In the article, we will show how to solve fractions with simple clear examples. Let's understand what a fraction is and consider solving fractions!
concept fractions is introduced into the course of mathematics starting from the 6th grade of secondary school.
Fractions look like: ±X / Y, where Y is the denominator, it tells how many parts the whole was divided into, and X is the numerator, it tells how many such parts were taken. For clarity, let's take an example with a cake:
In the first case, the cake was cut equally and one half was taken, i.e. 1/2. In the second case, the cake was cut into 7 parts, from which 4 parts were taken, i.e. 4/7.
If the part of dividing one number by another is not a whole number, it is written as a fraction.
For example, the expression 4:2 \u003d 2 gives an integer, but 4:7 is not completely divisible, so this expression is written as a fraction 4/7.
In other words fraction is an expression that denotes the division of two numbers or expressions, and which is written with a slash.
If the numerator is less than the denominator, the fraction is correct, if vice versa, it is incorrect. A fraction can contain an integer.
For example, 5 whole 3/4.
This entry means that in order to get the whole 6, one part of four is not enough.
If you want to remember how to solve fractions for 6th grade you need to understand that solving fractions basically comes down to understanding a few simple things.
- A fraction is essentially an expression for a fraction. That is, a numerical expression of what part a given value is from one whole. For example, the fraction 3/5 expresses that if we divide something whole into 5 parts and the number of parts or parts of this whole is three.
- A fraction can be less than 1, for example 1/2 (or essentially half), then it is correct. If the fraction is greater than 1, for example 3/2 (three halves or one and a half), then it is incorrect and to simplify the solution, it is better for us to select the whole part 3/2= 1 whole 1/2.
- Fractions are the same numbers as 1, 3, 10, and even 100, only the numbers are not whole, but fractional. With them, you can perform all the same operations as with numbers. Counting fractions is not more difficult, and further we will show this with specific examples.
How to solve fractions. Examples.
A variety of arithmetic operations are applicable to fractions.
Bringing a fraction to a common denominator
For example, you need to compare the fractions 3/4 and 4/5.
To solve the problem, we first find the lowest common denominator, i.e. the smallest number that is divisible without remainder by each of the denominators of the fractions
Least common denominator(4.5) = 20
Then the denominator of both fractions is reduced to the lowest common denominator
Answer: 15/20
Addition and subtraction of fractions
If it is necessary to calculate the sum of two fractions, they are first brought to a common denominator, then the numerators are added, while the denominator remains unchanged. The difference of fractions is considered in a similar way, the only difference is that the numerators are subtracted.
For example, you need to find the sum of fractions 1/2 and 1/3
Now find the difference between the fractions 1/2 and 1/4
Multiplication and division of fractions
Here the solution of fractions is simple, everything is quite simple here:
- Multiplication - numerators and denominators of fractions are multiplied among themselves;
- Division - first we get a fraction, the reciprocal of the second fraction, i.e. swap its numerator and denominator, after which we multiply the resulting fractions.
For example:
On this about how to solve fractions, all. If you have any questions about solving fractions, something is not clear, then write in the comments and we will answer you.
If you are a teacher, then it is possible to download a presentation for an elementary school (http://school-box.ru/nachalnaya-shkola/prezentazii-po-matematike.html) which will come in handy.
Fractions
Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")
Fractions in high school are not very annoying. For the time being. Until you come across exponents with rational exponents and logarithms. And there…. You press, you press the calculator, and it shows all the full scoreboard of some numbers. You have to think with your head, like in the third grade.
Let's deal with fractions, finally! Well, how much can you get confused in them!? Moreover, it is all simple and logical. So, what are fractions?
Types of fractions. Transformations.
Fractions are of three types.
1. Common fractions , for example:
Sometimes, instead of a horizontal line, they put a slash: 1/2, 3/4, 19/5, well, and so on. Here we will often use this spelling. The top number is called numerator, lower - denominator. If you constantly confuse these names (it happens ...), tell yourself the phrase with the expression: " Zzzzz remember! Zzzzz denominator - out zzzz u!" Look, everything will be remembered.)
A dash, which is horizontal, which is oblique, means division top number (numerator) to bottom number (denominator). And that's it! Instead of a dash, it is quite possible to put a division sign - two dots.
When the division is possible entirely, it must be done. So, instead of the fraction "32/8" it is much more pleasant to write the number "4". Those. 32 is simply divided by 8.
32/8 = 32: 8 = 4
I'm not talking about the fraction "4/1". Which is also just "4". And if it doesn’t divide completely, we leave it as a fraction. Sometimes you have to do the reverse. Make a fraction from a whole number. But more on that later.
2. Decimals , for example:
It is in this form that it will be necessary to write down the answers to tasks "B".
3. mixed numbers , for example:
Mixed numbers are practically not used in high school. In order to work with them, they must be converted to ordinary fractions. But you definitely need to know how to do it! And then such a number will come across in the puzzle and hang ... From scratch. But we remember this procedure! A little lower.
Most versatile common fractions. Let's start with them. By the way, if there are all sorts of logarithms, sines and other letters in the fraction, this does not change anything. In the sense that everything actions with fractional expressions are no different from actions with ordinary fractions!
Basic property of a fraction.
So let's go! First of all, I will surprise you. The whole variety of fraction transformations is provided by a single property! That's what it's called basic property of a fraction. Remember: If the numerator and denominator of a fraction are multiplied (divided) by the same number, the fraction will not change. Those:
It is clear that you can write further, until you are blue in the face. Do not let sines and logarithms confuse you, we will deal with them further. The main thing to understand is that all these various expressions are the same fraction . 2/3.
And we need it, all these transformations? And how! Now you will see for yourself. First, let's use the basic property of a fraction for fraction abbreviations. It would seem that the thing is elementary. We divide the numerator and denominator by the same number and that's it! It's impossible to go wrong! But... man is a creative being. You can make mistakes everywhere! Especially if you have to reduce not a fraction like 5/10, but a fractional expression with all sorts of letters.
How to reduce fractions correctly and quickly without doing unnecessary work can be found in special Section 555.
A normal student does not bother dividing the numerator and denominator by the same number (or expression)! He just crosses out everything the same from above and below! This is where a typical mistake lurks, a blunder, if you like.
For example, you need to simplify the expression:
There is nothing to think about, we cross out the letter "a" from above and the deuce from below! We get:
Everything is correct. But really you shared the whole numerator and the whole denominator "a". If you are used to just cross out, then, in a hurry, you can cross out the "a" in the expression
and get again
Which would be categorically wrong. Because here the whole numerator on "a" already not shared! This fraction cannot be reduced. By the way, such an abbreviation is, um ... a serious challenge to the teacher. This is not forgiven! Remember? When reducing, it is necessary to divide the whole numerator and the whole denominator!
Reducing fractions makes life a lot easier. You will get a fraction somewhere, for example 375/1000. And how to work with her now? Without a calculator? Multiply, say, add, square!? And if you are not too lazy, but carefully reduce by five, and even by five, and even ... while it is being reduced, in short. We get 3/8! Much nicer, right?
The basic property of a fraction allows you to convert ordinary fractions to decimals and vice versa without calculator! This is important for the exam, right?
How to convert fractions from one form to another.
It's easy with decimals. As it is heard, so it is written! Let's say 0.25. It's zero point, twenty-five hundredths. So we write: 25/100. We reduce (divide the numerator and denominator by 25), we get the usual fraction: 1/4. Everything. It happens, and nothing is reduced. Like 0.3. This is three tenths, i.e. 3/10.
What if integers are non-zero? It's OK. Write down the whole fraction without any commas in the numerator, and in the denominator - what is heard. For example: 3.17. This is three whole, seventeen hundredths. We write 317 in the numerator, and 100 in the denominator. We get 317/100. Nothing is reduced, that means everything. This is the answer. Elementary Watson! From all of the above, a useful conclusion: any decimal fraction can be converted to a common fraction .
But the reverse conversion, ordinary to decimal, some cannot do without a calculator. But you must! How will you write down the answer on the exam!? We carefully read and master this process.
What is a decimal fraction? She has in the denominator always is worth 10 or 100 or 1000 or 10000 and so on. If your usual fraction has such a denominator, there is no problem. For example, 4/10 = 0.4. Or 7/100 = 0.07. Or 12/10 = 1.2. And if in the answer to the task of section "B" it turned out 1/2? What will we write in response? Decimals are required...
We remember basic property of a fraction ! Mathematics favorably allows you to multiply the numerator and denominator by the same number. For anyone, by the way! Except zero, of course. Let's use this feature to our advantage! What can the denominator be multiplied by, i.e. 2 so that it becomes 10, or 100, or 1000 (smaller is better, of course...)? 5, obviously. Feel free to multiply the denominator (this is us necessary) by 5. But, then the numerator must also be multiplied by 5. This is already maths demands! We get 1/2 \u003d 1x5 / 2x5 \u003d 5/10 \u003d 0.5. That's all.
However, all sorts of denominators come across. For example, the fraction 3/16 will fall. Try it, figure out what to multiply 16 by to get 100, or 1000... Doesn't work? Then you can simply divide 3 by 16. In the absence of a calculator, you will have to divide in a corner, on a piece of paper, as they taught in elementary grades. We get 0.1875.
And there are some very bad denominators. For example, the fraction 1/3 cannot be turned into a good decimal. Both on a calculator and on a piece of paper, we get 0.3333333 ... This means that 1/3 into an exact decimal fraction does not translate. Just like 1/7, 5/6 and so on. Many of them are untranslatable. Hence another useful conclusion. Not every common fraction converts to a decimal. !
By the way, this is useful information for self-examination. In section "B" in response, you need to write down a decimal fraction. And you got, for example, 4/3. This fraction is not converted to decimal. This means that somewhere along the way you made a mistake! Come back, check the solution.
So, with ordinary and decimal fractions sorted out. It remains to deal with mixed numbers. To work with them, they all need to be converted to ordinary fractions. How to do it? You can catch a sixth grader and ask him. But not always a sixth grader will be at hand ... We will have to do it ourselves. This is not difficult. Multiply the denominator of the fractional part by the integer part and add the numerator of the fractional part. This will be the numerator of a common fraction. What about the denominator? The denominator will remain the same. It sounds complicated, but it's actually quite simple. Let's see an example.
Let in the problem you saw with horror the number:
Calmly, without panic, we understand. The whole part is 1. One. The fractional part is 3/7. Therefore, the denominator of the fractional part is 7. This denominator will be the denominator of the ordinary fraction. We count the numerator. We multiply 7 by 1 (the integer part) and add 3 (the numerator of the fractional part). We get 10. This will be the numerator of an ordinary fraction. That's all. It looks even simpler in mathematical notation:
Clearly? Then secure your success! Convert to common fractions. You should get 10/7, 7/2, 23/10 and 21/4.
The reverse operation - converting an improper fraction into a mixed number - is rarely required in high school. Well, if... And if you - not in high school - you can look into the special Section 555. In the same place, by the way, you will learn about improper fractions.
Well, almost everything. You remembered the types of fractions and understood how convert them from one type to another. The question remains: why do it? Where and when to apply this deep knowledge?
I answer. Any example itself suggests the necessary actions. If in the example ordinary fractions, decimals, and even mixed numbers are mixed into a bunch, we translate everything into ordinary fractions. It can always be done. Well, if something like 0.8 + 0.3 is written, then we think so, without any translation. Why do we need extra work? We choose the solution that is convenient us !
If the task is full of decimal fractions, but um ... some kind of evil ones, go to ordinary ones, try it! Look, everything will be fine. For example, you have to square the number 0.125. Not so easy if you have not lost the habit of the calculator! Not only do you need to multiply the numbers in a column, but also think about where to insert the comma! It certainly doesn't work in my mind! And if you go to an ordinary fraction?
0.125 = 125/1000. We reduce by 5 (this is for starters). We get 25/200. Once again on 5. We get 5/40. Oh, it's shrinking! Back to 5! We get 1/8. Easily square (in your mind!) and get 1/64. Everything!
Let's summarize this lesson.
1. There are three types of fractions. Ordinary, decimal and mixed numbers.
2. Decimals and mixed numbers always can be converted to common fractions. Reverse Translation not always available.
3. The choice of the type of fractions for working with the task depends on this very task. If there are different types of fractions in one task, the most reliable thing is to switch to ordinary fractions.
Now you can practice. First, convert these decimal fractions to ordinary ones:
3,8; 0,75; 0,15; 1,4; 0,725; 0,012
You should get answers like this (in a mess!):
On this we will finish. In this lesson, we brushed up on the key points on fractions. It happens, however, that there is nothing special to refresh ...) If someone has completely forgotten, or has not mastered it yet ... Those can go to a special Section 555. All the basics are detailed there. Many suddenly understand everything are starting. And they solve fractions on the fly).
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)
you can get acquainted with functions and derivatives.
Math-Calculator-Online v.1.0
The calculator performs the following operations: addition, subtraction, multiplication, division, working with decimals, extracting the root, raising to a power, calculating percentages, and other operations.
Solution:
How to use the math calculator
| Key | Designation | Explanation |
|---|---|---|
| 5 | numbers 0-9 | Arabic numerals. Enter natural integers, zero. To get a negative integer, press the +/- key |
| . | semicolon) | A decimal separator. If there is no digit before the dot (comma), the calculator will automatically substitute a zero before the dot. For example: .5 - 0.5 will be written |
| + | plus sign | Addition of numbers (whole, decimal fractions) |
| - | minus sign | Subtraction of numbers (whole, decimal fractions) |
| ÷ | division sign | Division of numbers (whole, decimal fractions) |
| X | multiplication sign | Multiplication of numbers (integers, decimals) |
| √ | root | Extracting the root from a number. When you press the "root" button again, the root is calculated from the result. For example: square root of 16 = 4; square root of 4 = 2 |
| x2 | squaring | Squaring a number. When you press the "squaring" button again, the result is squared. For example: square 2 = 4; square 4 = 16 |
| 1/x | fraction | Output to decimals. In the numerator 1, in the denominator the input number |
| % | percent | Get a percentage of a number. To work, you must enter: the number from which the percentage will be calculated, the sign (plus, minus, divide, multiply), how many percent in numerical form, the "%" button |
| ( | open bracket | An open parenthesis to set the evaluation priority. A closed parenthesis is required. Example: (2+3)*2=10 |
| ) | closed bracket | A closed parenthesis to set the evaluation priority. Mandatory open bracket |
| ± | plus minus | Changes sign to opposite |
| = | equals | Displays the result of the solution. Also, intermediate calculations and the result are displayed above the calculator in the "Solution" field. |
| ← | deleting a character | Deletes the last character |
| FROM | reset | Reset button. Completely resets the calculator to "0" |
The algorithm of the online calculator with examples
Addition.
Addition of whole natural numbers ( 5 + 7 = 12 )
Addition of whole natural and negative numbers ( 5 + (-2) = 3 )
Adding decimal fractional numbers ( 0.3 + 5.2 = 5.5 )
Subtraction.
Subtraction of whole natural numbers ( 7 - 5 = 2 )
Subtraction of whole natural and negative numbers ( 5 - (-2) = 7 )
Subtraction of decimal fractional numbers ( 6.5 - 1.2 = 4.3 )
Multiplication.
Product of whole natural numbers ( 3 * 7 = 21 )
Product of whole natural and negative numbers ( 5 * (-3) = -15 )
Product of decimal fractional numbers ( 0.5 * 0.6 = 0.3 )
Division.
Division of whole natural numbers ( 27 / 3 = 9 )
Division of whole natural and negative numbers ( 15 / (-3) = -5 )
Division of decimal fractional numbers ( 6.2 / 2 = 3.1 )
Extracting the root from a number.
Extracting the root of an integer ( root(9) = 3 )
Extracting the root of decimals ( root(2.5) = 1.58 )
Extracting the root from the sum of numbers ( root(56 + 25) = 9 )
Extracting the root of the difference in numbers ( root (32 - 7) = 5 )
Squaring a number.
Squaring an integer ( (3) 2 = 9 )
Squaring decimals ( (2.2) 2 = 4.84 )
Convert to decimal fractions.
Calculating percentages of a number
Increase 230 by 15% ( 230 + 230 * 0.15 = 264.5 )
Decrease the number 510 by 35% ( 510 - 510 * 0.35 = 331.5 )
18% of the number 140 is ( 140 * 0.18 = 25.2 )
Division by a decimal is the same as division by a natural number.
Rule for dividing a number by a decimal fraction
To divide a number by a decimal fraction, it is necessary both in the dividend and in the divisor to move the comma as many digits to the right as there are in the divisor after the decimal point. After that, divide by a natural number.
Examples.
Perform division by decimal:
To divide by a decimal fraction, you need to move the comma as many digits to the right in both the dividend and the divisor as there are after the decimal point in the divisor, that is, by one sign. We get: 35.1: 1.8 \u003d 351: 18. Now we perform division by a corner. As a result, we get: 35.1: 1.8 = 19.5.
2) 14,76: 3,6
To perform the division of decimal fractions, both in the dividend and in the divisor, we move the comma to the right by one sign: 14.76: 3.6 \u003d 147.6: 36. Now we perform on a natural number. Result: 14.76: 3.6 = 4.1.
To perform division by a decimal fraction of a natural number, it is necessary both in the dividend and in the divisor to move as many characters to the right as there are in the divisor after the decimal point. Since the comma is not written in the divisor in this case, we fill in the missing number of characters with zeros: 70: 1.75 \u003d 7000: 175. We divide the resulting natural numbers with a corner: 70: 1.75 \u003d 7000: 175 \u003d 40.
4) 0,1218: 0,058
To divide one decimal fraction into another, we move the comma to the right both in the dividend and in the divisor by as many digits as there are in the divisor after the decimal point, that is, by three digits. Thus, 0.1218: 0.058 \u003d 121.8: 58. Division by a decimal fraction was replaced by division by a natural number. We share a corner. We have: 0.1218: 0.058 = 121.8: 58 = 2.1.
5) 0,0456: 3,8
Decimal fractions are the same ordinary fractions, but in the so-called decimal notation. Decimal notation is used for fractions with denominators 10, 100, 1000, etc. In this case, instead of fractions 1/10; 1/100; 1/1000; ... write 0.1; 0.01; 0.001;... .
For example, 0.7 ( zero point seven) is a fraction 7/10; 5.43 ( five point forty-three hundredths) is a mixed fraction 5 43/100 (or, equivalently, an improper fraction 543/100).
It may happen that there is one or more zeros immediately after the decimal point: 1.03 is the fraction 1 3/100; 17.0087 is the fraction 1787/10000. The general rule is: there must be as many zeros in the denominator of an ordinary fraction as there are digits after the decimal point in the decimal fraction.
A decimal can end in one or more zeros. It turns out that these zeros are “extra” - they can simply be removed: 1.30 = 1.3; 5.4600 = 5.46; 3,000 = 3. Can you figure out why this is so?
Decimals naturally arise when dividing by "round" numbers - 10, 100, 1000, ... Be sure to understand the following examples:
27:10 = 27/10 = 2 7/10 = 2,7;
579:100 = 579/100 = 5 79/100 = 5,79;
33791:1000 = 33791/1000 = 33 791/1000 = 33,791;
34,9:10 = 349/10:10 = 349/100 = 3,49;
6,35:100 = 635/100:100 = 635/10000 = 0,0635.
Do you notice a pattern here? Try to formulate it. What happens if you multiply a decimal by 10, 100, 1000?
To convert an ordinary fraction to a decimal, you need to bring it to some kind of "round" denominator:
2/5 = 4/10 = 0.4; 11/20 = 55/100 = 0.55; 9/2 = 45/10 = 4.5 etc.
Adding decimal fractions is much more convenient than ordinary fractions. Addition is performed in the same way as with ordinary numbers - according to the corresponding digits. When adding in a column, the terms must be written so that their commas are on the same vertical. The sum comma will also appear on the same vertical. The subtraction of decimal fractions is performed in exactly the same way.
If, when adding or subtracting in one of the fractions, the number of digits after the decimal point is less than in the other, then the required number of zeros should be added at the end of this fraction. You can not add these zeros, but simply imagine them in your mind.
When multiplying decimal fractions, they should again be multiplied as ordinary numbers (in this case, it is no longer necessary to write a comma under a comma). In the result obtained, you need to separate with a comma the number of characters equal to the total number of decimal places in both factors.
When dividing decimal fractions, you can simultaneously move the comma to the right by the same number of digits in the dividend and divisor: the quotient will not change from this:
2,8:1,4 = 2,8/1,4 = 28/14 = 2;
4,2:0,7 = 4,2/0,7 = 42/7 = 6;
6:1,2 = 6,0/1,2 = 60/12 = 5.
Explain why this is so?
- Draw a 10x10 square. Paint over some part of it equal to: a) 0.02; b) 0.7; c) 0.57; d) 0.91; e) 0.135 of the area of the whole square.
- What is 2.43 squares? Draw in the picture.
- Divide 37 by 10; 795; four; 2.3; 65.27; 0.48 and write the result as a decimal fraction. Divide these numbers by 100 and 1000.
- Multiply by 10 the numbers 4.6; 6.52; 23.095; 0.01999. Multiply these numbers by 100 and 1000.
- Express the decimal as a fraction and reduce it:
a) 0.5; 0.2; 0.4; 0.6; 0.8;
b) 0.25; 0.75; 0.05; 0.35; 0.025;
c) 0.125; 0.375; 0.625; 0.875;
d) 0.44; 0.26; 0.92; 0.78; 0.666; 0.848. - Imagine as a mixed fraction: 1.5; 3.2; 6.6; 2.25; 10.75; 4.125; 23.005; 7.0125.
- Write a common fraction as a decimal:
a) 1/2; 3/2; 7/2; 15/2; 1/5; 3/5; 4/5; 18/5;
b) 1/4; 3/4; 5/4; 19/4; 1/20; 7/20; 49/20; 1/25; 13/25; 77/25; 1/50; 17/50; 137/50;
c) 1/8; 3/8; 5/8; 7/8; 11/8; 125/8; 1/16; 5/16; 9/16; 23/16;
d) 1/500; 3/250; 71/200; 9/125; 27/2500; 1999/2000. - Find the sum: a) 7.3 + 12.8; b) 65.14+49.76; c) 3.762+12.85; d) 85.4+129.756; e) 1.44+2.56.
- Think of a unit as the sum of two decimals. Find twenty more ways to do this.
- Find the difference: a) 13.4–8.7; b) 74.52–27.04; c) 49.736–43.45; d) 127.24–93.883; e) 67–52.07; f) 35.24–34.9975.
- Find the product: a) 7.6 3.8; b) 4.8 12.5; c) 2.39 7.4; d) 3.74 9.65.
