Hunting shot is a component for loading cartridges that has long become an integral part of the life of any hunter. It is with its help that game (roe deer, duck, wood grouse, black grouse, pheasant) is often killed. Unlike other cartridge components, production and appearance This ammunition has not actually changed in the 150 years that have passed since its invention.
Types of fractions
So what is a fraction? These are small lead balls (up to 5 mm in size) used for hunting a variety of animals (for example, black grouse, wood grouse, hare, pheasant). However, there are many types of it:
Material
According to the material from which it is made:
- Lead. The use of lead is very widespread, since this material has all the necessary qualities- heavy, cheap, fusible. It is easy to do it yourself at home. However, such pellets are too soft, in addition, lead is toxic and disrupts the environment. In the West, similar types of shot for hunting under pressure from the “greens” are actually no longer used today.
- Steel. Such ammunition does not deform, but loses speed faster and damages the bore.
- Red-hot. The same shot is lead, but tin, arsenic, antimony or some other chemicals are added to it.
- Clad. Lead shot coated with nickel or cupronickel. Currently the best in terms of characteristics and the most expensive option on the market.
Diameter
Remember that diameter classification varies depending on the country of origin (see below Russian table, and to get acquainted with foreign classification, it is recommended to refer to the materials provided by the country of origin).
Numbering of fractions in the Russian classification:
| Size | |
|---|---|
| Fraction 0000 (4/0) size | 5mm diameter |
| 000 (3/0) size | 4.75mm diameter |
| 00 (2/0) size | 4.5mm diameter |
| 0 size | 4.25mm diameter |
| 1 size | 4mm diameter |
| size 2 | 3.75mm diameter |
| Size 3 | 3.5mm diameter |
| size 4 | 3.25mm diameter |
| size 5 | 3mm diameter |
| size 6 | 2.75mm diameter |
| size 7 | 2.5mm diameter |
| size 8 | 2.25mm diameter |
| size 9 | 2mm diameter |
| size 10 | 1.75mm diameter |
| size 11 | 1.50mm diameter |
| size 12 | 1.25 mm diameter - the smallest shot |
As you'll notice, the millimeter of this ammunition decreases by a quarter (0.25) millimeter as the size goes down.
This classification is too cumbersome, so you can sort the fraction differently:
- Small (10-6 number);
- Average (5-1 number);
- Large(0, 00,000, 000);
Shot, buckshot or bullet?
Many new hunters often confuse these concepts, so it would be nice to make the difference more obvious:
Small, centered balls whose shape is close to a sphere. Excellent for small game.
Ammunition larger than 5 mm (used for hunting larger game, for example, roe deer).
Full metal projectile. There are many varieties of them, but they are used, like buckshot, for hunting roe deer, wild boars and other large game.
Which shot should I use for which game?
Many hunters ask who (goose, black grouse, pheasant, hare, wood grouse) needs to be killed and with what kind of shells? For information about who needs to be hit and with what, see below:
When determining the required shot number, remember that about 4-5 pellets should hit the game, therefore, when shooting at small targets (goose, duck, hare, pheasant, capercaillie) with buckshot at best case scenario 1-2 pellets will hit, which means you will be left wounded. On the other hand, if the shot fall is still satisfactory, then the game (duck, wood grouse, black grouse, pheasant, hare) will simply be torn to pieces and lose all its value.
On the other hand, if you shoot projectiles that are too small, you will not penetrate the plumage of a grouse or goose, as well as the skin of a roe deer, so you will shoot in vain.
How to improve combat accuracy with hunting shot?
Many people ask what is the point of making ammunition with my own hands, if there are good store hangers? If you make shot at home, it will be much cheaper, even if it is inferior in quality to the factory one. In addition, many old hunters prefer to make their own ammunition (depending on who they are hunting: black grouse, duck, wood grouse, hare or goose) to be sure of the quality of the fight. Casting usually produces buckshot or medium/large numbers. Lead is taken from either cable lead or battery lead (terminals) and mixed in a ratio of 1/3.
There are different ways to make shot at home, but all options are related to casting to one degree or another. Here is one of these methods:
- It all starts with a shotgun die, which needs to be done once, and then used for a lifetime. It looks like two pieces of metal with grooves that are connected by a hinge with handles. In both halves we make recesses for different sizes of pellets (from buckshot to number 2). The resulting hemispherical recesses are connected to each other by grooves. All the grooves, gathered together, go into the gutter. The better the grooves are made, the higher the quality of the buckshot.
- We pour molten shot lead (according to the recipe above) into the gutter, and after casting, the pellets are simply cut off from each other with metal scissors.
Ready! Before shooting someone with it, it is recommended to roll it on a shot roller, otherwise the accuracy and range of the fire will suffer (hunting roe deer, wood grouse, duck, goose or black grouse is out of the question).
Examples with fractions are one of the basic elements of mathematics. There are many different types equations with fractions. Below is detailed instructions for solving examples of this type.
How to solve examples with fractions - general rules
To solve examples with fractions of any type, be it 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 a second fractional expression (with the same denominator) from one fraction, you need to subtract their numerators and leave the denominator the same.
- To add or subtract fractional expressions with different denominators, you need to find the lowest common denominator.
- In order to find a fractional product, you need to multiply the numerators and denominators, and, if possible, reduce.
- To divide a fraction by a fraction, you multiply the first fraction by the second fraction reversed.
How to solve examples with fractions - practice
Rule 1, example 1:
Calculate 3/4 +1/4.
According to Rule 1, if fractions have two (or more) 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 equal 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 lowest common denominator. The lowest common denominator is a number that is divisible by the denominators of all fractional expressions example. Thus, we need to find the minimum number that will be divisible by both 4 and 6. This number is 12. We write 12 as the denominator. Divide 12 by the denominator of the first fraction, we get 3, multiply by 3, write 3 in the numerator *3 and + sign. Divide 12 by the denominator of the second fraction, we get 2, multiply 2 by 1, write 2*1 in the numerator. So, we get 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 steps, but in the numerator instead of the + sign, we write a 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 canceled.
2 cancels from 4. 10 cancels 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 examples with fractions - 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.
Let's look at an example:
Solve the equation 15/3x+5 = 3
Let us remember 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 purpose, there is an OA (permissible value range).
So 3x+5 ≠ 0.
Hence: 3x ≠ 5.
x ≠ 5/3
At x = 5/3 the equation simply has no solution.
Having indicated the ODZ, in the best possible way Solving this equation will get rid of the fractions. To do this, we first represent all non-fractional values in the form of a fraction, in in this case number 3. We get: 15/(3x+5) = 3/1. To get rid of fractions you need to multiply each of them by the lowest common denominator. In this case it will be (3x+5)*1. Sequencing:
- Multiply 15/(3x+5) by (3x+5)*1 = 15*(3x+5).
- Open the brackets: 15*(3x+5) = 45x + 75.
- We do the same with the right side of the equation: 3*(3x+5) = 9x + 15.
- We equate left and right side: 45x + 75 = 9x +15
- Move the 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 number axis. Let's look at this example.
Sequencing:
- We 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 number axis, writing the resulting values on it.
- Draw a circle under the value. There are two types of circles - filled and empty. A filled circle means that given value is included in the range of solutions. An empty circle indicates that this value is not included in the solution range.
- Since the denominator cannot be equal to 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)= -4. the value is negative, which means we write a minus above the area after the two. Then substitute for X any value of the interval from 5/3 to 2, for example 1. The value is again negative. We write a minus. We repeat the same with the area located up to 5/3. We substitute any number less than 5/3, for example 1. Again, minus.
- Since we are interested in the values of x at which the expression will be greater than or equal to 0, and there are no such values (there are minuses everywhere), this inequality has no solution, that is, x = Ø (an empty set).
Answer: x = Ø
We come across fractions in life much earlier than we begin studying them at school. If we cut a whole apple in half, we get ½ of the fruit. Let's cut it again - it will be ¼. These are fractions. And everything seemed simple. For an adult. For the child (and this topic start studying at the end junior school) abstract mathematical concepts are still frighteningly incomprehensible, and the teacher must clearly explain what a proper fraction and an improper fraction, an ordinary and a decimal are, what operations can be performed with them and, most importantly, what all this is needed for.
What are fractions?
Getting to know new topic at school it starts with ordinary fractions. They are easily recognized by the horizontal line separating the two numbers - above and below. The top one is called the numerator, the bottom one is the denominator. There is also a lowercase option for writing improper and proper ordinary fractions - through a slash, for example: ½, 4/9, 384/183. This option is used when the line height is limited and it is not possible to use a “two-story” entry form. Why? Yes, because it is more convenient. We'll see this a little later.
In addition to ordinary fractions, there are also decimal fractions. It is very simple to distinguish them: if in one case a horizontal or slash is used, in the other a comma is used to separate sequences of numbers. Let's look at an example: 2.9; 163.34; 1.953. We intentionally used a semicolon as a separator to delimit the numbers. The first of them will read like this: “two point nine.”
New concepts
Let's return to ordinary fractions. They come in two types.
The definition of a proper fraction is as follows: it is a fraction whose numerator is less than its denominator. Why is it important? We'll see now!
You have several apples, halved. Total - 5 parts. How would you say: do you have “two and a half” or “five and a half” apples? Of course, the first option sounds more natural, and we will use it when talking with friends. But if we need to calculate how many fruits each person will get, if there are five people in the company, we will write down the number 5/2 and divide it by 5 - from a mathematical point of view, this will be more clear.
So, for naming proper and improper fractions, the rule is this: if a whole part can be distinguished in a fraction (14/5, 2/1, 173/16, 3/3), then it is improper. If this cannot be done, as in the case of ½, 13/16, 9/10, it will be correct.
The main property of a fraction
If the numerator and denominator of a fraction are simultaneously multiplied or divided by the same number, its value does not change. Imagine: they cut the cake into 4 equal parts and gave you one. They cut the same cake into eight pieces and gave you two. Does it really matter? After all, ¼ and 2/8 are the same thing!
Reduction
Authors of problems and examples in mathematics textbooks often seek to confuse students by offering fractions that are cumbersome to write but can actually be abbreviated. Here is an example of a proper fraction: 167/334, which, it would seem, looks very “scary”. But we can actually write it as ½. The number 334 is divisible by 167 without a remainder - after performing this operation, we get 2.
Mixed numbers
An improper fraction can be represented as a mixed number. This is when the whole part is brought forward and written at the level of the horizontal line. In fact, the expression takes the form of a sum: 11/2 = 5 + ½; 13/6 = 2 + 1/6 and so on.
To take out the whole part, you need to divide the numerator by the denominator. Write the remainder of the division on top, above the line, and the whole part - before the expression. Thus, we get two structural parts: whole units + proper fraction.
You can also carry out the inverse operation - to do this, you need to multiply the integer part by the denominator and add the resulting value to the numerator. Nothing complicated.
Multiplication and division
Oddly enough, multiplying fractions is easier than adding. All that is required is to extend the horizontal line: (2/3) * (3/5) = 2*3 / 3*5 = 2/5.
With division, everything is also simple: you need to multiply the fractions crosswise: (7/8) / (14/15) = 7*15 / 8*14 = 15/16.
Adding Fractions
What to do if you need to perform addition or their denominator is different numbers? It will not work to do the same as with multiplication - here you should understand the definition of a proper fraction and its essence. It is necessary to bring the terms to a common denominator, that is, the lower part of both fractions must have the same numbers.
To do this, you should use the basic property of a fraction: multiply both parts by the same number. For example, 2/5 + 1/10 = (2*2)/(5*2) + 1/10 = 5/10 = ½.
How to choose which denominator to reduce the terms to? This must be the minimum number that is a multiple of both numbers in the denominators of the fractions: for 1/3 and 1/9 it will be 9; for ½ and 1/7 - 14, because there is no smaller value divisible by 2 and 7 without a remainder.
Usage
What are improper fractions used for? After all, it is much more convenient to immediately select the whole part and get mixed number- and that's the end of it! It turns out that if you need to multiply or divide two fractions, it is more profitable to use irregular ones.
Let's take the following example: (2 + 3/17) / (37 / 68).
It would seem that there is nothing to cut at all. But what if we write the addition result in the first parentheses as an improper fraction? Look: (37/17) / (37/68)
Now everything falls into place! Let's write the example in such a way that everything becomes obvious: (37*68) / (17*37).
Let's cancel 37 in the numerator and denominator and finally divide the top and bottom by 17. Do you remember the basic rule for proper and improper fractions? We can multiply and divide them by any number as long as we do it for the numerator and denominator at the same time.
So, we get the answer: 4. The example looked complicated, but the answer contains only one number. This happens often in mathematics. The main thing is not to be afraid and follow simple rules.
Common Mistakes
When implementing, a student can easily make one of the common mistakes. Usually they occur due to inattention, and sometimes due to the fact that the material studied has not yet been properly stored in the head.
Often the sum of numbers in the numerator makes you want to reduce its individual components. Let’s say in the example: (13 + 2) / 13, written without parentheses (with a horizontal line), many students, due to inexperience, cross out 13 above and below. But this should not be done under any circumstances, because this is a gross mistake! If instead of addition there was a multiplication sign, we would get the number 2 in the answer. But when performing addition, no operations with one of the terms are allowed, only with the entire sum.
Guys also often make mistakes when dividing fractions. Let's take two proper irreducible fractions and divide by each other: (5/6) / (25/33). The student can mix it up and write the resulting expression as (5*25) / (6*33). But this would happen with multiplication, but in our case everything will be somewhat different: (5*33) / (6*25). We reduce what is possible, and the answer will be 11/10. We write the resulting improper fraction as a decimal - 1.1.
Brackets
Remember that in any mathematical expression the order of operations is determined by the precedence of the operation signs and the presence of parentheses. All other things being equal, the order of actions is counted from left to right. This is also true for fractions - the expression in the numerator or denominator is calculated strictly according to this rule.
After all, this is the result of dividing one number by another. If they are not evenly divided, it becomes a fraction - that's all.
How to write a fraction on a computer
Since standard tools do not always allow creating a fraction consisting of two “tiers,” students sometimes resort to various tricks. For example, they copy the numerators and denominators into the Paint graphic editor and glue them together, drawing between them horizontal line. Of course, there is a simpler option, which, by the way, provides a lot of additional features, which will be useful to you in the future.
Open Microsoft Word. One of the panels at the top of the screen is called “Insert” - click it. On the right, on the side where the close and minimize window icons are located, there is a “Formula” button. This is exactly what we need!
If you use this function, a rectangular area will appear on the screen in which you can use any mathematical signs, missing from the keyboard, and also write fractions in classic form. That is, dividing the numerator and denominator with a horizontal line. You might even be surprised that such a proper fraction is so easy to write.
Learn math
If you are in grades 5-6, then soon knowledge of mathematics (including the ability to work with fractions!) will be required in many school subjects. In almost any problem in physics, when measuring the mass of substances in chemistry, in geometry and trigonometry, you cannot do without fractions. Soon you will learn to calculate everything in your mind, without even writing down expressions on paper, but more and more complex examples. Therefore, learn what a proper fraction is and how to work with it, keep up with curriculum, do your homework on time and you will succeed.
Do you want to feel like a sapper? Then this lesson is for you! Because now we will study fractions - these are such simple and harmless mathematical objects that, in their ability to “blow the mind”, surpass the rest of the algebra course.
The main danger of fractions is that they occur in real life. This is how they differ, for example, from polynomials and logarithms, which you can study and easily forget after the exam. Therefore, the material presented in this lesson can, without exaggeration, be called explosive.
A number fraction (or just a fraction) is a pair of integers written separated by a slash or a horizontal bar.
Fractions written through a horizontal line:
The same fractions written with a slash:
5/7; 9/(−30); 64/11; (−1)/4; 12/1.
Fractions are usually written through a horizontal line - it’s easier to work with them this way, and they look better. The number written on top is called the numerator of the fraction, and the number written below is called the denominator.
Any integer can be represented as a fraction with a denominator of 1. For example, 12 = 12/1 is the fraction from the example above.
In general, you can put any whole number into the numerator and denominator of a fraction. The only limitation is that the denominator must be different from zero. Remember the good old rule: “You can’t divide by zero!”
If the denominator still has a zero, the fraction is called an indefinite fraction. Such a record is meaningless and cannot be used in calculations.
The main property of a fraction
Fractions a /b and c /d are said to be equal if ad = bc.
From this definition it follows that the same fraction can be written in different ways. For example, 1/2 = 2/4, since 1 4 = 2 2. Of course, there are many fractions that are not equal to each other. For example, 1/3 ≠ 5/4, since 1 4 ≠ 3 5.
A reasonable question arises: how to find all fractions equal to a given one? We give the answer in the form of a definition:
The main property of a fraction is that the numerator and denominator can be multiplied by the same number other than zero. This will result in a fraction equal to the given one.
This is very important property- remember it. Using the basic property of a fraction, you can simplify and shorten many expressions. In the future, it will constantly “pop up” in the form of various properties and theorems.
Improper fractions. Selecting a whole part
If the numerator is less than the denominator, it is called a proper fraction. Otherwise (i.e., when the numerator is greater than or at least equal to the denominator), the fraction is called improper, and an integer part can be distinguished in it.
The whole part is written with a large number in front of the fraction and looks like this (marked in red):
To isolate the whole part of an improper fraction, you need to follow three simple steps:
- Find how many times the denominator fits in the numerator. In other words, find the maximum integer that, when multiplied by the denominator, will still be less than the numerator (at most, equal). This number will be the integer part, so we write it in front;
- Multiply the denominator by the integer part found in the previous step, and subtract the result from the numerator. The resulting “stub” is called the remainder of the division; it will always be positive (in extreme cases, zero). We write it in the numerator of the new fraction;
- We rewrite the denominator without changes.
Well, is it difficult? At first glance, it may be difficult. But with a little practice, you will be able to do it almost orally. In the meantime, take a look at the examples:
Task. Select the whole part in the indicated fractions:
In all examples, the whole part is highlighted in red, and the remainder of the division is highlighted in green.
Pay attention to the last fraction, where the remainder of the division turns out to be zero. It turns out that the numerator is completely divided by the denominator. This is quite logical, because 24: 6 = 4 is a hard fact from the multiplication table.
If everything is done correctly, the numerator of the new fraction will definitely be less than the denominator, i.e. the fraction will become correct. I will also note that it is better to highlight the whole part at the very end of the problem, before writing down the answer. Otherwise, the calculations can be significantly complicated.
Going to an improper fraction
There is also a reverse operation, when we get rid of the whole part. This is called the improper fraction transition and is much more common because working with improper fractions is much easier.
The transition to an improper fraction is also performed in three steps:
- Multiply the whole part by the denominator. The result can be quite large numbers, but this should not bother us;
- Add the resulting number to the numerator of the original fraction. Write the result in the numerator of the improper fraction;
- Rewrite the denominator - again, without changes.
Here are specific examples:
Task. Convert to improper fraction:
For clarity, the integer part is again highlighted in red, and the numerator of the original fraction is highlighted in green.
Consider the case when the numerator or denominator of the fraction contains a negative number. For example:
In principle, there is nothing criminal in this. However, working with such fractions can be inconvenient. Therefore, in mathematics it is customary to place minuses as fraction signs.
This is very easy to do if you remember the rules:
- “Plus for minus gives minus.” Therefore, if the numerator contains a negative number, and the denominator contains a positive number (or vice versa), feel free to cross out the minus and put it in front of the entire fraction;
- "Two negatives make an affirmative". When there is a minus in both the numerator and the denominator, we simply cross them out - no additional actions are required.
Of course, these rules can also be applied in the opposite direction, i.e. You can enter a minus sign under the fraction sign (most often in the numerator).
We deliberately do not consider the “plus on plus” case - with it, I think, everything is clear. Let's see how these rules work in practice:
Task. Take out the negatives of the four fractions written above.
Pay attention to the last fraction: there is already a minus sign in front of it. However, it is “burned” according to the rule “minus for minus gives a plus.”
Also, do not move minuses in fractions with the whole part highlighted. These fractions are first converted to improper fractions - and only then do calculations begin.
While studying the queen of all sciences - mathematics, at some point everyone comes across fractions. Although this concept (like the types of fractions themselves or mathematical operations with them) is not at all complicated, you need to treat it carefully, because in real life outside of school it will be very useful. So, let's refresh our knowledge about fractions: what they are, what they are for, what types they are and how to do different things with them arithmetic operations.
Her Majesty fraction: what is it
In mathematics, fractions are numbers, each of which consists of one or more parts of a unit. Such fractions are also called ordinary or simple. As a rule, they are written in the form of two numbers that are separated by a horizontal or slash line, it is called a “fractional” line. For example: ½, ¾.
The upper, or first, of these numbers is the numerator (shows how many parts are taken from the number), and the lower, or second, is the denominator (demonstrates how many parts the unit is divided into).
The fraction bar actually functions as a division sign. For example, 7:9=7/9
Traditionally, common fractions are less than one. While decimals can be larger than it.
What are fractions for? Yes for everything, because in real world Not all numbers are integers. For example, two schoolgirls in the cafeteria bought one delicious chocolate bar together. When they were about to share dessert, they met a friend and decided to treat her too. However, now it is necessary to correctly divide the chocolate bar, considering that it consists of 12 squares.
At first, the girls wanted to divide everything equally, and then each would get four pieces. But, after thinking it over, they decided to treat their friend, not 1/3, but 1/4 of the chocolate. And since the schoolgirls did not study fractions well, they did not take into account that in such a situation they would end up with 9 pieces, which are very difficult to divide into two. This fairly simple example shows how important it is to be able to correctly find a part of a number. But in life there are many more such cases.
Types of fractions: ordinary and decimal
All mathematical fractions are divided into two large categories: ordinary and decimal. The features of the first of them were described in the previous paragraph, so now it’s worth paying attention to the second.
Decimal is a positional notation of a fraction of a number, which is written in writing separated by a comma, without a dash or slash. For example: 0.75, 0.5.
In fact, a decimal fraction is identical to an ordinary fraction, however, its denominator is always one followed by zeros - hence its name.
The number preceding the comma is an integer part, and everything after it is a fraction. Any simple fraction can be converted to a decimal. Thus, the decimal fractions indicated in the previous example can be written as usual: ¾ and ½.
It is worth noting that both decimal and ordinary fractions can be either positive or negative. If they are preceded by a “-” sign, this fraction is negative, if “+” is a positive fraction.
Subtypes of ordinary fractions
There are these types of simple fractions.
Subtypes of decimal fraction
Unlike a simple fraction, a decimal fraction is divided into only 2 types.
- Final - received this name due to the fact that after the decimal point it has a limited (finite) number of digits: 19.25.
- An infinite fraction is a number with an infinite number of digits after the decimal point. For example, when dividing 10 by 3, the result will be an infinite fraction 3.333...
Adding Fractions
Carrying out various arithmetic manipulations with fractions is a little more difficult than with ordinary numbers. However, if you understand the basic rules, solving any example with them will not be difficult.
For example: 2/3+3/4. The least common multiple for them will be 12, therefore, it is necessary that this number be in each denominator. To do this, we multiply the numerator and denominator of the first fraction by 4, it turns out 8/12, we do the same with the second term, but only multiply by 3 - 9/12. Now you can easily solve the example: 8/12+9/12= 17/12. The resulting fraction is an incorrect unit because the numerator is greater than the denominator. It can and should be transformed into a correct mixed one by dividing 17:12 = 1 and 5/12.
When mixed fractions are added, operations are performed first with whole numbers, and then with fractions.
If the example contains a decimal fraction and a regular fraction, it is necessary to make both simple, then bring them to the same denominator and add them. For example 3.1+1/2. The number 3.1 can be written as mixed fraction 3 and 1/10 or as incorrect - 31/10. The common denominator for the terms will be 10, so you need to multiply the numerator and denominator of 1/2 by 5 alternately, you get 5/10. Then you can easily calculate everything: 31/10+5/10=35/10. The result obtained is an improper reducible fraction, we reduce it to normal look, reducing by 5: 7/2 = 3 and 1/2, or decimal - 3.5.
When adding 2 decimal fractions, it is important that there are the same number of digits after the decimal point. If this is not the case, you just need to add required amount zeros, because in decimal this can be done painlessly. For example, 3.5+3.005. To solve this problem, you need to add 2 zeros to the first number and then add one by one: 3.500+3.005=3.505.
Subtracting Fractions
When subtracting fractions, you should do the same as when adding: reduce to a common denominator, subtract one numerator from another, and, if necessary, convert the result to a mixed fraction.
For example: 16/20-5/10. The common denominator will be 20. You need to bring the second fraction to this denominator by multiplying both its parts by 2, you get 10/20. Now you can solve the example: 16/20-10/20= 6/20. However, this result applies to reducible fractions, so it is worth dividing both sides by 2 and the result is 3/10.
Multiplying fractions
Dividing and multiplying fractions are much simpler operations than addition and subtraction. The fact is that when performing these tasks, there is no need to look for a common denominator.
To multiply fractions, you simply need to multiply both numerators one by one, and then both denominators. Reduce the resulting result if the fraction is a reducible quantity.
For example: 4/9x5/8. After alternate multiplication, the result is 4x5/9x8=20/72. This fraction can be reduced by 4, so the final answer in the example is 5/18.
How to divide fractions
Dividing fractions is also a simple operation; in fact, it still comes down to multiplying them. To divide one fraction by another, you need to invert the second and multiply by the first.
For example, dividing the fractions 5/19 and 5/7. To solve the example, you need to swap the denominator and numerator of the second fraction and multiply: 5/19x7/5=35/95. The result can be reduced by 5 - it turns out 7/19.
If you need to divide a fraction by a prime number, the technique is slightly different. Initially, you should write this number as an improper fraction, and then divide according to the same scheme. For example, 2/13:5 should be written as 2/13: 5/1. Now you need to turn over 5/1 and multiply the resulting fractions: 2/13x1/5= 2/65.
Sometimes you have to divide mixed fractions. You need to treat them as you would with whole numbers: turn them into improper fractions, reverse the divisor and multiply everything. For example, 8 ½: 3. Convert everything into improper fractions: 17/2: 3/1. This is followed by a 3/1 flip and multiplication: 17/2x1/3= 17/6. Now you should convert the improper fraction to the correct one - 2 whole and 5/6.
So, having figured out what fractions are and how you can perform various arithmetic operations with them, you need to try not to forget about it. After all, people are always more inclined to divide something into parts than to add, so you need to be able to do it correctly.