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How to find the nose of three fractions. Reducing fractions to the lowest common denominator, rule, examples, solutions

I originally wanted to include common denominator techniques in the Adding and Subtracting Fractions section. But there turned out to be 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 basic property of a fraction comes to the rescue, which, let me remind you, sounds like this:

A fraction will not change if its numerator and denominator are multiplied by the same number other than zero.

Thus, if you choose the factors correctly, the denominators of the fractions will become equal - this process is called reduction to a common denominator. And the required numbers, “evening out” the denominators, are called additional factors.

Why do we need to reduce fractions to a common denominator? Here are just a few reasons:

  1. Adding and subtracting fractions with different denominators. There is no other way to perform this operation;
  2. Comparing fractions. Sometimes reduction to a common denominator greatly simplifies this task;
  3. Solving problems involving fractions and percentages. Percentages are, in fact, ordinary expressions that contain fractions.

There are many ways to find numbers that, when multiplied by them, will make the denominators of fractions equal. We will consider only three of them - in order of increasing complexity and, in a sense, effectiveness.

Criss-cross multiplication

The simplest and reliable way, which is guaranteed to equalize the denominators. We will act “in a headlong manner”: 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 study fractions, it is better to work using 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 do a lot of counting, because the denominators are multiplied “over and over”, and the result can be very big numbers. This is the price to pay for reliability.

Common Divisor Method

This technique helps to significantly reduce calculations, but, unfortunately, it is used quite rarely. The method is as follows:

  1. Before you go straight ahead (i.e., using the criss-cross method), take a look at the denominators. Perhaps one of them (the one that is larger) is divided into the other.
  2. The number resulting from this division will be an additional factor for the fraction with a smaller denominator.
  3. In this case, a fraction with a large denominator does not need to be multiplied by anything at all - this is where the savings lie. At the same time, the probability of error is sharply reduced.

Task. Find the meanings of the expressions:

Note that 84: 21 = 4; 72: 12 = 6. Since in both cases one denominator is divided 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 cut the amount of computation in half!

By the way, I didn’t take the fractions in this example by chance. If you're interested, try counting them using the criss-cross method. After reduction, the answers will be the same, but there will be much more work.

This is the power of the common divisors method, but, again, it can only be used when one of the denominators is divisible 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 denominator. 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 is assumed in the “criss-cross” 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 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 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:

Task. Find the meanings of the expressions:

Note that 234 = 117 2; 351 = 117 3. Factors 2 and 3 are coprime (have no common factors other than 1), and factor 117 is common. Therefore LCM(234, 351) = 117 2 3 = 702.

Likewise, 15 = 5 3; 20 = 5 · 4. Factors 3 and 4 are coprime, and factor 5 is common. Therefore LCM(15, 20) = 5 3 4 = 60.

Now let's reduce the fractions to common denominators:

Notice how useful it was to factorize the original denominators:

  1. Having discovered identical factors, we immediately arrived at the least common multiple, which, generally speaking, is a non-trivial problem;
  2. From the resulting expansion you can find out which factors are “missing” in each fraction. For example, 234 · 3 = 702, therefore, for the first fraction the additional factor is 3.

To appreciate how much of a difference the least common multiple method makes, try calculating these same examples using the criss-cross method. Of course, without a calculator. I think after this comments will be unnecessary.

Don't think that there won't be such complex fractions 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 can be found in a few seconds, literally “by eye,” but in general this is a complex computational task that requires separate consideration. We won't touch on that here.

To solve examples with fractions, you must be able to find the smallest common denominator. Below are detailed instructions.

How to find the lowest common denominator - concept

Least common denominator (LCD) in simple words is the minimum number that is divisible by the denominators of all fractions this 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 smallest number, which 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.


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 ordinary fractions is any natural number, which is divisible by all 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 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 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.

Content:

To add or subtract fractions with unlike denominators (the numbers below the fraction line), you first need to find their least common denominator (LCD). This number will be the smallest multiple that appears in the list of multiples of each denominator, that is, a number that is evenly divisible by each denominator. You can also calculate the least common multiple (LCM) of two or more denominators. In any case, we are talking about integers, the methods for finding which are very similar. Once you have determined the NOS, you can reduce fractions to a common denominator, which in turn allows you to add and subtract them.

Steps

1 Listing multiples

  1. 1 List the multiples of each denominator. Make a list of multiples of each denominator in the equation. Each list must consist of the product of the denominator by 1, 2, 3, 4, and so on.
    • Example: 1/2 + 1/3 + 1/5
    • Multiples of 2: 2 * 1 = 2; 2 * 2 = 4; 2 * 3 = 6; 2 * 4 = 8; 2 * 5 = 10; 2 * 6 = 12; 2 * 7 = 14; and so on.
    • Multiples of 3: 3 * 1 = 3; 3 * 2 = 6; 3 *3 = 9; 3 * 4 = 12; 3 * 5 = 15; 3 * 6 = 18; 3 * 7 = 21; and so on.
    • Multiples of 5: 5 * 1 = 5; 5 * 2 = 10; 5 * 3 = 15; 5 * 4 = 20; 5 * 5 = 25; 5 * 6 = 30; 5 * 7 = 35; and so on.
  2. 2 Determine the least common multiple. Go through each list and note any multiples that are common to all denominators. After identifying common multiples, determine the lowest denominator.
    • Note that if a common denominator is not found, you may need to continue writing out multiples until a common multiple appears.
    • It is better (and easier) to use this method when the denominators contain small numbers.
    • In our example, the common multiple of all denominators is the number 30: 2 * 15 = 30 ; 3 * 10 = 30 ; 5 * 6 = 30
    • NOZ = 30
  3. 3 In order to bring fractions to a common denominator without changing their meaning, multiply each numerator (the number above the fraction line) by a number equal to the quotient of NZ divided by the corresponding denominator.
    • Example: (15/15) * (1/2); (10/10) * (1/3); (6/6) * (1/5)
    • New equation: 15/30 + 10/30 + 6/30
  4. 4 Solve the resulting equation. After finding the NOS and changing the corresponding fractions, simply solve the resulting equation. Don't forget to simplify your answer (if possible).
    • Example: 15/30 + 10/30 + 6/30 = 31/30 = 1 1/30

2 Using the greatest common divisor

  1. 1 List the divisors of each denominator. A divisor is an integer that divides by a whole given number. For example, the divisors of the number 6 are the numbers 6, 3, 2, 1. The divisor of any number is 1, because any number is divisible by one.
    • Example: 3/8 + 5/12
    • Divisors 8: 1, 2, 4 , 8
    • Divisors 12: 1, 2, 3, 4 , 6, 12
  2. 2 Find the greatest common divisor (GCD) of both denominators. After listing the factors of each denominator, note all common factors. The greatest common factor is the largest common factor you will need to solve the problem.
    • In our example common divisors for the denominators of 8 and 12 the numbers are 1, 2, 4.
    • GCD = 4.
  3. 3 Multiply the denominators together. If you want to use GCD to solve a problem, first multiply the denominators together.
    • Example: 8 * 12 = 96
  4. 4 Divide the resulting value by GCD. Having received the result of multiplying the denominators, divide it by the gcd you calculated. The resulting number will be the lowest common denominator (LCD).
    • Example: 96 / 4 = 24
  5. 5
    • Example: 24 / 8 = 3; 24 / 12 = 2
    • (3/3) * (3/8) = 9/24; (2/2) * (5/12) = 10/24
    • 9/24 + 10/24
  6. 6 Solve the resulting equation.
    • Example: 9/24 + 10/24 = 19/24

3 Factoring each denominator into prime factors

  1. 1 Factor each denominator into prime factors. Factor each denominator into prime factors, that is prime numbers, which when multiplied give the original denominator. Recall that prime factors are numbers that are divisible only by 1 or themselves.
    • Example: 1/4 + 1/5 + 1/12
    • Prime factors 4: 2 * 2
    • Prime factors 5: 5
    • Prime factors of 12: 2 * 2 * 3
  2. 2 Count the number of times each prime factor is present in each denominator. That is, determine how many times each prime factor appears in the list of factors of each denominator.
    • Example: There are two 2 for denominator 4; zero 2 for 5; two 2 for 12
    • There is a zero 3 for 4 and 5; one 3 for 12
    • There is a zero 5 for 4 and 12; one 5 for 5
  3. 3 Take only the greatest number of times for each prime factor. Determine the greatest number of times each prime factor appears in any denominator.
    • For example: the greatest number of times for a multiplier 2 - 2 times; For 3 – 1 time; For 5 – 1 time.
  4. 4 Write down the prime factors found in the previous step in order. Do not write down the number of times each prime factor appears in all the original denominators - do this taking into account the largest number times (as described in the previous step).
    • Example: 2, 2, 3, 5
  5. 5 Multiply these numbers. The result of the product of these numbers is equal to NOS.
    • Example: 2 * 2 * 3 * 5 = 60
    • NOZ = 60
  6. 6 Divide the NOZ by the original denominator. To calculate the multiplier required to reduce fractions to a common denominator, divide the NCD you found by the original denominator. Multiply the numerator and denominator of each fraction by this factor. You will get fractions with a common denominator.
    • Example: 60/4 = 15; 60/5 = 12; 60/12 = 5
    • 15 * (1/4) = 15/60; 12 * (1/5) = 12/60; 5 * (1/12) = 5/60
    • 15/60 + 12/60 + 5/60
  7. 7 Solve the resulting equation. NOZ found; You can now add or subtract fractions. Don't forget to simplify your answer (if possible).
    • Example: 15/60 + 12/60 + 5/60 = 32/60 = 8/15

4 Working with mixed numbers

  1. 1 Convert each mixed number to an improper fraction. To do this, multiply the whole part mixed number to the denominator and add it to the numerator - this will be the numerator of the improper fraction. Convert the whole number to a fraction too (just put 1 in the denominator).
    • Example: 8 + 2 1/4 + 2/3
    • 8 = 8/1
    • 2 1/4, 2 * 4 + 1 = 8 + 1 = 9; 9/4
    • Rewritten equation: 8/1 + 9/4 + 2/3
  2. 2 Find the lowest common denominator. Calculate the NVA using any method described in the previous sections. For this example, we will use the "listing multiples" method, in which multiples of each denominator are written down and the NOC is calculated based on them.
    • Note that you don't need to list multiples for 1 , since any number multiplied by 1 , equal to itself; in other words, every number is a multiple of 1 .
    • Example: 4 * 1 = 4; 4 * 2 = 8; 4 * 3 = 12 ; 4 * 4 = 16; etc.
    • 3 * 1 = 3; 3 * 2 = 6; 3 * 3 = 9; 3 * 4 = 12 ; etc.
    • NOZ = 12
  3. 3 Rewrite the original equation. Multiply the numerators and denominators of the original fractions by a number equal to the quotient of dividing the NZ by the corresponding denominator.
    • For example: (12/12) * (8/1) = 96/12; (3/3) * (9/4) = 27/12; (4/4) * (2/3) = 8/12
    • 96/12 + 27/12 + 8/12
  4. 4 Solve the equation. NOZ found; You can now add or subtract fractions. Don't forget to simplify your answer (if possible).
    • Example: 96/12 + 27/12 + 8/12 = 131/12 = 10 11/12

What you will need

  • Pencil
  • Paper
  • Calculator (optional)

To reduce fractions to the least common denominator, you need to: 1) find the least common multiple of the denominators of the given fractions, it will be the least common denominator. 2) find an additional factor for each fraction by dividing the new denominator by the denominator of each fraction. 3) multiply the numerator and denominator of each fraction by its additional factor.

Examples. Reduce the following fractions to their lowest common denominator.

We find the least common multiple of the denominators: LCM(5; 4) = 20, since 20 is the smallest number that is divisible by both 5 and 4. Find for the 1st fraction an additional factor 4 (20 : 5=4). For the 2nd fraction the additional factor is 5 (20 : 4=5). We multiply the numerator and denominator of the 1st fraction by 4, and the numerator and denominator of the 2nd fraction by 5. We have reduced these fractions to the lowest common denominator ( 20 ).

The lowest common denominator of these fractions is the number 8, since 8 is divisible by 4 and itself. There will be no additional factor for the 1st fraction (or we can say that it is equal to one), for the 2nd fraction the additional factor is 2 (8 : 4=2). We multiply the numerator and denominator of the 2nd fraction by 2. We have reduced these fractions to the lowest common denominator ( 8 ).

These fractions are not irreducible.

Let's reduce the 1st fraction by 4, and reduce the 2nd fraction by 2. ( see examples on reducing ordinary fractions: Sitemap → 5.4.2. Examples of reducing common fractions). Find the LOC(16 ; 20)=2 4 · 5=16· 5=80. The additional multiplier for the 1st fraction is 5 (80 : 16=5). The additional factor for the 2nd fraction is 4 (80 : 20=4). We multiply the numerator and denominator of the 1st fraction by 5, and the numerator and denominator of the 2nd fraction by 4. We have reduced these fractions to the lowest common denominator ( 80 ).

We find the lowest common denominator NCD(5 ; 6 and 15)=NOK(5 ; 6 and 15)=30. The additional factor to the 1st fraction is 6 (30 : 5=6), the additional factor to the 2nd fraction is 5 (30 : 6=5), the additional factor to the 3rd fraction is 2 (30 : 15=2). We multiply the numerator and denominator of the 1st fraction by 6, the numerator and denominator of the 2nd fraction by 5, the numerator and denominator of the 3rd fraction by 2. We have reduced these fractions to the lowest common denominator ( 30 ).

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