Rounding numbers to tenths. How to round numbers into a big and smaller side of Excel features

Today we will consider a rather boring topic, without an understanding of which it is not possible to move further. This topic is called "rounding numbers" or differently "approximated numbers".

Design of lesson

Approximated values

Approximate (or approximate) values \u200b\u200bare used when it is impossible to find the exact value of something, or this value is not important for the subject under study.

For example, in words, we can say that half a million people live in the city, but this statement will not be true, since the number of people in the city changes - people come and leave, are born and dying. Therefore, it will be more correct to say that in the city there lives about Half a million man.

Another example. At nine in the morning there are classes. We left home at 8:30. After some time, on the road, we met our comrade, who asked from us how much time. When we left the house was 8:30, on the road we spent some unknown time. We do not know how long it is, so I answer the friend: "Now about about nine o'clock.

In mathematics, approximate values \u200b\u200bare specified using a special sign. It looks like this:

Reads as "approximately equal."

To indicate the approximate value of something, resort to such an operation as rounding the numbers.

Rounding numbers

To find an approximate value, this operation is used as rounding numbers.

The word "rounding" speaks for itself. Round the number means to make it round. Round is called the number that ends with zero. For example, the following numbers are round

10, 20, 30, 100, 300, 700, 1000

Any number can be made round. The procedure at which the number is made round is called rounding number.

We have already been engaged in the "rounding" of numbers when divided big numbers. Recall that for this we left without changing the figure forming the elder discharge, and the remaining numbers were replaced with zeros. But these were just the sketches that we did to facilitate division. A kind of lifehak. In fact, it was not even rounding numbers. That is why at the beginning of this paragraph we took the word rounding in quotes.

In fact, the essence of rounding is to find the nearest importance from the original. At the same time, the number can be rounded to a certain discharge - to discharge of dozens, discharge of hundreds, discharge of thousands.

Consider a simple example of rounding. The number 17 is given. It is required to round it up to the discharge of tens.

Do not pass forward to try to understand what "round up to the discharge of dozens means." When they say rounded the number 17, we require to find the nearest round number for the number 17. At the same time, during this search, it is possible to change the changes and the numbers that are in the discharge of dozens of 17 (i h).

Imagine that all numbers from 10 to 20 lie on a straight line:

The figure shows that for the number 17 the nearest round number is 20. So the answer to the task is so: 17 approximately 20

17 ≈ 20

We found an approximate value for 17, that is, rounded it to the discharge of tens. It can be seen that after rounding in the discharge of dozens, a new figure 2 appeared.

Let's try to find an approximate number for the number 12. To do this, imagine that all numbers from 10 to 20 lie on a straight line:

The figure shows that the nearest round number for 12 is the number 10. So the answer to the task is so: 12 approximately 10

12 ≈ 10

We found an approximate value for 12, that is, rounded it to the discharge of tens. This time, a number 1, which stood in the category of dozens of among 12, was not injured from rounding. Why so happened we will look later.

Let's try to find the nearest number for the number 15. We will again imagine that all numbers from 10 to 20 lie on a straight line:

The figure shows that the number 15 is equally removed from the circle numbers 10 and 20. The question arises: which from these round numbers will be an approximate value for the number 15? For such cases, it was agreed to take a larger number for approximated. 20 more than 10, so the approximate value for 15 will be the number 20

15 ≈ 20

You can round up and large numbers. Naturally, for them to draw a straight line and depicting numbers is not possible. For them, there is a way. For example, rounded the number 1456 to the discharge of dozens.

We must round down 1456 to the discharge of dozens. The discharge of dozens begins on the top five:

Now about the existence of the first digits 1 and 4 temporarily forget. The number 56 remains

Now we look at which round number is closer to the number 56. It is obvious that the nearest round number for 56 is the number 60. So replace the number 56 by number 60

It means when rounding the number 1456 to the discharge of dozens we get 1460

1456 ≈ 1460

It can be seen that after rounding the number 1456 to the discharge of dozens, the changes touched and the very discharge of tens. In the new received number in the discharge of dozens now there is a number 6, and not 5.

It is possible to round the numbers not only to the discharge of dozens. You can also round up to the discharge of hundreds, thousands, tens of thousands.

After it becomes clear that rounding is nothing, as the search for the nearest number, you can apply ready-made rules that greatly facilitate rounding numbers.

The first rule of rounding

From previous examples, it became clear that he had a rounded number to a certain discharge, the younger discharge were replaced with zeros. Numbers that are replaced by zeros call discarded numbers.

The first rounding rule is as follows:

If, when rounding the numbers, the first of the discarded numbers 0, 1, 2, 3 or 4, then the saved figure remains unchanged.

For example, rounded the number 123 to the discharge of tens.

First, we find the saved digit. To do this, you need to read the assignment itself. In the discharge, which is stated in the task and the saved digit is located. The task says: round number 123 to discharge dozens.

We see that in the discharge of dozens there is a two. It means the dual-saved digit 2

Now we find the first of the discarded numbers. The first of the discarded numbers is the figure that follows after the last digit. We see that the first digit after two is the figure 3. So the figure 3 is the first discarded digit.

Now we use the rounding rule. It says that if, when rounding the numbers, the first of the discarded numbers 0, 1, 2, 3 or 4, the saved figure remains unchanged.

And do it. We remain unchanged the saved figure, and all the youngest discharges replace zeros. In other words, all that follows after the number 2 replace zeros (more precisely zero):

123 ≈ 120

It means when rounding the number 123 to the discharge of dozens, we obtain the number 120 approximated to it.

Now let's try to round the same number 123, but already before sothen discharge.

We need to round the number 123 to the discharge of hundreds. We are looking for a saved figure again. This time, the dual-saved digit is 1, as we round the number to the discharge of hundreds.

Now we find the first of the discarded numbers. The first of the discarded numbers is the figure that follows after the last digit. See that the first digit after the unit is figure 2. So the figure 2 is the first discarded number:

Now apply a rule. It says that if, when rounding the numbers, the first of the discarded numbers 0, 1, 2, 3 or 4, the saved figure remains unchanged.

And do it. We remain unchanged the saved figure, and all the youngest discharges replace zeros. In other words, all that follows after the figure 1 replace zeros:

123 ≈ 100

Hence when rounding the number 123 to the discharge of hundreds, we obtain the number 100 approximate.

Example 3. Round the number 1234 to the discharge of dozens.

Here the retainable digit is 3. A first discarded digit is 4.

So we leave the saved figure 3 unchanged, and all that is located after it replaced with zero:

1234 ≈ 1230

Example 4. Round the number 1234 to the discharge of hundreds.

Here, the saved digit is 2. And the first discarded digit is 3. According to the rule, if the number of the numbers is the first of the discarded numbers 0, 1, 2, 3 or 4, the saved figure remains unchanged.

It means that we leave the saved number 2 unchanged, and all that is located after it replaced with zeros:

1234 ≈ 1200

Example 3. Round the number 1234 to the discharge of thousands.

Here, the saved digit is 1. And the first discarding digit is 2. According to the rule, if the number of the numbers from the discarded numbers 0, 1, 2, 3 or 4, the saved figure remains unchanged.

It means that we leave the saved number 1 unchanged, and all that is located after it replaced with zeros:

1234 ≈ 1000

The second rule of rounding

The second rounding rule is as follows:

If, when rounding the numbers, the first of the discarded numbers 5, 6, 7, 8, or 9, then the retained digit increases by one.

For example, rounded the number 675 to the discharge of tens.

First, we find the saved digit. To do this, you need to read the assignment itself. In the discharge, which is stated in the task and the saved digit is located. The task says: Round the number 675 to discharge dozens.

We see that there is a seven in the category of dozens. It means the dual-saved digit 7

Now we find the first of the discarded numbers. The first of the discarded numbers is the figure that follows after the last digit. We see that the first digit after the seven is a figure 5. So the figure 5 is the first discarded digit.

Our first of the discarded numbers is 5. So we must increase the one by the unit 7, and all that should be replaced after it zero:

675 ≈ 680

It means when rounding the number of 675 to the discharge of dozens, we get the number 680 approximated.

Now let's try to round the same number 675, but already before sothen discharge.

We need to round the number 675 to the discharge of hundreds. We are looking for a saved figure again. This time, the last digit is 6, since we are rounded the number to the discharge of hundreds:

Now we find the first of the discarded numbers. The first of the discarded numbers is the figure that follows after the last digit. We see that the first digit after the seisters is a figure 7. So the figure 7 is the first discarded number:

Now we use the second rounding rule. It says that if the first of the numbers are the first of the discarded numbers 5, 6, 7, 8 or 9, the saved digit increases by one.

We have the first of the discarded numbers 7. So we must increase the unit 6 per unit, and all that should be replaced after it with zeros:

675 ≈ 700

So when rounding the number of 675 to the discharge of hundreds, we obtain the number 700 approximated.

Example 3. Round the number 9876 to the discharge of dozens.

Here, the saved digit is 7. And the first discarded number is 6.

So increase the unit to the unit 7, and all that is located after it replaced with zero:

9876 ≈ 9880

Example 4. Round out the number 9876 to the discharge of hundreds.

Here, the saved digit is 8. And the first discarded digit is 7. According to the rule, if the first of the numbers are rounded from the discarded numbers 5, 6, 7, 8 or 9, then the saved digit increases by one.

It means that we increase the stored number 8 per unit, and all that is located after it replaced with zeros:

9876 ≈ 9900

Example 5. Round out the number 9876 to the discharge of thousands.

Here, the retainable digit is 9. And the first discarded digit is 8. According to the rule, if the number of the numbers is the first of the discarded numbers 5, 6, 7, 8, or 9, then the saved digit increases by one.

So increase the unit 9 to the unit 9, and all that is located after it replaced with zeros:

9876 ≈ 10000

Example 6. Round out the number 2971 to hundreds.

When rounding this number to hundreds should be attentive, since the saved digit is 9, and the first discarded number is 7. So the figure 9 should increase by one. But the fact is that after the increase in the nine per unit will turn out to be 10, and this figure does not fit in the category of hundreds of a new number.

In this case, in the discharge of hundreds of the new number it is necessary to write 0, and move the unit to the next discharge and folded with the digit that is located there. Next, replace all the numbers after the stored zeros:

2971 ≈ 3000

Rounding decimal fractions

When rounding decimal fractions should be particularly attentive, since the decimal fraction consists of an entire fractional part. And each of these two parts has its discharges:

Discharges of the whole part:

• discharge units
• discharge tens
• the discharge of hundreds
• the discharge of thousands

Children's discharges:

• discharge of tenths
• the discharge of hundredths
• the discharge of thousands

Consider decimal fraction 123,456 - one hundred and twenty three whole four hundred fifty-six thousandths. Here whole part This is 123, and the fractional part 456. At the same time, each of these parts have its discharges. It is very important not to confuse them:

For the whole part, the same rounding rules apply as for ordinary numbers. The difference is that after rounding the whole part and replacing all the numbers after the digit remains, the fractional part is completely discarded.

For example, rounded the fraction 123.456 to discharge dozens.It is before disclave dozens, but not discharge of tenths. It is very important not to confuse these discharges. Discharge dozens Located in the whole part, and the discharge tenths In fractional.

We must round down 123.456 to the discharge of dozens. Saved digit here is 2, and the first of the discarded numbers is 3

According to the rule, if the first numbers from the discarded numbers 0, 1, 2, 3 or 4, the saved digit remains unchanged.

So the saved figure will remain unchanged, and everything else will be replaced with zero. And what to do with the fractional part? It is simply discarded (cleaned):

123,456 ≈ 120

Now let's try to round the same fraction 123.456 to discharge units. The saved digit here will be 3, and the first of the discarded numbers is 4, which is in the fractional part:

According to the rule, if the first numbers from the discarded numbers 0, 1, 2, 3 or 4, the saved digit remains unchanged.

So the saved figure will remain unchanged, and everything else will be replaced with zero. The remaining fractional part will be discarded:

123,456 ≈ 123,0

Zero, which remained after the comma, can also be discarded. So the final answer will look like this:

123,456 ≈ 123,0 ≈ 123

Now we will deal with rounding fractional parts. For rounding fractional parts, the same rules are valid as for rounding integer parts. Let's try to round the fraction 123.456 to discharge of the tenths.In the discharge of the tenths there is a digit 4, it means it is a dual-saved digit, and the first discarded figure is 5, which is in the discharge of celloes:

According to the rule, if the number rounding numbers from the discarded numbers 5, 6, 7, 8, or 9, the saved digit increases by one.

It means the saved figure 4 will increase by one, and the rest will be zero

123,456 ≈ 123,500

Let's try to round the same fraction of 123.456 to the discharge of hundredths. Saved digit here is 5, and the first of the discarded digits is 6, which is in the discharge of thousands:

According to the rule, if the number rounding numbers from the discarded numbers 5, 6, 7, 8, or 9, the saved digit increases by one.

It means the saved figure 5 will increase by one, and the rest will be zero

123,456 ≈ 123,460

Did you like the lesson?
Join our new group VKontakte and start receiving notifications about new lessons

Rounding the numbers in life accounts for more often than it seems to many. This is especially true for people of those professions related to finance. This procedure people working in this area are well trained. But B. everyday life process bringing values \u200b\u200bto a whole mind Not unusual. Many people were safely forgotten how to round numbers, immediately after school bench. Recall the main points of this action.

In contact with

Round number

Before moving to the rules of rounding values, it is worth understanding what is a round number. If a we are talking About integers, it is necessarily ends with zero.

To the question where such skills are useful in everyday life, you can safely answer - with elementary shopping campaigns.

Using the rule of approximate counting, you can estimate how much purchases will cost and what amount must be taken with you.

It is with round numbers it is easier to calculate, without using the calculator.

For example, if in the supermarket or on the market we buy vegetables weighing 2 kg of 750 g, then in a simple conversation with the interlocutor, often does not call the exact weight, but they say that they purchased 3 kg of vegetables. When determining the distance between settlements Also apply the word "about". This means bringing the result to a convenient mind.

It should be noted that with some calculations in mathematics and task solutions, accurate values \u200b\u200bare also not always used. This is especially true in cases where they receive in response infinite periodic fraction. Let us give a few examples when approximate values \u200b\u200bare used:

• some values \u200b\u200bof permanent values \u200b\u200bare subordinate to the rounded form (the number "PI" and so on);
• tabletny values \u200b\u200bof sinus, cosine, tangent, catangens, which are rounded to a certain discharge.

Note!As practice shows, the approximation of values \u200b\u200bto the whole, of course, gives an error, but sucks insignificant. The higher the category, the more accurate the result will be.

Receipt of approximate values

This mathematical action is carried out according to certain rules.

But for each number of numbers they are different. It is noted that it is possible to round whole numbers and decimal.

But with ordinary fractions, the action is not performed.

First need them translate to decimal fractionsAnd then proceed to the procedure in the required context.

Rules of approximation of values \u200b\u200bare as follows:

• for integers - replacement of discharges following rounded, zeros;
• for decimal fractions - discard all the numbers that are for a rounded discharge.

For example, rounding 303,434 to thousands, it is necessary to replace hundreds, tens and units with zeros, that is, 303,000. In decimal fractions 3,3333 county to tenthx, simply discard all the subsequent numbers and get the result of 3.3.

Accurate rounding rules of numbers

When rounding decimal fractions is not enough enough discard numbers after rounded discharge. You can make sure this is possible on this example. If the store purchased 2 kg of 150 g of sweets, then they say that about 2 kg of sweets are purchased. If the weight is 2 kg of 850 g, then they produce rounding to the most side, that is, about 3 kg. That is, it can be seen that sometimes the rounded discharge is changed. When and how they do, the exact rules will be able to answer:

1. If, after the rounded discharge it follows the number 0, 1, 2, 3 or 4, then the rounded is left unchanged, and all subsequent numbers are discarded.
2. If after the rounded discharge it follows the number 5, 6, 7, 8 or 9, then the rounded increases per unit, and all subsequent numbers are also discarded.

For example, how correctly 7.41 closer to units. Determine the figure that follows the discharge. In this case, this is 4. Therefore, according to the rule, the number 7 is left unchanged, and the numbers 4 and 1 are discarded. That is, we get 7.

If the fraction of 7.62 is rounded, then after units it follows the number 6. According to the rule, 7 must be increased by 1, and the numbers 6 and 2 to discard. That is, the result will be 8.

The examples presented are shown how to round decimal fractions to units.

Approach to integers

It is noted that it is possible to round up to units in the same way as well. The principle is the same. Let us dwell more on rounding decimal fractions to a certain discharge in the whole part of the fraction. Imagine an example of approaching 756.247 to dozen. In the discharge of the tenths there is a digit 5. After the rounded discharge it follows the number 6. Consequently, according to the rules it is necessary to perform next steps:

• rounding in a large side of tens per unit;
• in the discharge of units, the number 6 is replaced;
• the numbers in the fractional part of the number are discarded;
• as a result, 760 are obtained.

Let's pay attention to some values \u200b\u200bin which the process of mathematical rounding to the entire rules does not display an objective picture. If you take a fraction of 8,499, then by converting it according to the rule, we get 8.

But in essence it is not quite so. If it is frozen to round up to whole, then first we get 8.5, and then we discard 5 after the comma, and we carry out rounding to the most side.

To consider the feature of rounding one or another, it is necessary to analyze specific examples And some basic information.

How to round numbers to hundredths

• To round the number to the cells, it is necessary to leave the two digits after the comma, the rest, of course, are discarded. If the first figure, which is discarded, is 0, 1, 2, 3 or 4, the previous digit remains unchanged.
• If the discarded digit is 5, 6, 7, 8, or 9, then you need to increase the previous number per unit.
• For example, if you need to round number 75.748, then after rounding we get 75.75. If we have 19,912, then as a result of rounding, or rather, in the absence of the need for its use, we get 19.91. In the case of 19.912, the figure that comes after the hundredths is not rounded, so it is simply discarded.
• If we are talking about number 18,4893, then rounding to the cells is as follows: the first figure that needs to be discarded is 3, so no changes occur. It turns out 18.48.
• In the case of a number of 0.2254, we have the first digit that is discarded when rounding to the hundredths. This is a five, which indicates that the previous number needs to be increased by one. That is, we get 0.23.
• There are cases when rounding changes all the numbers among the number. For example, to round up to the cells number 64.9972, we see that the number 7 is rounded the previous one. We get 65.00.

How to round numbers to whole

When rounding the numbers to the whole situation is the same. If we have, for example, 25.5, then after rounding we get 26. In case of sufficient number Diggers after the comma rounding occurs in this way: after rounding, 4,371251 we get 4.

The rounding to the tenths occurs in the same way as in the case of hundreds. For example, if you need to round the number 45.21618, then we get 45.2. If the second digit after the tenth is 5 or more, the previous digit increases by one. As an example, 13,6734 can be rounded, and in the end it turns out 13.7.

It is important to pay attention to the figure that is located in front of the one that is cut off. For example, if we have the number of 1.450, then after rounding we get 1.4. However, in the case of 4.851, it is advisable to round up to 4.9, since after the five there is still a unit.

To round the number to any discharge - we emphasize the figure of this discharge, and then all the numbers behind the underlined, replace zeros, and if they are after the comma, we discard. If the first replaced by zero or discarded digit is equal 0, 1, 2, 3 or 4, then the emphasized digit We leave unchanged . If the first replaced by zero or discarded digit is equal 5, 6, 7, 8 or 9, then the emphasized digit Increase by 1.

Examples.

Round up to the whole:

1) 12,5; 2) 28,49; 3) 0,672; 4) 547,96; 5) 3,71.

Decision. We emphasize the figure standing in the discharge of units (whole) and look at the figure for it. If this is a figure of 0, 1, 2, 3 or 4, then we leave an underlined digit, and all the numbers after it are discarding. If there is a digit 5 \u200b\u200bor 6 or 7 or 8 or 9 behind the underlined digit, then an underlined figure will increase by one.

1) 12 ,5≈13;

2) 28 ,49≈28;

3) 0 ,672≈1;

4) 547 ,96≈548;

5) 3 ,71≈4.

Round up to the tenths:

6) 0, 246; 7) 41,253; 8) 3,81; 9) 123,4567; 10) 18,962.

Decision. We emphasize the figure that stands in the discharge of the tenths, and then we do according to the rule: all those who stand after the underlined figures will throw out. If there was a digit of 0 or 1 or 2 or 3 or 4 behind the underlined digit, then we do not change the currency. If a digit 5 \u200b\u200bor 6 or 7 or 8 or 9, then an underlined digit will be increasing to 1 in the underlined digit.

6) 0, 2 46≈0,2;

7) 41,2 53≈41,3;

8) 3,8 1≈3,8;

9) 123,4 567≈123,5;

10) 18.9 62≈19.0. For the nine there is a six, therefore, we increase the nine to 1. (9 + 1 \u003d 10) zero write, 1 goes to the next discharge and will be 19. Just 19 We can not write in response, as it should be clear that we are rounded to The tenths - the figure in the discharge of the tenths should be. Therefore, the answer: 19.0.

Round up to hundredths:

11) 2, 045; 12) 32,093; 13) 0, 7689; 14) 543, 008; 15) 67, 382.

Decision. We emphasize the figure in the discharge of hundredths and, depending on which digit it is after the underlined, we leave an underlined figure without changing (if it is 0, 1, 2, 3, or 4) or increase the underlined digit to 1 (if it costs 5, 6, 7, 8, or 9).

11) 2, 04 5≈2,05;

12) 32,09 3≈32,09;

13) 0, 76 89≈0,77;

14) 543, 00 8≈543,01;

15) 67, 38 2≈67,38.

Important: In response, the latter should stand a figure in the category of which you are rounded.

Mathematics. 6 Class. Test 5 . Option 1 .

1. Infinite decimal non-periodic fractions are called ... numbers.

BUT) positive; IN) irrational; FROM) even; D) odd; E) rational.

2 . When rounding the number to any discharge, all the numbers following this discharge are replaced with zeros, and if they are after the comma - discard. If the first replaced by zero or discarded digit is 0, 1, 2, 3 or 4, then the number that stands before it does not change. If the first replaced by zero or discarded digit is 5, 6, 7, 8, or 9, then the number that stands in front of it increases per unit.Round up 9,974.

A) 10,0; B) 9,9; C) 9,0; D) 10; E) 9,97.

3. Round up to dozen number 264,85 .

A) 270; B) 260; C) 260,85; D) 300; E) 264,9.

4 . Round up to integer 52,71.

A) 52; B) 52,7; C) 53,7; D) 53; E) 50.

5. Round up to thousandth numbers 3, 2573 .

A) 3,257; B) 3,258; C) 3,28; D) 3,3; E) 3.

6. Round up to hundreds 49,583 .

A) 50; B) 0; C) 100; D) 49,58; E) 49.

7. The infinite periodic decimal fraction is equal to an ordinary fraction, in a numerator of which the difference between the difference after the comma and the number after the comma to the period; And the denominator consists of nine and zeros, moreover, nine as much as numbers in the period, and zeros as much as the numbers after the comma to the period. 0,58 (3) In ordinary.

8. Draw an infinite periodic decimal fraction 0,3 (12) In ordinary.

9. Draw an infinite periodic decimal fraction 1,5 (3) In a mixed number.

10. Draw an infinite periodic decimal fraction 5,2 (144) In a mixed number.

11. Anyone rational number can be recorded Record number 3 In the form of an infinite periodic decimal fraction.

BUT) 3,0 (0); IN) 3,(0); FROM) 3; D) 2,(9); E) 2,9 (0).

12 . Record ordinary fraction ½ In the form of an infinite periodic decimal fraction.

A) 0,5; B) 0,4 (9); C) 0,5 (0); D) 0,5 (00); E) 0,(5).

You will find answers to tests on the "Answers" page.

Page 1 of 1 1