Problems and examples for all operations with decimals. Decimal fractions. Decimal concept

fractional number.

Decimal notation of a fractional number is a set of two or more digits from $0$ to $9$, between which there is a so-called \textit (decimal point).

Example 1

For example, $35.02$; $100.7$; $123\456.5$; $54.89$.

The leftmost digit in the decimal notation of a number cannot be zero, the only exception being when the decimal point is immediately after the first digit $0$.

Example 2

For example, $0.357$; $0.064$.

Often the decimal point is replaced with a decimal point. For example, $35.02$; $100.7$; $123\456.5$; $54.89$.

Decimal definition

Definition 1

Decimals-- these are fractional numbers that are represented in decimal notation.

For example, $121.05; $67.9$; $345.6700$.

Decimals are used to more compactly write proper fractions, the denominators of which are the numbers $10$, $100$, $1\000$, etc. and mixed numbers, the denominators of the fractional part of which are the numbers $10$, $100$, $1\000$, etc.

For example, the common fraction $\frac(8)(10)$ can be written as a decimal $0.8$, and the mixed number $405\frac(8)(100)$ can be written as a decimal $405.08$.

Reading Decimals

Decimals, which correspond to regular fractions, are read in the same way as ordinary fractions, only the phrase “zero integers” is added in front. For example, the common fraction $\frac(25)(100)$ (read “twenty-five hundredths”) corresponds to the decimal fraction $0.25$ (read “zero point twenty-five hundredths”).

Decimal fractions that correspond to mixed numbers are read the same way as mixed numbers. For example, the mixed number $43\frac(15)(1000)$ corresponds to the decimal fraction $43.015$ (read “forty-three point fifteen thousandths”).

Places in decimals

In writing a decimal fraction, the meaning of each digit depends on its position. Those. in decimal fractions the concept also applies category.

Places in decimal fractions up to the decimal point are called the same as places in natural numbers. The decimal places after the decimal point are listed in the table:

Figure 1.

Example 3

For example, in the decimal fraction $56.328$, the digit $5$ is in the tens place, $6$ is in the units place, $3$ is in the tenths place, $2$ is in the hundredths place, $8$ is in the thousandths place.

Places in decimal fractions are distinguished by precedence. When reading a decimal fraction, move from left to right - from senior rank to younger.

Example 4

For example, in the decimal fraction $56.328$, the most significant (highest) place is the tens place, and the low (lowest) place is the thousandths place.

A decimal fraction can be expanded into digits similar to the digit decomposition of a natural number.

Example 5

For example, let's break down the decimal fraction $37.851$ into digits:

$37,851=30+7+0,8+0,05+0,001$

Ending decimals

Definition 2

Ending decimals are called decimal fractions, the records of which contain a finite number of characters (digits).

For example, $0.138$; $5.34$; $56.123456$; $350,972.54.

Any finite decimal fraction can be converted to a fraction or a mixed number.

Example 6

For example, the final decimal fraction $7.39$ corresponds to the fractional number $7\frac(39)(100)$, and the final decimal fraction $0.5$ corresponds to the proper common fraction $\frac(5)(10)$ (or any fraction which is equal to it, for example, $\frac(1)(2)$ or $\frac(10)(20)$.

Converting a fraction to a decimal

Converting fractions with denominators $10, 100, \dots$ to decimals

Before converting some proper fractions to decimals, they must first be “prepared.” The result of such preparation should be the same number of digits in the numerator and the same number of zeros in the denominator.

The essence of “preliminary preparation” of proper ordinary fractions for conversion to decimal fractions is adding such a number of zeros to the left in the numerator that the total number of digits becomes equal to the number of zeros in the denominator.

Example 7

For example, let's prepare the fraction $\frac(43)(1000)$ for conversion to a decimal and get $\frac(043)(1000)$. And the ordinary fraction $\frac(83)(100)$ does not need any preparation.

Let's formulate rule for converting a proper common fraction with a denominator of $10$, or $100$, or $1\000$, $\dots$ into a decimal fraction:

write $0$;

after it put a decimal point;

write down the number from the numerator (along with added zeros after preparation, if necessary).

Example 8

Convert the proper fraction $\frac(23)(100)$ to a decimal.

Solution.

The denominator contains the number $100$, which contains $2$ and two zeros. The numerator contains the number $23$, which is written with $2$.digits. This means that there is no need to prepare this fraction for conversion to a decimal.

Let's write $0$, put a decimal point and write down the number $23$ from the numerator. We get the decimal fraction $0.23$.

Answer: $0,23$.

Example 9

Write the proper fraction $\frac(351)(100000)$ as a decimal.

Solution.

The numerator of this fraction contains $3$ digits, and the number of zeros in the denominator is $5$, so this ordinary fraction must be prepared for conversion to a decimal. To do this, you need to add $5-3=2$ zeros to the left in the numerator: $\frac(00351)(100000)$.

Now we can form the desired decimal fraction. To do this, write down $0$, then add a comma and write down the number from the numerator. We get the decimal fraction $0.00351$.

Answer: $0,00351$.

Let's formulate rule for converting improper fractions with denominators $10$, $100$, $\dots$ into decimal fractions:

write down the number from the numerator;

Use a decimal point to separate as many digits on the right as there are zeros in the denominator of the original fraction.

Example 10

Convert the improper fraction $\frac(12756)(100)$ to a decimal.

Solution.

Let's write down the number from the numerator $12756$, then separate the $2$ digits on the right with a decimal point, because the denominator of the original fraction $2$ is zero. We get the decimal fraction $127.56$.

The sewing workshop had 5 colors of ribbon. There was more red tape than blue by 2.4 meters, but less than green by 3.8 meters. There was more white tape than black tape by 1.5 meters, but less than green tape by 1.9 meters. How many meters of tape were there in total in the workshop if the white one was 7.3 meters?

Solution
• 1) 7.3 + 1.9 = 9.2 (m) of green tape was in the workshop;
• 2) 7.3 – 1.5 = 5.8 (m) of black tape;
• 3) 9.2 – 3.8 = 5.4 (m) of red ribbon;
• 4) 5.4 - 2.4 = 3 (m) blue ribbon;
• 5) 7.3 + 9.2 + 5.8 + 5.4 + 3 = 30.7 (m).
• Answer: there was a total of 30.7 meters of tape in the workshop.

Problem 2

The length of the rectangular section is 19.4 meters and the width is 2.8 meters less. Calculate the perimeter of the site.

Solution
• 1) 19.4 – 2.8 = 16.6 (m) width of the area;
• 2) 16.6 * 2 + 19.4 * 2 = 33.2 + 38.8 = 72(m).
• Answer: the perimeter of the site is 72 meters.

Problem 3

The length of a kangaroo's jump can reach 13.5 meters in length. The world record for a person is 8.95 meters. How much further can a kangaroo jump?

Solution
• 1) 13.5 – 8.95 = 4.55 (m).
• 2) Answer: the kangaroo jumps 4.55 meters further.

Problem 4

The lowest temperature on the planet was recorded at Vostok station in Antarctica, in the summer of July 21, 1983 and was -89.2 ° C, and the hottest in the town of Al-Aziziya, on September 13, 1922 was +57.8 ° C. Calculate the difference between temperatures.

Solution
• 1) 89.2 + 57.8 = 147° C.
• Answer: The difference between temperatures is 147°C.

Problem 5

The carrying capacity of the Gazelle van is 1.5 tons, and the BelAZ mining dump truck is 24 times more. Calculate the carrying capacity of the BelAZ dump truck.

Solution
• 1) 1.5 * 24 = 36 (tons).
• Answer: BelAZ's carrying capacity is 36 tons.

Problem 6

The maximum speed of the Earth in its orbit is 30.27 km/sec, and the speed of Mercury is 17.73 km higher. At what speed does Mercury move in its orbit?

Solution
• 1) 30.27 + 17.73 = 48 (km/sec).
• Answer: Mercury's orbital speed is 48 km/sec.

Problem 7

The depth of the Mariana Trench is 11.023 km, and the height of the highest mountain in the world - Chomolungma is 8.848 km above sea level. Calculate the difference between these two points.

Solution
• 1) 11.023 + 8.848 = 19.871(km).
• Answer: 19,871 km.

Problem 8

For Kolya, like for any healthy person, the normal body temperature is 36.6 ° C, and for his four-legged friend Sharik it is 2.2 ° C higher. What temperature is considered normal for Sharik?

Solution
• 1) 36.6 + 2.2 = 38.8° C.
• Answer: Sharik’s normal body temperature is 38.8° C.

Problem 9

The painter painted 18.6 m² of fence in 1 day, and his assistant painted 4.4 m² less. How many square meters of fence will the painter and his assistant paint in a working week, if it is five days?

Solution
• 1) 18.6 – 4.4 = 14.2 (m²) will be painted by a painter’s assistant in 1 day;
• 2) 14.2 + 18.6 = 32.8 (m²) will be painted in 1 day together;
• 3) 32.8 *5 = 164 (m²).
• Answer: in a working week, the painter and his assistant will paint 164 m² of fence together.

Problem 10

Two boats simultaneously departed from two piers towards each other. The speed of one boat is 42.2 km/h, the second is 6 km/h more. What will be the distance between the boats after 2.5 hours if the distance between the piers is 140.5 km?

Solution
• 1) 42.2 + 6 = 48.2 (km/h) speed of the second boat;
• 2) 42.2 * 2.5 = 105.5 (km) will be covered by the first boat in 2.5 hours;
• 3) 48.2 * 2.5 = 120.5 (km) will be covered by the second boat in 2.5 hours;
• 4) 140.5 – 105.5 = 35 (km) distance from the first boat to the opposite pier;
• 5) 140.5 – 120. 5 = 20 (km) distance from the second boat to the opposite pier;
• 6) 35 + 20 = 55 (km);
• 7) 140 – 55 = 85 (km).
• Answer: there will be 85 km between the boats.

Problem 11

Every day a cyclist covers 30.2 km. A motorcyclist, if he spent the same amount of time, would cover a distance 2.5 times greater than a cyclist. How far can a motorcyclist cover in 4 days?

Solution
• 1) 30.2 * 2.5 = 75.5 (km) a motorcyclist will cover in 1 day;
• 2) 75.5 * 4 = 302 (km).
• Answer: a motorcyclist can cover 302 km in 4 days.

Problem 12

In 1 day, the store sold 18.3 kg of cookies, and 2.4 kg less candy. How many candies and cookies together were sold in the store that day?

Solution
• 1) 18.3 – 2.4 = 15.9 (kg) of sweets were sold in the store;
• 2) 15.9 + 18.3 = 34.2 (kg).
• Answer: a total of 34.2 kg of sweets and cookies were sold.

When adding decimal fractions, you need to write them one under the other so that the same digits are under each other, and the comma is under the comma, and add the fractions the same way you add natural numbers. Let's add, for example, the fractions 12.7 and 3.442. The first fraction contains one decimal place, and the second fraction contains three. To perform addition, we transform the first fraction so that there are three digits after the decimal point: , then

The subtraction of decimal fractions is performed in the same way. Let's find the difference between the numbers 13.1 and 0.37:

When multiplying decimal fractions, it is enough to multiply the given numbers, not paying attention to commas (like natural numbers), and then, as a result, separate as many digits from the right with a comma as there are after the decimal point in both factors in total.

For example, let's multiply 2.7 by 1.3. We have. We use a comma to separate two digits on the right (the sum of the digits of the factors after the decimal point is two). As a result, we get 2.7 1.3 = 3.51.

If the product contains fewer digits than must be separated by a comma, then the missing zeros are written in front, for example:

Let's consider multiplying a decimal fraction by 10, 100, 1000, etc. Let's say we need to multiply the fraction 12.733 by 10. We have . Separating three digits to the right with a comma, we get But. Means,

12 733 10=127.33. Thus, multiplying a decimal fraction by 10 is reduced to moving the decimal point one digit to the right.

In general, to multiply a decimal fraction by 10, 100, 1000, you need to move the decimal point in this fraction 1, 2, 3 digits to the right, adding, if necessary, a certain number of zeros to the fraction on the right). For example,

Dividing a decimal fraction by a natural number is performed in the same way as dividing a natural number by a natural number, and the comma in the quotient is placed after the division of the integer part is completed. Let us divide 22.1 by 13:

If the integer part of the dividend is less than the divisor, then the answer is zero integers, for example:

Let us now consider dividing a decimal by a decimal. Let's say we need to divide 2.576 by 1.12. To do this, in both the dividend and the divisor, move the comma to the right by as many digits as there are after the decimal point in the divisor (in this example, two). In other words, if we multiply the dividend and the divisor by 100, the quotient will not change. Then you need to divide the fraction 257.6 by the natural number 112, i.e. the problem reduces to the case already considered:

To divide a decimal fraction by, you need to move the decimal point in this fraction to the left (and, if necessary, add the required number of zeros to the left). For example, .

Just as division is not always feasible for natural numbers, it is not always feasible for decimal fractions. For example, divide 2.8 by 0.09:

The result is a so-called infinite decimal fraction. In such cases, we move on to ordinary fractions. For example:

It may turn out that some numbers are written in the form of ordinary fractions, others - in the form of mixed numbers, and others - in the form of decimal fractions. When performing operations on such numbers, you can do different things: either convert decimals to ordinary fractions and apply the rules for operating with ordinary fractions, or convert ordinary fractions and mixed numbers into decimals (if possible) and apply the rules for operating with decimals.

Fractions written in the form 0.8; 0.13; 2.856; 5.2; 0.04 is called decimal. In fact, decimals are a simplified notation for ordinary fractions. This notation is convenient to use for all fractions whose denominators are 10, 100, 1000, and so on.

Let's look at examples (0.5 is read as zero point five);

(0.15 read as, zero point fifteen);

(5.3 read as, five point three).

Please note that in the notation of a decimal fraction, a comma separates the integer part of a number from the fractional part, the integer part of a proper fraction is 0. The notation of the fractional part of a decimal fraction contains as many digits as there are zeros in the notation of the denominator of the corresponding ordinary fraction.

Let's look at an example, , , .

In some cases, it may be necessary to treat a natural number as a decimal whose fractional part is zero. It is customary to write that 5 = 5.0; 245 = 245.0 and so on. Note that in the decimal notation of a natural number, the unit of the least significant digit is 10 times less than the unit of the adjacent most significant digit. Writing decimal fractions has the same property. Therefore, immediately after the decimal point there is a place of tenths, then a place of hundredths, then a place of thousandths, and so on. Below are the names of the digits of the number 31.85431, the first two columns are the integer part, the remaining columns are the fractional part.

This fraction is read as thirty-one point eighty-five thousand four hundred and thirty-one hundred thousandths.

Adding and subtracting decimals

The first way is to convert decimal fractions into ordinary fractions and perform addition.

As can be seen from the example, this method is very inconvenient and it is better to use the second method, which is more correct, without converting decimal fractions into ordinary ones. In order to add two decimal fractions, you need to:

• equalize the number of digits after the decimal point in the terms;
• write the terms one below the other so that each digit of the second term is under the corresponding digit of the first term;
• add the resulting numbers the same way you add natural numbers;
• Place a comma in the resulting sum under the commas in the terms.

Let's look at examples:

• equalize the number of digits after the decimal point in the minuend and subtrahend;
• write the subtrahend under the minuend so that each digit of the subtrahend is under the corresponding digit of the minuend;
• perform subtraction in the same way as natural numbers are subtracted;
• put a comma in the resulting difference under the commas in the minuend and subtrahend.

Let's look at examples:

In the examples discussed above, it can be seen that the addition and subtraction of decimal fractions was performed bit by bit, that is, in the same way as we performed similar operations with natural numbers. This is the main advantage of the decimal form of writing fractions.

Multiplying Decimals

In order to multiply a decimal fraction by 10, 100, 1000, and so on, you need to move the decimal point in this fraction to the right by 1, 2, 3, and so on, respectively. Therefore, if the comma is moved to the right by 1, 2, 3 and so on digits, then the fraction will increase accordingly by 10, 100, 1000 and so on times. In order to multiply two decimal fractions, you need to:

• multiply them as natural numbers, ignoring commas;
• in the resulting product, separate with a comma on the right as many digits as there are after the commas in both factors together.

There are cases when a product contains fewer digits than is required to be separated by a comma; the required number of zeros are added to the left before this product, and then the comma is moved to the left by the required number of digits.

Let's look at examples: 2 * 4 = 8, then 0.2 * 0.4 = 0.08; 23 * 35 = 805, then 0.023 * 0.35 = 0.00805.

There are cases when one of the multipliers is equal to 0.1; 0.01; 0.001 and so on, it is more convenient to use the following rule.

• To multiply a decimal by 0.1; 0.01; 0.001 and so on, in this decimal fraction you need to move the decimal point to the left by 1, 2, 3, and so on, respectively.

Let's look at examples: 2.65 * 0.1 = 0.265; 457.6 * 0.01 = 4.576.

The properties of multiplication of natural numbers also apply to decimal fractions.

• ab = ba- commutative property of multiplication;
• (ab) c = a (bc)- the associative property of multiplication;
• a (b + c) = ab + ac is a distributive property of multiplication relative to addition.

Decimal division

It is known that if you divide a natural number a to a natural number b means to find such a natural number c, which when multiplied by b gives a number a. This rule remains true if at least one of the numbers a, b, c is a decimal fraction.

Let's look at an example: you need to divide 43.52 by 17 with a corner, ignoring the comma. In this case, the comma in the quotient should be placed immediately before the first digit after the decimal point in the dividend is used.

There are cases when the dividend is less than the divisor, then the integer part of the quotient is equal to zero. Let's look at an example:

Let's look at another interesting example.

The division process has stopped because the digits of the dividend have run out and the remainder does not have a zero. It is known that a decimal fraction will not change if any number of zeros are added to it on the right. Then it becomes clear that the numbers of the dividend cannot end.

In order to divide a decimal fraction by 10, 100, 1000, and so on, you need to move the decimal point in this fraction to the left by 1, 2, 3, and so on digits. Let's look at an example: 5.14: 10 = 0.514; 2: 100 = 0.02; 37.51: 1000 = 0.03751.

If the dividend and divisor are increased simultaneously by 10, 100, 1000, and so on times, then the quotient will not change.

Consider an example: 39.44: 1.6 = 24.65, increase the dividend and divisor by 10 times 394.4: 16 = 24.65 It is fair to note that dividing a decimal fraction by a natural number in the second example is easier.

In order to divide a decimal fraction by a decimal, you need to:

• move the commas in the dividend and divisor to the right by as many digits as there are after the decimal point in the divisor;
• divide by a natural number.

Let's consider an example: 23.6: 0.02, note that the divisor has two decimal places, therefore we multiply both numbers by 100 and get 2360: 2 = 1180, divide the result by 100 and get the answer 11.80 or 23.6: 0, 02 = 11.8.

Comparison of decimals

There are two ways to compare decimals. Method one, you need to compare two decimal fractions 4.321 and 4.32, equalize the number of decimal places and start comparing place by place, tenths with tenths, hundredths with hundredths, and so on, in the end we get 4.321 > 4.320.

The second way to compare decimal fractions is done using multiplication; multiply the above example by 1000 and compare 4321 > 4320. Which method is more convenient, everyone chooses for themselves.

We have already said that there are fractions ordinary And decimal. At this point, we've learned a little about fractions. We learned that there are proper and improper fractions. We also learned that common fractions can be reduced, added, subtracted, multiplied and divided. And we also learned that there are so-called mixed numbers, which consist of an integer and a fractional part.

We haven't fully explored ordinary fractions yet. There are many subtleties and details that should be talked about, but today we will begin to study decimal fractions, since ordinary and decimal fractions often have to be combined. That is, when solving problems you have to use both types of fractions.

This lesson may seem complicated and confusing. This is quite normal. These kinds of lessons require that they be studied, and not skimmed superficially.

Expressing quantities in fractional form

Sometimes it is convenient to show something in fractional form. For example, one tenth of a decimeter is written like this:

This expression means that one decimeter was divided into ten parts, and from these ten parts one part was taken:

As you can see in the figure, one tenth of a decimeter is one centimeter.

Consider the following example. Show 6 cm and another 3 mm in centimeters in fractional form.

So, you need to express 6 cm and 3 mm in centimeters, but in fractional form. We already have 6 whole centimeters:

but there are still 3 millimeters left. How to show these 3 millimeters, and in centimeters? Fractions come to the rescue. 3 millimeters is the third part of a centimeter. And the third part of a centimeter is written as cm

A fraction means that one centimeter was divided into ten equal parts, and from these ten parts three parts were taken (three out of ten).

As a result, we have six whole centimeters and three tenths of a centimeter:

In this case, 6 shows the number of whole centimeters, and the fraction shows the number of fractional centimeters. This fraction is read as "six point three centimeters".

Fractions whose denominator contains the numbers 10, 100, 1000 can be written without a denominator. First write the whole part, and then the numerator of the fractional part. The integer part is separated from the numerator of the fractional part by a comma.

For example, let's write it without a denominator. To do this, let's first write down the whole part. The integer part is the number 6. First we write down this number:

The whole part is recorded. Immediately after writing the whole part we put a comma:

And now we write down the numerator of the fractional part. In a mixed number, the numerator of the fractional part is the number 3. We write a three after the decimal point:

Any number that is represented in this form is called decimal.

Therefore, you can show 6 cm and another 3 mm in centimeters using a decimal fraction:

6.3 cm

It will look like this:

In fact, decimals are the same as ordinary fractions and mixed numbers. The peculiarity of such fractions is that the denominator of their fractional part contains the numbers 10, 100, 1000 or 10000.

Like a mixed number, a decimal fraction has an integer part and a fractional part. For example, in a mixed number the integer part is 6, and the fractional part is .

In the decimal fraction 6.3, the integer part is the number 6, and the fractional part is the numerator of the fraction, that is, the number 3.

It also happens that ordinary fractions in the denominator of which the numbers 10, 100, 1000 are given without an integer part. For example, a fraction is given without a whole part. To write such a fraction as a decimal, first write 0, then put a comma and write the numerator of the fraction. A fraction without a denominator will be written as follows:

Reads like "zero point five".

Converting mixed numbers to decimals

When we write mixed numbers without a denominator, we thereby convert them to decimal fractions. When converting fractions to decimals, there are a few things you need to know, which we'll talk about now.

After the whole part is written down, it is necessary to count the number of zeros in the denominator of the fractional part, since the number of zeros of the fractional part and the number of digits after the decimal point in the decimal fraction must be the same. What does it mean? Consider the following example:

At first

And you could immediately write down the numerator of the fractional part and the decimal fraction is ready, but you definitely need to count the number of zeros in the denominator of the fractional part.

So, we count the number of zeros in the fractional part of a mixed number. The denominator of the fractional part has one zero. This means that in a decimal fraction there will be one digit after the decimal point and this digit will be the numerator of the fractional part of the mixed number, that is, the number 2

Thus, when converted to a decimal fraction, a mixed number becomes 3.2.

This decimal fraction reads like this:

"Three point two"

“Tenths” because the number 10 is in the fractional part of a mixed number.

Example 2. Convert a mixed number to a decimal.

Write down the whole part and put a comma:

And you could immediately write down the numerator of the fractional part and get the decimal fraction 5.3, but the rule says that after the decimal point there should be as many digits as there are zeros in the denominator of the fractional part of the mixed number. And we see that the denominator of the fractional part has two zeros. This means that our decimal fraction must have two digits after the decimal point, not one.

In such cases, the numerator of the fractional part needs to be slightly modified: add a zero before the numerator, that is, before the number 3

Now you can convert this mixed number to a decimal fraction. Write down the whole part and put a comma:

And write down the numerator of the fractional part:

The decimal fraction 5.03 is read as follows:

"Five point three"

“Hundreds” because the denominator of the fractional part of a mixed number contains the number 100.

Example 3. Convert a mixed number to a decimal.

From previous examples, we learned that to successfully convert a mixed number to a decimal, the number of digits in the numerator of the fraction and the number of zeros in the denominator of the fraction must be the same.

Before converting a mixed number to a decimal fraction, its fractional part needs to be slightly modified, namely, to make sure that the number of digits in the numerator of the fractional part and the number of zeros in the denominator of the fractional part are the same.

First of all, we look at the number of zeros in the denominator of the fractional part. We see that there are three zeros:

Our task is to organize three digits in the numerator of the fractional part. We already have one digit - this is the number 2. It remains to add two more digits. They will be two zeros. Add them before the number 2. As a result, the number of zeros in the denominator and the number of digits in the numerator will be the same:

Now you can start converting this mixed number to a decimal fraction. First we write down the whole part and put a comma:

and immediately write down the numerator of the fractional part

3,002

We see that the number of digits after the decimal point and the number of zeros in the denominator of the fractional part of the mixed number are the same.

The decimal fraction 3.002 is read as follows:

"Three point two thousandths"

“Thousandths” because the denominator of the fractional part of the mixed number contains the number 1000.

Converting fractions to decimals

Common fractions with denominators of 10, 100, 1000, or 10000 can also be converted to decimals. Since an ordinary fraction does not have an integer part, first write down 0, then put a comma and write down the numerator of the fractional part.

Here, too, the number of zeros in the denominator and the number of digits in the numerator must be the same. Therefore, you should be careful.

Example 1.

The whole part is missing, so first we write 0 and put a comma:

Now we look at the number of zeros in the denominator. We see that there is one zero. And the numerator has one digit. This means you can safely continue the decimal fraction by writing the number 5 after the decimal point

In the resulting decimal fraction 0.5, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. This means the fraction is translated correctly.

The decimal fraction 0.5 is read as follows:

"Zero point five"

Example 2. Convert a fraction to a decimal.

A whole part is missing. First we write 0 and put a comma:

Now we look at the number of zeros in the denominator. We see that there are two zeros. And the numerator has only one digit. To make the number of digits and the number of zeros the same, add one zero in the numerator before the number 2. Then the fraction will take the form . Now the number of zeros in the denominator and the number of digits in the numerator are the same. So you can continue the decimal fraction:

In the resulting decimal fraction 0.02, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. This means the fraction is translated correctly.

The decimal fraction 0.02 is read as follows:

“Zero point two.”

Example 3. Convert a fraction to a decimal.

Write 0 and put a comma:

Now we count the number of zeros in the denominator of the fraction. We see that there are five zeros, and there is only one digit in the numerator. To make the number of zeros in the denominator and the number of digits in the numerator the same, you need to add four zeros in the numerator before the number 5:

Now the number of zeros in the denominator and the number of digits in the numerator are the same. So we can continue with the decimal fraction. Write the numerator of the fraction after the decimal point

In the resulting decimal fraction 0.00005, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. This means the fraction is translated correctly.

The decimal fraction 0.00005 is read as follows:

“Zero point five hundred thousandths.”

Converting improper fractions to decimals

An improper fraction is a fraction in which the numerator is greater than the denominator. There are improper fractions in which the denominator is the numbers 10, 100, 1000 or 10000. Such fractions can be converted to decimals. But before converting to a decimal fraction, such fractions must be separated into the whole part.

Example 1.

The fraction is an improper fraction. To convert such a fraction to a decimal fraction, you must first select the whole part of it. Let's remember how to isolate the whole part of improper fractions. If you have forgotten, we advise you to return to and study it.

So, let's highlight the whole part in the improper fraction. Recall that a fraction means division - in this case, dividing the number 112 by the number 10

Let's look at this picture and assemble a new mixed number, like a children's construction set. The number 11 will be the integer part, the number 2 will be the numerator of the fractional part, and the number 10 will be the denominator of the fractional part.

We got a mixed number. Let's convert it to a decimal fraction. And we already know how to convert such numbers into decimal fractions. First, write down the whole part and put a comma:

Now we count the number of zeros in the denominator of the fractional part. We see that there is one zero. And the numerator of the fractional part has one digit. This means that the number of zeros in the denominator of the fractional part and the number of digits in the numerator of the fractional part are the same. This gives us the opportunity to immediately write down the numerator of the fractional part after the decimal point:

In the resulting decimal fraction 11.2, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. This means the fraction is translated correctly.

This means that an improper fraction becomes 11.2 when converted to a decimal.

The decimal fraction 11.2 is read as follows:

"Eleven point two."

Example 2. Convert improper fraction to decimal.

It is an improper fraction because the numerator is greater than the denominator. But it can be converted to a decimal fraction, since the denominator contains the number 100.

First of all, let's select the whole part of this fraction. To do this, divide 450 by 100 with a corner:

Let's collect a new mixed number - we get . And we already know how to convert mixed numbers into decimal fractions.

Write down the whole part and put a comma:

Now we count the number of zeros in the denominator of the fractional part and the number of digits in the numerator of the fractional part. We see that the number of zeros in the denominator and the number of digits in the numerator are the same. This gives us the opportunity to immediately write down the numerator of the fractional part after the decimal point:

In the resulting decimal fraction 4.50, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. This means the fraction is translated correctly.

This means that an improper fraction becomes 4.50 when converted to a decimal.

When solving problems, if there are zeros at the end of the decimal fraction, they can be discarded. Let's also drop the zero in our answer. Then we get 4.5

This is one of the interesting things about decimals. It lies in the fact that the zeros that appear at the end of a fraction do not give this fraction any weight. In other words, the decimals 4.50 and 4.5 are equal. Let's put an equal sign between them:

4,50 = 4,5

The question arises: why does this happen? After all, 4.50 and 4.5 look like different fractions. The whole secret lies in the basic property of fractions, which we studied earlier. We will try to prove why the decimal fractions 4.50 and 4.5 are equal, but after studying the next topic, which is called “converting a decimal fraction to a mixed number.”

Converting a decimal to a mixed number

Any decimal fraction can be converted back to a mixed number. To do this, it is enough to be able to read decimal fractions. For example, let's convert 6.3 to a mixed number. 6.3 is six point three. First we write down six integers:

and next to three tenths:

Example 2. Convert decimal 3.002 to mixed number

3.002 is three whole and two thousandths. First we write down three integers

and next to it we write two thousandths:

Example 3. Convert decimal 4.50 to mixed number

4.50 is four point fifty. Write down four integers

and next fifty hundredths:

By the way, let's remember the last example from the previous topic. We said that the decimals 4.50 and 4.5 are equal. We also said that the zero can be discarded. Let's try to prove that the decimals 4.50 and 4.5 are equal. To do this, we convert both decimal fractions into mixed numbers.

When converted to a mixed number, the decimal 4.50 becomes , and the decimal 4.5 becomes

We have two mixed numbers and . Let's convert these mixed numbers to improper fractions:

Now we have two fractions and . It's time to remember the basic property of a fraction, which says that when you multiply (or divide) the numerator and denominator of a fraction by the same number, the value of the fraction does not change.

Let's divide the first fraction by 10

We got , and this is the second fraction. This means that both are equal to each other and equal to the same value:

Try using a calculator to divide first 450 by 100, and then 45 by 10. It will be a funny thing.

Converting a decimal fraction to a fraction

Any decimal fraction can be converted back to a fraction. To do this, again, it is enough to be able to read decimal fractions. For example, let's convert 0.3 to a common fraction. 0.3 is zero point three. First we write down zero integers:

and next to three tenths 0. Zero is traditionally not written down, so the final answer will not be 0, but simply .

Example 2. Convert the decimal fraction 0.02 to a fraction.

0.02 is zero point two. We don’t write down zero, so we immediately write down two hundredths

Example 3. Convert 0.00005 to fraction

0.00005 is zero point five. We don’t write down zero, so we immediately write down five hundred thousandths

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