Bringing fractions to the denominator. How to bring fractions to a common denominator
I originally wanted to include the common denominator methods in the "Adding and Subtracting Fractions" paragraph. But there was so much information, and its importance is so great (after all, not only numerical fractions have common denominators), that it is better to study this issue separately.
So let's say we have two fractions with different denominators. And we want to make sure that the denominators become the same. The main property of a fraction comes to the rescue, which, let me remind you, sounds like this:
A fraction does not change if its numerator and denominator are multiplied by the same non-zero number.
Thus, if you choose the factors correctly, the denominators of the fractions will be equal - this process is called reduction to a common denominator. And the desired numbers, "leveling" the denominators, are called additional factors.
Why do you need to bring fractions to a common denominator? Here are just a few reasons:
- Addition and subtraction of fractions with different denominators. There is no other way to perform this operation;
- Fraction comparison. Sometimes reduction to a common denominator greatly simplifies this task;
- Solving problems on shares and percentages. Percentages are, in fact, ordinary expressions that contain fractions.
There are many ways to find numbers that make the denominators equal when multiplied. We will consider only three of them - in order of increasing complexity and, in a sense, efficiency.
Multiplication "criss-cross"
The simplest and most reliable way, which is guaranteed to equalize the denominators. We will act "ahead": we multiply the first fraction by the denominator of the second fraction, and the second by the denominator of the first. As a result, the denominators of both fractions will become equal to the product of the original denominators. Take a look:
As additional factors, consider the denominators of neighboring fractions. We get:
Yes, it's that simple. If you are just starting to learn fractions, it is better to work with this method - this way you will insure yourself against many mistakes and are guaranteed to get the result.
The only drawback of this method is that you have to count a lot, because the denominators are multiplied "ahead", and as a result, very large numbers can be obtained. That's the price of reliability.
Common divisor method
This technique helps to greatly reduce the calculations, but, unfortunately, it is rarely used. The method is as follows:
- Look at the denominators before you go "thru" (i.e., "criss-cross"). Perhaps one of them (the one that is larger) is divisible by the other.
- The number resulting from such a division will be an additional factor for a fraction with a smaller denominator.
- At the same time, a fraction with a large denominator does not need to be multiplied by anything at all - this is the savings. At the same time, the probability of error is sharply reduced.
A task. Find expression values:
Note that 84: 21 = 4; 72:12 = 6. Since in both cases one denominator is divisible without a remainder by the other, we use the method of common factors. We have:
Note that the second fraction was not multiplied by anything at all. In fact, we have cut the amount of calculations in half!
By the way, I took the fractions in this example for a reason. If you're interested, try counting them using the criss-cross method. After the reduction, the answers will be the same, but there will be much more work.
This is the strength of the method of common divisors, but, again, it can only be applied when one of the denominators is divided by the other without a remainder. Which happens quite rarely.
Least common multiple method
When we reduce fractions to a common denominator, we are essentially trying to find a number that is divisible by each of the denominators. Then we bring the denominators of both fractions to this number.
There are a lot of such numbers, and the smallest of them will not necessarily equal the direct product of the denominators of the original fractions, as is assumed in the "cross-wise" method.
For example, for denominators 8 and 12, the number 24 is quite suitable, since 24: 8 = 3; 24:12 = 2. This number is much less than the product 8 12 = 96 .
The smallest number that is divisible by each of the denominators is called their least common multiple (LCM).
Notation: The least common multiple of a and b is denoted by LCM(a ; b ) . For example, LCM(16; 24) = 48 ; LCM(8; 12) = 24 .
If you manage to find such a number, the total amount of calculations will be minimal. Look at the examples:
A task. Find expression values:
Note that 234 = 117 2; 351 = 117 3 . Factors 2 and 3 are coprime (have no common divisors except 1), and factor 117 is common. Therefore LCM(234; 351) = 117 2 3 = 702.
Similarly, 15 = 5 3; 20 = 5 4 . Factors 3 and 4 are relatively prime, and factor 5 is common. Therefore LCM(15; 20) = 5 3 4 = 60.
Now let's bring the fractions to common denominators:
Note how useful the factorization of the original denominators turned out to be:
- Having found the same factors, we immediately reached the least common multiple, which, generally speaking, is a non-trivial problem;
- From the resulting expansion, you can find out which factors are “missing” for each of the fractions. For example, 234 3 \u003d 702, therefore, for the first fraction, the additional factor is 3.
To appreciate how much of a win the least common multiple method gives, try calculating the same examples using the criss-cross method. Of course, without a calculator. I think after that comments will be redundant.
Do not think that such complex fractions will not be in real examples. They meet all the time, and the above tasks are not the limit!
The only problem is how to find this NOC. Sometimes everything is found in a few seconds, literally “by eye”, but in general this is a complex computational problem that requires separate consideration. Here we will not touch on this.
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.
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 sought as smallest number, which is divisible by other 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 casting 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:
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 you get a fraction equal to it.
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 expanding 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. etc. Mathematics: Interlocutor textbook for grades 5-6 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
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 to worry about, 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:
\(\begin(align)&\frac(1)(4) = \frac(1 \times \color(red) (4))(4 \times \color(red) (4)) = \frac(4 )(16)\\\\&\frac(9)(16) = \frac(9 \times \color(red) (1))(16 \times \color(red) (1)) = \frac( 9)(16)\\\\ \end(align)\)