In this material we will look at how to correctly convert fractions to a new denominator, what an additional factor is and how to find it. After this, we will formulate the basic rule for reducing fractions to new denominators and illustrate it with examples of problems.
The concept of reducing a fraction to another denominator
Let us recall the basic property of a fraction. According to him, an 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 number). 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 reduce a fraction to another 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 will be a fraction equal to the original one.
Let's illustrate this with an example.
Example 1
Convert the fraction 11 25 to the new denominator.
Solution
Let's take an arbitrary natural number 4 and multiply both sides of the original fraction by it. We count: 11 · 4 = 44 and 25 · 4 = 100. The result is a fraction of 44 100.
All calculations can be written in this form: 11 25 = 11 4 25 4 = 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. Review the basic concepts of division—multiples and divisors. If the number is not a multiple of b, but it cannot be a divisor of the new fraction. Let us illustrate our idea with an example of solving a problem.
Example 2
Calculate whether it is possible to reduce the fraction 5 9 to the denominators 54 and 21.
Solution
54 is a multiple of nine, which is in the denominator of the new fraction (i.e. 54 can be divided by 9). This means that such a reduction is possible. But we cannot divide 21 by 9, so this 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 sides of a fraction are multiplied to bring it to a new denominator.
Those. when we do this with a fraction, we take an additional factor for it. For example, to convert the fraction 7 10 to the form 21 30, we need an additional factor of 3. And you can get the fraction 15 40 from 3 8 using the multiplier 5.
Accordingly, if we know the denominator to which a fraction needs to be reduced, then we can calculate an additional factor for it. Let's figure out how to do this.
We have a fraction a b, which can be reduced to a certain denominator c; Let's 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 conditions of the problem b · m = c. Let's remember how multiplication and division are related to each other. This connection will prompt us to the following conclusion: the additional factor is nothing more than the quotient of dividing c by b, in other words, m = c: b.
Thus, to find the additional factor, we need to divide the required denominator by the original one.
Example 3
Find the additional factor with which the fraction 17 4 was reduced to the denominator 124.
Solution
Using the rule above, we simply divide 124 by the denominator of the original fraction, four.
We count: 124: 4 = 31.
This type of calculation is often required when converting fractions to a common denominator.
The rule for reducing fractions to the specified denominator
Let's move on to defining the basic rule with which you can reduce fractions to the specified denominator. So,
Definition 2
To reduce a fraction to the specified denominator you need:
- determine an additional factor;
- multiply both the numerator and denominator of the original fraction by it.
How to apply this rule in practice? Let's give an example of solving the problem.
Example 4
Reduce the fraction 7 16 to the denominator 336.
Solution
Let's start by calculating the additional multiplier. Divide: 336: 16 = 21.
We multiply the resulting answer by both parts of the original fraction: 7 16 = 7 · 21 16 · 21 = 147 336. So we brought the original fraction to the desired denominator 336.
Answer: 7 16 = 147 336.
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This article explains how to find the lowest common denominator And how to reduce fractions to a common denominator. First, the definitions of common denominator of fractions and least common denominator are given, and it is shown how to find the common denominator of fractions. Below is a rule for reducing fractions to a common denominator and examples of the application of this rule are considered. In conclusion, examples of bringing three or more fractions to a common denominator are discussed.
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What is called reducing fractions to a common denominator?
Now we can say what it is to reduce fractions to a common denominator. Reducing fractions to a common denominator- This is the multiplication of the numerators and denominators of given fractions by such additional factors that the result is fractions with the same denominators.
Common denominator, definition, examples
Now it's time to define the common denominator of fractions.
In other words, the common denominator of a certain set of ordinary fractions is any natural number that is divisible by all the denominators of these fractions.
From the stated definition it follows that a given set of fractions has infinitely many common denominators, since there is an infinite number of common multiples of all denominators of the original set of fractions.
Determining the common denominator of fractions allows you to find the common denominators of given fractions. Let, for example, given the fractions 1/4 and 5/6, their denominators are 4 and 6, respectively. Positive common multiples of the numbers 4 and 6 are the numbers 12, 24, 36, 48, ... Any of these numbers is a common denominator of the fractions 1/4 and 5/6.
To consolidate the material, consider the solution to the following example.
Example.
Can the fractions 2/3, 23/6 and 7/12 be reduced to a common denominator of 150?
Solution.
To answer the question posed, we need to find out whether the number 150 is a common multiple of the denominators 3, 6 and 12. To do this, let’s check whether 150 is divisible by each of these numbers (if necessary, see the rules and examples of dividing natural numbers, as well as the rules and examples of dividing natural numbers with a remainder): 150:3=50, 150:6=25, 150: 12=12 (remaining 6) .
So, 150 is not evenly divisible by 12, therefore 150 is not a common multiple of 3, 6, and 12. Therefore, the number 150 cannot be the common denominator of the original fractions.
Answer:
It is forbidden.
Lowest common denominator, how to find it?
In the set of numbers that are common denominators of given fractions, there is a smallest natural number, which is called the least common denominator. Let us formulate the definition of the lowest common denominator of these fractions.
Definition.
Lowest common denominator is the smallest number of all the common denominators of these fractions.
It remains to deal with the question of how to find the least common divisor.
Since is the least positive common divisor of a given set of numbers, the LCM of the denominators of the given fractions represents the least common denominator of the given fractions.
Thus, finding the lowest common denominator of fractions comes down to the denominators of those fractions. Let's look at the solution to the example.
Example.
Find the lowest common denominator of the fractions 3/10 and 277/28.
Solution.
The denominators of these fractions are 10 and 28. The desired lowest common denominator is found as the LCM of the numbers 10 and 28. In our case it’s easy: since 10=2·5, and 28=2·2·7, then LCM(15, 28)=2·2·5·7=140.
Answer:
140 .
How to reduce fractions to a common denominator? Rule, examples, solutions
Common fractions usually result in a lowest common denominator. We will now write down a rule that explains how to reduce fractions to their lowest common denominator.
Rule for reducing fractions to lowest common denominator consists of three steps:
- First, find the lowest common denominator of the fractions.
- Second, an additional factor is calculated for each fraction by dividing the lowest common denominator by the denominator of each fraction.
- Third, the numerator and denominator of each fraction are multiplied by its additional factor.
Let us apply the stated rule to solve the following example.
Example.
Reduce the fractions 5/14 and 7/18 to their lowest common denominator.
Solution.
Let's perform all the steps of the algorithm for reducing fractions to the lowest common denominator.
First we find the least common denominator, which is equal to the least common multiple of the numbers 14 and 18. Since 14=2·7 and 18=2·3·3, then LCM(14, 18)=2·3·3·7=126.
Now we calculate additional factors with the help of which the fractions 5/14 and 7/18 will be reduced to the denominator 126. For the fraction 5/14 the additional factor is 126:14=9, and for the fraction 7/18 the additional factor is 126:18=7.
It remains to multiply the numerators and denominators of the fractions 5/14 and 7/18 by additional factors of 9 and 7, respectively. We have and 
.
So, reducing the fractions 5/14 and 7/18 to the lowest common denominator is complete. The resulting fractions were 45/126 and 49/126.
To solve examples with fractions, you need to be able to find the lowest common denominator. Below are detailed instructions.
How to find the lowest common denominator - concept
The least common denominator (LCD), in simple words, is the minimum number that is divisible by the denominators of all fractions in a given example. In other words, it is called the Least Common Multiple (LCM). NOS is used only if the denominators of the fractions are different.
How to find the lowest common denominator - examples
Let's look at examples of finding NOCs.
Calculate: 3/5 + 2/15.
Solution (Sequence of actions):
- We look at the denominators of the fractions, make sure that they are different and that the expressions are as abbreviated as possible.
- We find the smallest number that is divisible by both 5 and 15. This number will be 15. Thus, 3/5 + 2/15 = ?/15.
- We figured out the denominator. What will be in the numerator? An additional multiplier will help us figure this out. An additional factor is the number obtained by dividing the NZ by the denominator of a particular fraction. For 3/5, the additional factor is 3, since 15/5 = 3. For the second fraction, the additional factor is 1, since 15/15 = 1.
- Having found out the additional factor, we multiply it by the numerators of the fractions and add the resulting values. 3/5 + 2/15 = (3*3+2*1)/15 = (9+2)/15 = 11/15.
Answer: 3/5 + 2/15 = 11/15.
If in the example not 2, but 3 or more fractions are added or subtracted, then the NCD must be searched for as many fractions as are given.
Calculate: 1/2 – 5/12 + 3/6
Solution (sequence of actions):
- Finding the lowest common denominator. The minimum number divisible by 2, 12 and 6 is 12.
- We get: 1/2 – 5/12 + 3/6 = ?/12.
- We are looking for additional multipliers. For 1/2 – 6; for 5/12 – 1; for 3/6 – 2.
- We multiply by the numerators and assign the corresponding signs: 1/2 – 5/12 + 3/6 = (1*6 – 5*1 + 2*3)/12 = 7/12.
Answer: 1/2 – 5/12 + 3/6 = 7/12.
The denominator of the arithmetic fraction a / b is the number b, which shows the size of the fractions of a unit from which the fraction is composed. The denominator of an algebraic fraction A / B is the algebraic expression B. To perform arithmetic operations with fractions, they must be reduced to the lowest common denominator.
You will need
- To work with algebraic fractions and find the lowest common denominator, you need to know how to factor polynomials.
Instructions
Let's consider reducing two arithmetic fractions n/m and s/t to the least common denominator, where n, m, s, t are integers. It is clear that these two fractions can be reduced to any denominator divisible by m and t. But they try to bring it to the lowest common denominator. It is equal to the least common multiple of the denominators m and t of the given fractions. The least multiple (LMK) of a number is the smallest divisible by all given numbers at the same time. Those. in our case, we need to find the least common multiple of the numbers m and t. Denoted as LCM (m, t). Next, the fractions are multiplied by the corresponding ones: (n/m) * (LCM (m, t) / m), (s/t) * (LCM (m, t) / t).
Let's find the lowest common denominator of three fractions: 4/5, 7/8, 11/14. First, let's expand the denominators 5, 8, 14: 5 = 1 * 5, 8 = 2 * 2 * 2 = 2^3, 14 = 2 * 7. Next, calculate the LCM (5, 8, 14) by multiplying all the numbers included into at least one of the expansions. LCM (5, 8, 14) = 5 * 2^3 * 7 = 280. Note that if a factor occurs in the expansion of several numbers (factor 2 in the expansion of denominators 8 and 14), then we take the factor to a greater degree (2^3 in our case).
So, the general one is received. It is equal to 280 = 5 * 56 = 8 * 35 = 14 * 20. Here we get the numbers by which we need to multiply the fractions with the corresponding denominators in order to bring them to the lowest common denominator. We get 4/5 = 56 * (4/5) = 224/280, 7/8 = 35 * (7/8) = 245/280, 11/14 = 20 * (11/14) = 220/280.
Reduction of algebraic fractions to the lowest common denominator is carried out by analogy with arithmetic ones. For clarity, let's look at the problem using an example. Let two fractions (2 * x) / (9 * y^2 + 6 * y + 1) and (x^2 + 1) / (3 * y^2 + 4 * y + 1) be given. Let's factorize both denominators. Note that the denominator of the first fraction is a perfect square: 9 * y^2 + 6 * y + 1 = (3 * y + 1)^2. For