Without knowing how to reduce a fraction and having a stable skill in solving such examples, it is very difficult to study algebra in school. The further you go, the more new information is superimposed on the basic knowledge about reducing ordinary fractions. First, powers appear, then factors, which later become polynomials.
How can you avoid getting confused here? Thoroughly consolidate skills in previous topics and gradually prepare for knowledge of how to reduce a fraction, which becomes more complex from year to year.
Basic knowledge
Without them, you will not be able to cope with tasks of any level. To understand, you need to understand two simple points. First: you can only reduce factors. This nuance turns out to be very important when polynomials appear in the numerator or denominator. Then you need to clearly distinguish where the multiplier is and where the addend is.
The second point says that any number can be represented in the form of factors. Moreover, the result of reduction is a fraction whose numerator and denominator can no longer be reduced.
Rules for reducing common fractions
First, you should check whether the numerator is divisible by the denominator or vice versa. Then it is precisely this number that needs to be reduced. This is the simplest option.
The second is the analysis of the appearance of numbers. If both end in one or more zeros, then they can be shortened by 10, 100 or a thousand. Here you can notice whether the numbers are even. If yes, then you can safely cut it by two.
The third rule for reducing a fraction is to factor the numerator and denominator into prime factors. At this time, you need to actively use all your knowledge about the signs of divisibility of numbers. After this decomposition, all that remains is to find all the repeating ones, multiply them and reduce them by the resulting number.
What if there is an algebraic expression in a fraction?
This is where the first difficulties appear. Because this is where terms appear that can be identical to factors. I really want to reduce them, but I can’t. Before you can reduce an algebraic fraction, it must be converted so that it has factors.
To do this, you will need to perform several steps. You may need to go through all of them, or maybe the first one will provide a suitable option.
Check whether the numerator and denominator or any expression in them differ by sign. In this case, you just need to put minus one out of brackets. This produces equal factors that can be reduced.
See if it is possible to remove the common factor from the polynomial out of brackets. Perhaps this will result in a parenthesis, which can also be shortened, or it will be a removed monomial.
Try to group the monomials in order to then add a common factor to them. After this, it may turn out that there will be factors that can be reduced, or again the bracketing of common elements will be repeated.
Try to consider abbreviated multiplication formulas in writing. With their help, you can easily convert polynomials into factors.
Sequence of operations with fractions with powers
In order to easily understand the question of how to reduce a fraction with powers, you need to firmly remember the basic operations with them. The first of these is related to the multiplication of powers. In this case, if the bases are the same, the indicators must be added.
The second is division. Again, for those that have the same reasons, the indicators will need to be subtracted. Moreover, you need to subtract from the number that is in the dividend, and not vice versa.
The third is exponentiation. In this situation, the indicators are multiplied.
Successful reduction will also require the ability to reduce powers to equal bases. That is, to see that four is two squared. Or 27 - the cube of three. Because reducing 9 squared and 3 cubed is difficult. But if we transform the first expression as (3 2) 2, then the reduction will be successful.
If we need to divide 497 by 4, then when dividing we will see that 497 is not evenly divisible by 4, i.e. the remainder of the division remains. In such cases it is said that it is completed division with remainder, and the solution is written as follows:
497: 4 = 124 (1 remainder).
The division components on the left side of the equality are called the same as in division without a remainder: 497 - dividend, 4 - divider. The result of division when divided with a remainder is called incomplete private. In our case, this is the number 124. And finally, the last component, which is not in ordinary division, is remainder. In cases where there is no remainder, one number is said to be divided by another without a trace, or completely. It is believed that with such a division the remainder is zero. In our case, the remainder is 1.
The remainder is always less than the divisor.
Division can be checked by multiplication. If, for example, there is an equality 64: 32 = 2, then the check can be done like this: 64 = 32 * 2.
Often in cases where division with a remainder is performed, it is convenient to use the equality
a = b * n + r,
where a is the dividend, b is the divisor, n is the partial quotient, r is the remainder.
The quotient of natural numbers can be written as a fraction.
The numerator of a fraction is the dividend, and the denominator is the divisor.
Since the numerator of a fraction is the dividend and the denominator is the divisor, believe that the line of a fraction means the action of division. Sometimes it is convenient to write division as a fraction without using the ":" sign.
The quotient of the division of natural numbers m and n can be written as a fraction \(\frac(m)(n)\), where the numerator m is the dividend, and the denominator n is the divisor:
\(m:n = \frac(m)(n)\)
The following rules are true:
To get the fraction \(\frac(m)(n)\), you need to divide the unit into n equal parts (shares) and take m such parts.
To get the fraction \(\frac(m)(n)\), you need to divide the number m by the number n.
To find a part of a whole, you need to divide the number corresponding to the whole by the denominator and multiply the result by the numerator of the fraction that expresses this part.
To find a whole from its part, you need to divide the number corresponding to this part by the numerator and multiply the result by the denominator of the fraction that expresses this part.
If both the numerator and denominator of a fraction are multiplied by the same number (except zero), the value of the fraction will not change:
\(\large \frac(a)(b) = \frac(a \cdot n)(b \cdot n) \)
If both the numerator and denominator of a fraction are divided by the same number (except zero), the value of the fraction will not change:
\(\large \frac(a)(b) = \frac(a: m)(b: m) \)
This property is called main property of a fraction.
The last two transformations are called reducing a fraction.
If fractions need to be represented as fractions with the same denominator, then this action is called reducing fractions to a common denominator.
Proper and improper fractions. Mixed numbers
You already know that a fraction can be obtained by dividing a whole into equal parts and taking several such parts. For example, the fraction \(\frac(3)(4)\) means three-quarters of one. In many of the problems in the previous paragraph, fractions were used to represent parts of a whole. Common sense dictates that the part should always be less than the whole, but what about fractions such as \(\frac(5)(5)\) or \(\frac(8)(5)\)? It is clear that this is no longer part of the unit. This is probably why fractions whose numerator is greater than or equal to the denominator are called improper fractions. The remaining fractions, i.e. fractions whose numerator is less than the denominator, are called correct fractions.
As you know, any common fraction, both proper and improper, can be thought of as the result of dividing the numerator by the denominator. Therefore, in mathematics, unlike ordinary language, the term “improper fraction” does not mean that we did something wrong, but only that the numerator of this fraction is greater than or equal to the denominator.
If a number consists of an integer part and a fraction, then such fractions are called mixed.
For example:
\(5:3 = 1\frac(2)(3) \) : 1 is the integer part, and \(\frac(2)(3) \) is the fractional part.
If the numerator of the fraction \(\frac(a)(b)\) is divisible by a natural number n, then in order to divide this fraction by n, its numerator must be divided by this number:
\(\large \frac(a)(b) : n = \frac(a:n)(b) \)
If the numerator of the fraction \(\frac(a)(b)\) is not divisible by a natural number n, then to divide this fraction by n, you need to multiply its denominator by this number:
\(\large \frac(a)(b) : n = \frac(a)(bn) \)
Note that the second rule is also true when the numerator is divisible by n. Therefore, we can use it when it is difficult to determine at first glance whether the numerator of a fraction is divisible by n or not.
Actions with fractions. Adding fractions.
You can perform arithmetic operations with fractional numbers, just like with natural numbers. Let's look at adding fractions first. It's easy to add fractions with like denominators. Let us find, for example, the sum of \(\frac(2)(7)\) and \(\frac(3)(7)\). It is easy to understand that \(\frac(2)(7) + \frac(2)(7) = \frac(5)(7) \)
To add fractions with the same denominators, you need to add their numerators and leave the denominator the same.
Using letters, the rule for adding fractions with like denominators can be written as follows:
\(\large \frac(a)(c) + \frac(b)(c) = \frac(a+b)(c) \)
If you need to add fractions with different denominators, they must first be reduced to a common denominator. For example:
\(\large \frac(2)(3)+\frac(4)(5) = \frac(2\cdot 5)(3\cdot 5)+\frac(4\cdot 3)(5\cdot 3 ) = \frac(10)(15)+\frac(12)(15) = \frac(10+12)(15) = \frac(22)(15) \)
For fractions, as for natural numbers, the commutative and associative properties of addition are valid.
Adding mixed fractions
Notations such as \(2\frac(2)(3)\) are called mixed fractions. In this case, the number 2 is called whole part mixed fraction, and the number \(\frac(2)(3)\) is its fractional part. The entry \(2\frac(2)(3)\) is read as follows: “two and two thirds.”
When dividing the number 8 by the number 3, you can get two answers: \(\frac(8)(3)\) and \(2\frac(2)(3)\). They express the same fractional number, i.e. \(\frac(8)(3) = 2 \frac(2)(3)\)
Thus, the improper fraction \(\frac(8)(3)\) is represented as a mixed fraction \(2\frac(2)(3)\). In such cases they say that from an improper fraction highlighted the whole part.
Subtracting fractions (fractional numbers)
Subtraction of fractional numbers, like natural numbers, is determined on the basis of the action of addition: subtracting another from one number means finding a number that, when added to the second, gives the first. For example:
\(\frac(8)(9)-\frac(1)(9) = \frac(7)(9) \) since \(\frac(7)(9)+\frac(1)(9 ) = \frac(8)(9)\)
The rule for subtracting fractions with like denominators is similar to the rule for adding such fractions:
To find the difference between fractions with the same denominators, you need to subtract the numerator of the second from the numerator of the first fraction, and leave the denominator the same.
Using letters, this rule is written like this:
\(\large \frac(a)(c)-\frac(b)(c) = \frac(a-b)(c) \)
Multiplying fractions
To multiply a fraction by a fraction, you need to multiply their numerators and denominators and write the first product as the numerator, and the second as the denominator.
Using letters, the rule for multiplying fractions can be written as follows:
\(\large \frac(a)(b) \cdot \frac(c)(d) = \frac(a \cdot c)(b \cdot d) \)
Using the formulated rule, you can multiply a fraction by a natural number, by a mixed fraction, and also multiply mixed fractions. To do this, you need to write a natural number as a fraction with a denominator of 1, a mixed fraction - as an improper fraction.
The result of multiplication should be simplified (if possible) by reducing the fraction and isolating the whole part of the improper fraction.
For fractions, as for natural numbers, the commutative and combinative properties of multiplication, as well as the distributive property of multiplication relative to addition, are valid.
Division of fractions
Let's take the fraction \(\frac(2)(3)\) and “flip” it, swapping the numerator and denominator. We get the fraction \(\frac(3)(2)\). This fraction is called reverse fractions \(\frac(2)(3)\).
If we now “reverse” the fraction \(\frac(3)(2)\), we will get the original fraction \(\frac(2)(3)\). Therefore, fractions such as \(\frac(2)(3)\) and \(\frac(3)(2)\) are called mutually inverse.
For example, the fractions \(\frac(6)(5) \) and \(\frac(5)(6) \), \(\frac(7)(18) \) and \(\frac (18)(7)\).
Using letters, reciprocal fractions can be written as follows: \(\frac(a)(b) \) and \(\frac(b)(a) \)
It is clear that the product of reciprocal fractions is equal to 1. For example: \(\frac(2)(3) \cdot \frac(3)(2) =1 \)
Using reciprocal fractions, you can reduce division of fractions to multiplication.
The rule for dividing a fraction by a fraction is:
To divide one fraction by another, you need to multiply the dividend by the reciprocal of the divisor.
Using letters, the rule for dividing fractions can be written as follows:
\(\large \frac(a)(b) : \frac(c)(d) = \frac(a)(b) \cdot \frac(d)(c) \)
If the dividend or divisor is a natural number or a mixed fraction, then in order to use the rule for dividing fractions, it must first be represented as an improper fraction.
Children at school learn the rules of reducing fractions in 6th grade. In this article, we will first tell you what this action means, then we will explain how to convert a reducible fraction into an irreducible fraction. The next point will be the rules for reducing fractions, and then we will gradually get to the examples.
What does it mean to "reduce a fraction"?
So, we all know that ordinary fractions are divided into two groups: reducible and irreducible. Already by the names you can understand that those that are contractible are contracted, and those that are irreducible are not contracted.
- To reduce a fraction means to divide its denominator and numerator by their (other than one) positive divisor. The result, of course, is a new fraction with a smaller denominator and numerator. The resulting fraction will be equal to the original fraction.
It is worth noting that in mathematics books with the task “reduce a fraction,” this means that you need to reduce the original fraction to this irreducible form. In simple terms, dividing the denominator and numerator by their greatest common divisor is a reduction.
How to reduce a fraction. Rules for reducing fractions (grade 6)
So there are only two rules here.
- The first rule of reducing fractions is to first find the greatest common factor of the denominator and numerator of your fraction.
- The second rule: divide the denominator and numerator by the greatest common divisor, ultimately obtaining an irreducible fraction.
How to reduce an improper fraction?
The rules for reducing fractions are identical to the rules for reducing improper fractions.
In order to reduce an improper fraction, you will first need to factor the denominator and numerator into prime factors, and only then reduce the common factors.
Reducing mixed fractions
The rules for reducing fractions also apply to reducing mixed fractions. There is only a small difference: we can not touch the whole part, but reduce the fraction or convert the mixed fraction into an improper fraction, then reduce it and again convert it into a proper fraction.
There are two ways to reduce mixed fractions.
First: write the fractional part into prime factors and then leave the whole part alone.
The second way: first convert it to an improper fraction, write it into ordinary factors, then reduce the fraction. Convert the already obtained improper fraction into a proper fraction.
Examples can be seen in the photo above.
We really hope that we were able to help you and your children. After all, they are often inattentive in class, so they have to study more intensively at home on their own.
Reducing fractions is necessary in order to reduce the fraction to a simpler form, for example, in the answer obtained as a result of solving an expression.
Reducing fractions, definition and formula.
What is reducing fractions? What does it mean to reduce a fraction?
Definition:
Reducing Fractions- this is the division of a fraction's numerator and denominator by the same positive number not equal to zero and one. As a result of the reduction, a fraction with a smaller numerator and denominator is obtained, equal to the previous fraction according to.
Formula for reducing fractions basic properties of rational numbers.
\(\frac(p \times n)(q \times n)=\frac(p)(q)\)
Let's look at an example:
Reduce the fraction \(\frac(9)(15)\)
Solution:
We can factor a fraction into prime factors and cancel common factors.
\(\frac(9)(15)=\frac(3 \times 3)(5 \times 3)=\frac(3)(5) \times \color(red) (\frac(3)(3) )=\frac(3)(5) \times 1=\frac(3)(5)\)
Answer: after reduction we got the fraction \(\frac(3)(5)\). According to the basic property of rational numbers, the original and resulting fractions are equal.
\(\frac(9)(15)=\frac(3)(5)\)
How to reduce fractions? Reducing a fraction to its irreducible form.
To get an irreducible fraction as a result, we need find the greatest common divisor (GCD) for the numerator and denominator of the fraction.
There are several ways to find GCD; in the example we will use the decomposition of numbers into prime factors.
Get the irreducible fraction \(\frac(48)(136)\).
Solution:
Let's find GCD(48, 136). Let's write the numbers 48 and 136 into prime factors.
48=2⋅2⋅2⋅2⋅3
136=2⋅2⋅2⋅17
GCD(48, 136)= 2⋅2⋅2=6
\(\frac(48)(136)=\frac(\color(red) (2 \times 2 \times 2) \times 2 \times 3)(\color(red) (2 \times 2 \times 2) \times 17)=\frac(\color(red) (6) \times 2 \times 3)(\color(red) (6) \times 17)=\frac(2 \times 3)(17)=\ frac(6)(17)\)
The rule for reducing a fraction to an irreducible form.
- You need to find the greatest common divisor for the numerator and denominator.
- You need to divide the numerator and denominator by the greatest common divisor to obtain an irreducible fraction as a result of division.
Example:
Reduce the fraction \(\frac(152)(168)\).
Solution:
Let's find GCD(152, 168). Let's write the numbers 152 and 168 into prime factors.
152=2⋅2⋅2⋅19
168=2⋅2⋅2⋅3⋅7
GCD(152, 168)= 2⋅2⋅2=6
\(\frac(152)(168)=\frac(\color(red) (6) \times 19)(\color(red) (6) \times 21)=\frac(19)(21)\)
Answer: \(\frac(19)(21)\) is an irreducible fraction.
Reducing improper fractions.
How to reduce an improper fraction?
The rules for reducing fractions are the same for proper and improper fractions.
Let's look at an example:
Reduce the improper fraction \(\frac(44)(32)\).
Solution:
Let's write the numerator and denominator into simple factors. And then we’ll reduce the common factors.
\(\frac(44)(32)=\frac(\color(red) (2 \times 2 ) \times 11)(\color(red) (2 \times 2 ) \times 2 \times 2 \times 2 )=\frac(11)(2 \times 2 \times 2)=\frac(11)(8)\)
Reducing mixed fractions.
Mixed fractions follow the same rules as ordinary fractions. The only difference is that we can do not touch the whole part, but reduce the fractional part or Convert a mixed fraction to an improper fraction, reduce it and convert it back to a proper fraction.
Let's look at an example:
Cancel the mixed fraction \(2\frac(30)(45)\).
Solution:
Let's solve it in two ways:
First way:
Let's write the fractional part into simple factors, but we won't touch the whole part.
\(2\frac(30)(45)=2\frac(2 \times \color(red) (5 \times 3))(3 \times \color(red) (5 \times 3))=2\ frac(2)(3)\)
Second way:
Let's first convert it to an improper fraction, and then write it into prime factors and reduce. Let's convert the resulting improper fraction into a proper fraction.
\(2\frac(30)(45)=\frac(45 \times 2 + 30)(45)=\frac(120)(45)=\frac(2 \times \color(red) (5 \times 3) \times 2 \times 2)(3 \times \color(red) (3 \times 5))=\frac(2 \times 2 \times 2)(3)=\frac(8)(3)= 2\frac(2)(3)\)
Related questions:
Can you reduce fractions when adding or subtracting?
Answer: no, you must first add or subtract fractions according to the rules, and only then reduce them. Let's look at an example:
Evaluate the expression \(\frac(50+20-10)(20)\) .
Solution:
They often make the mistake of reducing the same numbers in the numerator and denominator, in our case the number 20, but they cannot be reduced until you have completed the addition and subtraction.
\(\frac(50+\color(red) (20)-10)(\color(red) (20))=\frac(60)(20)=\frac(3 \times 20)(20)= \frac(3)(1)=3\)
What numbers can you reduce a fraction by?
Answer: You can reduce a fraction by the greatest common factor or the common divisor of the numerator and denominator. For example, the fraction \(\frac(100)(150)\).
Let's write the numbers 100 and 150 into prime factors.
100=2⋅2⋅5⋅5
150=2⋅5⋅5⋅3
The greatest common divisor will be the number gcd(100, 150)= 2⋅5⋅5=50
\(\frac(100)(150)=\frac(2 \times 50)(3 \times 50)=\frac(2)(3)\)
We got the irreducible fraction \(\frac(2)(3)\).
But it is not necessary to always divide by gcd; an irreducible fraction is not always needed; you can reduce the fraction by a simple divisor of the numerator and denominator. For example, the number 100 and 150 have a common divisor of 2. Let's reduce the fraction \(\frac(100)(150)\) by 2.
\(\frac(100)(150)=\frac(2 \times 50)(2 \times 75)=\frac(50)(75)\)
We got the reducible fraction \(\frac(50)(75)\).
What fractions can be reduced?
Answer: You can reduce fractions in which the numerator and denominator have a common divisor. For example, the fraction \(\frac(4)(8)\). The number 4 and 8 have a number by which they are both divisible - the number 2. Therefore, such a fraction can be reduced by the number 2.
Example:
Compare the two fractions \(\frac(2)(3)\) and \(\frac(8)(12)\).
These two fractions are equal. Let's take a closer look at the fraction \(\frac(8)(12)\):
\(\frac(8)(12)=\frac(2 \times 4)(3 \times 4)=\frac(2)(3) \times \frac(4)(4)=\frac(2) (3)\times 1=\frac(2)(3)\)
From here we get, \(\frac(8)(12)=\frac(2)(3)\)
Two fractions are equal if and only if one of them is obtained by reducing the other fraction by the common factor of the numerator and denominator.
Example:
If possible, reduce the following fractions: a) \(\frac(90)(65)\) b) \(\frac(27)(63)\) c) \(\frac(17)(100)\) d) \(\frac(100)(250)\)
Solution:
a) \(\frac(90)(65)=\frac(2 \times \color(red) (5) \times 3 \times 3)(\color(red) (5) \times 13)=\frac (2 \times 3 \times 3)(13)=\frac(18)(13)\)
b) \(\frac(27)(63)=\frac(\color(red) (3 \times 3) \times 3)(\color(red) (3 \times 3) \times 7)=\frac (3)(7)\)
c) \(\frac(17)(100)\) irreducible fraction
d) \(\frac(100)(250)=\frac(\color(red) (2 \times 5 \times 5) \times 2)(\color(red) (2 \times 5 \times 5) \ times 5)=\frac(2)(5)\)
Many students make the same mistakes when working with fractions. And all because they forget the basic rules arithmetic. Today we will repeat these rules on specific tasks that I give in my classes.
Here is the task that I offer to everyone who is preparing for the Unified State Exam in mathematics:
Task. A porpoise eats 150 grams of food per day. But she grew up and began to eat 20% more. How many grams of feed does the pig eat now?
Wrong decision. This is a percentage problem that boils down to the equation:
Many (very many) reduce the number 100 in the numerator and denominator of a fraction:
This is the mistake my student made right on the day of writing this article. Numbers that have been truncated are marked in red.
Needless to say, the answer was wrong. Judge for yourself: the pig ate 150 grams, but began to eat 3150 grams. The increase is not 20%, but 21 times, i.e. by 2000%.
To avoid such misunderstandings, remember the basic rule:
Only multipliers can be reduced. The terms cannot be reduced!
Thus, the correct solution to the previous problem looks like this:
Numbers that are abbreviated in the numerator and denominator are marked in red. As you can see, the numerator is a product, the denominator is an ordinary number. Therefore, the reduction is completely legal.
Working with proportions
Another problem area is proportions. Especially when the variable is on both sides. For example:
Task. Solve the equation:
Wrong solution - some people are literally itching to shorten everything by m:
Reduced variables are shown in red. The expression 1/4 = 1/5 turns out to be complete nonsense, these numbers are never equal.
And now - the right decision. Essentially it's ordinary linear equation. It can be solved either by moving all elements to one side, or by the basic property of proportion:
Many readers will object: “Where is the mistake in the first solution?” Well, let's find out. Let's remember the rule for working with equations:
Any equation can be divided and multiplied by any number, non-zero.
Did you miss the trick? You can only divide by numbers non-zero. In particular, you can divide by a variable m only if m != 0. But what if m = 0? Let's substitute and check:
We received the correct numerical equality, i.e. m = 0 is the root of the equation. For the remaining m != 0 we obtain an expression of the form 1/4 = 1/5, which is naturally incorrect. Thus, there are no non-zero roots.
Conclusions: putting it all together
So, to solve fractional rational equations, remember three rules:
- Only multipliers can be reduced. Addends are not allowed. Therefore, learn to factor the numerator and denominator;
- The main property of proportion: the product of the extreme elements is equal to the product of the middle ones;
- Equations can only be multiplied and divided by numbers k other than zero. The case k = 0 must be checked separately.
Remember these rules and don't make mistakes.
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