Multiplying decimals: rules, examples, solutions. Fraction
To understand how to multiply decimals, let's look at specific examples.
Rule for multiplying decimals
1) Multiply without paying attention to the comma.
2) As a result, we separate as many digits after the decimal point as there are after the decimal points in both factors together.
Examples.
Find the product of decimal fractions:
To multiply decimal fractions, we multiply without paying attention to commas. That is, we multiply not 6.8 and 3.4, but 68 and 34. As a result, we separate as many digits after the decimal point as there are after the decimal points in both factors together. In the first factor there is one digit after the decimal point, in the second there is also one. In total, we separate two numbers after the decimal point. Thus, we got the final answer: 6.8∙3.4=23.12.
We multiply decimals without taking into account the decimal point. That is, in fact, instead of multiplying 36.85 by 1.14, we multiply 3685 by 14. We get 51590. Now in this result we need to separate as many digits with a comma as there are in both factors together. The first number has two digits after the decimal point, the second has one. In total, we separate three digits with a comma. Since there is a zero after the decimal point at the end of the entry, we do not write it in the answer: 36.85∙1.4=51.59.
To multiply these decimals, let's multiply the numbers without paying attention to the commas. That is, we multiply the natural numbers 2315 and 7. We get 16205. In this number, you need to separate four digits after the decimal point - as many as there are in both factors together (two in each). Final answer: 23.15∙0.07=1.6205.
Multiplying a decimal fraction by a natural number is done in the same way. We multiply the numbers without paying attention to the decimal point, that is, we multiply 75 by 16. The resulting result should contain the same number of signs after the decimal point as there are in both factors together - one. Thus, 75∙1.6=120.0=120.
We begin multiplying decimal fractions by multiplying natural numbers, since we do not pay attention to commas. After this, we separate as many digits after the decimal point as there are in both factors together. The first number has two decimal places, the second also has two. In total, the result should be four digits after the decimal point: 4.72∙5.04=23.7888.
You already know that a * 10 = a + a + a + a + a + a + a + a + a + a. For example, 0.2 * 10 = 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2. It is easy to guess that this sum is equal to 2, i.e. 0.2 * 10 = 2.
Similarly, you can verify that:
5,2 * 10 = 52 ;
0,27 * 10 = 2,7 ;
1,253 * 10 = 12,53 ;
64,95 * 10 = 649,5 .
You probably guessed that when multiplying a decimal fraction by 10, you need to move the decimal point in this fraction to the right by one digit.
How to multiply a decimal fraction by 100?
We have: a * 100 = a * 10 * 10. Then:
2,375 * 100 = 2,375 * 10 * 10 = 23,75 * 10 = 237,5 .
Reasoning similarly, we get that:
3,2 * 100 = 320 ;
28,431 * 100 = 2843,1 ;
0,57964 * 100 = 57,964 .
Multiply the fraction 7.1212 by the number 1,000.
We have: 7.1212 * 1,000 = 7.1212 * 100 * 10 = 712.12 * 10 = 7121.2.
These examples illustrate the following rule.
To multiply a decimal fraction by 10, 100, 1,000, etc., you need to move the decimal point in this fraction to the right by 1, 2, 3, etc., respectively. numbers.
So, if the comma is moved to the right by 1, 2, 3, etc. numbers, then the fraction will increase accordingly by 10, 100, 1,000, etc. once.
Hence, if the comma is moved to the left by 1, 2, 3, etc. numbers, then the fraction will decrease accordingly by 10, 100, 1,000, etc. once .
Let us show that the decimal form of writing fractions makes it possible to multiply them, guided by the rule of multiplication of natural numbers.
Let's find, for example, the product 3.4 * 1.23. Let's increase the first factor by 10 times, and the second by 100 times. This means that we have increased the product by 1,000 times.
Therefore, the product of the natural numbers 34 and 123 is 1,000 times greater than the desired product.
We have: 34 * 123 = 4182. Then to get the answer you need to reduce the number 4,182 by 1,000 times. Let's write: 4 182 = 4 182.0. Moving the decimal point in the number 4,182.0 three digits to the left, we get the number 4.182, which is 1,000 times smaller than the number 4,182. Therefore 3.4 * 1.23 = 4.182.
The same result can be obtained using the following rule.
To multiply two decimal fractions:
1) multiply them as natural numbers, ignoring commas;
2) in the resulting product, separate with a comma on the right as many digits as there are after the commas in both factors together.
In cases where the product contains fewer digits than required to be separated by a comma, the required number of zeros are added to the left before the product, and then the comma is moved to the left by the required number of digits.
For example, 2 * 3 = 6, then 0.2 * 3 = 0.006; 25 * 33 = 825, then 0.025 * 0.33 = 0.00825.
In cases where one of the multipliers is 0.1; 0.01; 0.001, etc., it is convenient to use the following rule.
To multiply a decimal by 0.1; 0.01; 0.001, etc., you need to move the decimal point in this fraction to the left, respectively, to 1, 2, 3, etc. numbers.
For example, 1.58 * 0.1 = 0.158 ; 324.7 * 0.01 = 3.247.
The properties of multiplication of natural numbers also apply to fractional numbers:
ab = ba is the commutative property of multiplication,
(ab) с = a(b с) – associative property of multiplication,
a(b + c) = ab + ac is the distributive property of multiplication relative to addition.
Back forward
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The purpose of the lesson:
- In a fun way, introduce to students the rule for multiplying a decimal fraction by a natural number, by a place value unit, and the rule for expressing a decimal fraction as a percentage. Develop the ability to apply acquired knowledge when solving examples and problems.
- To develop and activate students’ logical thinking, the ability to identify patterns and generalize them, strengthen memory, the ability to cooperate, provide assistance, evaluate their own work and the work of each other.
- Cultivate interest in mathematics, activity, mobility, and communication skills.
Equipment: interactive whiteboard, poster with a cyphergram, posters with statements by mathematicians.
During the classes
- Organizing time.
- Oral arithmetic – generalization of previously studied material, preparation for studying new material.
- Explanation of new material.
- Homework assignment.
- Mathematical physical education.
- Generalization and systematization of acquired knowledge in a playful way using a computer.
- Grading.
2. Guys, today our lesson will be somewhat unusual, because I will not be teaching it alone, but with my friend. And my friend is also unusual, you will see him now. (A cartoon computer appears on the screen.) My friend has a name and he can talk. What's your name, buddy? Komposha replies: “My name is Komposha.” Are you ready to help me today? YES! Well then, let's start the lesson.
Today I received an encrypted cyphergram, guys, which we must solve and decipher together. (A poster is hung on the board with an oral calculation for adding and subtracting decimal fractions, as a result of which the children receive the following code 523914687. )
| 5 | 2 | 3 | 9 | 1 | 4 | 6 | 8 | 7 |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Komposha helps decipher the received code. The result of decoding is the word MULTIPLICATION. Multiplication is the key word of the topic of today's lesson. The topic of the lesson is displayed on the monitor: “Multiplying a decimal fraction by a natural number”
Guys, we know how to multiply natural numbers. Today we will look at multiplying decimal numbers by a natural number. Multiplying a decimal fraction by a natural number can be considered as a sum of terms, each of which is equal to this decimal fraction, and the number of terms is equal to this natural number. For example: 5.21 ·3 = 5.21 + 5.21 + 5.21 = 15.63 This means 5.21·3 = 15.63.
Presenting 5.21 as a common fraction to a natural number, we get
And in this case we got the same result: 15.63. Now, ignoring the comma, instead of the number 5.21, take the number 521 and multiply it by this natural number. Here we must remember that in one of the factors the comma has been moved two places to the right. When multiplying the numbers 5, 21 and 3, we get a product equal to 15.63. Now in this example we move the comma to the left two places. Thus, by how many times one of the factors was increased, by how many times the product was decreased. Based on the similarities of these methods, we will draw a conclusion.
To multiply a decimal fraction by a natural number, you need to:
1) without paying attention to the comma, multiply natural numbers;
2) in the resulting product, separate as many digits from the right with a comma as there are in the decimal fraction. The following examples are displayed on the monitor, which we analyze together with Komposha and the guys: 5.21·3 = 15.63 and 7.624·15 = 114.34.
Then I show multiplication by a round number 12.6·50 = 630. Next, I move on to multiplying a decimal fraction by a place value unit. I show the following examples: 7.423
·100 = 742.3 and 5.2·1000 = 5200. So, I introduce the rule for multiplying a decimal fraction by a digit unit:
To multiply a decimal fraction by digit units 10, 100, 1000, etc., you need to move the decimal point in this fraction to the right by as many places as there are zeros in the digit unit.
I finish my explanation by expressing the decimal fraction as a percentage. I introduce the rule:
4. To express a decimal fraction as a percentage, you must multiply it by 100 and add the % sign. I’ll give an example on a computer: 0.5 100 = 50 or 0.5 = 50%. At the end of the explanation I give the guys № 1030, № 1034, № 1032.
5. In order for the guys to rest a little, we are doing a mathematical physical education session together with Komposha to consolidate the topic. Everyone stands up, shows the solved examples to the class, and they must answer whether the example was solved correctly or incorrectly. If the example is solved correctly, then they raise their arms above their heads and clap their palms. If the example is not solved correctly, the guys stretch their arms to the sides and stretch their fingers.
6. And now you have rested a little, you can solve the tasks. Open your textbook to page 205, № 1029. In this task you need to calculate the value of the expressions:
The tasks appear on the computer. As they are solved, a picture appears with the image of a boat that floats away when fully assembled.
No. 1031 Calculate:
By solving this task on a computer, the rocket gradually folds up; after solving the last example, the rocket flies away. The teacher gives a little information to the students: “Every year, spaceships take off from the Baikonur Cosmodrome from Kazakhstan’s soil to the stars. Kazakhstan is building its new Baiterek cosmodrome near Baikonur.
No. 1035. Problem.
How far will a passenger car travel in 4 hours if the speed of the passenger car is 74.8 km/h.
This task is accompanied by sound design and a brief condition of the task displayed on the monitor. If the problem is solved, correctly, then the car begins to move forward until the finish flag.
№ 1033. Write the decimals as percentages.
0,2 = 20%; 0,5 = 50%; 0,75 = 75%; 0,92 = 92%; 1,24 =1 24%; 3,5 = 350%; 5,61= 561%.
By solving each example, when the answer appears, a letter appears, resulting in a word Well done.
The teacher asks Komposha why this word would appear? Komposha replies: “Well done, guys!” and says goodbye to everyone.
The teacher sums up the lesson and gives grades.
In the last lesson, we learned how to add and subtract decimals (see lesson “Adding and subtracting decimals”). At the same time, we assessed how much calculations are simplified compared to ordinary “two-story” fractions.
Unfortunately, this effect does not occur with multiplying and dividing decimals. In some cases, decimal notation even complicates these operations.
First, let's introduce a new definition. We'll see him quite often, and not just in this lesson.
The significant part of a number is everything between the first and last non-zero digit, including the ends. It's about about numbers only, the decimal point is not taken into account.
The digits included in the significant part of a number are called significant digits. They can be repeated and even be equal to zero.
For example, consider several decimal fractions and write out the corresponding significant parts:
- 91.25 → 9125 (significant figures: 9; 1; 2; 5);
- 0.008241 → 8241 (significant figures: 8; 2; 4; 1);
- 15.0075 → 150075 (significant figures: 1; 5; 0; 0; 7; 5);
- 0.0304 → 304 (significant figures: 3; 0; 4);
- 3000 → 3 (there is only one significant figure: 3).
Please note: the zeros inside the significant part of the number do not go anywhere. We have already encountered something similar when we learned to convert decimal fractions to ordinary ones (see lesson “ Decimals”).
This point is so important, and mistakes are made here so often, that I will publish a test on this topic in the near future. Be sure to practice! And we, armed with the concept of the significant part, will proceed, in fact, to the topic of the lesson.
Multiplying Decimals
The multiplication operation consists of three successive steps:
- For each fraction, write down the significant part. You will get two ordinary integers - without any denominators and decimal points;
- Multiply these numbers in any convenient way. Directly, if the numbers are small, or in a column. We obtain the significant part of the desired fraction;
- Find out where and by how many digits the decimal point in the original fractions is shifted to obtain the corresponding significant part. Perform reverse shifts for the significant part obtained in the previous step.
Let me remind you once again that zeros on the sides of the significant part are never taken into account. Ignoring this rule leads to errors.
- 0.28 12.5;
- 6.3 · 1.08;
- 132.5 · 0.0034;
- 0.0108 1600.5;
- 5.25 · 10,000.
We work with the first expression: 0.28 · 12.5.
- Let's write down the significant parts for the numbers from this expression: 28 and 125;
- Their product: 28 · 125 = 3500;
- In the first factor the decimal point is shifted 2 digits to the right (0.28 → 28), and in the second it is shifted by 1 more digit. In total, you need a shift to the left by three digits: 3500 → 3,500 = 3.5.
Now let's look at the expression 6.3 · 1.08.
- Let's write down the significant parts: 63 and 108;
- Their product: 63 · 108 = 6804;
- Again, two shifts to the right: by 2 and 1 digit, respectively. Total - again 3 digits to the right, so the reverse shift will be 3 digits to the left: 6804 → 6.804. This time there are no trailing zeros.
We reached the third expression: 132.5 · 0.0034.
- Significant parts: 1325 and 34;
- Their product: 1325 · 34 = 45,050;
- In the first fraction, the decimal point moves to the right by 1 digit, and in the second - by as many as 4. Total: 5 to the right. We shift by 5 to the left: 45,050 → .45050 = 0.4505. The zero was removed at the end, and added at the front so as not to leave a “bare” decimal point.
The following expression is: 0.0108 · 1600.5.
- We write the significant parts: 108 and 16 005;
- We multiply them: 108 · 16,005 = 1,728,540;
- We count the numbers after the decimal point: in the first number there are 4, in the second there are 1. The total is again 5. We have: 1,728,540 → 17.28540 = 17.2854. At the end, the “extra” zero was removed.
Finally, the last expression: 5.25 10,000.
- Significant parts: 525 and 1;
- We multiply them: 525 · 1 = 525;
- The first fraction is shifted 2 digits to the right, and the second fraction is shifted 4 digits to the left (10,000 → 1.0000 = 1). Total 4 − 2 = 2 digits to the left. We perform a reverse shift by 2 digits to the right: 525, → 52,500 (we had to add zeros).
Note the last example: since the decimal point is moved to different directions, the total shift is found through the difference. This is very important point! Here's another example:
Consider the numbers 1.5 and 12,500. We have: 1.5 → 15 (shift by 1 to the right); 12,500 → 125 (shift 2 to the left). We “step” 1 digit to the right, and then 2 to the left. As a result, we stepped 2 − 1 = 1 digit to the left.
Decimal division
Division is perhaps the most difficult operation. Of course, here you can act by analogy with multiplication: divide the significant parts, and then “move” the decimal point. But in this case there are many subtleties that negate potential savings.
Therefore, let's look at a universal algorithm, which is a little longer, but much more reliable:
- Convert all decimal fractions to ordinary fractions. With a little practice, this step will take you a matter of seconds;
- Divide the resulting fractions in the classical way. In other words, multiply the first fraction by the “inverted” second (see lesson “Multiplying and dividing numerical fractions");
- If possible, present the result again as a decimal fraction. This step is also quick, since the denominator is often already a power of ten.
Task. Find the meaning of the expression:
- 3,51: 3,9;
- 1,47: 2,1;
- 6,4: 25,6:
- 0,0425: 2,5;
- 0,25: 0,002.
Let's consider the first expression. First, let's convert fractions to decimals:
Let's do the same with the second expression. The numerator of the first fraction will again be factorized:
There is an important point in the third and fourth examples: after getting rid of the decimal notation, reducible fractions appear. However, we will not perform this reduction.
The last example is interesting because the numerator of the second fraction contains a prime number. There’s simply nothing to factorize here, so we consider it straight ahead:
Sometimes division results in an integer (I'm talking about the last example). In this case, the third step is not performed at all.
In addition, when dividing, “ugly” fractions often arise that cannot be converted to decimals. This distinguishes division from multiplication, where the results are always represented in decimal form. Of course, in this case the last step is again not performed.
Pay also attention to the 3rd and 4th examples. In them, we deliberately do not reduce ordinary fractions obtained from decimals. Otherwise, this will complicate the inverse task - representing the final answer again in decimal form.
Remember: the basic property of a fraction (like any other rule in mathematics) in itself does not mean that it must be applied everywhere and always, at every opportunity.
Just like regular numbers.
2. We count the number of decimal places for the 1st decimal fraction and for the 2nd. We add up their numbers.
3. In the final result, count from right to left the same number of digits as in the paragraph above, and put a comma.
Rules for multiplying decimal fractions.
1. Multiply without paying attention to the comma.
2. In the product, we separate the same number of digits after the decimal point as there are after the decimal points in both factors together.
When multiplying a decimal fraction by a natural number, you need to:
1. Multiply numbers without paying attention to the comma;
2. As a result, we place the comma so that there are as many digits to the right of it as there are in the decimal fraction.
Multiplying decimal fractions by column.
Let's look at an example:
We write the decimal fractions in a column and multiply them as natural numbers, not paying attention to the commas. Those. We consider 3.11 as 311, and 0.01 as 1.
The result is 311. Next, we count the number of signs (digits) after the decimal point for both fractions. The first decimal has 2 digits and the 2nd has 2. Total number digits after decimal points:
2 + 2 = 4
We count from right to left four digits of the result. The final result contains fewer numbers than need to be separated by a comma. In this case, you need to add the missing number of zeros to the left.
In our case, the first digit is missing, so we add 1 zero to the left.
Note:
When multiplying any decimal fraction by 10, 100, 1000, and so on, the decimal point in the decimal fraction is moved to the right by as many places as there are zeros after the one.
For example:
70,1 . 10 = 701
0,023 . 100 = 2,3
5,6 . 1 000 = 5 600
Note:
To multiply a decimal by 0.1; 0.01; 0.001; and so on, you need to move the decimal point in this fraction to the left by as many places as there are zeros before the one.
We count even zero!
For example:
12 . 0,1 = 1,2
0,05 . 0,1 = 0,005
1,256 . 0,01 = 0,012 56
