Fractions in brackets with different denominators. How to add fractions with different denominators. The specifics of working with multi-storey fractions

Instruction

First, remember that a fraction is just a conditional notation for dividing one number by another. In addition and multiplication, dividing two integers does not always result in an integer. So call these two "divisible" numbers. The number that is being divided is the numerator, and the number that is being divided is the denominator.

To write a fraction, first write its numerator, then draw a horizontal line under this number, and write the denominator under the line. The horizontal line separating the numerator and denominator is called a fractional bar. Sometimes it is depicted as a slash "/" or "∕". In this case, the numerator is written to the left of the line, and the denominator to the right. So, for example, the fraction "two-thirds" will be written as 2/3. For clarity, the numerator is usually written at the top of the line, and the denominator at the bottom, that is, instead of 2/3, you can find: ⅔.

If the numerator of a fraction is greater than its denominator, then such an "improper" fraction is usually written as a "mixed" fraction. To get a mixed fraction from an improper fraction, simply divide the numerator by the denominator and write down the resulting quotient. Then put the remainder of the division in the numerator of the fraction and write this fraction to the right of the quotient (do not touch the denominator). For example, 7/3 = 2⅓.

To add two fractions with the same denominator, simply add their numerators (leave the denominators). For example, 2/7 + 3/7 = (2+3)/7 = 5/7. Similarly, subtract two fractions (the numerators are subtracted). For example, 6/7 - 2/7 = (6-2)/7 = 4/7.

To add two fractions with different denominators, 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. As a result, you will get the sum of two fractions with the same denominators, the addition of which is described in the previous paragraph.

For example, 3/4 + 2/3 = (3*3)/(4*3) + (2*4)/(3*4) = 9/12 + 8/12 = (9+8)/12 = 17/12 = 15/12.

If the denominators of fractions have common divisors, that is, they are divisible by the same number, choose as the common denominator the smallest number divisible by the first and second denominators at the same time. So, for example, if the first denominator is 6 and the second 8, then take as a common denominator not their product (48), but the number 24, which is divisible by both 6 and 8. The numerators of the fractions are then multiplied by the quotient of dividing the common denominator by the denominator of each fraction. For example, for the denominator 6, this number will be 4 - (24/6), and for the denominator 8 - 3 (24/8). This process is more clearly seen in a specific example:

5/6 + 3/8 = (5*4)/24 + (3*3)/24 = 20/24 + 9/24 = 29/24 = 1 5/24.

Subtraction of fractions with different denominators is done in exactly the same way.

) and the denominator by the denominator (we get the denominator of the product).

Fraction multiplication formula:

For example:

Before proceeding with the multiplication of numerators and denominators, it is necessary to check for the possibility of fraction reduction. If you manage to reduce the fraction, then it will be easier for you to continue to make calculations.

Division of an ordinary fraction by a fraction.

Division of fractions involving a natural number.

It's not as scary as it seems. As in the case of addition, we convert an integer into a fraction with a unit in the denominator. For example:

Multiplication of mixed fractions.

Rules for multiplying fractions (mixed):

• convert mixed fractions to improper;
• multiply the numerators and denominators of fractions;
• we reduce the fraction;
• if we get an improper fraction, then we convert the improper fraction to a mixed one.

Note! To multiply a mixed fraction by another mixed fraction, you first need to bring them to the form of improper fractions, and then multiply according to the rule for multiplying ordinary fractions.

The second way to multiply a fraction by a natural number.

It is more convenient to use the second method of multiplying an ordinary fraction by a number.

Note! To multiply a fraction by a natural number, it is necessary to divide the denominator of the fraction by this number, and leave the numerator unchanged.

From the above example, it is clear that this option is more convenient to use when the denominator of a fraction is divided without a remainder by a natural number.

Multilevel fractions.

In high school, three-story (or more) fractions are often found. Example:

To bring such a fraction to its usual form, division through 2 points is used:

Note! When dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.

Note, For example:

When dividing one by any fraction, the result will be the same fraction, only inverted:

Practical tips for multiplying and dividing fractions:

1. The most important thing in working with fractional expressions is accuracy and attentiveness. Do all calculations carefully and accurately, concentratedly and clearly. It is better to write down a few extra lines in a draft than to get confused in the calculations in your head.

2. In tasks with different types of fractions - go to the type of ordinary fractions.

3. We reduce all fractions until it is no longer possible to reduce.

4. We bring multi-level fractional expressions into ordinary ones, using division through 2 points.

5. We divide the unit into a fraction in our mind, simply by turning the fraction over.

Fractions are ordinary numbers, they can also be added and subtracted. But due to the fact that they have a denominator, more complex rules are required here than for integers.

Consider the simplest case, when there are two fractions with the same denominators. Then:

To add fractions with the same denominators, add their numerators and leave the denominator unchanged.

To subtract fractions with the same denominators, it is necessary to subtract the numerator of the second from the numerator of the first fraction, and again leave the denominator unchanged.

Within each expression, the denominators of the fractions are equal. By definition of addition and subtraction of fractions, we get:

As you can see, nothing complicated: just add or subtract the numerators - and that's it.

But even in such simple actions, people manage to make mistakes. Most often they forget that the denominator does not change. For example, when adding them, they also begin to add up, and this is fundamentally wrong.

Getting rid of the bad habit of adding denominators is quite simple. Try to do the same when subtracting. As a result, the denominator will be zero, and the fraction (suddenly!) will lose its meaning.

Therefore, remember once and for all: when adding and subtracting, the denominator does not change!

Also, many people make mistakes when adding several negative fractions. There is confusion with the signs: where to put a minus, and where - a plus.

This problem is also very easy to solve. It is enough to remember that the minus before the fraction sign can always be transferred to the numerator - and vice versa. And of course, do not forget two simple rules:

1. Plus times minus gives minus;
2. Two negatives make an affirmative.

Let's analyze all this with specific examples:

Task. Find the value of the expression:

In the first case, everything is simple, and in the second, we will add minuses to the numerators of fractions:

What if the denominators are different

You cannot directly add fractions with different denominators. At least, this method is unknown to me. However, the original fractions can always be rewritten so that the denominators become the same.

There are many ways to convert fractions. Three of them are discussed in the lesson " Bringing fractions to a common denominator", so we will not dwell on them here. Let's take a look at some examples:

Task. Find the value of the expression:

In the first case, we bring the fractions to a common denominator using the "cross-wise" method. In the second, we will look for the LCM. Note that 6 = 2 3; 9 = 3 · 3. The last factors in these expansions are equal, and the first ones are coprime. Therefore, LCM(6; 9) = 2 3 3 = 18.

What if the fraction has an integer part

I can please you: different denominators of fractions are not the greatest evil. Much more errors occur when the whole part is highlighted in the fractional terms.

Of course, for such fractions there are own addition and subtraction algorithms, but they are rather complicated and require a long study. Better use the simple diagram below:

1. Convert all fractions containing an integer part to improper. We get normal terms (even if with different denominators), which are calculated according to the rules discussed above;
2. Actually, calculate the sum or difference of the resulting fractions. As a result, we will practically find the answer;
3. If this is all that was required in the task, we perform the inverse transformation, i.e. we get rid of the improper fraction, highlighting the integer part in it.

The rules for switching to improper fractions and highlighting the integer part are described in detail in the lesson "What is a numerical fraction". If you don't remember, be sure to repeat. Examples:

Task. Find the value of the expression:

Everything is simple here. The denominators inside each expression are equal, so it remains to convert all fractions to improper ones and count. We have:

To simplify the calculations, I skipped some obvious steps in the last examples.

A small note to the last two examples, where fractions with a highlighted integer part are subtracted. The minus before the second fraction means that it is the whole fraction that is subtracted, and not just its whole part.

Reread this sentence again, look at the examples, and think about it. This is where beginners make a lot of mistakes. They like to give such tasks at control work. You will also meet them repeatedly in the tests for this lesson, which will be published shortly.

Summary: General Scheme of Computing

In conclusion, I will give a general algorithm that will help you find the sum or difference of two or more fractions:

1. If an integer part is highlighted in one or more fractions, convert these fractions to improper ones;
2. Bring all the fractions to a common denominator in any way convenient for you (unless, of course, the compilers of the problems did this);
3. Add or subtract the resulting numbers according to the rules for adding and subtracting fractions with the same denominators;
4. Reduce the result if possible. If the fraction turned out to be incorrect, select the whole part.

Remember that it is better to highlight the whole part at the very end of the task, just before writing the answer.

Let's go to battle with math homework! The enemy is recalcitrant fractions. Grade 5 program. A strategically important task is to explain fractions to the child. Let's change roles with the teacher and try to do it "with little blood", without nerves and in an accessible form. It is much easier to train one soldier than a company...

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How to explain fractions to a child

Don't wait until your child is in 5th grade and encounters fractions on the pages of a math textbook. We recommend looking for the answer to the question “How to explain fractions to a child” in the kitchen! And do it right now! Even if your kid is only 4-5 years old, he is able to understand the meaning of the concept of “fractions” and can even learn the simplest actions with fractions.

We shared an orange.
There are many of us, and he is one
This slice for a hedgehog, this slice for a siskin...
And for a wolf - peel.

Remember the poem? Here is the most illustrative example and the most effective guide to action! It is easiest to explain fractions to a child using food as an example: we cut an apple into halves and quarters, we divide pizza between family members, we cut a loaf of bread before dinner, etc. Most importantly, before you eat the "visual aid" do not forget to voice what part of the whole you are "destroying".

• Enter the concept of "share".

Emphasize that a WHOLE orange (apple, chocolate bar, watermelon, etc.) is 1 (denoted by the number 1).

• Enter the concept of "fraction".

We divide an orange or a chocolate bar, you can also say “crush” into several parts.

Show your child a well-known object - a ruler. Explain that there are intermediate values ​​between numbers - parts.

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• Explain how to write fractions: what the numerator means and what the denominator indicates.

The meaning of the concept of “fractions” and the correct notation can be easily shown using the example of a constructor. In the numerator ABOVE the line we write which part, and in the denominator UNDER the line - into how many such parts the whole was divided.

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Be sure to use a good example to show the difference between fractions with the same numerator but different denominators.

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Using the example of 4 squares of the same size, show how you can divide them into the same / different number of parts. Let the child cut the paper blanks with scissors, and then write down the results using fractions.

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• Explain how to write a whole as a fraction.

Remember the square and how we divided it into 4 parts. A square is a whole, we can write it as 1. But how to write it as a fraction: what is in the numerator, what is in the denominator? If we divided the square into 4 parts, then the whole square is 4/4. If we divided the square into 8 parts, then the whole square is 8/8. But it's still a square, i.e. 1. Both 4/4 and 8/8 are a unit, a whole!

How to explain fractions to a child: ask the RIGHT questions

In order for a grade 5 student to understand the topic “Fractions” and learn how to perform calculations with fractions, let's look at the methodology. It is important for us, parents, to understand how the teacher at school explains fractions to children, otherwise we can completely confuse our “soldier”.

A fraction is a number that is part of a whole object. It is always less than one.

Example 1 An apple is a whole, and a half is one half. Is it smaller than a whole apple? Divide the halves in half again. Each slice is one-fourth of a whole apple, and it is less than one-half.

A fraction is the number of parts of a whole.

Example 2 For example, a new product was brought to a clothing store: 30 shirts. Sellers managed to lay out and hang out only one third of all shirts from the new collection. How many shirts did they hang?
The child will easily verbally calculate that a third (one third) is 10 shirts, i.e. 10 were hung up and taken to the trading floor, and another 20 remained in the warehouse.

CONCLUSION: Anything can be measured with fractions, not only slices of pizza, but also liters in barrels, the number of wild animals in the forest, area, etc.

Give a variety of examples from life so that a 5th grade child understands the ESSENCE of fractions: this will help in the future in solving problems and performing calculations with proper and improper fractions, and learning in 5th grade will not be a burden, but a joy.

How to make sure that the child has learned that in the recording of fractions the numbers in the numerator and in the denominator are denoted?

Example 3 Ask what does 5 mean in the fraction 4/5?

- This is how many parts it was divided into.
- What does 4 mean?
- This is how much they took.

Comparing fractions is perhaps the most difficult topic.

Example 4 Invite the child to say which fraction is larger: 3/10 or 3/20? It seems that since 10 is less than 20, then the answer is obvious, but it's not! Remember the squares that we cut into pieces. If two squares of the same size are cut - one into 10, the second into 20 parts - is the answer obvious? So which fraction is bigger?

Actions with fractions

If you see that the child has well mastered the meaning of writing in the form of a fraction, you can proceed to simple arithmetic operations with fractions. On the example of the constructor, you can do this very clearly.

Example 5

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Example 6 Mathematical lotto on the topic "Fractions".

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Dear readers, if you know other effective methods for explaining fractions to a child, share them in the comments. We are happy to replenish our piggy bank of practical school tips.

Fraction- a form of representation of a number in mathematics. The slash indicates the division operation. numerator fractions is called the dividend, and denominator- divider. For example, in a fraction, the numerator is 5 and the denominator is 7.

Correct A fraction is called if the modulus of the numerator is greater than the modulus of the denominator. If the fraction is correct, then the modulus of its value is always less than 1. All other fractions are wrong.

Fraction is called mixed if it is written as an integer and a fraction. This is the same as the sum of this number and a fraction:

Basic property of a fraction

If the numerator and denominator of a fraction are multiplied by the same number, then the value of the fraction will not change, that is, for example,

Bringing fractions to a common denominator

To bring two fractions to a common denominator, you need:

1. Multiply the numerator of the first fraction by the denominator of the second
2. Multiply the numerator of the second fraction by the denominator of the first
3. Replace the denominators of both fractions with their product

Actions with fractions

Addition. To add two fractions, you need

1. Add new numerators of both fractions, and leave the denominator unchanged

Example:

Subtraction. To subtract one fraction from another,

1. Bring fractions to a common denominator
2. Subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged

Example:

Multiplication. To multiply one fraction by another, multiply their numerators and denominators.