The original contains such shares fraction, shows the numerator - in the given example there are three of them, which means the percentage expression of one share (25%) should be tripled 25*3=75. The resulting value will be the desired value. Conclusion: to find the percentage equivalent expressed in ordinary fraction Well, divide one hundred by the denominator and multiply by the numerator.
For improper fractions, use the same calculation algorithm. The only distinctive feature of this case is that the resulting value will always be more than one hundred percent. For example, to convert the fraction 7/4, you need to divide 100 by 4 and multiply the result by 7: 100/4*7 = 175%.
If necessary, round the result to the required number of decimal places. The rounding rules are as follows: if the highest digit to be deleted contains a digit from 0 to 4, then the next highest digit (which is not deleted) does not change, and if the digit is from 5 to 9, it increases by one. If the last of these operations is subjected to the digit with the number 9, the unit is transferred to another, even more senior digit, like a column. Please note that rounding to the available number of familiar places does not always carry out this operation. Sometimes there are hidden bits in its memory that are not displayed on the indicator. Logarithmic, having low accuracy (up to two decimal places), often handles rounding in the right direction better.
If you find that a certain sequence of numbers is repeated after a decimal point, place that sequence in parentheses. They say about it that it is located "" because it repeats periodically. For example, number 53.7854785478547854... can be written as 53,(7854).
A proper fraction, the value of which is greater than one, consists of two parts: an integer and a fraction. First, divide the numerator of the fraction by its denominator. Then add the result of division to the whole part. After this, if necessary, round the result to the required number of decimal places or find the periodicity and highlight it in brackets.
All measurements are expressed by numbers, for example, length, area and volume in geometry, distance and speed in physics, etc. The result does not always turn out to be whole; this is how fractions appear. There are various actions with them and ways to convert them, in particular, you can turn a regular fraction into a decimal.
Instructions
A fraction is a notation of the form m/n, where m belongs to the set of integers, and n belongs to the set of natural numbers. Moreover, if m>n, then the fraction is improper, and a whole part can be separated from it. When multiplying the numerator m and the denominator n by the same number, the result remains unchanged. All transformation operations are based on this rule. Thus, you can turn by selecting the appropriate multiplier.
Choose a number so that the result of multiplying it by the denominator is 10. Reason backwards: is it possible to turn the number 4 into 10? Answer: no, because 10 is not divisible by 4. Then 100? Yes, 100 is divided by 4 without a remainder, the result is 25. Multiply the numerator and denominator by 25 and write the answer in decimal form:
¼ = 25/100 = 0.25.
It is not always possible to use the selection method; there are two more ways. Their principle is practically the same, only the recording differs. One of them is the gradual allocation of decimal places. Example: convert the fraction 1/8.
Think about it this way:
1/8 does not have an integer part, therefore it is equal to 0. Write down this number and place a comma after it;
Multiply 1/8 by 10 to get 10/8. From this fraction you can select an integer part equal to 1. Write it after the decimal point. Continue working with the resulting 2/8 residue;
2/8*10 = 20/8. The whole part is equal to 2, – 4/8. Subtotal – 0.12;
4/8*10 = 40/8. From the multiplication table it follows that 40 is divisible by 8. This completes your calculations, the final answer is 0.125 or 125/1000.
And finally, the third method is column division. Every time you have to divide a smaller number by a larger one, drop the zero “on top” (see figure).
In dry mathematical language, a fraction is a number that is represented as a part of one. Fractions are widely used in human life: we use fractions to indicate proportions in culinary recipes, give decimal scores in competitions, or use them to calculate discounts in stores.
Representation of fractions
There are at least two forms of writing one fractional number: in decimal form or in the form of an ordinary fraction. In decimal form, the numbers look like 0.5; 0.25 or 1.375. We can represent any of these values as an ordinary fraction:
- 0,5 = 1/2;
- 0,25 = 1/4;
- 1,375 = 11/8.
And if we easily convert 0.5 and 0.25 from an ordinary fraction to a decimal and back, then in the case of the number 1.375 everything is not obvious. How to quickly convert any decimal number to a fraction? There are three simple ways.
Getting rid of the comma
The simplest algorithm involves multiplying a number by 10 until the comma disappears from the numerator. This transformation is carried out in three steps:
Step 1: To begin with, we write the decimal number as a fraction “number/1”, that is, we get 0.5/1; 0.25/1 and 1.375/1.
Step 2: After this, multiply the numerator and denominator of the new fractions until the comma disappears from the numerators:
- 0,5/1 = 5/10;
- 0,25/1 = 2,5/10 = 25/100;
- 1,375/1 = 13,75/10 = 137,5/100 = 1375/1000.
Step 3: We reduce the resulting fractions to a digestible form:
- 5/10 = 1 × 5 / 2 × 5 = 1/2;
- 25/100 = 1 × 25 / 4 × 25 = 1/4;
- 1375/1000 = 11 × 125 / 8 × 125 = 11/8.
The number 1.375 had to be multiplied by 10 three times, which is no longer very convenient, but what do we have to do if we need to convert the number 0.000625? In this situation, we use the following method of converting fractions.
Getting rid of commas even easier
The first method describes in detail the algorithm for “removing” a comma from a decimal, but we can simplify this process. Again, we follow three steps.
Step 1: We count how many digits are after the decimal point. For example, the number 1.375 has three such digits, and 0.000625 has six. We will denote this quantity by the letter n.
Step 2: Now we just need to represent the fraction in the form C/10 n, where C are the significant digits of the fraction (without zeros, if any), and n is the number of digits after the decimal point. Eg:
- for the number 1.375 C = 1375, n = 3, the final fraction according to the formula 1375/10 3 = 1375/1000;
- for the number 0.000625 C = 625, n = 6, the final fraction according to the formula 625/10 6 = 625/1000000.
Essentially, 10n is a 1 with n zeros, so you don't have to bother raising the ten to the power - just 1 with n zeros. After this, it is advisable to reduce a fraction so rich in zeros.
Step 3: We reduce the zeros and get the final result:
- 1375/1000 = 11 × 125 / 8 × 125 = 11/8;
- 625/1000000 = 1 × 625/ 1600 × 625 = 1/1600.
The fraction 11/8 is an improper fraction because its numerator is greater than its denominator, which means we can isolate the whole part. In this situation, we subtract the whole part of 8/8 from 11/8 and get the remainder 3/8, therefore the fraction looks like 1 and 3/8.
Conversion by ear
For those who can read decimals correctly, the easiest way to convert them is by hearing. If you read 0.025 not as “zero, zero, twenty-five” but as “25 thousandths,” then you will have no problem converting decimals to fractions.
0,025 = 25/1000 = 1/40
Thus, reading a decimal number correctly allows you to immediately write it down as a fraction and reduce it if necessary.
Examples of using fractions in everyday life
At first glance, ordinary fractions are practically not used in everyday life or at work, and it is difficult to imagine a situation when you need to convert a decimal fraction into a regular fraction outside of school tasks. Let's look at a couple of examples.
Job
So, you work in a candy store and sell halva by weight. To make the product easier to sell, you divide the halva into kilogram briquettes, but few buyers are willing to purchase a whole kilogram. Therefore, you have to divide the treat into pieces each time. And if the next buyer asks you for 0.4 kg of halva, you will sell him the required portion without any problems.
0,4 = 4/10 = 2/5
Life
For example, you need to make a 12% solution to paint the model in the shade you want. To do this, you need to mix paint and solvent, but how to do it correctly? 12% is a decimal fraction of 0.12. Convert the number to a common fraction and get:
0,12 = 12/100 = 3/25
Knowing the fractions will help you mix the ingredients correctly and get the color you want.
Conclusion
Fractions are commonly used in everyday life, so if you frequently need to convert decimals to fractions, you'll want to use an online calculator that can instantly get your result as a reduced fraction.
Percentage is one of the interesting and often used tools in practice. Percentages are partially or fully used in any science, in any job, and even in everyday communication. A person who is good at percentages gives the impression of being smart and educated. In this lesson we will learn what a percentage is and what actions you can perform with it.
Lesson contentWhat is percentage?
Fractions are most common in everyday life. They even got their own names: half, third and quarter, respectively.
But there is another fraction that also occurs frequently. This is a fraction (one hundredth). This fraction is called percent. What does the fraction one hundredth mean? This fraction means that something is divided into one hundred parts and one part is taken from there. So a percentage is one hundredth of something.
A percentage is one hundredth of something
For example, one meter is 1 cm. One meter is divided into one hundred parts, and one part is taken (remember that 1 meter is 100 cm). And one part of these hundred parts is 1 cm. This means that one percent of one meter is 1 cm.
One meter is already 2 centimeters. This time, one meter was divided into one hundred parts and not one, but two parts were taken from there. And two parts out of a hundred are two centimeters. So two percent of one meter is 2 centimeters.
Another example: one ruble equals one kopeck. The ruble was divided into one hundred parts, and one part was taken from there. And one part of these hundred parts is one kopeck. This means that one percent of one ruble is one kopeck.
Percentages were so common that people replaced the fraction with a special icon that looks like this:
This entry reads "one percent." It replaces a fraction. It also replaces the decimal fraction 0.01 because if we convert a regular fraction to a decimal fraction, we get 0.01. Therefore, between these three expressions we can put an equal sign:
1% = = 0,01
Two percent in fractional form will be written as , in decimal form as 0.02, and using a special icon, two percent is written as 2%.
2% = = 0,02
How to find the percentage?
The principle of finding a percentage is the same as the usual finding of a fraction from a number. To find a percentage of something, you need to divide it into 100 parts and multiply the resulting number by the desired percentage.
For example, find 2% of 10 cm.
What does the entry 2% mean? The 2% entry replaces the . If we translate this task into a more understandable language, it will look like this:
Find from 10 cm
And we already know how to solve such tasks. This is the usual way of finding a fraction from a number. To find a fraction of a number, you need to divide this number by the denominator of the fraction, and multiply the resulting result by the numerator of the fraction.
So, divide the number 10 by the denominator of the fraction
We got 0.1. Now we multiply 0.1 by the numerator of the fraction
0.1 × 2 = 0.2
We received an answer of 0.2. This means that 2% of 10 cm is 0.2 cm. And if , then we get 2 millimeters:
0.2 cm = 2 mm
This means that 2% of 10 cm is 2 mm.
Example 2. Find 50% of 300 rubles.
To find 50% of 300 rubles, you need to divide these 300 rubles by 100, and multiply the resulting result by 50.
So, we divide 300 rubles 100
300: 100 = 3
Now multiply the result by 50
3 × 50 = 150 rub.
This means that 50% of 300 rubles is 150 rubles.
If at first it is difficult to get used to the notation with the % sign, you can replace this notation with a regular fractional notation.
For example, the same 50% can be replaced with the entry . Then the task will look like this: Find from 300 rubles, but solving such problems is still easier for us
300: 100 = 3
3 × 50 = 150
In principle, there is nothing complicated here. If difficulties arise, we advise you to stop and re-examine and.
Example 3. The garment factory produced 1,200 suits. Of these, 32% are suits of a new style. How many new style suits did the factory produce?
Here you need to find 32% of 1200. The found number will be the answer to the problem. Let's use the rule for finding percentage. Let's divide 1200 by 100 and multiply the resulting result by the desired percentage, i.e. at 32
1200: 100 = 12
12 × 32 = 384
Answer: The factory produced 384 suits of a new style.
Second way to find percentage
The second method of finding the percentage is much simpler and more convenient. It lies in the fact that the number from which the percentage is being sought will immediately be multiplied by the desired percentage, expressed as a decimal fraction.
For example, let's solve the previous problem using this method. Find 50% of 300 rubles.
The entry 50% replaces the entry , and if we convert these to a decimal fraction, we get 0.5
Now, to find 50% of 300, it will be enough to multiply the number 300 by the decimal fraction 0.5
300 × 0.5 = 150
By the way, the mechanism for finding percentage on calculators works on the same principle. To find a percentage using a calculator, you need to enter into the calculator the number from which the percentage is being sought, then press the multiplication key and enter the desired percentage. Then press the percentage key %
Finding a number by its percentage
Knowing the percentage of a number, you can find out the whole number. For example, an enterprise paid us 60,000 rubles for work, and this amounts to 2% of the total profit received by the enterprise. Knowing our share and what percentage it is, we can find out the total profit.
First you need to find out how many rubles make up one percent. How to do it? Try to guess by carefully studying the following figure:
If two percent of the total profit is 60 thousand rubles, then it is easy to guess that one percent is 30 thousand rubles. And to get these 30 thousand rubles, you need to divide 60 thousand by 2
60 000: 2 = 30 000
We found one percent of the total profit, i.e. . If one part is 30 thousand, then to determine one hundred parts, you need to multiply 30 thousand by 100
30,000 × 100 = 3,000,000
We found the total profit. It is three million.
Let's try to formulate a rule for finding a number by its percentage.
To find a number by its percentage, you need to divide the known number by the given percentage, and multiply the resulting result by 100.
Example 2. The number 35 is 7% of some unknown number. Find this unknown number.
Let's read the first part of the rule:
To find a number by its percentage, you need to divide the known number by the given percentage.
Our known number is 35, and the given percentage is 7. Divide 35 by 7
35: 7 = 5
Read the second part of the rule:
and multiply the result by 100
Our result is the number 5. Multiply 5 by 100
5 × 100 = 500
500 is an unknown number that needed to be found. You can do a check. To do this, we find 7% of 500. If we did everything correctly, we should get 35
500: 100 = 5
5 × 7 = 35
We got 35. So the problem was solved correctly.
The principle of finding a number by its percentage is the same as the usual finding of a whole number by its fraction. If percentages are confusing and confusing at first, then the percentage entry can be replaced with a fractional entry.
For example, the previous problem can be stated as follows: the number 35 is from some unknown number. Find this unknown number. We already know how to solve such problems. This is finding a number using a fraction. To find a number using a fraction, we divide this number by the numerator of the fraction and multiply the resulting result by the denominator of the fraction. In our example, the number 35 must be divided by 7 and the resulting result multiplied by 100
35: 7 = 5
5 × 100 = 500
In the future we will solve problems involving percentages, some of which will be difficult. In order not to complicate learning at first, it is enough to be able to find the percentage of a number, and the number by percentage.
Tasks for independent solution
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