9 addition and subtraction of ordinary fractions. Subtracting fractions with different denominators. Adding and subtracting ordinary fractions
As we know from mathematics, a fractional number consists of a numerator and a denominator. The numerator is at the top and the denominator is at the bottom.
It is quite simple to perform mathematical operations of adding or subtracting fractional quantities with the same denominator. You just need to be able to add or subtract the numbers in the numerator (above), and the same bottom number remains unchanged.
For example, let's take the fractional number 7/9, here:
- the number “seven” on top is the numerator;
- the number “nine” below is the denominator.
Example 1. Addition:
5/49 + 4/49 = (5+4) / 49 =9/49.
Example 2. Subtraction:
6/35−3/35 = (6−3) / 35 = 3/35.
Subtracting simple fractional values that have different denominators
To perform the mathematical operation of subtracting quantities that have different denominators, you must first reduce them to a single denominator. When performing this task, you must adhere to the rule that this common denominator should be the least of all possible options.
Example 3
Given two simple quantities With different denominators(lower numbers): 7/8 and 2/9.
It is necessary to subtract the second from the first value.
The solution consists of several steps:
1. Find the common lower number, i.e. something that is divisible by both the lower value of the first fraction and the second. This will be the number 72, since it is a multiple of the numbers eight and nine.
2. The bottom digit of each fraction has increased:
- the number “eight” in the fraction 7/8 has increased ninefold - 8*9=72;
- the number “nine” in the fraction 2/9 has increased eightfold - 9*8=72.
3. If the denominator (lower digit) has changed, then the numerator (upper digit) must also change. According to the existing mathematical rule, the top number must be increased by exactly the same amount as the bottom one. That is:
- the numerator “seven” in the first fraction (7/8) is multiplied by the number “nine” - 7*9=63;
- We multiply the numerator “two” in the second fraction (2/9) by the number “eight” - 2*8=16.
4. As a result of our actions, we got two new quantities, which, however, are identical to the original ones.
- first: 7/8 = 7*9 / 8*9 = 63/72;
- second: 2/9 = 2*8 / 9*8 = 16/72.
5. Now it is possible to subtract one fractional number from another:
7/8−2/9 = 63/72−16/72 =?
6. Carrying out this action, we return to the topic of subtracting fractions with the same lower digits (denominators). This means that the subtraction action will be carried out on top, in the numerator, and the bottom digit will be transferred without changes.
63/72−16/72 = (63−16) / 72 = 47/72.
7/8−2/9 = 47/72.
Example 4
Let's complicate the problem by taking several fractions with different but multiple numbers at the bottom to solve.
The values given are: 5/6; 1/3; 1/12; 7/24.
They must be taken away from each other in this sequence.
1. We bring the fractions using the above method to a common denominator, which will be the number “24”:
- 5/6 = 5*4 / 6*4 = 20/24;
- 1/3 = 1*8 / 3*8 = 8/24;
- 1/12 = 1*2 / 12*2 = 2/24.
7/24 - we leave this last value unchanged, since the denominator is total number"24".
2. We subtract all quantities:
20/24−8/2−2/24−7/24 = (20−8−2−7)/24 = 3/24.
3. Since the numerator and denominator of the resulting fraction are divisible by one number, they can be reduced by dividing by the number “three”:
3:3 / 24:3 = 1/8.
4. We write the answer like this:
5/6−1/3−1/12−7/24 = 1/8.
Example 5
Three fractions with non-multiple denominators are given: 3/4; 2/7; 1/13.
You need to find the difference.
1. We bring the first two numbers to a common denominator, it will be the number “28”:
- ¾ = 3*7 / 4*7 = 21/28;
- 2/7 = 2*4 / 7*4 = 8/28.
2. Subtract the first two fractions from each other:
¾−2/7 = 21/28−8/28 = (21−8) / 28 = 13/28.
3. Subtract the third given fraction from the resulting value:
4. We bring the numbers to a common denominator. If it is not possible to select the same denominator more the easy way, then you just need to perform the actions by multiplying all the denominators by each other in sequence, not forgetting to increase the value of the numerator by the same figure. In this example we do this:
- 13/28 = 13*13 / 28*13 = 169/364, where 13 is the lower digit of 5/13;
- 5/13 = 5*28 / 13*28 = 140/364, where 28 is the lower number from 13/28.
5. Subtract the resulting fractions:
13/28−5/13 = 169/364−140/364 = (169−140) / 364 = 29/364.
Answer: ¾−2/7−5/13 = 29/364.
Mixed fractions
In the examples discussed above, only proper fractions were used.
As an example:
- 8/9 is a proper fraction;
- 9/8 is incorrect.
It is impossible to turn an improper fraction into a proper fraction, but it is possible to turn it into mixed. Why do you divide the top number (numerator) by the bottom (denominator) and get a number with a remainder? The integer resulting from division is written down like this, the remainder is written in the numerator at the top, and the denominator at the bottom remains the same. To make it clearer, let's consider specific example:
Example 6
We translate improper fraction 9/8 is correct.
To do this, divide the number “nine” by “eight”, resulting in a mixed fraction with an integer and a remainder:
9: 8 = 1 and 1/8 (this can be written differently as 1+1/8), where:
- number 1 is the integer resulting from division;
- another number 1 is the remainder;
- the number 8 is the denominator, which remains unchanged.
An integer is also called a natural number.
The remainder and denominator are a new, but proper fraction.
When writing the number 1, it is written before the proper fraction 1/8.
Subtracting mixed numbers with different denominators
From the above, we give the definition of a mixed fractional number: "Mixed number - this is a quantity that is equal to the sum of a whole number and a proper ordinary fraction. In this case, the whole part is called natural number, and the number that is left is his fractional part».
Example 7
Given: two mixed fractional quantities consisting of a whole number and a proper fraction:
- the first value is 9 and 4/7, that is (9+4/7);
- the second value is 3 and 5/21, that is (3+5/21).
It is required to find the difference between these quantities.
1. To subtract 3+5/21 from 9+4/7, you must first subtract integer values from each other:
4/7−5/21 = 4*3 / 7*3−5/21 =12/21−5/21 = (12−5) / 21 = 7/21.
3. The resulting result of the difference between two mixed numbers will consist of the natural (integer) number 6 and the proper fraction 7/21 = 1/3:
(9 + 4/7) - (3 + 5/21) = 6 + 1/3.
Mathematicians from all countries have agreed that the “+” sign when writing mixed quantities can be omitted and only the whole number left before the fraction without any sign.
Instructions
It is customary to separate ordinary and decimal fractions, acquaintance with which begins in high school. There is currently no area of knowledge where this is not applied. Even in we say the first 17th century, and all at once, which means 1600-1625. You also often have to deal with elementary actions on, as well as their transformation from one type to another.
Reducing fractions to a common denominator is perhaps the most important operation on. This is the basis for absolutely all calculations. So, let's say there are two fractions a/b and c/d. Then, in order to bring them to a common denominator, you need to find the least common multiple (M) of the numbers b and d, and then multiply the numerator of the first fractions by (M/b), and the second numerator by (M/d).
Comparing fractions is another important task. In order to do this, give the given simple fractions to a common denominator and then compare the numerators, whose numerator is greater, that fraction and greater.
To perform addition or subtraction ordinary fractions, you need to bring them to a common denominator, and then perform the necessary mathematical calculations from these fractions. The denominator remains unchanged. Let's say you need to subtract c/d from a/b. To do this, you need to find the least common multiple of M numbers b and d, and then subtract the other from one numerator without changing the denominator: (a*(M/b)-(c*(M/d))/M
It is enough to simply multiply one fraction by another; to do this, simply multiply their numerators and denominators:
(a/b)*(c/d)=(a*c)/(b*d)To divide one fraction by another, you need to multiply the fraction of the dividend by the reciprocal fraction of the divisor. (a/b)/(c/d)=(a*d)/(b*c)
It is worth recalling that in order to obtain a reciprocal fraction, you need to swap the numerator and denominator.
Mixed fractions, just like simple fractions, can be subtracted. To take away mixed numbers fractions you need to know a few rules for subtraction. Let's study these rules with examples.
Subtracting mixed fractions with like denominators.
Let's consider an example with the condition that the integer being reduced and the fractional part are greater than the integer and fractional parts being subtracted, respectively. Under such conditions, subtraction occurs separately. We subtract the integer part from the whole part, and the fractional part from the fractional part.
Let's look at an example:
Subtract mixed fractions \(5\frac(3)(7)\) and \(1\frac(1)(7)\).
\(5\frac(3)(7)-1\frac(1)(7) = (5-1) + (\frac(3)(7)-\frac(1)(7)) = 4\ frac(2)(7)\)
The correctness of the subtraction is checked by addition. Let's check the subtraction:
\(4\frac(2)(7)+1\frac(1)(7) = (4 + 1) + (\frac(2)(7) + \frac(1)(7)) = 5\ frac(3)(7)\)
Let's consider an example with the condition when the fractional part of the minuend is less than the corresponding fractional part of the subtrahend. In this case, we borrow one from the whole in the minuend.
Let's look at an example:
Subtract mixed fractions \(6\frac(1)(4)\) and \(3\frac(3)(4)\).
The minuend \(6\frac(1)(4)\) has a smaller fractional part than the fractional part of the subtrahend \(3\frac(3)(4)\). That is, \(\frac(1)(4)< \frac{1}{3}\), поэтому сразу отнять мы не сможем. Займем у целой части у 6 единицу, а потом выполним вычитание. Единицу мы запишем как \(\frac{4}{4} = 1\)
\(\begin(align)&6\frac(1)(4)-3\frac(3)(4) = (6 + \frac(1)(4))-3\frac(3)(4) = (5 + \color(red) (1) + \frac(1)(4))-3\frac(3)(4) = (5 + \color(red) (\frac(4)(4)) + \frac(1)(4))-3\frac(3)(4) = (5 + \frac(5)(4))-3\frac(3)(4) = \\\\ &= 5\frac(5)(4)-3\frac(3)(4) = 2\frac(2)(4) = 2\frac(1)(4)\\\\ \end(align)\)
Next example:
\(7\frac(8)(19)-3 = 4\frac(8)(19)\)
Subtracting a mixed fraction from a whole number.
Example: \(3-1\frac(2)(5)\)
The minuend 3 does not have a fractional part, so we cannot immediately subtract. Let's borrow one from the whole part of 3, and then do the subtraction. We will write the unit as \(3 = 2 + 1 = 2 + \frac(5)(5) = 2\frac(5)(5)\)
\(3-1\frac(2)(5)= (2 + \color(red) (1))-1\frac(2)(5) = (2 + \color(red) (\frac(5 )(5)))-1\frac(2)(5) = 2\frac(5)(5)-1\frac(2)(5) = 1\frac(3)(5)\)
Subtracting mixed fractions with unlike denominators.
Let's consider an example with the condition that the fractional parts of the minuend and subtrahend have different denominators. You need to bring it to a common denominator, and then perform subtraction.
Subtract two mixed fractions with different denominators \(2\frac(2)(3)\) and \(1\frac(1)(4)\).
The common denominator will be the number 12.
\(2\frac(2)(3)-1\frac(1)(4) = 2\frac(2 \times \color(red) (4))(3 \times \color(red) (4) )-1\frac(1 \times \color(red) (3))(4 \times \color(red) (3)) = 2\frac(8)(12)-1\frac(3)(12 ) = 1\frac(5)(12)\)
Questions on the topic:
How to subtract mixed fractions? How to solve mixed fractions?
Answer: you need to decide what type the expression belongs to and apply the solution algorithm based on the type of expression. From the integer part we subtract the integer, from the fractional part we subtract the fractional part.
How to subtract a fraction from a whole number? How to subtract a fraction from a whole number?
Answer: you need to take a unit from an integer and write this unit as a fraction
\(4 = 3 + 1 = 3 + \frac(7)(7) = 3\frac(7)(7)\),
and then subtract the whole from the whole, subtract the fractional part from the fractional part. Example:
\(4-2\frac(3)(7) = (3 + \color(red) (1))-2\frac(3)(7) = (3 + \color(red) (\frac(7 )(7)))-2\frac(3)(7) = 3\frac(7)(7)-2\frac(3)(7) = 1\frac(4)(7)\)
Example #1:
Subtract a proper fraction from one: a) \(1-\frac(8)(33)\) b) \(1-\frac(6)(7)\)
Solution:
a) Let's imagine one as a fraction with a denominator 33. We get \(1 = \frac(33)(33)\)
\(1-\frac(8)(33) = \frac(33)(33)-\frac(8)(33) = \frac(25)(33)\)
b) Let's imagine one as a fraction with a denominator 7. We get \(1 = \frac(7)(7)\)
\(1-\frac(6)(7) = \frac(7)(7)-\frac(6)(7) = \frac(7-6)(7) = \frac(1)(7) \)
Example #2:
Perform a subtraction mixed fraction from an integer: a) \(21-10\frac(4)(5)\) b) \(2-1\frac(1)(3)\)
Solution:
a) Let’s borrow 21 units from the integer and write it like this \(21 = 20 + 1 = 20 + \frac(5)(5) = 20\frac(5)(5)\)
\(21-10\frac(4)(5) = (20 + 1)-10\frac(4)(5) = (20 + \frac(5)(5))-10\frac(4)( 5) = 20\frac(5)(5)-10\frac(4)(5) = 10\frac(1)(5)\\\\\)
b) Let's take one from the integer 2 and write it like this \(2 = 1 + 1 = 1 + \frac(3)(3) = 1\frac(3)(3)\)
\(2-1\frac(1)(3) = (1 + 1)-1\frac(1)(3) = (1 + \frac(3)(3))-1\frac(1)( 3) = 1\frac(3)(3)-1\frac(1)(3) = \frac(2)(3)\\\\\)
Example #3:
Subtract an integer from a mixed fraction: a) \(15\frac(6)(17)-4\) b) \(23\frac(1)(2)-12\)
a) \(15\frac(6)(17)-4 = 11\frac(6)(17)\)
b) \(23\frac(1)(2)-12 = 11\frac(1)(2)\)
Example #4:
Subtract a proper fraction from a mixed fraction: a) \(1\frac(4)(5)-\frac(4)(5)\)
\(1\frac(4)(5)-\frac(4)(5) = 1\\\\\)
Example #5:
Calculate \(5\frac(5)(16)-3\frac(3)(8)\)
\(\begin(align)&5\frac(5)(16)-3\frac(3)(8) = 5\frac(5)(16)-3\frac(3 \times \color(red) ( 2))(8 \times \color(red) (2)) = 5\frac(5)(16)-3\frac(6)(16) = (5 + \frac(5)(16))- 3\frac(6)(16) = (4 + \color(red) (1) + \frac(5)(16))-3\frac(6)(16) = \\\\ &= (4 + \color(red) (\frac(16)(16)) + \frac(5)(16))-3\frac(6)(16) = (4 + \color(red) (\frac(21 )(16)))-3\frac(3)(8) = 4\frac(21)(16)-3\frac(6)(16) = 1\frac(15)(16)\\\\ \end(align)\)
Adding and subtracting fractions with like denominators
Let's start by looking at the simplest example - adding and subtracting fractions with same denominators. In this case, you just need to perform operations with the numerators - add them or subtract them.
When adding and subtracting fractions with the same denominators, the denominator does not change!
The main thing is not to perform any addition or subtraction operations in the denominator, but some schoolchildren forget about this. To better understand this rule, let us resort to the principle of visualization, or speaking in simple words, let's look at a real-life example:
You have half an apple - that's ½ of the whole apple. They give you another half, that is, another ½. Obviously, now you have a whole apple (not counting the fact that it is cut :)). Therefore ½ + ½ = 1, and not something else like 2/4. Or this half is taken away from you: ½ - ½ = 0. In the case of subtraction with identical denominators, a special case arises altogether - when subtracting identical denominators, we get 0, but we cannot divide by 0, and this fraction will not make sense.
Let's give one last example:
Adding and subtracting fractions with different denominators
What to do if the denominators are different? To do this, we first need to reduce the fractions to the same denominator, and then act as I indicated above.
There are two ways to reduce a fraction to a common denominator. All methods use one rule - When multiplying the numerator and denominator by the same number, the fraction does not change .
There are two ways. The first is the simplest - the so-called “criss-cross”. It consists in the fact that we multiply the first fraction by the denominator of the second fraction (both the numerator and the denominator), and we multiply the second fraction by the denominator of the first (similarly, the numerator and denominator). After this, we proceed as in the case with identical denominators - now they are really the same!
The previous method is universal, but in most cases the denominators of fractions can be found least common multiple - a number that divides both the first denominator and the second, and the smallest one. In this method, you need to be able to see such LOCs, because their special search is quite capacious and inferior in speed to the “criss-cross” method. But in most cases, NOCs are quite visible if you keep your eyes open and practice enough.
I hope that you are now fluent in adding and subtracting fractions!
This lesson will cover addition and subtraction. algebraic fractions with different denominators. We already know how to add and subtract common fractions with different denominators. To do this, the fractions must be reduced to a common denominator. It turns out that algebraic fractions follow the same rules. At the same time, we already know how to reduce algebraic fractions to a common denominator. Adding and subtracting fractions with different denominators is one of the most important and difficult topics in the 8th grade course. Moreover, this topic will appear in many topics in the algebra course that you will study in the future. As part of the lesson, we will study the rules for adding and subtracting algebraic fractions with different denominators, and also analyze whole line typical examples.
Let's consider simplest example for ordinary fractions.
Example 1. Add fractions: .
Solution:
Let's remember the rule for adding fractions. To begin, fractions must be reduced to a common denominator. The common denominator for ordinary fractions is least common multiple(LCM) of the original denominators.
Definition
Least natural number, which is simultaneously divisible by the numbers and .
To find the LCM, you need to factor the denominators into prime factors, and then select all the prime factors that are included in the expansion of both denominators.
; . Then the LCM of numbers must include two twos and two threes: .
After finding the common denominator, you need to find an additional factor for each fraction (in fact, divide the common denominator by the denominator of the corresponding fraction).
Each fraction is then multiplied by the resulting additional factor. We get fractions with the same denominators, which we learned to add and subtract in previous lessons.
We get: 
.
Answer:.
Let us now consider the addition of algebraic fractions with different denominators. First, let's look at fractions whose denominators are numbers.
Example 2. Add fractions: .
Solution:
The solution algorithm is absolutely similar to the previous example. It is easy to find the common denominator of these fractions: and additional factors for each of them.
.
Answer:.
So, let's formulate algorithm for adding and subtracting algebraic fractions with different denominators:
1. Find the lowest common denominator of fractions.
2. Find additional factors for each of the fractions (by dividing the common denominator by the denominator of the given fraction).
3. Multiply the numerators by the corresponding additional factors.
4. Add or subtract fractions using the rules for adding and subtracting fractions with like denominators.
Let us now consider an example with fractions whose denominator contains literal expressions.
Example 3. Add fractions: .
Solution:
Since the letter expressions in both denominators are the same, you should find a common denominator for the numbers. The final common denominator will look like: . Thus, the solution to this example looks like:.
Answer:.
Example 4. Subtract fractions: .
Solution:
If you can’t “cheat” when choosing a common denominator (you can’t factor it or use abbreviated multiplication formulas), then you have to take the product of the denominators of both fractions as the common denominator.
Answer:.
In general, when deciding similar examples, the most difficult task is finding the common denominator.
Let's look at a more complex example.
Example 5. Simplify: .
Solution:
When finding a common denominator, you must first try to factor the denominators of the original fractions (to simplify the common denominator).
In this particular case:
Then it is easy to determine the common denominator: 
.
We determine additional factors and solve this example:
Answer:.
Now let's establish the rules for adding and subtracting fractions with different denominators.
Example 6. Simplify: .
Solution:
Answer:.
Example 7. Simplify: .
Solution:
.
Answer:.
Let us now consider an example in which not two, but three fractions are added (after all, the rules of addition and subtraction for more fractions remain the same).
Example 8. Simplify: .
