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 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.
\(\bf \frac(a)(b) \times \frac(c)(d) = \frac(a \times c)(b \times d)\\\)
Consider 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 \(\frac(12)(21) = \frac(4 \times 3)(7 \times 3) = \frac(4)(7)\\\) has been reduced by 3.
Multiplying a fraction by a number.
Let's start with the rule any number can be represented as a fraction \(\bf n = \frac(n)(1)\) .
Let's use this rule for multiplication.
\(5 \times \frac(4)(7) = \frac(5)(1) \times \frac(4)(7) = \frac(5 \times 4)(1 \times 7) = \frac (20)(7) = 2\frac(6)(7)\\\)
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, multiply the number by the numerator and leave the denominator unchanged. Example:
\(\frac(2)(5) \times 3 = \frac(2 \times 3)(5) = \frac(6)(5) = 1\frac(1)(5)\\\\\) \(\bf \frac(a)(b) \times c = \frac(a \times c)(b)\\\)
Multiplication of mixed fractions.
To multiply mixed fractions, you must first represent each mixed fraction as an improper fraction, and then use the multiplication rule. The numerator is multiplied with the numerator, the denominator is multiplied 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 reciprocals. The product of reciprocal fractions is 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 the numerator with the numerator, the denominator with the denominator. To get the product of mixed fractions, you need to convert them to 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 to an improper fraction and then find the product according to the rules of multiplication.
How to multiply a number by a fraction?
Answer: We multiply the number with the numerator, and 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)\ )
Decision:
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 product of a number and a fraction: a) \(3 \times \frac(17)(23)\) b) \(\frac(2)(3) \times 11\)
Decision:
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 \(\frac(1)(3)\)?
Answer: \(\frac(3)(1) = 3\)
Example #4:
Calculate the product of two reciprocal fractions: a) \(\frac(104)(215) \times \frac(215)(104)\)
Decision:
a) \(\frac(104)(215) \times \frac(215)(104) = 1\)
Example #5:
Can mutually inverse fractions be:
a) both proper fractions;
b) simultaneously improper fractions;
c) natural numbers at the same time?
Decision:
a) Let's use an example to answer the first question. The fraction \(\frac(2)(3)\) is proper, its reciprocal will be equal to \(\frac(3)(2)\) - not proper 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 an improper fraction at the same time. For example, the improper fraction is \(\frac(3)(3)\) , its reciprocal is \(\frac(3)(3)\). We get two improper fractions. Answer: not always under certain conditions, when the numerator and denominator are equal.
c) natural numbers are the numbers that we use when counting, for example, 1, 2, 3, .... If we take the number \(3 = \frac(3)(1)\), then its reciprocal will be \(\frac(1)(3)\). The fraction \(\frac(1)(3)\) is not a natural number. If we go through all the numbers, the reciprocal is always a fraction, except for 1. If we take the number 1, then its reciprocal will be \(\frac(1)(1) = \frac(1)(1) = 1\). Number 1 natural number. Answer: they can be simultaneously natural numbers only in one case, if this number is 1.
Example #6:
Perform the product of mixed fractions: a) \(4 \times 2\frac(4)(5)\) b) \(1\frac(1)(4) \times 3\frac(2)(7)\)
Decision:
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 mutually reciprocals be at the same time mixed numbers?
Let's look at an example. Take a mixed fraction \(1\frac(1)(2)\), find for it reciprocal, for this we translate it into an improper fraction \(1\frac(1)(2) = \frac(3)(2)\) . Its reciprocal will be equal to \(\frac(2)(3)\) . The fraction \(\frac(2)(3)\) is a proper fraction. Answer: Two mutually inverse fractions cannot be mixed numbers at the same time.
Multiplication and division of fractions.
Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")
This operation is much nicer than addition-subtraction! Because it's easier. I remind you: to multiply a fraction by a fraction, you need to multiply the numerators (this will be the numerator of the result) and the denominators (this will be the denominator). I.e:
For example:
Everything is extremely simple. And please don't look for a common denominator! Don't need it here...
To divide a fraction by a fraction, you need to flip second(this is important!) fraction and multiply them, i.e.:
For example:
If multiplication or division with integers and fractions is caught, it's okay. As with addition, we make a fraction from a whole number with a unit in the denominator - and go! For example:
In high school, you often have to deal with three-story (or even four-story!) fractions. For example:
How to bring this fraction to a decent form? Yes, very easy! Use division through two points:
But don't forget about the division order! Unlike multiplication, this is very important here! Of course, we will not confuse 4:2 or 2:4. But in a three-story fraction it is easy to make a mistake. Please note, for example:
In the first case (expression on the left):
In the second (expression on the right):
Feel the difference? 4 and 1/9!
What is the order of division? Or brackets, or (as here) the length of horizontal dashes. Develop an eye. And if there are no brackets or dashes, like:
then divide-multiply in order, left to right!
And another very simple and important trick. In actions with degrees, it will come in handy for you! Let's divide the unit by any fraction, for example, by 13/15:
The shot has turned over! And it always happens. When dividing 1 by any fraction, the result is the same fraction, only inverted.
That's all the actions with fractions. The thing is quite simple, but gives more than enough errors. Note practical advice, and they (errors) will be less!
Practical Tips:
1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not common words, not good wishes! This is a severe need! Do all the calculations on the exam as a full-fledged task, with concentration and clarity. It is better to write two extra lines in a draft than to mess up when calculating in your head.
2. In the examples with different types fractions - go to ordinary fractions.
3. We reduce all fractions to the stop.
4. Multi-storey fractional expressions we reduce to ordinary ones using division through two points (we follow the order of division!).
5. We divide the unit into a fraction in our mind, simply by turning the fraction over.
Here are the tasks you need to complete. Answers are given after all tasks. Use the materials of this topic and practical advice. Estimate how many examples you could solve correctly. The first time! Without a calculator! And draw the right conclusions...
Remember the correct answer obtained from the second (especially the third) time - does not count! Such is the harsh life.
So, solve in exam mode ! This is preparation for the exam, by the way. We solve an example, we check, we solve the following. We decided everything - we checked again from the first to the last. Only after look at the answers.
Calculate:
Did you decide?
Looking for answers that match yours. I specifically wrote them down in a mess, away from the temptation, so to speak ... Here they are, the answers, written down with a semicolon.
0; 17/22; 3/4; 2/5; 1; 25.
And now we draw conclusions. If everything worked out - happy for you! Elementary calculations with fractions are not your problem! You can do more serious things. If not...
So you have one of two problems. Or both at once.) Lack of knowledge and (or) inattention. But this solvable Problems.
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)
you can get acquainted with functions and derivatives.
Multiplying a whole number by a fraction is a simple task. But there are subtleties that you probably understood at school, but have since forgotten.
How to multiply an integer by a fraction - a few terms
If you remember what the numerator and denominator are and how a proper fraction differs from an improper one, 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 the bottom one. This is what we share.
A proper fraction is one whose numerator is less than the denominator. An improper fraction is a fraction whose numerator is greater than or equal to the 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, and do not touch the denominator. For example: two multiplied by one fifth - we get two fifths. Four times three sixteenths is twelve sixteenths.
Reduction
In the second example, the resulting fraction can be reduced.
What does it mean? Note that both the numerator and denominator of this fraction are divisible by four. Divide both numbers by common divisor and is called - reduce the fraction. We get three quarters.
Improper fractions
But suppose we multiply four times two fifths. Got eight fifths. This is the wrong fraction.
It must be brought to correct form. To do this, you need to select a whole part from it.
Here you need to use division with a remainder. We get one and three in the remainder.
One whole and three fifths is our proper fraction.
Correcting thirty-five eighths is a bit more difficult. The closest number to thirty-seven that is divisible by eight is thirty-two. When divided, we get four. We subtract thirty-two from thirty-five - we get three. Outcome: four whole and three eighths.
Equality of the numerator and denominator. And here everything is very simple and beautiful. When the numerator and denominator are equal, the result is just one.
) and the denominator by the denominator (we get the denominator of the product).
Fraction multiplication formula:
For example:
Before proceeding with the multiplication of numerators and denominators, it is necessary to check for the possibility of fraction reduction. If you manage to reduce the fraction, then it will be easier for you to continue to make calculations.
Division of an ordinary fraction by a fraction.
Division of fractions involving a natural number.
It's not as scary as it seems. As in the case of addition, we convert an integer into a fraction with a unit in the denominator. For example:
Multiplication of mixed fractions.
Rules for multiplying fractions (mixed):
- convert mixed fractions to improper;
- multiply the numerators and denominators of fractions;
- we reduce the fraction;
- if we get an improper fraction, then we convert the improper fraction to a mixed one.
Note! To multiply a mixed fraction by another mixed fraction, you first need to bring them to the form improper fractions, and then multiply by the rule of multiplication of ordinary fractions.
The second way to multiply a fraction by a natural number.
It is more convenient to use the second method of multiplication common fraction to the number.
Note! To multiply a fraction by a natural number, it is necessary to divide the denominator of the fraction by this number, and leave the numerator unchanged.
From the above example, 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.
Multilevel fractions.
In high school, three-story (or more) fractions are often found. Example:
To bring such a fraction to its usual form, division through 2 points is used:
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 in working with fractional expressions is accuracy and attentiveness. Do all calculations carefully and accurately, concentratedly and clearly. It is better to write down a few extra lines in a draft than to get confused in the calculations in your head.
2. In tasks with different types of fractions - go to the type of ordinary fractions.
3. We reduce all fractions until it is no longer possible to reduce.
4. We bring multi-level fractional expressions into ordinary ones, using division through 2 points.
5. We divide the unit into a fraction in our mind, simply by turning the fraction over.