Mixed numbers, converting a mixed number to an improper fraction and vice versa. How to convert a mixed number to an improper fraction
has a higher numerator than the denominator. Such fractions are called incorrect.Remember!
An improper fraction has the numerator equal to or greater than the denominator. therefore improper fraction or equal to one or greater than one.
Any incorrect fraction is always more correct.
How to select a whole part
You can select the whole part of an incorrect fraction. Let's see how this can be done.
To select a whole part from an incorrect fraction, you need to:
- divide the numerator by the denominator with the remainder;
- the resulting incomplete quotient is written in the whole part of the fraction;
- the remainder is written in the numerator of the fraction;
- the divisor is written into the denominator of the fraction.
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Remember!
The resulting number above, containing an integer and fractional part, is called mixed number.
We got a mixed number from an improper fraction, but you can do the opposite, that is represent a mixed number as an improper fraction.
To represent a mixed number as an improper fraction, you need to:
- multiply its integer part by the denominator of the fractional part;
- add the numerator of the fractional part to the resulting product;
- write the resulting amount from paragraph 2 into the numerator of the fraction, and leave the denominator of the fractional part the same.
Example. Let's represent the mixed number as an improper fraction.
In this article, we will analyze such a concept as mixed numbers. Let's start, as always, with a definition and small examples, then we will explain the connection between mixed numbers and irregular fractions. After that, we will learn how to correctly extract the whole part from a fraction and get an integer as a result.
Mixed number concept
If we take the sum n + a b, where the value of n can be any natural number, and a b is a regular fraction, then we can write the same thing without using a plus: n a b. Let's take specific numbers for clarity: so, 28 + 5 7 is the same as 28 5 7. Writing a fraction next to an integer is usually called a mixed number.
Definition 1
Mixed number is a number that is equal to the sum of a natural number n with a regular fraction a b. In this case, n is the integer part of the number, and a b is its fractional part.
It follows from the definition that any mixed number is equal to what will result from the addition of its integer and fractional parts. Thus, the equality n a b = n + a b will be fulfilled.
It can also be written as n + a b = n a b.
What are some examples of mixed numbers? So, they include 5 1 8, while the five is its whole part, and one eighth is the fractional part. More examples: 1 1 2, 234 34 53, 34000 6 25.
Above we wrote that in the fractional part of the mixed number there should be only a regular fraction. Sometimes you can find records of the form 5 22 3, 75 7 2. They are not mixed numbers as they are. their fractional part is incorrect. They should be understood as the sum of the whole and fractional parts. Such numbers can be reduced to the standard form of notation for mixed numbers by separating the whole part from the improper fraction and adding it to 5 and 75 in these examples, respectively.
Numbers like 0 3 14 are also not mixed. The first part of the condition is not met here: the integer part must be represented only by a natural number, and zero is not.
How improper fractions and mixed numbers relate to each other
The easiest way to trace this connection is with a specific example.
Example 1
Take a whole cake and three quarters of the same. According to the rules of addition, we have 1 + 3 4 cakes on our table. This amount can be represented as a mixed number like 1 3 4 of a cake. If we take a whole cake and also cut it into four equal parts, then we will have 7 4 cakes on the table. Obviously, the amount did not increase from cutting, and 1 3 4 = 7 4.
Our example proves that any number can be represented as a mixed number. improper fraction.
Let's go back to our 7 4 cakes left on the table. Put one cake back out of its pieces (1 + 3 4). We will have 1 3 4 again.
Answer: 7 4 = 1 3 4 .
We figured out how to convert an improper fraction to a mixed number. If the numerator of the improper fraction contains such a number that can be divided by the denominator without a remainder, then you can do this, and then our improper fraction will become a natural number.
Example 2
For example,
8 4 = 2, since 8: 4 = 2.
How to convert a mixed number to an improper fraction
To successfully solve problems, it is useful to be able to perform the opposite action, that is, to make improper fractions out of mixed numbers. In this paragraph, we will analyze how to do it correctly.
To do this, you need to reproduce the following sequence of actions:
1. To begin with, we present the available mixed number n a b as the sum of the integer and fractional parts. It turns out n + a b
3. After that, we perform the already familiar action - add two ordinary fractions n 1 and a b. The resulting improper fraction will be equal to the mixed number given in the condition.
Let's analyze this action with a specific example.
Example 3
Imagine 5 3 7 as an improper fraction.
Decision
We carry out sequentially the steps of the above algorithm. Our number 5 3 7 is the sum of the integer and fractional parts, that is, 5 + 3 7. Now we will write the five in the form 5 1. We got the sum 5 1 + 3 7.
The last step is to add fractions with different denominators:
5 1 + 3 7 = 35 7 + 3 7 = 38 7
The entire solution to the short form can be written as 5 3 7 = 5 + 3 7 = 5 1 + 3 7 = 35 7 + 3 7 = 38 7.
Answer: 5 3 7 = 38 7 .
Thus, using the above chain of actions, we can convert any mixed number n a b into an improper fraction. We got the formula n a b = n b + a b, which we will take to solve further problems.
Example 4
Imagine 15 2 5 as an improper fraction.
Decision
Let's take the specified formula and substitute the required values into it. We have n = 15, a = 2, b = 5, therefore, 15 2 5 = 15 5 + 2 5 = 77 5.
Answer: 15 2 5 = 77 5 .
We usually do not use an incorrect fraction as the final answer. It is customary to bring the calculations to the end and replace it with either a natural number (dividing the numerator by the denominator), or a mixed number. As a rule, the first method is used when it is possible to divide the numerator by the denominator without a remainder, and the second - if such an action is impossible.
When we extract the whole part from an improper fraction, we simply replace it with an equal mixed number.
Let's see how exactly this is done.
Definition 2
Let us give a proof of this statement.
We need to explain why q r b = a b. For this, the mixed number q r b must be represented as an improper fraction, following all the steps of the algorithm from the previous paragraph. Since is an incomplete quotient, and r is the remainder of dividing a by b, then the equality a = b q + r must hold.
Thus, q b + r b = a b therefore q r b = a b. This is the proof of our statement. Let's summarize:
Definition 3
The separation of the whole part from the improper fraction a b is carried out as follows:
1) divide a by b with a remainder and write down the incomplete quotient q and the remainder r separately.
2) We write the results in the form q r b. This is our mixed number, equal to the original improper fraction.
Example 5
Think of 107 4 as a mixed number.
Decision
We divide 104 by 7 with a column:
Dividing the numerator a = 118 by the denominator b = 7 gives us as a result an incomplete quotient q = 16 and a remainder r = 6.
As a result, we get that the improper fraction 118 7 is equal to the mixed number q r b = 16 6 7.
Answer: 118 7 = 16 6 7 .
It remains for us to see how to replace an improper fraction with a natural number (provided that its numerator is divisible by the denominator without remainder).
To do this, let's remember what connection exists between ordinary fractions and division. From this we can deduce the equalities: a b = a: b = c. It turns out that the improper fraction a b can be replaced with a natural number c.
Example 6
For example, if the answer turned out to be an incorrect fraction 27 3, then we can write 9 instead of it, since 27 3 = 27: 3 = 9.
Answer: 27 3 = 9 .
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Depending on whether the numerator is less than the denominator and whether the integer part is highlighted in the number, there are three forms of fractional notation of non-integers:
- regular fractions, for example $ \ frac (1) (2), \ frac (4) (5) $;
- improper fractions such as $ \ frac (156) (27), \ frac (48) (5) $;
- mixed numbers: $ 5 \ frac (15) (27), 9 \ frac (4) (85) $.
Definition 1
An irregular fraction is a form of writing a number in which the numerator is greater than or equal to the denominator.
The advantage of this form is that it can be quickly obtained by calculating the numerator and denominator separately. The disadvantage is low visibility: you have to make additional efforts to distinguish between the whole and fractional parts.
An improper fraction can always be converted to a mixed number, because when the numerator is divided by the denominator, the result is greater than one. For such a transformation, it is necessary to divide the numerator by the denominator, move the whole part of the result before the fraction, and leave the remainder in the numerator, for example:
$ \ frac (17) (5) = 3 \ cdot 5 + 2 = 3 \ frac (2) (5) $
Converting to an improper fraction
Sometimes it is required to perform the opposite operation - to convert the mixed number into an improper fraction. In this case, on the contrary, the whole part, expressed in fractions indicated by the denominator, should be returned to the numerator.
Example 1
How to convert the mixed number $ 2 \ frac (5) (6) $ to an improper fraction?.
The whole part here corresponds to the number of sixths in a unit taken 2 times, i.e. to transfer it to the numerator, we calculate
$ 2 \ cdot 6 = $ 12.
To this result, you need to add the remainder, which was originally in the numerator:
Answer:
$ \ frac (12 + 5) (6) = \ frac (17) (6) $
