Rules for subtracting fractions with different denominators. Drawing up a system of equations

Actions with fractions. In this article, we will analyze examples, everything is detailed with explanations. We will consider ordinary fractions. In the future, we will analyze decimals. I recommend to watch the whole and study sequentially.

1. Sum of fractions, difference of fractions.

Rule: when adding fractions with equal denominators, the result is a fraction - the denominator of which remains the same, and its numerator will be equal to the sum of the numerators of the fractions.

Rule: when calculating the difference of fractions with the same denominators, we get a fraction - the denominator remains the same, and the numerator of the second is subtracted from the numerator of the first fraction.

Formal notation of the sum and difference of fractions with equal denominators:

Examples (1):

It is clear that when ordinary fractions are given, then everything is simple, but if they are mixed? Nothing complicated...

Option 1- you can convert them into ordinary ones and then calculate them.

Option 2- you can separately "work" with the integer and fractional parts.

Examples (2):

More:

And if the difference of two mixed fractions is given and the numerator of the first fraction is less than the numerator of the second? It can also be done in two ways.

Examples (3):

* Translated into ordinary fractions, calculated the difference, converted the resulting improper fraction into a mixed one.

* Divided into integer and fractional parts, got three, then presented 3 as the sum of 2 and 1, with the unit presented as 11/11, then found the difference between 11/11 and 7/11 and calculated the result. The meaning of the above transformations is to take (select) a unit and present it as a fraction with the denominator we need, then from this fraction we can already subtract another.

Another example:

Conclusion: there is a universal approach - in order to calculate the sum (difference) of mixed fractions with equal denominators, they can always be converted into improper ones, then perform the necessary action. After that, if as a result we get an improper fraction, we translate it into a mixed one.

Above, we looked at examples with fractions that have equal denominators. What if the denominators differ? In this case, the fractions are reduced to the same denominator and the specified action is performed. To change (transform) a fraction, the main property of the fraction is used.

Consider simple examples:

In these examples, we immediately see how one of the fractions can be converted to get equal denominators.

If we designate ways to reduce fractions to one denominator, then this one will be called METHOD ONE.

That is, immediately when “evaluating” the fraction, you need to figure out whether such an approach will work - we check whether the larger denominator is divisible by the smaller one. And if it is divided, then we perform the transformation - we multiply the numerator and denominator so that the denominators of both fractions become equal.

Now look at these examples:

This approach does not apply to them. There are other ways to reduce fractions to a common denominator, consider them.

Method SECOND.

Multiply the numerator and denominator of the first fraction by the denominator of the second, and the numerator and denominator of the second fraction by the denominator of the first:

*In fact, we bring fractions to the form when the denominators become equal. Next, we use the rule of adding timid with equal denominators.

Example:

*This method can be called universal, and it always works. The only negative is that after the calculations, a fraction may turn out that will need to be further reduced.

Consider an example:

It can be seen that the numerator and denominator are divisible by 5:

Method THIRD.

Find the least common multiple (LCM) of the denominators. This will be the common denominator. What is this number? This is the smallest natural number that is divisible by each of the numbers.

Look, here are two numbers: 3 and 4, there are many numbers that are divisible by them - these are 12, 24, 36, ... The smallest of them is 12. Or 6 and 15, 30, 60, 90 are divisible by them .... Least 30. Question - how to determine this least common multiple?

There is a clear algorithm, but often this can be done immediately without calculations. For example, according to the above examples (3 and 4, 6 and 15), no algorithm is needed, we took large numbers (4 and 15), doubled them and saw that they are divisible by the second number, but pairs of numbers can be others, such as 51 and 119.

Algorithm. In order to determine the least common multiple of several numbers, you must:

- decompose each of the numbers into SIMPLE factors

- write out the decomposition of the BIGGER of them

- multiply it by the MISSING factors of other numbers

Consider examples:

50 and 60 50 = 2∙5∙5 60 = 2∙2∙3∙5

in the expansion of a larger number, one five is missing

=> LCM(50,60) = 2∙2∙3∙5∙5 = 300

48 and 72 48 = 2∙2∙2∙2∙3 72 = 2∙2∙2∙3∙3

in the expansion of a larger number, two and three are missing

=> LCM(48,72) = 2∙2∙2∙2∙3∙3 = 144

* Least common multiple of two prime numbers equal to their product

Question! And why is it useful to find the least common multiple, because you can use the second method and simply reduce the resulting fraction? Yes, you can, but it's not always convenient. See what the denominator will be for the numbers 48 and 72 if you simply multiply them 48∙72 = 3456. Agree that it is more pleasant to work with smaller numbers.

Consider examples:

*51 = 3∙17 119 = 7∙17

in the expansion of a larger number, a triple is missing

=> LCM(51,119) = 3∙7∙17

And now we apply the first method:

* Look at the difference in the calculations, in the first case there is a minimum of them, and in the second you need to work separately on a piece of paper, and even the fraction that you got needs to be reduced. Finding the LCM simplifies the work considerably.

More examples:

* In the second example, it is already clear that the smallest number that is divisible by 40 and 60 is 120.

TOTAL! GENERAL CALCULATION ALGORITHM!

- we bring fractions to ordinary ones, if there is an integer part.

- we bring the fractions to a common denominator (first we look to see if one denominator is divisible by another, if it is divisible, then we multiply the numerator and denominator of this other fraction; if it is not divisible, we act using the other methods indicated above).

- having received fractions with equal denominators, we perform actions (addition, subtraction).

- if necessary, we reduce the result.

- if necessary, select the whole part.

2. Product of fractions.

The rule is simple. When multiplying fractions, their numerators and denominators are multiplied:

Examples:

You can perform various actions with fractions, for example, adding fractions. Addition of fractions can be divided into several types. Each type of addition of fractions has its own rules and algorithm of actions. Let's take a closer look at each type of addition.

Adding fractions with the same denominators.

For example, let's see how to add fractions with a common denominator.

The hikers went on a hike from point A to point E. On the first day, they walked from point A to B, or \(\frac(1)(5)\) all the way. On the second day they went from point B to D or \(\frac(2)(5)\) the whole way. How far did they travel from the beginning of the journey to point D?

To find the distance from point A to point D, add the fractions \(\frac(1)(5) + \frac(2)(5)\).

Adding fractions with the same denominators is that you need to add the numerators of these fractions, and the denominator will remain the same.

\(\frac(1)(5) + \frac(2)(5) = \frac(1 + 2)(5) = \frac(3)(5)\)

In literal form, the sum of fractions with the same denominators will look like this:

\(\bf \frac(a)(c) + \frac(b)(c) = \frac(a + b)(c)\)

Answer: the tourists traveled \(\frac(3)(5)\) all the way.

Adding fractions with different denominators.

Consider an example:

Add two fractions \(\frac(3)(4)\) and \(\frac(2)(7)\).

To add fractions with different denominators must first be found, and then use the rule for adding fractions with the same denominators.

For denominators 4 and 7, the common denominator is 28. The first fraction \(\frac(3)(4)\) must be multiplied by 7. The second fraction \(\frac(2)(7)\) must be multiplied by 4.

\(\frac(3)(4) + \frac(2)(7) = \frac(3 \times \color(red) (7) + 2 \times \color(red) (4))(4 \ times \color(red) (7)) = \frac(21 + 8)(28) = \frac(29)(28) = 1\frac(1)(28)\)

In literal form, we get the following formula:

\(\bf \frac(a)(b) + \frac(c)(d) = \frac(a \times d + c \times b)(b \times d)\)

Addition of mixed numbers or mixed fractions.

Addition occurs according to the law of addition.

For mixed fractions, add the integer parts to the integer parts and the fractional parts to the fractional parts.

If the fractional parts of mixed numbers have the same denominators, then add the numerators, and the denominator remains the same.

Add mixed numbers \(3\frac(6)(11)\) and \(1\frac(3)(11)\).

\(3\frac(6)(11) + 1\frac(3)(11) = (\color(red) (3) + \color(blue) (\frac(6)(11))) + ( \color(red) (1) + \color(blue) (\frac(3)(11))) = (\color(red) (3) + \color(red) (1)) + (\color( blue) (\frac(6)(11)) + \color(blue) (\frac(3)(11))) = \color(red)(4) + (\color(blue) (\frac(6 + 3)(11))) = \color(red)(4) + \color(blue) (\frac(9)(11)) = \color(red)(4) \color(blue) (\frac (9)(11))\)

If the fractional parts of mixed numbers have different denominators, then we find a common denominator.

Let's add mixed numbers \(7\frac(1)(8)\) and \(2\frac(1)(6)\).

The denominator is different, so you need to find a common denominator, it is equal to 24. Multiply the first fraction \(7\frac(1)(8)\) by an additional factor of 3, and the second fraction \(2\frac(1)(6)\) on 4.

\(7\frac(1)(8) + 2\frac(1)(6) = 7\frac(1 \times \color(red) (3))(8 \times \color(red) (3) ) = 2\frac(1 \times \color(red) (4))(6 \times \color(red) (4)) =7\frac(3)(24) + 2\frac(4)(24 ) = 9\frac(7)(24)\)

Related questions:
How to add fractions?
Answer: first you need to decide what type the expression belongs to: fractions have the same denominators, different denominators or mixed fractions. Depending on the type of expression, we proceed to the solution algorithm.

How to solve fractions with different denominators?
Answer: you need to find a common denominator, and then follow the rule of adding fractions with the same denominators.

How to solve mixed fractions?
Answer: Add integer parts to integer parts and fractional parts to fractional parts.

Example #1:
Can the sum of two result in a proper fraction? Wrong fraction? Give examples.

\(\frac(2)(7) + \frac(3)(7) = \frac(2 + 3)(7) = \frac(5)(7)\)

The fraction \(\frac(5)(7)\) is a proper fraction, it is the result of the sum of two proper fractions \(\frac(2)(7)\) and \(\frac(3)(7)\).

\(\frac(2)(5) + \frac(8)(9) = \frac(2 \times 9 + 8 \times 5)(5 \times 9) =\frac(18 + 40)(45) = \frac(58)(45)\)

The fraction \(\frac(58)(45)\) is an improper fraction, it is the result of the sum of the proper fractions \(\frac(2)(5)\) and \(\frac(8)(9)\).

Answer: The answer is yes to both questions.

Example #2:
Add fractions: a) \(\frac(3)(11) + \frac(5)(11)\) b) \(\frac(1)(3) + \frac(2)(9)\).

a) \(\frac(3)(11) + \frac(5)(11) = \frac(3 + 5)(11) = \frac(8)(11)\)

b) \(\frac(1)(3) + \frac(2)(9) = \frac(1 \times \color(red) (3))(3 \times \color(red) (3)) + \frac(2)(9) = \frac(3)(9) + \frac(2)(9) = \frac(5)(9)\)

Example #3:
Write the mixed fraction as the sum of a natural number and a proper fraction: a) \(1\frac(9)(47)\) b) \(5\frac(1)(3)\)

a) \(1\frac(9)(47) = 1 + \frac(9)(47)\)

b) \(5\frac(1)(3) = 5 + \frac(1)(3)\)

Example #4:
Calculate the sum: a) \(8\frac(5)(7) + 2\frac(1)(7)\) b) \(2\frac(9)(13) + \frac(2)(13) \) c) \(7\frac(2)(5) + 3\frac(4)(15)\)

a) \(8\frac(5)(7) + 2\frac(1)(7) = (8 + 2) + (\frac(5)(7) + \frac(1)(7)) = 10 + \frac(6)(7) = 10\frac(6)(7)\)

b) \(2\frac(9)(13) + \frac(2)(13) = 2 + (\frac(9)(13) + \frac(2)(13)) = 2\frac(11 )(thirteen) \)

c) \(7\frac(2)(5) + 3\frac(4)(15) = 7\frac(2 \times 3)(5 \times 3) + 3\frac(4)(15) = 7\frac(6)(15) + 3\frac(4)(15) = (7 + 3)+(\frac(6)(15) + \frac(4)(15)) = 10 + \frac (10)(15) = 10\frac(10)(15) = 10\frac(2)(3)\)

Task #1:
At dinner they ate \(\frac(8)(11)\) of the cake, and in the evening at dinner they ate \(\frac(3)(11)\). Do you think the cake was completely eaten or not?

Solution:
The denominator of the fraction is 11, it indicates how many parts the cake was divided into. At lunch, we ate 8 pieces of cake out of 11. At dinner, we ate 3 pieces of cake out of 11. Let's add 8 + 3 = 11, we ate pieces of cake out of 11, that is, the whole cake.

\(\frac(8)(11) + \frac(3)(11) = \frac(11)(11) = 1\)

Answer: They ate the whole cake.

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 numeric expression how much of 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 on concrete examples we will show it.

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 instance:

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, it is possible to download the presentation for elementary school(http://school-box.ru/nachalnaya-shkola/prezentazii-po-matematike.html) will come in handy.

Instruction

It is customary to separate ordinary and decimal fractions, acquaintance with which begins in high school. At present, there is no such field of knowledge where this would not be applied. Even in we are talking about the first 17th century, and all at once, which means 1600-1625. You also often have to deal with elementary operations on , as well as their transformation from one form to another.

Reducing fractions to a common denominator is perhaps the most important operation on. It is the basis of all calculations. So let's say there are two fractions a/b and c/d. Then, in order to bring them to a common denominator, you need to find the least common multiple (M) of the numbers b and d, and then multiply the numerator of the first fractions on (M/b), and the second numerator on (M/d).

Comparing fractions is another important task. To do this, give the given simple fractions to a common denominator and then compare the numerators, whose numerator is greater, that fraction is greater.

To perform addition or subtraction ordinary fractions, you need to bring them to a common denominator, and then produce the necessary mathematical from these fractions. The denominator remains unchanged. Suppose you need to subtract c/d from a/b. To do this, you need to find the least common multiple M of the numbers b and d, and then subtract the other from one numerator without changing the denominator: (a*(M/b)-(c*(M/d))/M

It is enough just to multiply one fraction by another, for this you just need to multiply their numerators and denominators:
(a / b) * (c / d) \u003d (a * c) / (b * d) To divide one fraction by another, you need to multiply the dividend fraction by the reciprocal of the divisor. (a/b)/(c/d)=(a*d)/(b*c)
It is worth recalling that in order to get a reciprocal, you need to swap the numerator and denominator.

The next action that can be performed with ordinary fractions is subtraction. As part of this material, we will consider how to correctly calculate the difference between fractions with the same and different denominators, how to subtract a fraction from a natural number and vice versa. All examples will be illustrated with tasks. Let us clarify in advance that we will analyze only cases where the difference of fractions results in a positive number.

Yandex.RTB R-A-339285-1

How to find the difference between fractions with the same denominator

Let's start right away with good example: let's say we have an apple that has been divided into eight parts. Let's leave five parts on the plate and take two of them. This action can be written like this:

We end up with 3 eighths because 5 − 2 = 3 . It turns out that 5 8 - 2 8 = 3 8 .

Thereby a simple example we have seen exactly how the subtraction rule works for fractions whose denominators are the same. Let's formulate it.

Definition 1

To find the difference between fractions with the same denominators, you need to subtract the numerator of one from the numerator of the other, and leave the denominator the same. This rule can be written as a b - c b = a - c b .

We will use this formula in what follows.

Let's take concrete examples.

Example 1

Subtract from the fraction 24 15 the common fraction 17 15 .

Solution

We see that these fractions have the same denominators. So all we have to do is subtract 17 from 24. We get 7 and add a denominator to it, we get 7 15 .

Our calculations can be written like this: 24 15 - 17 15 \u003d 24 - 17 15 \u003d 7 15

If necessary, you can reduce a complex fraction or separate the whole part from an improper one to make it more convenient to count.

Example 2

Find the difference 37 12 - 15 12 .

Solution

Let's use the formula described above and calculate: 37 12 - 15 12 = 37 - 15 12 = 22 12

It is easy to see that the numerator and denominator can be divided by 2 (we already talked about this earlier when we analyzed the signs of divisibility). Reducing the answer, we get 11 6 . This is an improper fraction, from which we will select the whole part: 11 6 \u003d 1 5 6.

How to find the difference between fractions with different denominators

Such a mathematical operation can be reduced to what we have already described above. To do this, simply bring the desired fractions to the same denominator. Let's formulate the definition:

Definition 2

To find the difference between fractions that have different denominators, you need to bring them to the same denominator and find the difference between the numerators.

Let's look at an example of how this is done.

Example 3

Subtract 1 15 from 2 9 .

Solution

The denominators are different, and you need to reduce them to the smallest common sense. In this case, the LCM is 45. For the first fraction, an additional factor of 5 is required, and for the second - 3.

Let's calculate: 2 9 = 2 5 9 5 = 10 45 1 15 = 1 3 15 3 = 3 45

We have two fractions with same denominator, and now we can easily find their difference using the algorithm described earlier: 10 45 - 3 45 = 10 - 3 45 = 7 45

A brief record of the solution looks like this: 2 9 - 1 15 \u003d 10 45 - 3 45 \u003d 10 - 3 45 \u003d 7 45.

Do not neglect the reduction of the result or the selection of a whole part from it, if necessary. In this example, we do not need to do this.

Example 4

Find the difference 19 9 - 7 36 .

Solution

We bring the fractions indicated in the condition to the lowest common denominator 36 and obtain 76 9 and 7 36 respectively.

We consider the answer: 76 36 - 7 36 \u003d 76 - 7 36 \u003d 69 36

The result can be reduced by 3 to get 23 12 . The numerator is greater than the denominator, which means we can extract the whole part. The final answer is 1 11 12 .

The summary of the whole solution is 19 9 - 7 36 = 1 11 12 .

How to subtract a natural number from a common fraction

Such an action can also be easily reduced to a simple subtraction of ordinary fractions. This can be done by representing a natural number as a fraction. Let's show an example.

Example 5

Find the difference 83 21 - 3 .

Solution

3 is the same as 3 1 . Then you can calculate like this: 83 21 - 3 \u003d 20 21.

If in the condition it is necessary to subtract an integer from an improper fraction, it is more convenient to first extract the integer from it, writing it as a mixed number. Then the previous example can be solved differently.

From the fraction 83 21, when you select the integer part, you get 83 21 \u003d 3 20 21.

Now just subtract 3 from it: 3 20 21 - 3 = 20 21 .

How to subtract a fraction from a natural number

This action is done similarly to the previous one: we rewrite a natural number as a fraction, bring both to a common denominator and find the difference. Let's illustrate this with an example.

Example 6

Find the difference: 7 - 5 3 .

Solution

Let's make 7 a fraction 7 1 . We do the subtraction and transform the final result, extracting the integer part from it: 7 - 5 3 = 5 1 3 .

There is another way to make calculations. It has some advantages that can be used in cases where the numerators and denominators of the fractions in the problem are large numbers.

Definition 3

If the fraction to be subtracted is correct, then the natural number from which we are subtracting must be represented as the sum of two numbers, one of which is equal to 1. After that, you need to subtract the desired fraction from unity and get the answer.

Example 7

Calculate the difference 1 065 - 13 62 .

Solution

The fraction to be subtracted is correct, because its numerator is less than the denominator. Therefore, we need to subtract one from 1065 and subtract the desired fraction from it: 1065 - 13 62 \u003d (1064 + 1) - 13 62

Now we need to find the answer. Using the properties of subtraction, the resulting expression can be written as 1064 + 1 - 13 62 . Let's calculate the difference in brackets. To do this, we represent the unit as a fraction 1 1 .

It turns out that 1 - 13 62 \u003d 1 1 - 13 62 \u003d 62 62 - 13 62 \u003d 49 62.

Now let's remember about 1064 and formulate the answer: 1064 49 62 .

We use old way to prove that it is less convenient. Here are the calculations we would get:

1065 - 13 62 = 1065 1 - 13 62 = 1065 62 1 62 - 13 62 = 66030 62 - 13 62 = = 66030 - 13 62 = 66017 62 = 1064 4 6

The answer is the same, but the calculations are obviously more cumbersome.

We considered the case when you need to subtract the correct fraction. If it's wrong, we replace it with a mixed number and subtract according to the familiar rules.

Example 8

Calculate the difference 644 - 73 5 .

Solution

The second fraction is improper, and the whole part must be separated from it.

Now we calculate similarly to the previous example: 630 - 3 5 = (629 + 1) - 3 5 = 629 + 1 - 3 5 = 629 + 2 5 = 629 2 5

Subtraction properties when working with fractions

The properties that the subtraction of natural numbers possesses also apply to the cases of subtracting ordinary fractions. Let's see how to use them when solving examples.

Example 9

Find the difference 24 4 - 3 2 - 5 6 .

Solution

We have already solved similar examples when we analyzed the subtraction of a sum from a number, so we act according to the already known algorithm. First, we calculate the difference 25 4 - 3 2, and then subtract the last fraction from it:

25 4 - 3 2 = 24 4 - 6 4 = 19 4 19 4 - 5 6 = 57 12 - 10 12 = 47 12

Let's transform the answer by extracting the integer part from it. The result is 3 11 12.

Brief summary of the whole solution:

25 4 - 3 2 - 5 6 = 25 4 - 3 2 - 5 6 = 25 4 - 6 4 - 5 6 = = 19 4 - 5 6 = 57 12 - 10 12 = 47 12 = 3 11 12

If the expression contains both fractions and integers, it is recommended to group them by type when calculating.

Example 10

Find the difference 98 + 17 20 - 5 + 3 5 .

Solution

Knowing the basic properties of subtraction and addition, we can group numbers as follows: 98 + 17 20 - 5 + 3 5 = 98 + 17 20 - 5 - 3 5 = 98 - 5 + 17 20 - 3 5

Let's complete the calculations: 98 - 5 + 17 20 - 3 5 = 93 + 17 20 - 12 20 = 93 + 5 20 = 93 + 1 4 = 93 1 4

If you notice a mistake in the text, please highlight it and press Ctrl+Enter