Examples with fractions are one of the basic elements of mathematics. There are many different types equations with fractions. Below is detailed instructions by solving examples of this type.
How to solve examples with fractions - general rules
To solve examples with fractions of any type, whether it be addition, subtraction, multiplication or division, you need to know the basic rules:
- In order to add fractional expressions with the same denominator (the denominator is the number at the bottom of the fraction, the numerator at the top), you need to add their numerators, and leave the denominator the same.
- In order to subtract from one fractional expression the second (with the same denominator), you need to subtract their numerators, and leave the denominator the same.
- To add or subtract fractional expressions with different denominators, we need to find the least common denominator.
- In order to find a fractional product, you need to multiply the numerators and denominators, while, if possible, reduce.
- To divide a fraction by a fraction, you need to multiply the first fraction by the reversed second.
How to solve examples with fractions - practice
Rule 1, example 1:
Calculate 3/4 +1/4.
According to Rule 1, if fractions of two (or more) have the same denominator, you just need to add their numerators. We get: 3/4 + 1/4 = 4/4. If a fraction has the same numerator and denominator, the fraction will be 1.
Answer: 3/4 + 1/4 = 4/4 = 1.
Rule 2, example 1:
Calculate: 3/4 - 1/4
Using rule number 2, to solve this equation, you need to subtract 1 from 3, and leave the denominator the same. We get 2/4. Since two 2 and 4 can be reduced, we reduce and get 1/2.
Answer: 3/4 - 1/4 = 2/4 = 1/2.
Rule 3, Example 1
Calculate: 3/4 + 1/6
Solution: Using the 3rd rule, we find the least common denominator. The least common denominator is the number that is divisible by the denominators of all fractional expressions in the example. Thus, we need to find such a minimum number that will be divisible by both 4 and 6. This number is 12. We write 12 as the denominator. We divide 12 by the denominator of the first fraction, we get 3, we multiply by 3, we write 3 in the numerator *3 and + sign. We divide 12 by the denominator of the second fraction, we get 2, we multiply 2 by 1, we write 2 * 1 in the numerator. So, we got a new fraction with a denominator equal to 12 and a numerator equal to 3*3+2*1=11. 11/12.
Answer: 11/12
Rule 3, Example 2:
Calculate 3/4 - 1/6. This example is very similar to the previous one. We do all the same actions, but in the numerator instead of the + sign, we write the minus sign. We get: 3*3-2*1/12 = 9-2/12 = 7/12.
Answer: 7/12
Rule 4, Example 1:
Calculate: 3/4 * 1/4
Using the fourth rule, we multiply the denominator of the first fraction by the denominator of the second and the numerator of the first fraction by the numerator of the second. 3*1/4*4 = 3/16.
Answer: 3/16
Rule 4, Example 2:
Calculate 2/5 * 10/4.
This fraction can be reduced. In the case of a product, the numerator of the first fraction and the denominator of the second and the numerator of the second fraction and the denominator of the first are reduced.
2 is reduced from 4. 10 is reduced from 5. we get 1 * 2/2 = 1 * 1 = 1.
Answer: 2/5 * 10/4 = 1
Rule 5, Example 1:
Calculate: 3/4: 5/6
Using the 5th rule, we get: 3/4: 5/6 = 3/4 * 6/5. We reduce the fraction according to the principle of the previous example and get 9/10.
Answer: 9/10.
How to Solve Fraction Examples - Fractional Equations
Fractional equations are examples where the denominator contains an unknown. In order to solve such an equation, you need to use certain rules.
Consider an example:
Solve equation 15/3x+5 = 3
Recall that you cannot divide by zero, i.e. the denominator value must not be zero. When solving such examples, this must be indicated. For this, there is an ODZ (area allowed values).
So 3x+5 ≠ 0.
Hence: 3x ≠ 5.
x ≠ 5/3
For x = 5/3, the equation simply has no solution.
By specifying the ODZ, in the best possible way solve this equation will get rid of the fractions. To do this, we first represent all non-fractional values as a fraction, in this case the number 3. We get: 15/(3x+5) = 3/1. To get rid of fractions, you need to multiply each of them by the smallest common denominator. In this case, that would be (3x+5)*1. Sequencing:
- Multiply 15/(3x+5) by (3x+5)*1 = 15*(3x+5).
- Expand the brackets: 15*(3x+5) = 45x + 75.
- We do the same with the right side of the equation: 3*(3x+5) = 9x + 15.
- Equate the left and right sides: 45x + 75 = 9x +15
- Move x's to the left, numbers to the right: 36x = -50
- Find x: x = -50/36.
- We reduce: -50/36 = -25/18
Answer: ODZ x ≠ 5/3. x = -25/18.
How to solve examples with fractions - fractional inequalities
Fractional inequalities of the type (3x-5)/(2-x)≥0 are solved using the numerical axis. Consider this example.
Sequencing:
- Equate the numerator and denominator to zero: 1. 3x-5=0 => 3x=5 => x=5/3
2. 2-x=0 => x=2 - We draw a numerical axis, painting the resulting values on it.
- Draw a circle under the value. The circle is of two types - filled and empty. A filled circle means that this value is included in the range of solutions. An empty circle indicates that this value is not included in the range of solutions.
- Since the denominator cannot be zero, there will be an empty circle under the 2nd.
- To determine the signs, we substitute any number greater than two into the equation, for example 3. (3 * 3-5) / (2-3) \u003d -4. the value is negative, so we write a minus over the area after the deuce. Then we substitute any value of the interval from 5/3 to 2 instead of x, for example 1. The value is again negative. We write minus. We repeat the same with the area up to 5/3. We substitute any number less than 5/3, for example 1. Minus again.
- Since we are interested in x values, at which the expression will be greater than or equal to 0, and there are no such values (cons everywhere), this inequality has no solution, i.e. x = Ø (empty set).
Answer: x = Ø
Students are introduced to fractions in 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, and so on. For several centuries, the examples were considered too complex. Now developed detailed rules on the conversion of fractions, addition, multiplication and other actions. It is enough to understand the material a little, and the solution will be given easily.
An ordinary fraction, which is 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.
Select proper fractions (m< n) а также неправильные (m >n).
A proper 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). An improper 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 matched (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 the same number(only not by zero), then the value of the fraction will not change (3/5 = 6/10 (just multiplied by 2).
- Reducing fractions is similar to expanding, but here they are divided by a number.
- Compare. If two fractions have the same numerator, then the fraction with the smaller denominator will be larger. 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 the upper parts, and the lower part does not change). For different ones, you will have to find a common denominator and additional factors.
- Multiply and divide fractions.
Examples of operations with fractions are considered below.
Reduced fractions Grade 6
To reduce means to divide the top and bottom of a fraction by some equal number.
The figure shows simple examples of reduction. 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 divisible by 2 in any way. Even numbers are 2, 4, 6 ... 32 8 (ends in even), etc.
In the second case, when dividing 6 by 18, it is immediately clear that the numbers are divisible by 2. Dividing, we get 3/9. This fraction is also divisible by 3. Then the answer is 1/3. If you multiply both divisors: 2 by 3, then 6 will come out. It turns out that the fraction was divided by six. This gradual division is called successive reduction of the fraction by common divisors.
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 the number consists of digits, the addition of which will result in a number divisible by 3, then the original can also be reduced by 3. Example: the number 341. Add the numbers: 3 + 4 + 1 = 8 (8 is not divisible by 3, so the number 341 cannot be reduced by 3 without a remainder). Another example: 264. Add: 2 + 6 + 4 = 12 (divided by 3). We get: 264: 3 = 88. This will simplify the reduction of large numbers.
In addition to the method of successive reduction of a fraction by common divisors, there are other ways.
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 match, if there are several of them (as in the picture below), then you need to multiply.
Mixed fractions grade 6
All improper fractions can be turned into mixed by highlighting the whole part in them. The integer is written on the left.
Often you have to make a mixed number from an improper fraction. The conversion process in the example below: 22/4 = 22 divided 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 reduced, we divide the upper and lower parts by 2.
A mixed number can easily be turned into a non proper 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 this is easy to do: add up the integer parts and 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 (NOD).
In the example below, for the numbers 9 and 6, the denominator will be 18. After that, additional factors are needed. To find them, you should divide 18 by 9, so an additional number is found - 2. We multiply it by the numerator 4, we get the fraction 8/18). The same is done with the second fraction. We already add the converted fractions (whole numbers and numerators separately, we do not change the denominator). In the example, the answer had to be converted to a proper fraction (initially, the numerator turned out to be greater than the denominator).
Please note that with the difference of fractions, the algorithm of actions 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, multiply the top and bottom parts and write down the answer. If it is clear that fractions can be reduced, then we reduce immediately.
In this example, we didn’t have to cut anything, we just wrote down the answer and highlighted the whole part.
In this example, I had to reduce the numbers under one line. Though it is possible to reduce also the ready answer.
When dividing, the algorithm is almost the same. First, we turn the mixed fraction into an improper one, then we write the numbers under one line, replacing the division with multiplication. Do not forget to swap the upper and lower parts of the second fraction (this rule division of fractions).
If necessary, we reduce the numbers (in the example below, they reduced it by five and two). We transform the improper fraction by highlighting the integer part.
Basic tasks for fractions Grade 6
The video shows a few more tasks. For clarity, we used graphic images solutions to help visualize fractions.
Examples of fraction multiplication 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 match. So you need to find a common one.
- For fractions, the common denominator is 12.
- We 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 numerators to convert fractions: 1 x 7 \u003d 7 (first fraction: 7/12); 4 x 2 = 8 (second fraction: 8/12).
- Now we can compare: 7/12 and 8/12. Turned out: 7/12< 8/12.
To represent fractions better, you can use drawings for clarity, where an 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 chosen. In the second, they divide into 3 parts and take 2. With the naked eye, it will be clear that 2/3 will be more than 4/7.
Examples with fractions grade 6 for training
As an exercise, you can perform the following tasks.
- Compare fractions
- do the multiplication
Tip: if it is difficult to find the lowest common denominator of 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, you get 72.
Solving equations with fractions Grade 6
In solving equations, you need to remember the actions with fractions: multiplication, division, subtraction and addition. If one of the factors is unknown, then the product (total) is divided by the 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, you need to divide the dividend by the quotient.
Imagine simple examples solving equations:
Here it is only required to produce the difference of fractions, without leading to a common denominator.
The answer is an improper fraction. It can be converted to 1 whole and 3/5.
In the second method, the numerator and denominator were multiplied by 4 to shorten the bottom rather than flip the denominator.
This article is a general look at operations with fractions. Here we formulate and justify the rules for addition, subtraction, multiplication, division and raising to the power of fractions of the general form A/B, where A and B are some numbers, numerical expressions or expressions with variables. As usual, we will supply the material with explanatory examples with detailed descriptions solutions.
Page navigation.
Rules for performing operations with numerical fractions of a general form
Let's agree on numbers general view understand fractions in which the numerator and / or denominator can be represented not only by natural numbers, but also by other numbers or numerical expressions. For clarity, here are a few examples of such fractions: 
.
We know the rules by which . By the same rules, you can perform operations with fractions of a general form:
Rationale for the rules
To justify the validity of the rules for performing actions with general numerical fractions, one can start from the following points:
- a fractional bar is essentially a division sign,
- division by some non-zero number can be considered as multiplication by the reciprocal of the divisor (this immediately explains the rule for dividing fractions),
- properties of actions with real numbers,
- and its generalized understanding,
They allow you to carry out the following transformations that justify the rules for adding, subtracting fractions with the same and different denominators, as well as the rule for multiplying fractions: 
Examples
Let us give examples of performing an action with fractions of a general form according to the rules learned in the previous paragraph. Let's say right away that usually, after performing operations with fractions, the resulting fraction requires simplification, and the process of simplifying a fraction is often more difficult than performing the previous actions. We will not dwell on the simplification of fractions (the corresponding transformations are discussed in the article Transformation of fractions), so as not to be distracted from the topic of interest to us.
Let's start with examples of addition and subtraction numeric fractions with the same denominators. Let's start by adding the fractions and . Obviously the denominators are equal. According to the corresponding rule, we write down a fraction whose numerator is equal to the sum of the numerators of the original fractions, and leave the denominator the same, we have . The addition is done, it remains to simplify the resulting fraction: 
. So, 
.
It was possible to carry out the decision in a different way: first, make the transition to ordinary fractions, and then carry out addition. With this approach, we have 
.
Now subtract from the fraction 
fraction 
. The denominators of fractions are equal, therefore, we act according to the rule for subtracting fractions with the same denominators: 
Let's move on to examples of adding and subtracting fractions with different denominators. Here the main difficulty lies in bringing the fractions to a common denominator. For fractions of a general form, this is a rather extensive topic, we will analyze it in detail in a separate article. reducing fractions to a common denominator. Now let's confine ourselves to a couple general recommendations, because in this moment we are more interested in the technique of performing operations with fractions.
In general, the process is similar to reduction to a common denominator of ordinary fractions. That is, the denominators are presented as products, then all the factors from the denominator of the first fraction are taken and the missing factors from the denominator of the second fraction are added to them.
When the denominators of the added or subtracted fractions do not have common factors, then it is logical to take their product as a common denominator. Let's take an example.
Let's say we need to add fractions and 1/2. Here, as a common denominator, it is logical to take the product of the denominators of the original fractions, that is, . In this case, the additional factor for the first fraction will be 2 . After multiplying the numerator and denominator by it, the fraction will take the form . And for the second fraction, the additional factor is the expression. With its help, the fraction 1/2 is reduced to the form. It remains to add the resulting fractions with the same denominators. Here is a summary of the entire solution: 
In the case of fractions of a general form, we are no longer talking about the least common denominator, to which ordinary fractions are usually reduced. Although in this matter it is still desirable to strive for some minimalism. By this we want to say that it is not necessary to immediately take the product of the denominators of the original fractions as a common denominator. For example, it is not at all necessary to take the common denominator of fractions and the product 
. Here, as a common denominator, we can take .
We turn to examples of multiplication of fractions of a general form. Multiply the fractions and . The rule for performing this action tells us to write down a fraction whose numerator is the product of the numerators of the original fractions, and the denominator is the product of the denominators. We have 
. Here, as in many other cases when multiplying fractions, you can reduce the fraction: 
.
The rule for dividing fractions allows you to move from division to multiplication by a reciprocal. Here you need to remember that in order to get a fraction reciprocal of a given one, you need to swap the numerator and denominator of this fraction. Here is an example of the transition from dividing general fractions to multiplication: 
. It remains to perform the multiplication and simplify the resulting fraction (if necessary, see the transformation of irrational expressions): 
Concluding the information of this paragraph, we recall that any number or numerical expression can be represented as a fraction with a denominator 1, therefore, addition, subtraction, multiplication and division of a number and a fraction can be considered as performing the corresponding action with fractions, one of which has a unit in the denominator . For example, replacing in the expression 
root of three fractions, we will proceed from multiplying a fraction by a number to multiplying two fractions: 
.
Performing operations with fractions containing variables
The rules from the first part of this article also apply to performing operations with fractions that contain variables. Let us justify the first of them - the rule of addition and subtraction of fractions with the same denominators, the rest are proved in exactly the same way.
Let us prove that for any expressions A , C and D (D is identically non-zero) we have the equality 
on its range of acceptable values of variables.
Let's take some set of variables from ODZ. Let the expressions A , C and D take the values a 0 , c 0 and d 0 for these values of the variables. Then substituting the values of variables from the selected set into the expression turns it into a sum (difference) of numerical fractions with the same denominators of the form , which, according to the rule of addition (subtraction) of numerical fractions with the same denominators, is equal to . But substituting the values of the variables from the selected set into the expression turns it into the same fraction. This means that for the selected set of variable values from the ODZ, the values of the expressions and are equal. It is clear that the values of the indicated expressions will be equal for any other set of values of variables from the ODZ, which means that the expressions and are identically equal, that is, the equality being proved is true .
Examples of addition and subtraction of fractions with variables
When the denominators of the fractions being added or subtracted are the same, then everything is quite simple - the numerators are added or subtracted, and the denominator remains the same. It is clear that the fraction obtained after this is simplified if necessary and possible.
Note that sometimes the denominators of fractions differ only at first glance, but in fact they are identical in equal terms, such as, 
and , or and . And sometimes it is enough to simplify the initial fractions so that their identical denominators “appear”.
Example.
, b) 
, in) 
.
Solution.
a) We need to subtract fractions with the same denominators. According to the corresponding rule, we leave the denominator the same and subtract the numerators, we have 
. Action done. But you can still open the brackets in the numerator and bring like terms: 
.
b) Obviously, the denominators of the added fractions are the same. Therefore, we add the numerators, and leave the denominator the same: . Addition completed. But it is easy to see that the resulting fraction can be reduced. Indeed, the numerator of the resulting fraction can be reduced by the square of the sum as (lgx + 2) 2 (see the abbreviated multiplication formulas), so the following transformations take place: 
.
c) Fractions in the sum have different denominators. But, by converting one of the fractions, you can proceed to adding fractions with the same denominators. We show two solutions.
First way. The denominator of the first fraction can be factored using the difference of squares formula, and then reduce this fraction: 
. In this way, . It doesn’t hurt to get rid of irrationality in the denominator of a fraction: 
.
The second way. Multiplying the numerator and denominator of the second fraction (this expression does not vanish for any values of the variable x from the DPV for the original expression) allows you to achieve two goals at once: get rid of irrationality and move on to adding fractions with the same denominators. We have 
Answer:
a) 
, b) 
, in) 
.
The last example brought us to the question of bringing fractions to a common denominator. There, we almost accidentally came to the same denominators, simplifying one of the added fractions. But in most cases, when adding and subtracting fractions with different denominators, one has to purposefully bring the fractions to a common denominator. To do this, the denominators of fractions are usually presented as products, all factors are taken from the denominator of the first fraction, and the missing factors from the denominator of the second fraction are added to them.
Example.
Perform actions with fractions: a) 
, b) , c) 
.
Solution.
a) There is no need to do anything with the denominators of the fractions. As a common denominator, we take the product 
. In this case, the additional factor for the first fraction is the expression, and for the second fraction - the number 3. These additional factors bring fractions to a common denominator, which further allows us to perform the action we need, we have 
b) In this example, the denominators are already presented as products, and no additional transformations are required. Obviously, the factors in the denominators differ only in exponents, therefore, as a common denominator, we take the product of the factors with the largest exponents, that is, 
. Then the additional factor for the first fraction will be x 4 , and for the second - ln(x+1) . Now we are ready to subtract fractions: 
c) And in this case, to begin with, we will work with the denominators of fractions. The formulas of the difference of squares and the square of the sum allow you to go from the original sum to the expression 
. Now it is clear that these fractions can be reduced to a common denominator 
. With this approach, the solution will be next view:
Answer:
a) 
b) 
in) 
Examples of multiplying fractions with variables
Multiplying fractions gives a fraction whose numerator is the product of the numerators of the original fractions, and the denominator is the product of the denominators. Here, as you can see, everything is familiar and simple, and we can only add that the fraction obtained as a result of this action is often reduced. In these cases, it is reduced, unless, of course, it is necessary and justified.
Actions with fractions. In this article, we will analyze examples, everything is detailed with explanations. We will consider ordinary fractions. In the future, we will analyze decimals. I recommend to watch the whole and study sequentially.
1. Sum of fractions, difference of fractions.
Rule: when adding fractions with equal denominators, the result is a fraction - the denominator of which remains the same, and its numerator will be equal to the sum of the numerators of the fractions.
Rule: when calculating the difference of fractions with the same denominators, we get a fraction - the denominator remains the same, and the numerator of the second is subtracted from the numerator of the first fraction.
Formal notation of the sum and difference of fractions with equal denominators:
Examples (1):
It is clear that when ordinary fractions are given, then everything is simple, but if they are mixed? Nothing complicated...
Option 1- you can convert them into ordinary ones and then calculate them.
Option 2- you can separately "work" with the integer and fractional parts.
Examples (2):
Yet:
And if the difference of two mixed fractions is given and the numerator of the first fraction is less than the numerator of the second? It can also be done in two ways.
Examples (3):
* Translated into ordinary fractions, calculated the difference, converted the resulting improper fraction into a mixed one.
* Divided into integer and fractional parts, got three, then presented 3 as the sum of 2 and 1, with the unit presented as 11/11, then found the difference between 11/11 and 7/11 and calculated the result. The meaning of the above transformations is to take (select) a unit and present it as a fraction with the denominator we need, then from this fraction we can already subtract another.
Another example:
Conclusion: there is a universal approach - in order to calculate the sum (difference) of mixed fractions with equal denominators, they can always be converted to improper ones, then execute required action. After that, if as a result we get an improper fraction, we translate it into a mixed one.
Above, we looked at examples with fractions that have equal denominators. What if the denominators differ? In this case, the fractions are reduced to the same denominator and the specified action is performed. To change (transform) a fraction, the main property of the fraction is used.
Consider simple examples:
In these examples, we immediately see how one of the fractions can be converted to get equal denominators.
If we designate ways to reduce fractions to one denominator, then this one will be called METHOD ONE.
That is, immediately when “evaluating” the fraction, you need to figure out whether such an approach will work - we check whether the larger denominator is divisible by the smaller one. And if it is divided, then we perform the transformation - we multiply the numerator and denominator so that the denominators of both fractions become equal.
Now look at these examples:
This approach does not apply to them. There are other ways to reduce fractions to a common denominator, consider them.
Method SECOND.
Multiply the numerator and denominator of the first fraction by the denominator of the second, and the numerator and denominator of the second fraction by the denominator of the first:
*In fact, we bring fractions to the form when the denominators become equal. Next, we use the rule of adding timid with equal denominators.
Example:
*This method can be called universal, and it always works. The only negative is that after the calculations, a fraction may turn out that will need to be further reduced.
Consider an example:
It can be seen that the numerator and denominator are divisible by 5:
Method THIRD.
Find the least common multiple (LCM) of the denominators. This will be the common denominator. What is this number? This is the smallest natural number that is divisible by each of the numbers.
Look, here are two numbers: 3 and 4, there are many numbers that are divisible by them - these are 12, 24, 36, ... The smallest of them is 12. Or 6 and 15, 30, 60, 90 are divisible by them .... Least 30. Question - how to determine this least common multiple?
There is a clear algorithm, but often this can be done immediately without calculations. For example, according to the above examples (3 and 4, 6 and 15), no algorithm is needed, we took large numbers (4 and 15), doubled them and saw that they are divisible by the second number, but pairs of numbers can be others, such as 51 and 119.
Algorithm. In order to determine the least common multiple of several numbers, you must:
- expand each of the numbers into SIMPLE multipliers
- write out the decomposition of the BIGGER of them
- multiply it by the MISSING factors of other numbers
Consider examples:
50 and 60 50 = 2∙5∙5 60 = 2∙2∙3∙5
in decomposition more missing one five
=> LCM(50,60) = 2∙2∙3∙5∙5 = 300
48 and 72 48 = 2∙2∙2∙2∙3 72 = 2∙2∙2∙3∙3
in the expansion of a larger number, two and three are missing
=> LCM(48,72) = 2∙2∙2∙2∙3∙3 = 144
* Least common multiple of two prime numbers equal to their product
Question! And why is it useful to find the least common multiple, because you can use the second method and simply reduce the resulting fraction? Yes, you can, but it's not always convenient. See what the denominator will be for the numbers 48 and 72 if you simply multiply them 48∙72 = 3456. Agree that it is more pleasant to work with smaller numbers.
Consider examples:
*51 = 3∙17 119 = 7∙17
in the expansion of a larger number, a triple is missing
=> LCM(51,119) = 3∙7∙17
And now we apply the first method:
* Look at the difference in the calculations, in the first case there is a minimum of them, and in the second you need to work separately on a piece of paper, and even the fraction that you got needs to be reduced. Finding the LCM simplifies the work considerably.
More examples:
* In the second example, it is clear that smallest number, which is divided by 40 and 60 is equal to 120.
TOTAL! GENERAL CALCULATION ALGORITHM!
- we bring fractions to ordinary ones, if there is an integer part.
- we bring the fractions to a common denominator (first we look to see if one denominator is divisible by another, if it is divisible, then we multiply the numerator and denominator of this other fraction; if it is not divisible, we act using the other methods indicated above).
- having received fractions with equal denominators, we perform actions (addition, subtraction).
- if necessary, we reduce the result.
- if necessary, select the whole part.
2. Product of fractions.
The rule is simple. When multiplying fractions, their numerators and denominators are multiplied:
Examples:
A task. 13 tons of vegetables were brought to the base. Potatoes make up ¾ of all imported vegetables. How many kilograms of potatoes were brought to the base?
Let's finish with the work.
*Earlier I promised you to give a formal explanation of the main property of the fraction through the product, please:
3. Division of fractions.
The division of fractions is reduced to their multiplication. It is important to remember here that the fraction that is a divisor (the one that is divided by) is turned over and the action changes to multiplication:
This action can be written as a so-called four-story fraction, because the division itself “:” can also be written as a fraction:
Examples:
That's all! Good luck to you!
Sincerely, Alexander Krutitskikh.
Arithmetic with ordinary fractions
1. Addition.
To add fractions with the same denominators, add their numerators and leave the denominator the same.
Example. .
To add fractions with different denominators, you need to bring them to the lowest common denominator, and then add the resulting numerators and sign the common denominator under the sum.
Example.
Briefly written like this:
To add mixed numbers, you need to separately find the sum of integers and the sum of fractional parts. The action is written like this:
2. Subtraction.
To subtract fractions with the same denominators, you need to subtract the numerator of the subtracted from the numerator of the minuend and leave the same denominator. The action is written like this:
To subtract fractions with different denominators, you must first bring them to the smallest common denominator, then subtract the numerator of the subtrahend from the numerator of the minuend and sign the common denominator under their difference. The action is written like this:
If you need to subtract one mixed number from another mixed number, then, if possible, subtract a fraction from a fraction, and a whole from a whole. The action is written like this:
If the fraction of the subtrahend is greater than the fraction of the minuend, then one unit is taken from the whole number of the minuend, it is divided into the appropriate shares and added to the fraction of the minuend, after which they proceed as described above. The action is written like this:
Do the same when you need to subtract a fractional number from a whole number.
Example. .
3. Extension of the properties of addition and subtraction to fractional numbers.All laws and properties of addition and subtraction of natural numbers are also valid for fractional numbers. Their use in many cases greatly simplifies the calculation process.
4. Multiplication.
To multiply a fraction by a fraction, you need to multiply the numerator by the numerator, and the denominator by the denominator and make the first product the numerator and the second product the denominator.
When multiplying, one should do (if possible) a reduction.
Example. .
If we take into account that an integer is a fraction with a denominator of 1, then multiplying a fraction by an integer and an integer by a fraction can be carried out according to the same rule.
Examples.
5. Multiplication mixed numbers.
To multiply mixed numbers, you must first convert them to improper fractions and then multiply according to the rule of multiplying fractions.
Example. .
6. Division of a fraction by a fraction.
To divide a fraction by a fraction, you need to multiply the numerator of the first fraction by the denominator of the second, and the denominator of the first by the numerator of the second and write the first product as the numerator, and the second as the denominator.
Example. .
By the same rule, you can divide a fraction by an integer and an integer by a fraction, if you represent an integer as a fraction with a denominator of 1.
Examples.
7. Division of mixed numbers.
To perform the division of mixed numbers, they are first converted to improper fractions and then divided according to the rule for dividing fractions.
Example. .
8. Replacement of division by multiplication.
If you swap the numerator and denominator in any fraction, you get a new fraction, the reciprocal of the given one. For example, for a fractionthe reciprocal will be.
Obviously, the product of two mutually reciprocal fractions equals 1.
- Finding a fraction of a number.
There are many problems in which you need to find a part or fraction of a given number. Such problems are solved by multiplication.
A task. The hostess had 20 rubles;she used them for shopping. How much do purchases cost?
Here you need to findnumber 20. You can do it like this:
Answer. The hostess spent 8 rubles.
Examples. Find from 30. Solution. .
Find from . Solution. .
- Finding a number by the known value of its fraction.
Sometimes it is required to determine the whole number from the known part of the number and the fraction expressing this part. Such tasks are solved by division.
A task. There are 12 Komsomol members in the class, which ispart of all the students in the class. How many students are in the class?
Solution. .
Answer. 20 students.
Example. Find a numberwhich is 34.
Solution. .
Answer. The desired number is.
- Finding the ratio of two numbers.
Let's consider the problem: A worker made 40 parts in a day. What part of the monthly task was completed by the worker if the monthly plan is 400 parts?
Solution. .
Answer. Worker completedpart of the monthly plan.
In this case, the part (40 parts) is expressed as fractions of the whole (400 parts). They also say that the ratio of the number of parts manufactured per day to the monthly plan has been found.
- Converting a decimal to a common fraction.
To convert decimal into an ordinary one, it is written with a denominator and, if possible, reduced:
Examples.
- Converting a fraction to a decimal.
There are several ways to convert a common fraction to a decimal.
First way. To convert a fraction to a decimal, you need to divide the numerator by the denominator.
Examples. .
The second way. To turn an ordinary fraction into a decimal, you need to multiply the numerator and denominator of this fraction by such a number that the denominator is one with zeros (if possible).
Example.
- Compare decimals by magnitude. To find out which of the two decimal fractions is greater, you need to compare their whole parts, tenths, hundredths, etc. If the whole parts are equal, the fraction with more tenths is greater; if integers and decimals are equal, the one with more hundredths is greater, etc.
Example. From three fractions 2.432; 2.41 and 2.4098 is the largest first, since it has the most hundredths, and whole and tenths are the same in all fractions.
Operations with decimals
- Multiplying and dividing a decimal by 10, 100, 1000, etc.
To multiply a decimal by 10, 100, 1000, etc. you need to move the comma, respectively, to one, two, three, etc. sign to the right. If at the same time there are not enough signs for the number, then zeros are assigned.
Example. 15.45 10 = 154.5; 32.3 100 = 3230.
To divide a decimal by 10, 100, 1000, etc., you need to move the comma to one, two, three, etc., respectively. sign to the left. If there are not enough signs to move the comma, their number is supplemented with the corresponding number of zeros on the left.
Examples. 184.35: 100 = 1.8435; 3.5: 100 = 0.035.
- Addition and subtraction of decimal fractions.
Decimals are added and subtracted in much the same way as they are added and subtracted. integers. The digit is written under the digit, the comma is written under the comma
Examples.
- Multiplying decimals.
To multiply two decimal fractions, it is enough, without paying attention to commas, to multiply them as integers and in the product to separate with a comma on the right as many decimal places as there were in the multiplicand and factor together.
Example 1. 2.064 0.05.
We multiply the integers 2064 5 = 10320. The first factor had three decimal places, the second - two. The product must have five decimal places. We separate them on the right and get 0.10320. The zero at the end can be discarded: 2.064 0.05 = 0.1032.
Example 2. 1.125 0.08; 1125 8 = 9000.
The number of decimal places should be 3 + 2 = 5. We assign zeros to the left of 9000 (009000) and separate five characters from the right. We get 1.125 0.08 = 0.09000 = 0.09.
- Division of decimals.
Two cases of division of decimal fractions without a remainder are considered: 1) division of a decimal fraction by an integer; 2) dividing a number (whole or fractional) by a decimal fraction.
Dividing a decimal by an integer is the same as dividing integers; the resulting remainders are divided sequentially into smaller decimal parts and division continues until the remainder is zero.
Examples.
Dividing a number (integer or fractional) by a decimal in all cases leads to division by an integer. To do this, increase the divisor by 10, 100, 1000, etc. times, and so that the quotient does not change, the dividend is increased by the same number of times, after which it is divided by an integer (as in the first case).
Example. 47.04: 0.0084 = 470400: 84 = 5600;
- Examples for joint actions with ordinary and decimal fractions.
Consider first an example for all actions with decimal fractions.
Example 1 Calculate:
Here they use the reduction of the dividend and the divisor to an integer, taking into account the fact that the quotient does not change. Then we have:
When solving examples for joint actions with ordinary and decimal fractions, some of the actions can be performed in decimal fractions, and some in ordinary ones. It must be borne in mind that not always an ordinary fraction can be turned into a final decimal fraction. Therefore, writing as a decimal fraction is possible only when it is verified that such a conversion is possible.
Example 2 Calculate:
Interest
The concept of interest.A percentage of a number is a hundredth of that number. For example, instead of saying "54 per cent of all the inhabitants of our country are women", you can say "54 percent of all the inhabitants of our country are women". Instead of the word "percentage" they also write the% sign, for example, 35% means 35 percent.
Since the percentage is a hundredth, it follows that the percentage is a fraction with a denominator of 100. Therefore, the fraction is 0.49, or, can be read as 49 percent and written without the denominator as 49%. In general, having determined how many hundredths are in a given decimal fraction, it is easy to write it down as a percentage. To do this, use the rule: to write a decimal fraction as a percentage, you need to move the comma in this fraction two decimal places to the right.
Examples. 0.33 = 33%; 1.25 = 125%; 0.002 = 0.2%; 21 = 2100%.
And vice versa: 7% = 0.07; 24.5% = 0.245; 0.1% = 0.001; 200% = 2.
1. Finding percentages of a given number
A task. According to the plan, the team of tractor drivers must use 9 tons of fuel. Tractor drivers took a social obligation to save 20% of fuel. Determine fuel savings in tons.
If in this problem instead of 20% we write the number 0.2 equal to it, we get a problem to find the fraction of a number. And such problems are solved by multiplication. From here comes the solution:
20% = 0.2; 9 0.2 = 1.8(m).
The calculations can also be written like this:
(m)
To find a few percent of a given number, it is enough to divide the given number by 100 and multiply the result by the number of percent.
A task. A worker in 1963 received 90 rubles a month, and in 1964 he began to receive 30% more. How much did he earn in 1964?
Solution (first method).
1) How many more rubles did the worker get?
(rub.)
90 + 27 = 117 (rub).
The second way.
1) What percentage of the previous earnings did the worker receive in 1964?
100% + 30% = 130%.
2) What was the monthly salary of a worker in 1964?
(rub.)
2. Finding a number from a given value of its percentage.
A task. On the collective farm, corn was sown on an area of 280 hectares, which is 14% of the total sown area. Determine the sown area of the collective farm.
If in this problem instead of 14% we write 0.14 or, then we get the problem of finding a number by the known value of its fraction. And such problems are solved by division.
Solution. 14% = 0.14; 280: 0.14 = 2000 (ha). You can make this decision like this:
(ha)
To find a number for a given value of several percent of it, it is enough to divide this value by the number of percent and multiply the result by 100.
A task. In March, the plant smelted 125.4 t metal, overfulfilling the plan by 4.5%. How many tons of metal the plant was supposed to smelt in March according to the plan?
Solution.
1) By what percentage did the plant fulfill the plan in March?
100% + 4,5% = 104,5%.
2) How many tons of metal the plant had to smelt?
(ha)
- Finding the percentage of two numbers.
A task. It is necessary to plow 300 hectares of land. On the first day, 120 hectares were plowed. What percentage of the task was plowed on the first day?
Solution.
First way. 300 ha is 100%, which means that 1% accounts for 3 ha. Having determined how many times 3 hectares, which are 1%, are contained in 120 hectares, we will find out how many percent of the task the land was plowed on the first day
120: 3 = 40(%).
The second way. Having determined what part of the land was plowed on the first day, we express this fraction as a percentage.
Let's write the calculation:
To calculate the percentage of a number a to number b , you need to find the ratio a to b and multiply it by 100.
