which, it turns out, there are not so many.
If at first it may seem that this is a difficult section of mathematics, then this is only a temporary impression - remember, once you thought this way about the multiplication table, and even earlier - about ordinary writing and counting from one to ten.
It is important to understand that fractions are used everywhere in everyday life. You will deal with money and engineering calculations, information technology and musical literacy, and everywhere - everywhere! - fractional numbers will appear. Therefore, do not be lazy and study this topic thoroughly - especially since it is not so difficult.
In this article, a tutor in mathematics and physics tells how to perform elementary operations with ordinary fractions: addition and subtraction, multiplication and division. Described how to represent a mixed number as an improper fraction and vice versa, as well as how to reduce fractions.
Adding and subtracting ordinary fractions
Recall that denominator fraction is the number that is from below, a numerator- the number that is above from the fractional line. For example, in a fraction, the number is the numerator and the number is the denominator.
The common denominator is the smallest possible number that is divisible by both the denominator of the first fraction and the denominator of the second fraction.
Example 1... Add two fractions:.
Let's use the algorithm described above:
1) The smallest number that is divisible by both the denominator of the first fraction and the denominator of the second fraction is. This number will be the common denominator. Now you need to bring both fractions to a common denominator.
2) Add the resulting fractions: 
.
Multiplication of ordinary fractions
In other words, for all real numbers,,,, the equality is true:
Example 2... Multiply fractions:.
To solve this problem, we will use the formula presented above: 
.
Division of ordinary fractions
In other words, for all real numbers,,,,, the equality is true:
Example 3... Divide fractions:.
To solve this problem, we will use the above formula: 
.
Improper fraction representation of a mixed number
Now let's figure out what to do if you want to perform any operation with fractions presented as mixed numbers. In this case, you first need to represent the mixed numbers as improper fractions, and then perform the necessary operation.
Recall that wrong is a fraction in which the numerator is greater than or equal to the denominator.
We also recall that the mixed number has fractional part and whole part... For example, for a mixed number, the fractional part is equal, and the integer part is equal.
Example 4... Present a mixed number as an improper fraction.
Let's use the algorithm presented above: 
.
Example 5... Present an improper fraction as a mixed number.
Students get acquainted with fractions in the 5th grade. Previously, people who knew how to perform actions with fractions were considered very smart. The first fraction was 1/2, that is, half, then 1/3 appeared, etc. For several centuries, examples were considered too complex. Now, detailed rules have been developed for converting fractions, addition, multiplication and other actions. It is enough to understand the material a little, and the decision will be easy.
An ordinary fraction, called a simple fraction, is written as a division of two numbers: m and n.
M is the dividend, that is, the numerator of the fraction, and the divisor n is called the denominator.
Allocate correct fractions (m< n) а также неправильные (m >n).
A regular fraction is less than one (for example, 5/6 - this means that 5 parts are taken from one; 2/8 - 2 parts are taken from one). The irregular fraction is equal to or greater than 1 (8/7 - the unit will be 7/7 and one more part is taken as a plus).
So, a unit is when the numerator and denominator coincide (3/3, 12/12, 100/100 and others).
Actions with ordinary fractions grade 6
With simple fractions, you can do the following:
- Expand fraction. If you multiply the upper and lower parts of the fraction by any of the same number (but not zero), then the value of the fraction will not change (3/5 = 6/10 (just multiplied by 2).
- Reducing fractions is similar to expansion, but here it is divided by some number.
- Compare. If two fractions have the same numerators, then the larger fraction will be the fraction with the lower denominator. If the denominators are the same, then the fraction with the largest numerator will be larger.
- Perform addition and subtraction. With the same denominators, this is easy to do (we sum up the upper parts, and the lower part does not change). For different, you will have to find a common denominator and additional factors.
- Multiply and divide fractions.
We will consider examples of actions with fractions below.
Reduced fractions grade 6
To abbreviate means to divide the upper and lower parts of the fraction by any of the same number.
The figure shows simple examples of abbreviation. In the first option, you can immediately guess that the numerator and denominator are divisible by 2.
On a note! If the number is even, then it is in any way divisible by 2. Even numbers are 2, 4, 6 ... 32 8 (ends with even), etc.
In the second case, when dividing 6 by 18, you can immediately see that the numbers are divisible by 2. Dividing, we get 3/9. This fraction is further divisible by 3. Then the answer is 1/3. If you multiply both divisors: 2 by 3, then you get 6. It turns out that the fraction was divided by six. This gradual division is called successive reduction of fractions by common factors.
Someone will immediately divide by 6, someone will need division by parts. The main thing is that at the end there is a fraction that cannot be reduced in any way.
Note that if a number consists of digits, adding up to a number divisible by 3, then the original can also be reduced by 3. Example: number 341. Add the numbers: 3 + 4 + 1 = 8 (8 is not divisible by 3, hence, the number 341 cannot be reduced by 3 without a remainder). Another example: 264. Add: 2 + 6 + 4 = 12 (divisible by 3). We get: 264: 3 = 88. This will simplify the reduction of large numbers.
In addition to the method of successive reduction of fractions by common factors, there are other methods.
GCD is the largest divisor for a number. Having found the GCD for the denominator and numerator, you can immediately reduce the fraction by the desired number. The search is carried out by gradually dividing each number. Next, they look at which divisors coincide, if there are several of them (as in the picture below), then you need to multiply.
Mixed fractions grade 6
All irregular fractions can be turned into mixed ones by highlighting the whole part in them. An integer is written to the left.
Often you have to make a mixed number out of an improper fraction. The transformation process in the example below: 22/4 = 22 we divide by 4, we get 5 integers (5 * 4 = 20). 22 - 20 = 2. We get 5 integers and 2/4 (the denominator does not change). Since the fraction can be canceled, we divide the upper and lower parts by 2.
It is easy to turn a mixed number into an improper fraction (this is necessary when dividing and multiplying fractions). To do this: multiply the whole number by the lower part of the fraction and add the numerator to this. Ready. The denominator does not change.
Calculations with fractions grade 6
Mixed numbers can be added. If the denominators are the same, then it is easy to do: add the whole parts and the numerators, the denominator remains in place.
When adding numbers with different denominators, the process is more complicated. First, we bring the numbers to one smallest denominator (NOZ).
In the example below, for numbers 9 and 6, the denominator is 18. After that, additional factors are needed. To find them, 18 should be divided by 9, so the additional number is found - 2. We multiply it by the numerator 4 to get the fraction 8/18). The same is done with the second fraction. We are already adding up the converted fractions (integers and numerators separately, we do not change the denominator). In the example, the answer had to be converted into a regular fraction (initially, the numerator was larger than the denominator).
Please note that for the difference of fractions, the procedure is the same.
When multiplying fractions, it is important to place both under the same line. If the number is mixed, then we turn it into a simple fraction. Next, we multiply the top and bottom and write down the answer. If it can be seen that the fractions can be reduced, then we reduce immediately.
In the above example, we didn't have to cut anything, we just wrote down the answer and selected the whole part.
In this example, I had to abbreviate the numbers below one line. Although you can shorten a ready-made answer.
When dividing, the algorithm is almost the same. First, we turn the mixed fraction into an irregular one, then write the numbers under one line, replacing division with multiplication. Do not forget to swap the upper and lower parts of the second fraction (this is the rule for dividing fractions).
If necessary, we reduce the numbers (in the example below, we have reduced them by five and two). We transform the irregular fraction by highlighting the whole part.
Basic problems for fractions grade 6
The video shows a few more tasks. For clarity, graphic images of solutions were used to help visualize fractions.
Examples of multiplication of a fraction grade 6 with explanations
Multiplying fractions are written under one line. After that, they are reduced by dividing by the same numbers (for example, 15 in the denominator and 5 in the numerator can be divided by five).
Comparison of fractions grade 6
To compare fractions, you need to remember two simple rules.
Rule 1. If the denominators are different
Rule 2. When the denominators are the same
For example, let's compare the fractions 7/12 and 2/3.
- We look at the denominators, they do not coincide. So you need to find a common one.
- For fractions, the common denominator is 12.
- Divide 12 first by the lower part of the first fraction: 12: 12 = 1 (this is an additional factor for the 1st fraction).
- Now we divide 12 by 3, we get 4 - add. multiplier of the 2nd fraction.
- We multiply the resulting numbers by the numerators to convert the fractions: 1 x 7 = 7 (first fraction: 7/12); 4 x 2 = 8 (second fraction: 8/12).
- Now we can compare: 7/12 and 8/12. Happened: 7/12< 8/12.
To represent the fractions better, you can use drawings for clarity, where the object is divided into parts (for example, a cake). If you want to compare 4/7 and 2/3, then in the first case, the cake is divided into 7 parts and 4 of them are selected. In the second, they divide it into 3 parts and take 2. It will be clear to the naked eye that 2/3 will be more than 4/7.
Examples with fractions grade 6 for training
As a workout, you can do the following tasks.
- Compare fractions
- perform multiplication
Tip: if it is difficult to find the lowest common denominator for fractions (especially if their values are small), then you can multiply the denominator of the first and second fractions. Example: 2/8 and 5/9. Finding their denominator is simple: multiply 8 by 9, we get 72.
Solving equations with fractions grade 6
In solving equations, you need to remember actions with fractions: multiplication, division, subtraction and addition. If one of the factors is unknown, then the product (total) is divided by a known factor, that is, the fractions are multiplied (the second is turned over).
If the dividend is unknown, then the denominator is multiplied by the divisor, and to find the divisor, the dividend must be divided by the quotient.
Let's present simple examples of solving equations:
Here it is only required to produce the difference of fractions without leading to a common denominator.
The answer came out as an incorrect fraction. It can be converted to 1 integer and 3/5.
In the second method, the numerator and denominator were multiplied by 4 to cancel out the bottom, rather than flip the denominator.
Actions with fractions.
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very ..."
And for those who are "very even ...")
So, what are fractions, types of fractions, transformations - we remembered. Let's get down 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 fractions are no different from operations with integers. Actually, that's why they are good, decimal. The only thing is that you need to put the comma correctly.
Mixed numbers, as I said, are of little use for most actions. They still need to be converted into fractions.
But the actions with ordinary fractions 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 on are no different from actions with ordinary fractions! Fractional operations are the foundation for all algebra. It is for this reason that we will analyze all this arithmetic in great detail here.
Addition and subtraction of fractions.
Everyone can add (subtract) fractions with the same denominators (I really hope!). Well, let me remind you completely forgetful: when adding (subtracting) the denominator does not change. The numerators are added (subtracted) to give the numerator of the result. Type:
In short, in general terms:
And if the denominators are different? Then, using the basic property of the fraction (here it came in handy again!), We make the denominators the same! For example:
Here we had to make 4/10 from the fraction 2/5. For the sole purpose of making the denominators the same. Note, just in case, that 2/5 and 4/10 are the same fraction! Only 2/5 is uncomfortable for us, and 4/10 is nothing at all.
By the way, this is the essence of solving any problems in mathematics. When we are from uncomfortable expressions do the same, but already convenient for solution.
Another example:
The situation is similar. Here we make 48 out of 16. By simple multiplication by 3. It's all clear. But here we came across something like:
How to be ?! It is difficult to make nine out of seven! But we are smart, we know the rules! We transform every fraction so that the denominators become the same. This is called "converting to a common denominator":
How! How did I know about 63? Very simple! 63 is a number that is evenly divisible by 7 and 9 at the same time. Such a number can always be obtained by multiplying the denominators. If we multiplied some 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, in steps. You just need to find a denominator common to all fractions, and bring each fraction to this very denominator. For example:
And what is the common denominator? You can, of course, multiply 2, 4, 8, and 16. We get 1024. Nightmare. It’s easier to figure out that the number 16 is perfectly divisible by 2, and 4, and 8. Therefore, from these numbers it is easy to get 16. This number will be the common denominator. 1/2 will turn into 8/16, 3/4 into 12/16, and so on.
By the way, if we take 1024 as the common denominator, everything will work out too, in the end everything will shrink. Only not everyone will get to this end, because of the calculations ...
Complete the example yourself. Not a logarithm ... It should be 29/16.
So, adding (subtracting) fractions is clear, I hope? Of course, it is easier to work in a shortened version, with additional factors. But this pleasure is available to those who honestly worked in the lower grades ... And have not forgotten anything.
And now we will do the same actions, but not with fractions, but with fractional expressions... There will be a new rake here, yes ...
So, we need to add two fractional expressions:
We need to make the denominators the same. And only with the help multiplication! So the basic property of a fraction dictates. Therefore, I cannot add one to the first fraction in the denominator to the x. (but it would be nice!). But if you multiply the denominators, you see, everything will grow together! So we write down, a line of the fraction, we leave an empty space on top, then we add it, and below we write the product of the denominators, 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 of the right side, we figure out: in order to get the denominator x (x + 1) in the first fraction, the numerator and denominator of this fraction must be multiplied by (x + 1). And in the second fraction - by x. Here's what happens:
Note! Brackets appeared here! This is the rake that many are stepping on. Not brackets, of course, but their absence. The parentheses appear because we are multiplying the whole numerator and the whole denominator! And not their separate pieces ...
In the numerator of the right side, we write the sum of the numerators, everything is like in numeric fractions, then we open the brackets in the numerator of the right side, i.e. we multiply everything and give similar ones. You don't need to open parentheses in denominators, you don't need to multiply something! In general, a work is always more pleasant in the denominators (any)! We get:
So we got the answer. The process seems long and difficult, but it depends on the practice. Solve examples, get used to it, everything will become simple. Those who have mastered fractions in due time do all these operations with one hand, on the machine!
And one more note. Many famously deal with fractions, but hang on examples with whole numbers. Like: 2 + 1/2 + 3/4 =? Where to fasten the deuce? You don't need to fasten it anywhere, you need to make a fraction out of two. It is not easy, but very simple! 2 = 2/1. Like this. Any integer 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, in addition - subtraction of fractions, knowledge has been refreshed. We repeated the conversion of fractions from one type to another. You can and check. Shall we solve 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 actions 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. Instant validation testing. Learning - with interest!)
you can get acquainted with functions and derivatives.
Now that we have learned how to add and multiply individual fractions, we can consider more complex designs. For example, what if the same problem contains addition, subtraction, and multiplication of fractions?
First of all, you need to translate all fractions into incorrect ones. Then we sequentially perform the required actions - in the same order as for ordinary numbers. Namely:
- The exponentiation is performed first - get rid of all expressions containing indicators;
- Then - division and multiplication;
- The last step is addition and subtraction.
Of course, if there are brackets in the expression, the order of actions changes - everything inside the brackets must be counted first. And remember about incorrect fractions: you need to select the whole part only when all other actions have already been completed.
Let's translate all fractions from the first expression into incorrect ones, and then perform the following actions:
Now let's find the value of the second expression. There are no fractions with an integer part here, but there are brackets, so first we perform addition, and only then - division. Note that 14 = 7 2. Then:
Finally, consider the third example. There are brackets and a degree here - it is better to count them separately. Taking into account that 9 = 3 3, we have:
Take a look at the last example. To raise a fraction to a power, you must separately raise the numerator to this power, and separately - the denominator.
You can decide in a different way. If we recall the definition of the degree, the problem will be reduced to the usual multiplication of fractions:
Multi-storey fractions
Until now, we have considered only "pure" fractions, when the numerator and denominator are ordinary numbers. This is quite consistent with the definition of a numeric fraction given in the very first lesson.
But what if a more complex object is placed in the numerator or denominator? For example, another number fraction? Such constructions occur quite often, especially when working with long expressions. Here are a couple of examples:
There is only one rule for working with multi-storey fractions: you must immediately get rid of them. Removing "extra" floors is quite simple, if you remember that the fractional bar means the standard division operation. Therefore, any fraction can be rewritten as follows:
Using this fact and observing the order of actions, we can easily reduce any multi-level fraction to a regular one. Take a look at examples:
Task. Convert multi-storey fractions to regular ones:
In each case, we rewrite the main fraction, replacing the dividing line with a division sign. Also, remember that any integer can be represented as a fraction with a denominator of 1. That is, 12 = 12/1; 3 = 3/1. We get:
In the last example, the fractions were canceled before the final multiplication.
The specifics of working with multi-level fractions
There is one subtlety in multi-storey fractions that must always be remembered, otherwise you can get the wrong answer, even if all the calculations were correct. Take a look:
- The numerator contains a single number 7, and the denominator contains the fraction 12/5;
- The numerator contains the fraction 7/12, and the denominator is the single number 5.
So, for one recording, we got two completely different interpretations. If you count, the answers will also be different:
To always read the entry unambiguously, use a simple rule: the separating line of the main fraction must be longer than the nested line. It is desirable - several times.
If you follow this rule, then the above fractions should be written as follows:
Yes, it might be ugly and take up too much space. But you will count correctly. Finally, a couple of examples where multi-storey fractions really arise:
Task. Find the values of the expressions:
So, we are working with the first example. Let's convert all fractions to irregular ones, and then perform addition and division operations:
Let's do the same with the second example. Let's translate all fractions into irregular ones and perform the required operations. In order not to tire the reader, I will omit some of the obvious calculations. We have:
Due to the fact that there are sums in the numerator and denominator of the main fractions, the rule for writing multi-storey fractions is observed automatically. Also, in the last example, we intentionally left 46/1 in fractional form to do division.
Also note that in both examples, the fractional bar actually replaces the parentheses: first of all, we found the sum, and only then - the quotient.
Some would argue that the transition to improper fractions in the second example was clearly redundant. Perhaps it is so. But with this we insure ourselves against mistakes, because next time the example may turn out to be much more complicated. Choose for yourself which is more important: speed or reliability.