your child brought homework from school and you don't know how to solve it? Then this mini tutorial is for you!
How to add decimals
It is more convenient to add decimal fractions in a column. To add decimals, you need to follow one simple rule:
- The digit must be under the digit, comma under the comma.
As you can see in the example, whole units are under each other, tenths and hundredths are under each other. Now we add the numbers, ignoring the comma. What to do with a comma? The comma is transferred to the place where it stood in the discharge of integers.
Adding fractions with equal denominators
To perform addition with a common denominator, you need to keep the denominator unchanged, find the sum of the numerators and get a fraction, which will be the total amount.
Adding fractions with different denominators by finding a common multiple
The first thing to pay attention to is the denominators. The denominators are different, are they not divisible by one another, are they prime numbers. First you need to bring to one common denominator, there are several ways to do this:
- 1/3 + 3/4 = 13/12, to solve this example, we need to find the least common multiple (LCM) that will be divisible by 2 denominators. To denote the smallest multiple of a and b - LCM (a; b). AT this example LCM (3;4)=12. Check: 12:3=4; 12:4=3.
- We multiply the factors and perform the addition of the resulting numbers, we get 13/12 - an improper fraction.
- In order to convert an improper fraction to a proper one, we divide the numerator by the denominator, we get the integer 1, the remainder 1 is the numerator and 12 is the denominator.
Adding fractions using cross multiplication
For adding fractions with different denominators, there is another way according to the “cross by cross” formula. This is a guaranteed way to equalize the denominators, for this you need to multiply the numerators with the denominator of one fraction and vice versa. If you are only on initial stage learning fractions, then this method is the easiest and most accurate, how to get the right result when adding fractions with different denominators.
Actions with fractions.
Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")
So, what are fractions, types of fractions, transformations - we remembered. Let's tackle the main question.
What can you do with fractions? Yes, everything is the same as with ordinary numbers. Add, subtract, multiply, divide.
All these actions with decimal operations with fractions are no different from operations with integers. Actually, this is what they are good for, decimal. The only thing is that you need to put the comma correctly.
mixed numbers, as I said, are of little use for most actions. They still need to be converted to ordinary fractions.
And here are the actions with ordinary fractions will be smarter. And much more important! Let me remind you: all actions with fractional expressions with letters, sines, unknowns, and so on and so forth are no different from actions with ordinary fractions! Operations with ordinary fractions are the basis for all algebra. It is for this reason that we will analyze all this arithmetic in great detail here.
Addition and subtraction of fractions.
Everyone can add (subtract) fractions with the same denominators (I really hope!). Well, let me remind you that I’m completely forgetful: when adding (subtracting), the denominator does not change. The numerators are added (subtracted) to give the numerator of the result. Type:
In short, in general view:
What if the denominators are different? Then, using the main property of the fraction (here it came in handy again!), We make the denominators the same! For example:
Here we had to make the fraction 4/10 from the fraction 2/5. Solely for the purpose of making the denominators the same. I note, just in case, that 2/5 and 4/10 are the same fraction! Only 2/5 is uncomfortable for us, and 4/10 is even nothing.
By the way, this is the essence of solving any tasks in mathematics. When we're out uncomfortable expressions do the same, but more convenient to solve.
Another example:
The situation is similar. Here we make 48 out of 16. By simple multiplication on 3. This is all clear. But here we come across something like:
How to be?! It's hard to make a nine out of a seven! But we are smart, we know the rules! Let's transform every fraction so that the denominators are the same. This is called "reduce to a common denominator":
How! How did I know about 63? Very simple! 63 is a number that is evenly divisible by 7 and 9 at the same time. Such a number can always be obtained by multiplying the denominators. If we multiply some number by 7, for example, then the result will certainly be divided by 7!
If you need to add (subtract) several fractions, there is no need to do it in pairs, step by step. You just need to find the denominator that is common to all fractions, and bring each fraction to this same denominator. For example:
And what will be the common denominator? You can, of course, multiply 2, 4, 8, and 16. We get 1024. Nightmare. It is easier to estimate that the number 16 is perfectly divisible by 2, 4, and 8. Therefore, it is easy to get 16 from these numbers. This number will be the common denominator. Let's turn 1/2 into 8/16, 3/4 into 12/16, and so on.
By the way, if we take 1024 as a common denominator, everything will work out too, in the end everything will be reduced. Only not everyone will get to this end, because of the calculations ...
Solve the example yourself. Not a logarithm... It should be 29/16.
So, with the addition (subtraction) of fractions is clear, I hope? Of course, it is easier to work in a shortened version, with additional multipliers. But this pleasure is available to those who honestly worked in the lower grades ... And did not forget anything.
And now we will do the same actions, but not with fractions, but with fractional expressions. New rakes will be found here, yes ...
So, we need to add two fractional expressions:
We need to make the denominators the same. And only with the help multiplication! So the main property of the fraction says. Therefore, I cannot add one to x in the first fraction in the denominator. (But that would be nice!). But if you multiply the denominators, you see, everything will grow together! So we write down, the line of the fraction, leave an empty space on top, then add it, and write the product of the denominators below, so as not to forget:
And, of course, we don’t multiply anything on the right side, we don’t open brackets! And now, looking at the common denominator of the right side, we think: in order to get the denominator x (x + 1) in the first fraction, we need to multiply the numerator and denominator of this fraction by (x + 1). And in the second fraction - x. You get this:
Note! Parentheses are here! This is the rake that many step on. Not brackets, of course, but their absence. Parentheses appear because we multiply the whole numerator and the whole denominator! And not their individual pieces ...
In the numerator of the right side we write the sum of the numerators, everything is as in fractions, then open the brackets in the numerator of the right side, i.e. multiply everything and give like. You don't need to open the brackets in the denominators, you don't need to multiply something! In general, in denominators (any) the product is always more pleasant! We get:
Here we got the answer. The process seems long and difficult, but it depends on practice. Solve examples, get used to it, everything will become simple. Those who have mastered the fractions in the allotted time, do all these operations with one hand, on the machine!
And one more note. Many famously deal with fractions, but hang on examples with whole numbers. Type: 2 + 1/2 + 3/4= ? Where to fasten a deuce? No need to fasten anywhere, you need to make a fraction out of a deuce. It's not easy, it's very simple! 2=2/1. Like this. Any whole number can be written as a fraction. The numerator is the number itself, the denominator is one. 7 is 7/1, 3 is 3/1 and so on. It's the same with letters. (a + b) \u003d (a + b) / 1, x \u003d x / 1, etc. And then we work with these fractions according to all the rules.
Well, on addition - subtraction of fractions, knowledge was refreshed. Transformations of fractions from one type to another - repeated. You can also check. Shall we settle a little?)
Calculate:
Answers (in disarray):
71/20; 3/5; 17/12; -5/4; 11/6
Multiplication / division of fractions - in the next lesson. There are also tasks for all actions with fractions.
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you can get acquainted with functions and derivatives.
Fractional expressions are difficult for a child to understand. Most people have difficulties with . When studying the topic "addition of fractions with integers", the child falls into a stupor, finding it difficult to solve the task. In many examples, a series of calculations must be performed before an action can be performed. For example, convert fractions or convert an improper fraction to a proper one.
Explain to the child clearly. Take three apples, two of which will be whole, and the third will be cut into 4 parts. Separate one slice from the cut apple, and put the remaining three next to two whole fruits. We get ¼ apples on one side and 2 ¾ on the other. If we combine them, we get three whole apples. Let's try to reduce 2 ¾ apples by ¼, that is, remove one more slice, we get 2 2/4 apples.
Let's take a closer look at actions with fractions, which include integers:
First, let's recall the calculation rule for fractional expressions with a common denominator:
At first glance, everything is easy and simple. But this applies only to expressions that do not require conversion.
How to find the value of an expression where the denominators are different
In some tasks, it is necessary to find the value of an expression where the denominators are different. Consider a specific case:
3 2/7+6 1/3
Find the value of this expression, for this we find a common denominator for two fractions.
For numbers 7 and 3, this is 21. We leave the integer parts the same, and reduce the fractional parts to 21, for this we multiply the first fraction by 3, the second by 7, we get:
6/21+7/21, do not forget that whole parts are not subject to conversion. As a result, we get two fractions with one denominator and calculate their sum:
3 6/21+6 7/21=9 15/21
What if the result of addition is an improper fraction that already has an integer part:
2 1/3+3 2/3
In this case, we add the integer parts and fractional parts, we get:
5 3/3, as you know, 3/3 is one, so 2 1/3+3 2/3=5 3/3=5+1=6
With finding the sum, everything is clear, let's analyze the subtraction:
From what has been said follows the rule of action on mixed numbers which sounds like this:
- If it is necessary to subtract an integer from a fractional expression, it is not necessary to represent the second number as a fraction, it is enough to operate only on integer parts.
Let's try to calculate the value of expressions on our own:
Let's take a closer look at the example under the letter "m":
4 5/11-2 8/11, the numerator of the first fraction is less than the second. To do this, we take one integer from the first fraction, we get,
3 5/11+11/11=3 whole 16/11, subtract the second from the first fraction:
3 16/11-2 8/11=1 whole 8/11
- Be careful when completing the task, do not forget to convert improper fractions to mixed ones, highlighting the whole part. To do this, it is necessary to divide the value of the numerator by the value of the denominator, then what happened takes the place of the integer part, the remainder will be the numerator, for example:
19/4=4 ¾, check: 4*4+3=19, in the denominator 4 remains unchanged.
Summarize:
Before proceeding with the task related to fractions, it is necessary to analyze what kind of expression it is, what transformations need to be performed on the fraction in order for the solution to be correct. Look for more rational solutions. Don't go the hard way. Plan all actions, decide first in draft version, then transfer to a school notebook.
To avoid confusion when solving fractional expressions, it is necessary to follow the sequence rule. Decide everything carefully, without rushing.
Ordinary fractional numbers first meet schoolchildren in the 5th grade and accompany them throughout their lives, since in everyday life it is often necessary to consider or use some object not entirely, but in separate pieces. The beginning of the study of this topic - share. Shares are equal parts into which an object is divided. After all, it is not always possible to express, for example, the length or price of a product as an integer; one should take into account parts or shares of any measure. Formed from the verb "to crush" - to divide into parts, and having Arabic roots, in the VIII century the word "fraction" itself appeared in Russian.
Fractional expressions have long been considered the most difficult section of mathematics. In the 17th century, when first textbooks in mathematics appeared, they were called "broken numbers", which was very difficult to display in people's understanding.
modern look simple fractional residues, parts of which are separated precisely by a horizontal line, were first contributed to Fibonacci - Leonardo of Pisa. His writings are dated 1202. But the purpose of this article is to simply and clearly explain to the reader how the multiplication of mixed fractions with different denominators occurs.
Multiplying fractions with different denominators
Initially, it is necessary to determine varieties of fractions:
- correct;
- wrong;
- mixed.
Next, you need to remember how fractional numbers with the same denominators are multiplied. The very rule of this process is easy to formulate independently: the result of multiplication simple fractions with the same denominators is a fractional expression, the numerator of which is the product of the numerators, and the denominator is the product of the denominators of the given fractions. That is, in fact, the new denominator is the square of one of the existing ones initially.
When multiplying simple fractions with different denominators for two or more factors, the rule does not change:
a/b * c/d = a*c / b*d.
The only difference is that the formed number under the fractional bar will be the product of different numbers and, naturally, it cannot be called the square of one numerical expression.
It is worth considering the multiplication of fractions with different denominators using examples:
- 8/ 9 * 6/ 7 = 8*6 / 9*7 = 48/ 63 = 16/2 1 ;
- 4/ 6 * 3/ 7 = 2/ 3 * 3/7 <> 2*3 / 3*7 = 6/ 21 .
The examples use ways to reduce fractional expressions. You can reduce only the numbers of the numerator with the numbers of the denominator; adjacent factors above or below the fractional bar cannot be reduced.
Along with simple fractional numbers, there is the concept of mixed fractions. A mixed number consists of an integer and a fractional part, that is, it is the sum of these numbers:
1 4/ 11 =1 + 4/ 11.
How does multiplication work?
Several examples are provided for consideration.
2 1/ 2 * 7 3/ 5 = 2 + 1/ 2 * 7 + 3/ 5 = 2*7 + 2* 3/ 5 + 1/ 2 * 7 + 1/ 2 * 3/ 5 = 14 + 6/5 + 7/ 2 + 3/ 10 = 14 + 12/ 10 + 35/ 10 + 3/ 10 = 14 + 50/ 10 = 14 + 5=19.
The example uses the multiplication of a number by ordinary fractional part, you can write down the rule for this action by the formula:
a * b/c = a*b /c.
In fact, such a product is the sum of identical fractional residues, and the number of terms indicates this natural number. special case:
4 * 12/ 15 = 12/ 15 + 12/ 15 + 12/ 15 + 12/ 15 = 48/ 15 = 3 1/ 5.
There is another option for solving the multiplication of a number by a fractional remainder. You just need to divide the denominator by this number:
d* e/f = e/f: d.
It is useful to use this technique when the denominator is divided by a natural number without a remainder or, as they say, completely.
Convert mixed numbers to improper fractions and get the product in the previously described way:
1 2/ 3 * 4 1/ 5 = 5/ 3 * 21/ 5 = 5*21 / 3*5 =7.
This example involves a representation method mixed fraction into the wrong one, it can also be represented as a general formula:
a bc = a*b+ c / c, where the denominator of the new fraction is formed by multiplying the integer part with the denominator and adding it to the numerator of the original fractional remainder, and the denominator remains the same.
This process also works in reverse. To select the integer part and the fractional remainder, you need to divide the numerator of an improper fraction by its denominator with a “corner”.
Multiplication improper fractions produced in the usual way. When the entry goes under a single fractional line, as necessary, you need to reduce the fractions in order to reduce the numbers using this method and it is easier to calculate the result.
There are many helpers on the Internet to solve even complex problems. math problems in various programs. A sufficient number of such services offer their help in counting the multiplication of fractions with different numbers in denominators - the so-called online calculators for calculating fractions. They are able not only to multiply, but also to perform all other simple arithmetic operations with ordinary fractions and mixed numbers. It is easy to work with it, the corresponding fields are filled in on the site page, the sign is selected mathematical action and click "calculate". The program counts automatically.
Topic arithmetic operations with fractional numbers is relevant throughout the education of middle and senior schoolchildren. In high school, they are no longer considering the simplest species, but integer fractional expressions, but the knowledge of the rules for transformation and calculations, obtained earlier, is applied in its original form. Well-learned basic knowledge gives full confidence in good decision most challenging tasks.
In conclusion, it makes sense to cite the words of Leo Tolstoy, who wrote: “Man is a fraction. It is not in the power of man to increase his numerator - his own merits, but anyone can decrease his denominator - his opinion of himself, and by this decrease come closer to his perfection.
The rules for adding fractions with different denominators are very simple.
Consider the rules for adding fractions with different denominators in steps:
1. Find the LCM (least common multiple) of the denominators. The resulting LCM will be the common denominator of the fractions;
2. Bring fractions to a common denominator;
3. Add fractions reduced to a common denominator.
On the simple example Learn how to add fractions with different denominators.
Example
An example of adding fractions with different denominators.
Add fractions with different denominators:
| 1 | + | 5 |
|---|---|---|
| 6 | 12 |
Let's decide step by step.
1. Find the LCM (least common multiple) of the denominators.
The number 12 is divisible by 6.
From this we conclude that 12 is the least common multiple of the numbers 6 and 12.
Answer: the nok of the numbers 6 and 12 is 12:
LCM(6, 12) = 12
The resulting NOC will be the common denominator of the two fractions 1/6 and 5/12.
2. Bring fractions to a common denominator.
In our example, only the first fraction needs to be reduced to a common denominator of 12, because the second fraction already has a denominator of 12.
Divide the common denominator of 12 by the denominator of the first fraction:
2 has an additional multiplier.
Multiply the numerator and denominator of the first fraction (1/6) by an additional factor of 2.
