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 unit segment is divided into 4 equal parts, label each part Latin letter, then the result can be excellent visual material. 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.
Under proper fraction understand a number whose numerator is less than its 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 - whole part, ½ - fractional. 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 to 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 integer part in 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 is 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 decimal 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.
Another operation that can be performed with ordinary fractions is multiplication. We will try to explain its basic rules when solving problems, show how an ordinary fraction is multiplied by a natural number and how to correctly multiply three ordinary fractions or more.
Let's first write down the basic rule:
Definition 1
If we multiply one ordinary fraction, then the numerator of the resulting fraction will be equal to the product of the numerators of the original fractions, and the denominator will be equal to the product of their denominators. In literal form, for two fractions a / b and c / d, this can be expressed as a b · c d = a · c b · d.
Let's look at an example of how to correctly apply this rule. Let's say we have a square whose side is equal to one numerical unit. Then the area of the figure will be 1 square. unit. If we divide the square into equal rectangles with sides equal to 1 4 and 1 8 numerical units, we get that it now consists of 32 rectangles (because 8 4 = 32). Accordingly, the area of each of them will be equal to 1 32 of the area of the entire figure, i.e. 1 32 sq. units.
We have a shaded fragment with sides equal to 5 8 numerical units and 3 4 numerical units. Accordingly, to calculate its area, you need to multiply the first fraction by the second. It will be equal to 5 8 · 3 4 sq. units. But we can simply count how many rectangles are included in the fragment: there are 15 of them, which means the total area is 15 32 square units.
Since 5 3 = 15 and 8 4 = 32, we can write the following equality:
5 8 3 4 = 5 3 8 4 = 15 32
It confirms the rule we formulated for multiplying ordinary fractions, which is expressed as a b · c d = a · c b · d. It works the same for both proper and improper fractions; It can be used to multiply fractions with both different and identical denominators.
Let's look at solutions to several problems involving multiplication of ordinary fractions.
Example 1
Multiply 7 11 by 9 8.
Solution
First, let's calculate the product of the numerators of the indicated fractions by multiplying 7 by 9. We got 63. Then we calculate the product of the denominators and get: 11 · 8 = 88. Let's compose two numbers and the answer is: 63 88.
The whole solution can be written like this:
7 11 9 8 = 7 9 11 8 = 63 88
Answer: 7 11 · 9 8 = 63 88.
If we get a reducible fraction in our answer, we need to complete the calculation and perform its reduction. If we get an improper fraction, we need to separate out the whole part from it.
Example 2
Calculate product of fractions 4 15 and 55 6 .
Solution
According to the rule studied above, we need to multiply the numerator by the numerator, and the denominator by the denominator. The solution record will look like this:
4 15 55 6 = 4 55 15 6 = 220 90
We got a reducible fraction, i.e. one that is divisible by 10.
Let's reduce the fraction: 220 90 gcd (220, 90) = 10, 220 90 = 220: 10 90: 10 = 22 9. As a result, we get an improper fraction, from which we select the whole part and get a mixed number: 22 9 = 2 4 9.
Answer: 4 15 55 6 = 2 4 9.
For ease of calculation, we can also reduce the original fractions before performing the multiplication operation, for which we need to reduce the fraction to the form a · c b · d. Let's decompose the values of the variables into simple factors and reduce the same ones.
Let's explain what this looks like using data from a specific task.
Example 3
Calculate the product 4 15 55 6.
Solution
Let's write down the calculations based on the multiplication rule. We will get:
4 15 55 6 = 4 55 15 6
Since 4 = 2 2, 55 = 5 11, 15 = 3 5 and 6 = 2 3, then 4 55 15 6 = 2 2 5 11 3 5 2 3.
2 11 3 3 = 22 9 = 2 4 9
Answer: 4 15 · 55 6 = 2 4 9 .
A numerical expression in which ordinary fractions are multiplied has a commutative property, that is, if necessary, we can change the order of the factors:
a b · c d = c d · a b = a · c b · d
How to multiply a fraction with a natural number
Let's write down the basic rule right away, and then try to explain it in practice.
Definition 2
To multiply a common fraction by a natural number, you need to multiply the numerator of that fraction by that number. In this case, the denominator of the final fraction will be equal to the denominator of the original ordinary fraction. Multiplication of a certain fraction a b by a natural number n can be written as the formula a b · n = a · n b.
It’s easy to understand this formula if you remember that any natural number can be represented as an ordinary fraction with a denominator equal to one, that is:
a b · n = a b · n 1 = a · n b · 1 = a · n b
Let us explain our idea with specific examples.
Example 4
Calculate the product 2 27 times 5.
Solution
As a result of multiplying the numerator of the original fraction by the second factor, we get 10. By virtue of the rule stated above, we will get 10 27 as a result. The entire solution is given in this post:
2 27 5 = 2 5 27 = 10 27
Answer: 2 27 5 = 10 27
When we multiply a natural number with a fraction, we often have to abbreviate the result or represent it as a mixed number.
Example 5
Condition: calculate the product 8 by 5 12.
Solution
According to the rule above, we multiply the natural number by the numerator. As a result, we get that 5 12 8 = 5 8 12 = 40 12. The final fraction has signs of divisibility by 2, so we need to reduce it:
LCM (40, 12) = 4, so 40 12 = 40: 4 12: 4 = 10 3
Now all we have to do is select the whole part and write down the ready answer: 10 3 = 3 1 3.
In this entry you can see the entire solution: 5 12 8 = 5 8 12 = 40 12 = 10 3 = 3 1 3.
We could also reduce the fraction by factoring the numerator and denominator into prime factors, and the result would be exactly the same.
Answer: 5 12 8 = 3 1 3.
A numerical expression in which a natural number is multiplied by a fraction also has the property of displacement, that is, the order of the factors does not affect the result:
a b · n = n · a b = a · n b
How to multiply three or more common fractions
We can extend to the action of multiplying ordinary fractions the same properties that are characteristic of multiplying natural numbers. This follows from the very definition of these concepts.
Thanks to knowledge of the combining and commutative properties, you can multiply three or more ordinary fractions. It is acceptable to rearrange the factors for greater convenience or arrange the brackets in a way that makes it easier to count.
Let's show with an example how this is done.
Example 6
Multiply the four common fractions 1 20, 12 5, 3 7 and 5 8.
Solution: First, let's record the work. We get 1 20 · 12 5 · 3 7 · 5 8 . We need to multiply all the numerators and all the denominators together: 1 20 · 12 5 · 3 7 · 5 8 = 1 · 12 · 3 · 5 20 · 5 · 7 · 8 .
Before we start multiplying, we can make things a little easier on ourselves and factor some numbers into prime factors for further reduction. This will be easier than reducing the resulting fraction that is already ready.
1 12 3 5 20 5 7 8 = 1 (2 2 3) 3 5 2 2 5 5 7 (2 2 2) = 3 3 5 7 2 2 2 = 9,280
Answer: 1 · 12 · 3 · 5 20 · 5 · 7 · 8 = 9,280.
Example 7
Multiply 5 numbers 7 8 · 12 · 8 · 5 36 · 10 .
Solution
For convenience, we can group the fraction 7 8 with the number 8, and the number 12 with the fraction 5 36, since future abbreviations will be obvious to us. As a result, we will get:
7 8 12 8 5 36 10 = 7 8 8 12 5 36 10 = 7 8 8 12 5 36 10 = 7 1 2 2 3 5 2 2 3 3 10 = 7 5 3 10 = 7 5 10 3 = 350 3 = 116 2 3
Answer: 7 8 12 8 5 36 10 = 116 2 3.
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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 fractions have the same or different denominators, multiplication occurs according to the rule of finding the product of a numerator with a numerator, a denominator with a 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 reciprocals be mixed numbers at the same time?
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.
Multiplying common fractions
Let's look at an example.
Let there be $\frac(1)(3)$ part of an apple on a plate. We need to find the $\frac(1)(2)$ part of it. Required part is the result of multiplying the fractions $\frac(1)(3)$ and $\frac(1)(2)$. The result of multiplying two common fractions is a common fraction.
Multiplying two ordinary fractions
Rule for multiplying ordinary fractions:
The result of multiplying a fraction by a fraction is a fraction whose numerator is equal to the product of the numerators of the fractions being multiplied, and the denominator is equal to the product of the denominators:
Example 1
Perform multiplication of common fractions $\frac(3)(7)$ and $\frac(5)(11)$.
Solution.
Let's use the rule for multiplying ordinary fractions:
\[\frac(3)(7)\cdot \frac(5)(11)=\frac(3\cdot 5)(7\cdot 11)=\frac(15)(77)\]
Answer:$\frac(15)(77)$
If multiplying fractions results in a reducible or improper fraction, you need to simplify it.
Example 2
Multiply the fractions $\frac(3)(8)$ and $\frac(1)(9)$.
Solution.
We use the rule for multiplying ordinary fractions:
\[\frac(3)(8)\cdot \frac(1)(9)=\frac(3\cdot 1)(8\cdot 9)=\frac(3)(72)\]
As a result, we got a reducible fraction (based on division by $3$. Divide the numerator and denominator of the fraction by $3$, we get:
\[\frac(3)(72)=\frac(3:3)(72:3)=\frac(1)(24)\]
Short solution:
\[\frac(3)(8)\cdot \frac(1)(9)=\frac(3\cdot 1)(8\cdot 9)=\frac(3)(72)=\frac(1) (24)\]
Answer:$\frac(1)(24).$
When multiplying fractions, you can reduce the numerators and denominators until you find their product. In this case, the numerator and denominator of the fraction are decomposed into simple factors, after which the repeating factors are canceled and the result is found.
Example 3
Calculate the product of the fractions $\frac(6)(75)$ and $\frac(15)(24)$.
Solution.
Let's use the formula for multiplying ordinary fractions:
\[\frac(6)(75)\cdot \frac(15)(24)=\frac(6\cdot 15)(75\cdot 24)\]
Obviously, the numerator and denominator contain numbers that can be reduced in pairs to the numbers $2$, $3$ and $5$. Let's factor the numerator and denominator into simple factors and make a reduction:
\[\frac(6\cdot 15)(75\cdot 24)=\frac(2\cdot 3\cdot 3\cdot 5)(3\cdot 5\cdot 5\cdot 2\cdot 2\cdot 2\cdot 3)=\frac(1)(5\cdot 2\cdot 2)=\frac(1)(20)\]
Answer:$\frac(1)(20).$
When multiplying fractions, you can apply the commutative law:
Multiplying a common fraction by a natural number
The rule for multiplying a common fraction by a natural number:
The result of multiplying a fraction by a natural number is a fraction in which the numerator is equal to the product of the numerator of the multiplied fraction by the natural number, and the denominator is equal to the denominator of the multiplied fraction:
where $\frac(a)(b)$ is an ordinary fraction, $n$ is a natural number.
Example 4
Multiply the fraction $\frac(3)(17)$ by $4$.
Solution.
Let's use the rule for multiplying an ordinary fraction by a natural number:
\[\frac(3)(17)\cdot 4=\frac(3\cdot 4)(17)=\frac(12)(17)\]
Answer:$\frac(12)(17).$
Do not forget to check the result of multiplication by the reducibility of a fraction or by an improper fraction.
Example 5
Multiply the fraction $\frac(7)(15)$ by the number $3$.
Solution.
Let's use the formula for multiplying a fraction by a natural number:
\[\frac(7)(15)\cdot 3=\frac(7\cdot 3)(15)=\frac(21)(15)\]
By dividing by the number $3$) we can determine that the resulting fraction can be reduced:
\[\frac(21)(15)=\frac(21:3)(15:3)=\frac(7)(5)\]
The result was an incorrect fraction. Let's select the whole part:
\[\frac(7)(5)=1\frac(2)(5)\]
Short solution:
\[\frac(7)(15)\cdot 3=\frac(7\cdot 3)(15)=\frac(21)(15)=\frac(7)(5)=1\frac(2) (5)\]
Fractions could also be reduced by replacing the numbers in the numerator and denominator with their factorizations into prime factors. In this case, the solution could be written as follows:
\[\frac(7)(15)\cdot 3=\frac(7\cdot 3)(15)=\frac(7\cdot 3)(3\cdot 5)=\frac(7)(5)= 1\frac(2)(5)\]
Answer:$1\frac(2)(5).$
When multiplying a fraction by a natural number, you can use the commutative law:
Dividing fractions
The division operation is the inverse of multiplication and its result is a fraction by which a known fraction must be multiplied to obtain the known product of two fractions.
Dividing two ordinary fractions
Rule for dividing ordinary fractions: Obviously, the numerator and denominator of the resulting fraction can be factorized and reduced:
\[\frac(8\cdot 35)(15\cdot 12)=\frac(2\cdot 2\cdot 2\cdot 5\cdot 7)(3\cdot 5\cdot 2\cdot 2\cdot 3)= \frac(2\cdot 7)(3\cdot 3)=\frac(14)(9)\]
As a result, we get an improper fraction, from which we select the whole part:
\[\frac(14)(9)=1\frac(5)(9)\]
Answer:$1\frac(5)(9).$
Last time we learned how to add and subtract fractions (see lesson “Adding and subtracting fractions”). Most difficult moment those actions involved 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 out of 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 pay attention to 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 problem, 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. Therefore, it is impossible to apply the main property of a fraction, since in this property we're talking about specifically about multiplying 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.
