This article explains how to bring fractions to a common denominator and how to find the lowest common denominator. Definitions are given, a rule for reducing fractions to a common denominator is given, and practical examples are considered.
What is common denominator reduction?
Common fractions have a numerator at the top and a denominator at the bottom. If fractions have the same denominator, they are said to be brought to a common denominator. For example, the fractions 11 14, 17 14, 9 14 have the same denominator 14. In other words, they are brought to a common denominator.
If the fractions have different denominators, then they can always be brought to a common denominator with the help of simple actions. To do this, you need to multiply the numerator and denominator by certain additional factors.
Obviously, the fractions 4 5 and 3 4 are not brought to a common denominator. To do this, you need to bring them to the denominator 20 using additional factors 5 and 4. How exactly to do this? Multiply the numerator and denominator of 4 5 by 4, and multiply the numerator and denominator of 3 4 by 5. Instead of fractions 4 5 and 3 4, we get 16 20 and 15 20, respectively.
Common denominator of fractions
Bringing fractions to a common denominator is multiplying the numerators and denominators of fractions by factors such that the result is identical fractions with the same denominator.
Common denominator: definition, examples
What is the common denominator?
Common denominator
The common denominator of a fraction is any positive number that is the common multiple of all given fractions.
In other words, the common denominator of a set of fractions will be a natural number that is evenly divisible by all the denominators of these fractions.
The range of natural numbers is infinite, and therefore, by definition, each set of ordinary fractions has an infinite set of common denominators. In other words, there are infinitely many common multiples for all the denominators of the original set of fractions.
The common denominator for multiple fractions is easy to find using the definition. Let there be fractions 1 6 and 3 5. The common denominator of the fractions is any positive common multiple of 6 and 5. These positive common multiples are 30, 60, 90, 120, 150, 180, 210, and so on.
Let's look at an example.
Example 1. Common denominator
Can the fraction 1 3, 21 6, 5 12 be reduced to a common denominator, which is 150?
To find out if this is so, you need to check if 150 is a common multiple for the denominators of fractions, that is, for the numbers 3, 6, 12. In other words, the number 150 must be divisible by 3, 6, 12 without a remainder. Let's check:
150 ÷ 3 = 50, 150 ÷ 6 = 25, 150 ÷ 12 = 12, 5
Hence, 150 is not the common denominator of these fractions.
Least common denominator
The smallest natural number from the set of common denominators of a set of fractions is called the lowest common denominator.
Least common denominator
The lowest common denominator of a fraction is the smallest number among all the common denominators of those fractions.
The least common divisor of a given set of numbers is the least common multiple (LCM). The LCM of all denominators of fractions is the lowest common denominator of those fractions.
How do you find the lowest common denominator? Finding it is reduced to finding the least common multiple of fractions. Let's look at an example:
Example 2. Find the least common denominator
Find the lowest common denominator for the fractions 1 10 and 127 28.
We are looking for the LCM of numbers 10 and 28. Let's decompose them into prime factors and get:
10 = 2 5 28 = 2 2 7 H O K (15, 28) = 2 2 5 7 = 140
How to bring fractions to the lowest common denominator
There is a rule that explains how to bring fractions to a common denominator. The rule consists of three points.
The rule for reducing fractions to a common denominator
- Find the lowest common denominator of the fractions.
- Find an additional factor for each fraction. To find the factor, you need to divide the lowest common denominator by the denominator of each fraction.
- Multiply the numerator and denominator by the additional factor found.
Let's consider the application of this rule with a specific example.
Example 3. Reducing fractions to a common denominator
There are fractions 3 14 and 5 18. Let's bring them to the lowest common denominator.
As a rule, we first find the LCM of the denominators of the fractions.
14 = 2 7 18 = 2 3 3 H O K (14, 18) = 2 3 3 7 = 126
We calculate additional factors for each fraction. For 3 14, the additional multiplier is 126 ÷ 14 = 9, and for the fraction 5 18, the additional multiplier will be 126 ÷ 18 = 7.
We multiply the numerator and denominator of fractions by additional factors and get:
3 9 14 9 = 27 126, 5 7 18 7 = 35 126.
Reducing multiple fractions to the lowest common denominator
According to the considered rule, not only pairs of fractions can be reduced to a common denominator, but also a larger number of them.
Let's give one more example.
Example 4. Reducing fractions to a common denominator
Reduce fractions 3 2, 5 6, 3 8, and 17 18 to the lowest common denominator.
Let's calculate the LCM of the denominators. We find the LCM of three or more numbers:
H O C (2, 6) = 6 H O C (6, 8) = 24 H O C (24, 18) = 72 H O C (2, 6, 8, 18) = 72
For 3 2 the additional multiplier is 72 ÷ 2 = 36, for 5 6 the additional multiplier is 72 ÷ 6 = 12, for 3 8 the additional multiplier is 72 ÷ 8 = 9, and finally for 17 18 the additional multiplier is 72 ÷ 18 = 4.
We multiply the fractions by additional factors and go to the lowest common denominator:
3 2 36 = 108 72 5 6 12 = 60 72 3 8 9 = 27 72 17 18 4 = 68 72
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Initially, I wanted to include common denominator methods in the Adding and Subtracting Fractions paragraph. But there was so much information, and its importance is so great (after all, common denominators are not only for numeric fractions) 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 basic property of a fraction comes to the rescue, which, recall, sounds like this:
The fraction will not change if its numerator and denominator are multiplied by the same nonzero number.
Thus, if you choose the right factors, the denominators of the fractions become equal - this process is called common denominator reduction. And the required numbers, "leveling" the denominators, are called additional factors.
Why do you even 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;
- Comparison of fractions. Sometimes converting to a common denominator makes this task much easier;
- Solving problems for shares and percentages. Percentages are, in fact, common expressions that contain fractions.
There are many ways to find numbers that, when multiplied by, make the denominators of fractions equal. We will consider only three of them - in ascending order of complexity and, in a sense, efficiency.
Cross-multiplication
The simplest and most reliable way that is guaranteed to align the denominators. We will go 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:
Consider the denominators of neighboring fractions as additional factors. We get:
Yes, it's that simple. If you are just starting to learn fractions, it is better to work with this particular 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 of time", and as a result, very large numbers can be obtained. This is the price to pay for reliability.
Common divisors method
This technique helps to greatly reduce calculations, but, unfortunately, it is rarely used. The method is as follows:
- Before you go ahead (that is, the criss-cross method), take a look at the denominators. Perhaps one of them (the one that is larger) is divided by the other.
- The number obtained as a result of such division will be an additional factor for the fraction with a lower denominator.
- In this case, the 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.
Task. Find the values of the expressions:
Note that 84: 21 = 4; 72: 12 = 6. Since in both cases one denominator is divisible by the other without a remainder, we apply the method of common factors. We have:
Note that the second fraction was never multiplied by anything at all. In fact, we have cut the amount of computation in half!
By the way, I took the fractions in this example for a reason. If you're curious, try counting them crosswise. After 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 be applied only when one of the denominators is divisible by the other without a remainder. Which is rare enough.
Least Common Multiple Method
When we bring 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 be equal to the direct product of the denominators of the original fractions, as it is assumed in the "criss-cross" method.
For example, for the denominators 8 and 12, the number 24 is fine, 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 can find such a number, the total amount of computation will be minimal. Take a look at examples:
Task. Find the values of the expressions:
Note that 234 = 117 · 2; 351 = 117 3. The factors 2 and 3 are relatively prime (they have no common divisors except 1), and the factor 117 is common. Therefore, the LCM (234; 351) = 117 2 3 = 702.
Similarly, 15 = 5 · 3; 20 = 5 4. The factors 3 and 4 are relatively prime, and the factor 5 is common. Therefore, LCM (15; 20) = 5 3 4 = 60.
Now we bring the fractions to common denominators:
Note how helpful factoring the original denominators was:
- Having found the same factors, we immediately arrived at the least common multiple, which, generally speaking, is a nontrivial problem;
- From the resulting expansion, you can find out which factors are "missing" for each of the fractions. For example, 234 3 = 702, therefore, for the first fraction, the additional factor is 3.
To estimate how colossal gains the least common multiple method gives, try calculating the same examples using the criss-cross method. Without a calculator, of course. I think after that comments will be superfluous.
Do not think that such complex fractions will not be in the real examples. They meet all the time, and the above tasks are not the limit!
The only problem is how to find this very NOC. Sometimes everything is found in a few seconds, literally "by eye", but on the whole this is a complex computational task that requires separate consideration. We will not touch on this here.
Common denominator of fractions
Fractions AND have the same denominator. They say they have common denominator 25. Fractions and have different denominators, but they can be brought to a common denominator using the basic property of fractions. To do this, we find a number that is divisible by 8 and 3, for example, 24. Let us bring the fractions to the denominator 24, for this we multiply the numerator and denominator of the fraction by additional factor 3. The additional factor is usually written on the left above the numerator:
Multiply the numerator and denominator of the fraction by an additional factor of 8:
Let us bring the fractions to a common denominator. Most often, fractions result in the lowest common denominator, which is the lowest common multiple of the fraction's denominator. Since the LCM (8, 12) = 24, the fractions can be reduced to the denominator 24. Find the additional factors of the fractions: 24: 8 = 3, 24:12 = 2. Then
Several fractions can be brought to a common denominator.
Example. Let us bring the fractions to a common denominator. Since 25 = 5 2, 10 = 2 5, 6 = 2 3, then LCM (25, 10, 6) = 2 3 5 2 = 150.
Let's find additional factors of fractions and bring them to the denominator 150:
Comparison of fractions
In fig. 4.7 shows a segment AB of length 1. It is divided into 7 equal parts. The segment AC has a length and the segment AD has a length.
The length of the segment AD is greater than the length of the segment AC, i.e. the fraction is greater than the fraction
Of the two fractions with a common denominator, the one with the larger numerator is larger, i.e.
For example, or
To compare any two fractions, they are brought to a common denominator, and then the rule for comparing fractions with a common denominator is applied.
Example. Compare fractions
Solution. LCM (8, 14) = 56. Then Since 21> 20, then
If the first fraction is less than the second, and the second is less than the third, then the first is less than the third.
Proof. Let three fractions be given. Let's bring them to a common denominator. Let after that they have the form Since the first fraction is less
second, then r< s. Так как вторая дробь меньше третьей, то s < t. Из полученных неравенств для натуральных чисел следует, что r < t, тогда первая дробь меньше третьей.
The fraction is called correct if its numerator is less than the denominator.
The fraction is called wrong if its numerator is greater than or equal to the denominator.
For example, fractions are correct and fractions are incorrect.
The correct fraction is less than 1 and the improper fraction is greater than or equal to 1.