In the course of secondary and high school Students studied the topic “Fractions”. However, this concept is much broader than what is given in the learning process. Today, the concept of a fraction is encountered quite often, and not everyone can calculate any expression, for example, multiplying fractions.
What is a fraction?
Historically, fractional numbers arose out of the need to measure. As practice shows, there are often examples of determining the length of a segment and the volume of a rectangular rectangle.
Initially, students are introduced to the concept of a share. For example, if you divide a watermelon into 8 parts, then each person will get one-eighth of the watermelon. This one part of eight is called a share.
A share equal to ½ of any value is called half; ⅓ - third; ¼ - a quarter. Records of the form 5/8, 4/5, 2/4 are called ordinary fractions. A common fraction is divided into a numerator and a denominator. Between them is the fraction bar, or fraction bar. The fractional line can be drawn as either a horizontal or an oblique line. IN in this case it represents the division sign.
The denominator represents how many equal parts the quantity or object is divided into; and the numerator is how many identical shares are taken. The numerator is written above the fraction line, the denominator is written below it.
It is most convenient to show ordinary fractions on a coordinate ray. If a single segment is divided into 4 equal parts, each part is designated by a Latin letter, then the result can be an excellent visual aid. So, point A shows a share equal to 1/4 of the entire unit segment, and point B marks 2/8 of a given segment.
Types of fractions
Fractions can be ordinary, decimal, and mixed numbers. In addition, fractions can be divided into proper and improper. This classification is more suitable for ordinary fractions.
A proper fraction is a number whose numerator is less than the denominator. Accordingly, an improper fraction is a number whose numerator is greater than its denominator. The second type is usually written as a mixed number. This expression consists of an integer and a fractional part. For example, 1½. 1 is an integer part, ½ is a fractional part. However, if you need to carry out some manipulations with the expression (dividing or multiplying fractions, reducing or converting them), the mixed number is converted into an improper fraction.
A correct fractional expression is always less than one, and an incorrect one is always greater than or equal to 1.
As for this expression, we mean a record in which any number is represented, the denominator of the fractional expression of which can be expressed in terms of one with several zeros. If the fraction is proper, then the whole part is decimal notation will be equal to zero.
To write a decimal fraction, you must first write the whole part, separate it from the fraction using a comma, and then write the fraction expression. It must be remembered that after the decimal point the numerator must contain the same number of digital characters as there are zeros in the denominator.
Example. Express the fraction 7 21 / 1000 in decimal notation.
Algorithm for converting an improper fraction to a mixed number and vice versa
It is incorrect to write an improper fraction in the answer to a problem, so it needs to be converted to a mixed number:
- divide the numerator by the existing denominator;
- V specific example incomplete quotient - whole;
- and the remainder is the numerator of the fractional part, with the denominator remaining unchanged.
Example. Convert improper fraction to mixed number: 47 / 5.
Solution. 47: 5. The partial quotient is 9, the remainder = 2. So, 47 / 5 = 9 2 / 5.
Sometimes you need to represent a mixed number as an improper fraction. Then you need to use the following algorithm:
- the integer part is multiplied by the denominator of the fractional expression;
- the resulting product is added to the numerator;
- the result is written in the numerator, the denominator remains unchanged.
Example. Present the number in mixed form as an improper fraction: 9 8 / 10.
Solution. 9 x 10 + 8 = 90 + 8 = 98 is the numerator.
Answer: 98 / 10.
Multiplying fractions
Various algebraic operations can be performed on ordinary fractions. To multiply two numbers, you need to multiply the numerator with the numerator, and the denominator with the denominator. Moreover, multiplying fractions with different denominators is no different from multiplying fractions with the same denominators.
It happens that after finding the result you need to reduce the fraction. It is imperative to simplify the resulting expression as much as possible. Of course, one cannot say that an improper fraction in an answer is an error, but it is also difficult to call it a correct answer.
Example. Find the product of two ordinary fractions: ½ and 20/18.
As can be seen from the example, after finding the product, a reducible fractional notation was obtained. Both the numerator and the denominator in this case are divided by 4, and the result is the answer 5 / 9.
Multiplying decimal fractions
The product of decimal fractions is quite different from the product of ordinary fractions in its principle. So, multiplying fractions is as follows:
- two decimal fractions must be written one under the other so that the rightmost digits are one under the other;
- you need to multiply the written numbers, despite the commas, that is, as natural numbers;
- count the number of digits after the decimal point in each number;
- in the result obtained after multiplication, you need to count from the right as many digital symbols as are contained in the sum in both factors after the decimal point, and put a separating sign;
- if there are fewer numbers in the product, then you need to write as many zeros in front of them to cover this number, put a comma and add the whole part equal to zero.
Example. Calculate the product of two decimal fractions: 2.25 and 3.6.
Solution.
Multiplying mixed fractions
To calculate the product of two mixed fractions, you need to use the rule for multiplying fractions:
- convert mixed numbers into improper fractions;
- find the product of the numerators;
- find the product of denominators;
- write down the result;
- simplify the expression as much as possible.
Example. Find the product of 4½ and 6 2/5.
Multiplying a number by a fraction (fractions by a number)
In addition to finding the product of two fractions and mixed numbers, there are tasks where you need to multiply by a fraction.
So, to find the product of a decimal fraction and a natural number, you need:
- write the number under the fraction so that the rightmost digits are one above the other;
- find the product despite the comma;
- in the resulting result, separate the integer part from the fractional part using a comma, counting from the right the number of digits that are located after the decimal point in the fraction.
To multiply a common fraction by a number, you need to find the product of the numerator and the natural factor. If the answer produces a fraction that can be reduced, it should be converted.
Example. Calculate the product of 5 / 8 and 12.
Solution. 5 / 8 * 12 = (5*12) / 8 = 60 / 8 = 30 / 4 = 15 / 2 = 7 1 / 2.
Answer: 7 1 / 2.
As you can see from the previous example, it was necessary to reduce the resulting result and convert the incorrect fractional expression into a mixed number.
Multiplication of fractions also concerns finding the product of a number in mixed form and a natural factor. To multiply these two numbers, you should multiply the whole part of the mixed factor by the number, multiply the numerator by the same value, and leave the denominator unchanged. If necessary, you need to simplify the resulting result as much as possible.
Example. Find the product of 9 5 / 6 and 9.
Solution. 9 5 / 6 x 9 = 9 x 9 + (5 x 9) / 6 = 81 + 45 / 6 = 81 + 7 3 / 6 = 88 1 / 2.
Answer: 88 1 / 2.
Multiplication by factors of 10, 100, 1000 or 0.1; 0.01; 0.001
The following rule follows from the previous paragraph. To multiply a decimal fraction by 10, 100, 1000, 10000, etc., you need to move the decimal point to the right by as many digits as there are zeros in the factor after the one.
Example 1. Find the product of 0.065 and 1000.
Solution. 0.065 x 1000 = 0065 = 65.
Answer: 65.
Example 2. Find the product of 3.9 and 1000.
Solution. 3.9 x 1000 = 3.900 x 1000 = 3900.
Answer: 3900.
If you need to multiply natural number and 0.1; 0.01; 0.001; 0.0001, etc., you should move the comma in the resulting product to the left by as many digit characters as there are zeros before one. If necessary, a sufficient number of zeros are written before the natural number.
Example 1. Find the product of 56 and 0.01.
Solution. 56 x 0.01 = 0056 = 0.56.
Answer: 0,56.
Example 2. Find the product of 4 and 0.001.
Solution. 4 x 0.001 = 0004 = 0.004.
Answer: 0,004.
So, finding the product of different fractions should not cause any difficulties, except perhaps calculating the result; in this case, you simply cannot do without a calculator.
To correctly multiply a fraction by a fraction or a fraction by a number, you need to know simple rules. We will now analyze these rules in detail.
Multiplying a common fraction by a fraction.
To multiply a fraction by a fraction, you need to calculate the product of the numerators and the product of the denominators of these fractions.
\(\bf \frac(a)(b) \times \frac(c)(d) = \frac(a \times c)(b \times d)\\\)
Let's look at an example:
We multiply the numerator of the first fraction with the numerator of the second fraction, and we also multiply the denominator of the first fraction with the denominator of the second fraction.
\(\frac(6)(7) \times \frac(2)(3) = \frac(6 \times 2)(7 \times 3) = \frac(12)(21) = \frac(4 \ times 3)(7 \times 3) = \frac(4)(7)\\\)
The fraction \(\frac(12)(21) = \frac(4 \times 3)(7 \times 3) = \frac(4)(7)\\\) was reduced by 3.
Multiplying a fraction by a number.
First, let's remember the rule, any number can be represented as a fraction \(\bf n = \frac(n)(1)\) .
Let's use this rule when multiplying.
\(5 \times \frac(4)(7) = \frac(5)(1) \times \frac(4)(7) = \frac(5 \times 4)(1 \times 7) = \frac (20)(7) = 2\frac(6)(7)\\\)
Improper fraction \(\frac(20)(7) = \frac(14 + 6)(7) = \frac(14)(7) + \frac(6)(7) = 2 + \frac(6)( 7)= 2\frac(6)(7)\\\) converted to a mixed fraction.
In other words, When multiplying a number by a fraction, we multiply the number by the numerator and leave the denominator unchanged. Example:
\(\frac(2)(5) \times 3 = \frac(2 \times 3)(5) = \frac(6)(5) = 1\frac(1)(5)\\\\\) \(\bf \frac(a)(b) \times c = \frac(a \times c)(b)\\\)
Multiplying mixed fractions.
To multiply mixed fractions, you must first represent each mixed fraction as an improper fraction, and then use the multiplication rule. We multiply the numerator with the numerator, and multiply the denominator with the denominator.
Example:
\(2\frac(1)(4) \times 3\frac(5)(6) = \frac(9)(4) \times \frac(23)(6) = \frac(9 \times 23) (4 \times 6) = \frac(3 \times \color(red) (3) \times 23)(4 \times 2 \times \color(red) (3)) = \frac(69)(8) = 8\frac(5)(8)\\\)
Multiplication of reciprocal fractions and numbers.
The fraction \(\bf \frac(a)(b)\) is the inverse of the fraction \(\bf \frac(b)(a)\), provided a≠0,b≠0.
The fractions \(\bf \frac(a)(b)\) and \(\bf \frac(b)(a)\) are called reciprocal fractions. The product of reciprocal fractions is equal to 1.
\(\bf \frac(a)(b) \times \frac(b)(a) = 1 \\\)
Example:
\(\frac(5)(9) \times \frac(9)(5) = \frac(45)(45) = 1\\\)
Related questions:
How to multiply a fraction by a fraction?
Answer: The product of ordinary fractions is the multiplication of a numerator with a numerator, a denominator with a denominator. To get the product of mixed fractions, you need to convert them into an improper fraction and multiply according to the rules.
How to multiply fractions with different denominators?
Answer: it doesn’t matter whether they are the same or different denominators For fractions, multiplication occurs according to the rule of finding the product of the numerator with the numerator, the denominator with the denominator.
How to multiply mixed fractions?
Answer: first of all, you need to convert the mixed fraction into an improper fraction and then find the product using the rules of multiplication.
How to multiply a number by a fraction?
Answer: we multiply the number with the numerator, but leave the denominator the same.
Example #1:
Calculate the product: a) \(\frac(8)(9) \times \frac(7)(11)\) b) \(\frac(2)(15) \times \frac(10)(13)\ )
Solution:
a) \(\frac(8)(9) \times \frac(7)(11) = \frac(8 \times 7)(9 \times 11) = \frac(56)(99)\\\\ \)
b) \(\frac(2)(15) \times \frac(10)(13) = \frac(2 \times 10)(15 \times 13) = \frac(2 \times 2 \times \color( red) (5))(3 \times \color(red) (5) \times 13) = \frac(4)(39)\)
Example #2:
Calculate the products of a number and a fraction: a) \(3 \times \frac(17)(23)\) b) \(\frac(2)(3) \times 11\)
Solution:
a) \(3 \times \frac(17)(23) = \frac(3)(1) \times \frac(17)(23) = \frac(3 \times 17)(1 \times 23) = \frac(51)(23) = 2\frac(5)(23)\\\\\)
b) \(\frac(2)(3) \times 11 = \frac(2)(3) \times \frac(11)(1) = \frac(2 \times 11)(3 \times 1) = \frac(22)(3) = 7\frac(1)(3)\)
Example #3:
Write the reciprocal of the fraction \(\frac(1)(3)\)?
Answer: \(\frac(3)(1) = 3\)
Example #4:
Calculate the product of two mutually inverse fractions: a) \(\frac(104)(215) \times \frac(215)(104)\)
Solution:
a) \(\frac(104)(215) \times \frac(215)(104) = 1\)
Example #5:
Can reciprocal fractions be:
a) simultaneously with proper fractions;
b) simultaneously improper fractions;
c) simultaneously natural numbers?
Solution:
a) to answer the first question, let's give an example. The fraction \(\frac(2)(3)\) is proper, its inverse fraction will be equal to \(\frac(3)(2)\) - an improper fraction. Answer: no.
b) in almost all enumerations of fractions this condition is not met, but there are some numbers that fulfill the condition of being simultaneously an improper fraction. For example, the improper fraction is \(\frac(3)(3)\), its inverse fraction is equal to \(\frac(3)(3)\). We get two improper fractions. Answer: not always under certain conditions when the numerator and denominator are equal.
c) natural numbers are numbers that we use when counting, for example, 1, 2, 3, …. If we take the number \(3 = \frac(3)(1)\), then its inverse fraction will be \(\frac(1)(3)\). The fraction \(\frac(1)(3)\) is not a natural number. If we go through all the numbers, the reciprocal of the number is always a fraction, except for 1. If we take the number 1, then its reciprocal fraction will be \(\frac(1)(1) = \frac(1)(1) = 1\). Number 1 is a natural number. Answer: they can simultaneously be natural numbers only in one case, if this is the number 1.
Example #6:
Do the product of mixed fractions: a) \(4 \times 2\frac(4)(5)\) b) \(1\frac(1)(4) \times 3\frac(2)(7)\)
Solution:
a) \(4 \times 2\frac(4)(5) = \frac(4)(1) \times \frac(14)(5) = \frac(56)(5) = 11\frac(1 )(5)\\\\ \)
b) \(1\frac(1)(4) \times 3\frac(2)(7) = \frac(5)(4) \times \frac(23)(7) = \frac(115)( 28) = 4\frac(3)(7)\)
Example #7:
Can two reciprocal numbers exist at the same time? mixed numbers?
Let's look at an example. Let's take a mixed fraction \(1\frac(1)(2)\), find its inverse fraction, to do this we convert it into an improper fraction \(1\frac(1)(2) = \frac(3)(2) \) . Its inverse fraction will be equal to \(\frac(2)(3)\) . The fraction \(\frac(2)(3)\) is a proper fraction. Answer: Two fractions that are mutually inverse cannot be mixed numbers at the same time.
Last time we learned how to add and subtract fractions (see lesson “Adding and subtracting fractions”). The most difficult part of those actions was bringing fractions to a common denominator.
Now it's time to deal with multiplication and division. The good news is that these operations are even simpler than addition and subtraction. First, let's consider the simplest case, when there are two positive fractions without a separated integer part.
To multiply two fractions, you must multiply their numerators and denominators separately. The first number will be the numerator of the new fraction, and the second will be the denominator.
To divide two fractions, you need to multiply the first fraction by the “inverted” second fraction.
Designation:
From the definition it follows that dividing fractions reduces to multiplication. To “flip” a fraction, just swap the numerator and denominator. Therefore, throughout the lesson we will mainly consider multiplication.
As a result of multiplication, a reducible fraction can arise (and often does arise) - it, of course, must be reduced. If after all the reductions the fraction turns out to be incorrect, the whole part should be highlighted. But what definitely won't happen with multiplication is reduction to a common denominator: no criss-cross methods, greatest factors and least common multiples.
By definition we have:
Multiplying fractions with whole parts and negative fractions
If fractions contain an integer part, they must be converted to improper ones - and only then multiplied according to the schemes outlined above.
If there is a minus in the numerator of a fraction, in the denominator or in front of it, it can be taken out of the multiplication or removed altogether according to the following rules:
- Plus by minus gives minus;
- Two negatives make an affirmative.
Until now, these rules have only been encountered when adding and subtracting negative fractions, when it was necessary to get rid of the whole part. For a work, they can be generalized in order to “burn” several disadvantages at once:
- We cross out the negatives in pairs until they completely disappear. In extreme cases, one minus can survive - the one for which there was no mate;
- If there are no minuses left, the operation is completed - you can start multiplying. If the last minus is not crossed out because there was no pair for it, we take it outside the limits of multiplication. The result is a negative fraction.
Task. Find the meaning of the expression:
We convert all fractions to improper ones, and then take the minuses outside the multiplication. We multiply what is left according to the usual rules. We get:
Let me remind you once again that the minus that appears in front of a fraction with a highlighted whole part refers specifically to the entire fraction, and not just to its whole part (this applies to the last two examples).
Also note negative numbers: When multiplying, they are enclosed in parentheses. This is done in order to separate the minuses from the multiplication signs and make the entire notation more accurate.
Reducing fractions on the fly
Multiplication is a very labor-intensive operation. The numbers here turn out to be quite large, and to simplify the task, you can try to reduce the fraction further before multiplication. Indeed, in essence, the numerators and denominators of fractions are ordinary factors, and, therefore, they can be reduced using the basic property of a fraction. Take a look at the examples:
Task. Find the meaning of the expression:
By definition we have:
In all examples, the numbers that have been reduced and what remains of them are marked in red.
Please note: in the first case, the multipliers were reduced completely. In their place there remain units that, generally speaking, need not be written. In the second example, it was not possible to achieve a complete reduction, but the total amount of calculations still decreased.
However, never use this technique when adding and subtracting fractions! Yes, sometimes there are similar numbers that you just want to reduce. Here, look:
You can't do that!
The error occurs because when adding, the numerator of a fraction produces a sum, not a product of numbers. Consequently, it is impossible to apply the basic property of a fraction, since this property deals specifically with the multiplication of numbers.
There are simply no other reasons for reducing fractions, so correct solution the previous task looks like this:
Correct solution:
As you can see, the correct answer turned out to be not so beautiful. In general, be careful.
) and denominator by denominator (we get the denominator of the product).
Formula for multiplying fractions:
For example:
Before you begin multiplying numerators and denominators, you need to check whether the fraction can be reduced. If you can reduce the fraction, it will be easier for you to make further calculations.
Dividing a common fraction by a fraction.
Dividing fractions involving natural numbers.
It's not as scary as it seems. As in the case of addition, we convert the integer into a fraction with one in the denominator. For example:
Multiplying mixed fractions.
Rules for multiplying fractions (mixed):
- convert mixed fractions to improper fractions;
- multiplying the numerators and denominators of fractions;
- reduce the fraction;
- If you get an improper fraction, then we convert the improper fraction into a mixed fraction.
Note! To multiply a mixed fraction by another mixed fraction, you first need to bring them to the form improper fractions, and then multiply according to the rule for multiplying ordinary fractions.
The second way to multiply a fraction by a natural number.
It may be more convenient to use the second method of multiplication common fraction per number.
Note! To multiply a fraction by a natural number, you must divide the denominator of the fraction by this number, and leave the numerator unchanged.
From the example given above, it is clear that this option is more convenient to use when the denominator of a fraction is divided without a remainder by a natural number.
Multistory fractions.
In high school, three-story (or more) fractions are often encountered. Example:
To bring such a fraction to its usual form, use division through 2 points:
Note! When dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.
Note, For example:
When dividing one by any fraction, the result will be the same fraction, only inverted:
Practical tips for multiplying and dividing fractions:
1. The most important thing when working with fractional expressions is accuracy and attentiveness. Do all calculations carefully and accurately, concentratedly and clearly. It's better to write a few extra lines in your draft than to get lost in mental calculations.
2. In tasks with different types fractions - go to the form of ordinary fractions.
3. We reduce all fractions until it is no longer possible to reduce.
4. We transform multi-level fractional expressions into ordinary ones using division through 2 points.
5. Divide a unit by a fraction in your head, simply turning the fraction over.
Multiplying a whole number by a fraction is not a difficult task. But there are subtleties that you probably understood at school, but have since forgotten.
How to multiply a whole number by a fraction - a few terms
If you remember what a numerator and denominator are and how a proper fraction differs from an improper fraction, skip this paragraph. It is for those who have completely forgotten the theory.
The numerator is top part fractions are what we divide. The denominator is lower. This is what we divide by.
A proper fraction is one whose numerator is less than its denominator. An improper fraction is one whose numerator is greater than or equal to its denominator.
How to multiply a whole number by a fraction
The rule for multiplying an integer by a fraction is very simple - we multiply the numerator by the integer, but do not touch the denominator. For example: two multiplied by one fifth - we get two fifths. Four multiplied by three sixteenths equals twelve sixteenths.
Reduction
In the second example, the resulting fraction can be reduced.
What does it mean? Please note that both the numerator and denominator of this fraction are divisible by four. Divide both numbers by common divisor and it’s called reducing a fraction. We get three quarters.
Improper fractions
But suppose we multiply four by two fifths. It turned out to be eight-fifths. This is an improper fraction.
She definitely needs to be brought to the right kind. To do this, you need to select an entire part from it.
Here you need to use division with a remainder. We get one and three as a remainder.
One whole and three fifths is our proper fraction.
Bringing thirty-five eighths to the correct form is a slightly more difficult task. The closest number to thirty-seven that is divisible by eight is thirty-two. When divided we get four. Subtract thirty-two from thirty-five and we get three. Result: four whole and three eighths.
Equality of numerator and denominator. And here everything is very simple and beautiful. If the numerator and denominator are equal, the result is simply one.
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