To understand how to add fractions with different denominators, let's first study the rule and then look at specific examples.
To add or subtract fractions with different denominators:
1) Find (NOZ) given fractions.
2) Find an additional factor for each fraction. To do this, the new denominator must be divided by the old.
3) Multiply the numerator and denominator of each fraction by an additional factor and add or subtract fractions with the same denominators.
4) Check if the resulting fraction is regular and irreducible.
In the following examples, you need to add or subtract fractions with different denominators:
1) To subtract fractions with different denominators, first look for the smallest common denominator of these fractions. We choose the larger of the numbers and check if it is divisible by the smaller one. 25 is not divisible by 20. We multiply 25 by 2. 50 is not divisible by 20. We multiply 25 by 3. 75 is not divisible by 20. We multiply 25 by 4. 100 is divisible by 20. So the least common denominator is 100.
2) To find an additional factor for each fraction, you need to divide the new denominator by the old one. 100:25=4, 100:20=5. Accordingly, to the first fraction an additional factor is 4, to the second - 5.
3) We multiply the numerator and denominator of each fraction by an additional factor and subtract the fractions according to the rule for subtracting fractions with the same denominators.
4) The resulting fraction is regular and irreducible. So this is the answer.
1) To add fractions with different denominators, first look for the smallest common denominator. 16 is not divisible by 12. 16∙2=32 is not divisible by 12. 16∙3=48 is divisible by 12. So 48 is NOZ.
2) 48:16=3, 48:12=4. These are additional factors to each fraction.
3) multiply the numerator and denominator of each fraction by an additional factor and add the new fractions.
4) The resulting fraction is regular and irreducible.
1) 30 is not divisible by 20. 30∙2=60 is divisible by 20. So 60 is the least common denominator of these fractions.
2) to find an additional factor for each fraction, you need to divide the new denominator by the old one: 60:20=3, 60:30=2.
3) multiply the numerator and denominator of each fraction by an additional factor and subtract new fractions.
4) the resulting fractional 5.
1) 8 is not divisible by 6. 8∙2=16 is not divisible by 6. 8∙3=24 is divisible by both 4 and 6. Hence, 24 is the NOZ.
2) to find an additional factor for each fraction, you need to divide the new denominator by the old one. 24:8=3, 24:4=6, 24:6=4. So 3, 6 and 4 are additional factors to the first, second and third fractions.
3) multiply the numerator and denominator of each dolby by an additional factor. We add and subtract. The resulting fraction is improper, so you need to select the whole part.
In this lesson, we will look at reducing fractions to a common denominator and solve problems on this topic. Let's give a definition of the concept of a common denominator and an additional factor, remember about coprime numbers. Let's define the concept of the least common denominator (LCD) and solve a number of problems to find it.
Topic: Adding and subtracting fractions with different denominators
Lesson: Reducing fractions to a common denominator
Repetition. Basic property of a fraction.
If the numerator and denominator of a fraction are multiplied or divided by the same natural number, then a fraction equal to it will be obtained.
For example, the numerator and denominator of a fraction can be divided by 2. We get a fraction. This operation is called fraction reduction. You can also perform the reverse transformation by multiplying the numerator and denominator of the fraction by 2. In this case, we say that we have reduced the fraction to a new denominator. The number 2 is called an additional factor.
Conclusion. A fraction can be reduced to any denominator that is a multiple of the denominator of the given fraction. In order to bring a fraction to a new denominator, its numerator and denominator are multiplied by an additional factor.
1. Bring the fraction to the denominator 35.
The number 35 is a multiple of 7, that is, 35 is divisible by 7 without a remainder. So this transformation is possible. Let's find an additional factor. To do this, we divide 35 by 7. We get 5. We multiply the numerator and denominator of the original fraction by 5.
2. Bring the fraction to the denominator 18.
Let's find an additional factor. To do this, we divide the new denominator by the original one. We get 3. We multiply the numerator and denominator of this fraction by 3.
3. Bring the fraction to the denominator 60.
By dividing 60 by 15, we get an additional multiplier. It is equal to 4. Let's multiply the numerator and denominator by 4.
4. Bring the fraction to the denominator 24
In simple cases, reduction to a new denominator is performed in the mind. It is customary to only indicate an additional factor behind the bracket a little to the right and above the original fraction.
A fraction can be reduced to a denominator of 15 and a fraction can be reduced to a denominator of 15. Fractions have a common denominator of 15.
The common denominator of fractions can be any common multiple of their denominators. For simplicity, fractions are reduced to the lowest common denominator. It is equal to the least common multiple of the denominators of the given fractions.
Example. Reduce to the least common denominator of the fraction and .
First, find the least common multiple of the denominators of these fractions. This number is 12. Let's find an additional factor for the first and second fractions. To do this, we divide 12 by 4 and by 6. Three is an additional factor for the first fraction, and two for the second. We bring the fractions to the denominator 12.
We reduced the fractions to a common denominator, that is, we found fractions that are equal to them and have the same denominator.
Rule. To bring fractions to the lowest common denominator,
First, find the least common multiple of the denominators of these fractions, which will be their least common denominator;
Secondly, divide the least common denominator by the denominators of these fractions, that is, find an additional factor for each fraction.
Thirdly, multiply the numerator and denominator of each fraction by its additional factor.
a) Reduce the fractions and to a common denominator.
The lowest common denominator is 12. The additional factor for the first fraction is 4, for the second - 3. We bring the fractions to the denominator 24.
b) Reduce the fractions and to a common denominator.
The lowest common denominator is 45. Dividing 45 by 9 by 15, we get 5 and 3, respectively. We bring the fractions to the denominator 45.
c) Reduce the fractions and to a common denominator.
The common denominator is 24. The additional factors are 2 and 3, respectively.
Sometimes it is difficult to verbally find the least common multiple for the denominators of given fractions. Then the common denominator and additional factors are found by factoring into prime factors.
Reduce to a common denominator of the fraction and .
Let's decompose the numbers 60 and 168 into prime factors. Let's write out the expansion of the number 60 and add the missing factors 2 and 7 from the second expansion. Multiply 60 by 14 and get a common denominator of 840. The additional factor for the first fraction is 14. The additional factor for the second fraction is 5. Let's reduce the fractions to a common denominator of 840.
Bibliography
1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S. and others. Mathematics 6. - M.: Mnemozina, 2012.
2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - Gymnasium, 2006.
3. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - Enlightenment, 1989.
4. Rurukin A.N., Chaikovsky I.V. Tasks for the course of mathematics grade 5-6. - ZSH MEPhI, 2011.
5. Rurukin A.N., Sochilov S.V., Chaikovsky K.G. Mathematics 5-6. A manual for students of the 6th grade of the MEPhI correspondence school. - ZSH MEPhI, 2011.
6. Shevrin L.N., Gein A.G., Koryakov I.O. and others. Mathematics: A textbook-interlocutor for grades 5-6 of high school. Library of the teacher of mathematics. - Enlightenment, 1989.
You can download the books specified in clause 1.2. this lesson.
Homework
Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S. and others. Mathematics 6. - M .: Mnemozina, 2012. (see link 1.2)
Homework: No. 297, No. 298, No. 300.
Other tasks: #270, #290
In this material, we will analyze how to correctly bring fractions to a new denominator, what an additional factor is and how to find it. After that, we formulate the basic rule for reducing fractions to new denominators and illustrate it with examples of problems.
The concept of reducing a fraction to a different denominator
Recall the basic property of a fraction. According to him, the ordinary fraction a b (where a and b are any numbers) has an infinite number of fractions that are equal to it. Such fractions can be obtained by multiplying the numerator and denominator by the same number m (natural). In other words, all ordinary fractions can be replaced by others of the form a m b m . This is the reduction of the original value to a fraction with the desired denominator.
You can bring a fraction to a different denominator by multiplying its numerator and denominator by any natural number. The main condition is that the multiplier must be the same for both parts of the fraction. The result is a fraction equal to the original.
Let's illustrate this with an example.
Example 1
Convert the fraction 11 25 to a new denominator.
Decision
Take an arbitrary natural number 4 and multiply both parts of the original fraction by it. We consider: 11 4 \u003d 44 and 25 4 \u003d 100. The result is a fraction of 44,100.
All calculations can be written in this form: 11 25 \u003d 11 4 25 4 \u003d 44 100
It turns out that any fraction can be reduced to a huge number of different denominators. Instead of four, we could take another natural number and get another fraction equivalent to the original one.
But not any number can become the denominator of a new fraction. So, for a b the denominator can only contain numbers b · m that are multiples of b . Recall the basic concepts of division - multiples and divisors. If the number is not a multiple of b, but it cannot be a divisor of a new fraction. Let us explain our idea with an example of solving the problem.
Example 2
Calculate whether it is possible to reduce the fraction 5 9 to the denominators 54 and 21.
Decision
54 is a multiple of nine, which is the denominator of the new fraction (i.e. 54 can be divided by 9). Hence, such a reduction is possible. And we cannot divide 21 by 9, so such an action cannot be performed for this fraction.
The concept of an additional multiplier
Let us formulate what an additional factor is.
Definition 1
Additional multiplier is a natural number by which both parts of a fraction are multiplied to bring it to a new denominator.
Those. when we perform this action on a fraction, we take an additional multiplier for it. For example, to reduce the fraction 7 10 to the form 21 30, we need an additional factor 3 . And you can get a fraction 15 40 out of 3 8 using a multiplier 5.
Accordingly, if we know the denominator to which the fraction must be reduced, then we can calculate an additional factor for it. Let's figure out how to do it.
We have a fraction a b , which can be reduced to some denominator c ; calculate the additional factor m . We need to multiply the denominator of the original fraction by m. We get b · m , and according to the condition of the problem b · m = c . Recall how multiplication and division are related. This connection will lead us to the following conclusion: the additional factor is nothing but the quotient of dividing c by b, in other words, m = c: b.
Thus, to find an additional factor, we need to divide the required denominator by the original one.
Example 3
Find the additional factor by which the fraction 17 4 was brought to the denominator 124 .
Decision
Using the rule above, we simply divide 124 by the denominator of the original fraction, four.
We consider: 124: 4 \u003d 31.
This type of calculation is often required when reducing fractions to a common denominator.
The rule for reducing fractions to a specified denominator
Let's move on to the definition of the basic rule, with which you can bring fractions to the specified denominator. So,
Definition 2
To bring a fraction to the specified denominator, you need:
- determine an additional multiplier;
- multiply by it both the numerator and the denominator of the original fraction.
How to apply this rule in practice? Let us give an example of solving the problem.
Example 4
Carry out the reduction of the fraction 7 16 to the denominator 336 .
Decision
Let's start by calculating the additional multiplier. Divide: 336: 16 = 21.
We multiply the received answer by both parts of the original fraction: 7 16 \u003d 7 21 16 21 \u003d 147 336. So we brought the original fraction to the desired denominator 336.
Answer: 7 16 = 147 336.
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How to bring algebraic (rational) fractions to a common denominator?
1) If the denominators of the fractions are polynomials, you need to try one of the known methods.
2) The lowest common denominator (LCD) consists of all multipliers taken in greatest degree.
The least common denominator for numbers is verbally searched for as the smallest number that is divisible by the rest of the numbers.
3) To find an additional factor for each fraction, you need to divide the new denominator by the old one.
4) The numerator and denominator of the original fraction are multiplied by an additional factor.
Consider examples of reducing algebraic fractions to a common denominator.
To find a common denominator for numbers, choose the larger number and check if it is divisible by the smaller one. 15 is not divisible by 9. We multiply 15 by 2 and check if the resulting number is divisible by 9. 30 is not divisible by 9. We multiply 15 by 3 and check if the resulting number is divisible by 9. 45 is divisible by 9, which means that the common denominator for the numbers is 45.
The lowest common denominator is the sum of all factors taken to the highest power. Thus, the common denominator of these fractions is 45 bc (letters are usually written in alphabetical order).
To find an additional factor for each fraction, you need to divide the new denominator by the old one. 45bc:(15b)=3c, 45bc:(9c)=5b. We multiply the numerator and denominator of each fraction by an additional factor:
First, we look for a common denominator for numbers: 8 is not divisible by 6, 8∙2=16 is not divisible by 6, 8∙3=24 is divisible by 6. Each of the variables must be included in the common denominator once. From the degrees we take the degree with a large exponent.
Thus, the common denominator of these fractions is 24a³bc.
To find an additional factor for each fraction, you need to divide the new denominator by the old one: 24a³bc:(6a³c)=4b, 24a³bc:(8a²bc)=3a.
We multiply the additional factor by the numerator and denominator:
The polynomials in the denominators of these fractions are needed. The denominator of the first fraction is the full square of the difference: x²-18x+81=(x-9)²; in the denominator of the second - the difference of squares: x²-81=(x-9)(x+9):
The common denominator consists of all factors taken to the greatest extent, that is, it is equal to (x-9)²(x+9). We find additional factors and multiply them by the numerator and denominator of each fraction:
Fractions have different or the same denominators. Same denominator or otherwise called common denominator at the fraction An example of a common denominator:
\(\frac(17)(5), \frac(1)(5)\)
An example of different denominators for fractions:
\(\frac(8)(3), \frac(2)(13)\)
How to find a common denominator of a fraction?
The first fraction has a denominator of 3, the second is 13. You need to find a number that is divisible by both 3 and 13. This number is 39.
The first fraction must be multiplied by additional multiplier 13. So that the fraction does not change, we must multiply both the numerator by 13 and the denominator.
\(\frac(8)(3) = \frac(8 \times \color(red) (13))(3 \times \color(red) (13)) = \frac(104)(39)\)
We multiply the second fraction by an additional factor of 3.
\(\frac(2)(13) = \frac(2 \times \color(red) (3))(13 \times \color(red) (3)) = \frac(6)(39)\)
We have reduced the common denominator of the fraction:
\(\frac(8)(3) = \frac(104)(39), \frac(2)(13) = \frac(6)(39)\)
Lowest common denominator.
Consider another example:
Let's bring the fractions \(\frac(5)(8)\) and \(\frac(7)(12)\) to a common denominator.
The common denominator for the numbers 8 and 12 can be the numbers 24, 48, 96, 120, ..., it is customary to choose lowest common denominator in our case, this number is 24.
Lowest common denominator is the smallest number that divides the denominator of the first and second fractions.
How to find the lowest common denominator?
By enumeration of numbers, by which the denominator of the first and second fractions is divided and choose the smallest of them.
We need to multiply the fraction with a denominator of 8 by 3, and multiply the fraction with a denominator of 12 by 2.
\(\begin(align)&\frac(5)(8) = \frac(5 \times \color(red) (3))(8 \times \color(red) (3)) = \frac(15 )(24)\\\\&\frac(7)(12) = \frac(7 \times \color(red) (2))(12 \times \color(red) (2)) = \frac( 14)(24)\\\\ \end(align)\)
If you can’t immediately bring the fractions to the lowest common denominator, there’s nothing wrong with that, in the future, when solving the example, you may have to get the answer
A common denominator can be found for any two fractions; it can be the product of the denominators of these fractions.
For example:
Reduce the fractions \(\frac(1)(4)\) and \(\frac(9)(16)\) to the lowest common denominator.
The easiest way to find the common denominator is to multiply the denominators 4⋅16=64. The number 64 is not the lowest common denominator. The task is to find the smallest common denominator. So we are looking further. We need a number that is divisible by both 4 and 16, this is the number 16. Let's reduce the fraction to a common denominator, multiply the fraction with a denominator of 4 by 4, and the fraction with a denominator of 16 by one. We get:
\(\begin(align)&\frac(1)(4) = \frac(1 \times \color(red) (4))(4 \times \color(red) (4)) = \frac(4 )(16)\\\\&\frac(9)(16) = \frac(9 \times \color(red) (1))(16 \times \color(red) (1)) = \frac( 9)(16)\\\\ \end(align)\)
