Finding the percentage of a given number. Finding a number by its percentage. Finding a part of a number and a number from its part
Percent is one hundredth of a number. It follows that two percent is two hundredths, twenty percent is twenty hundredths, and so on.
The word percentage is denoted by the % sign. So, 43% of a number means 43 percent, that is, of that number. However, it is worth noting that the % sign is not written in calculations; it can be written in the problem statement and in the final result.
The value from which percentages are calculated (for example, price, length, number of candies, etc.) is 100 of its hundredths, that is, 100%.
To find one percent of a number, you divide that number by 100.
Example 1. Find one percent of the number 300.
Solution:
Answer: One percent of 300 is equal to 3.
Example 2. Find one percent of the number 27.5
Solution:
27,5: 100 = 0,275
Answer: One percent of 27.5 is equal to 0.275.
Finding percentages of a number
To find some percentage of given number, you need to divide this number by 100 and multiply by the number of percent.
Task 1. That year, the store bought 200 Christmas trees for the New Year. This year the number of purchased Christmas trees has increased by 120%. How many Christmas trees did you buy this year?
Solution: First we need to find 120% of 200, for this we need to divide 200 by 100, so we find 1%, and then multiply the result by 120:
(200: 100) 120 = 240
The number 240 is 120% of 200. This means that this year the number of Christmas trees sold increased by 240 pieces. That is, the number of Christmas trees sold this year is equal to:
200 + 240 = 440 (trees)
Answer: This year we bought 440 Christmas trees.
Task 2. There are 28 candies in a box, 25% of candies with strawberry filling. How many candies with strawberry filling are in the box?
Solution:
Answer: The box contains 7 candies with strawberry filling.
Finding a number by its percentage
To find a number from a given percentage, you need to divide this value by the number of percentages and multiply by 100.
Task. The price of a meter of cloth decreased by 24 rubles, which was 15% of the price. How much did a meter of cloth cost before the reduction?
Solution:
Answer: A meter of cloth cost 160 rubles.
Percentage of two numbers
To find out what percentage the first number is of the second, you need to divide the first number by the second and multiply the result by 100.
Task. According to the annual plan, the plant must produce products worth 1,250,000 rubles. During the 1st quarter, he issued it in the amount of 450,000 rubles. By what percentage did the plant fulfill its annual plan for the 1st quarter?
Solution:
Answer: For the 1st quarter the plan was fulfilled by 36%.
Converting percentages to decimals
To convert interest to decimal, you need to divide the percentage by 100.
Example 1: Express 25% as a decimal.
Answer: 25% is 0.25.
Example 2: Express 100% as a decimal.
Answer: 100% is 1.
Example 3: Express 230% as a decimal.
Answer: 230% is 2.3.
From these examples it follows that To convert percentages to decimal fractions, you need to move the decimal point two places to the left in the number before the % sign..
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?
IN Everyday life fractions are the most common. 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 translate ordinary 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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Finding the percentage of a given number.
Task. Soybean seeds contain 20% oil. How much oil is contained in 700 kg of soybeans?
Solution.
The problem requires finding the specified portion (20%) of a known quantity (700 kg). Such problems can be solved by reduction to unity. The basic value of the value is 700 kg. We can take it as a conventional unit. And the conventional unit is 100%.
Briefly, the conditions of the problem can be written as follows:
700 kg - 100%
X kg - 20%.
Here X is taken to be the desired mass of oil. Let's find out what mass of soybeans accounts for 1%. Since 100% accounts for 700 kg, then 1% will account for a mass that is one hundred times smaller, that is, 700: 100 = 7 (kg). This means that 20% will account for 20 times more: 7 x 20 = 140 (kg). Therefore, 700 kg of soybeans contains 140 kg of oil.
This problem can be solved in another way. If in the condition of this problem instead
20% write the number equal to it 0.2, then we get a problem of finding a fraction of a number. And such problems are solved by multiplication. From here we get another solution:
1) 20% = 0.2; 2) 700 x 0.2 = 140 (kg).
To find a few percent of a number, you need to express the percentage as a fraction, and then find the fraction of the given number.
Finding a number by its percentage.
Task. Raw cotton produces 24% fiber. How much raw cotton does it take to get 480 kg of fiber?
Solution
480 kg of fiber constitutes 24% of a certain mass of raw cotton, which we take as X kg. We will assume that X kg is 100%. Now, briefly, the problem condition can be written as follows:
480 kg - 24%
X kg - 100%
Let's solve this problem by reducing to unity. Let's find out what mass of fiber is in 1%. Since 24% accounts for 480 kg, then, obviously, 1% will have a mass 24 times less, that is, 480: 24 = 20 (kg). Next, we reason like this: if 1% accounts for a mass of 20 kg, then 100% will account for a mass 100 times greater, that is, 20 x 100 = 2000 (kg)
2 (t). Therefore, to obtain 480 kg of fiber, you need to take 2 tons of raw cotton.
This problem can be solved in another way.
If in the conditions of this problem, instead of 24%, we write the number 0.24 equal to it, then we get a problem of finding a number from its known part (fraction). And such problems are solved by division. This leads to another solution:
1) 24% = 0.24; 2) 480: 0.24 = 2000 (kg) = 2 (t).
To find a number given its percentages, you need to express the percentages as a fraction and solve the problem of finding a number given its fraction.
Percentage relationship between two numbers.
Task 1. We need to plow a field of 500 hectares. On the first day, 150 hectares were plowed. What percentage of the plowed area is the total area?
Solution
To answer the question of the problem, you need to find the ratio (quotient) of the plowed part of the plot to the entire area of the plot and express its ratio as a percentage:
150/500 = 3/10 = 0,3 = 30 %
Thus, we found the percentage ratio, that is, what percentage one number (150) is from another number (500).
To find the percentage ratio of two numbers, you need to find the ratio of these numbers and express it as a percentage.
Problem 2. A worker produced 45 parts during a shift instead of 36 according to plan. What percentage of actual output is the planned output?
Solution
To answer the question of the problem, you need to find the ratio (quotient) of the number 45 to 36 and express it as a percentage:
45: 36 = 1,25 = 125 %.
In the process of solving problems 149–156, it is necessary to bring students to an understanding of the rule for finding part of a number:
To find the part of a number expressed as a fraction, you can divide this number by the denominator of the fraction and multiply the resulting result by its numerator.
Of course, students can formulate this rule only for specific situations: to find 3 / 4 number 24, you can divide this number by the denominator fractions 4 And multiply the resulting result by the numerator 3.
149 . a) 12 birds were sitting on a branch; 2/3 of their number flew away. How many birds flew away?
b) There are 32 students in the class; 3/4 of all students skied. How many students skied?
150 . a) The cyclists covered 48 in two days. km. On the first day they covered 2/3 of the entire route. How many kilometers did they travel on the second day?
b) Someone, having 350 rubles, spent 5/7 of his money. How much money does he have left?
c) The notebook has 24 pages. The girl wrote 5/8 of all pages of the notebook. How many unwritten pages are left?
151 . An ancient problem. Having bought a chest of drawers for 36 R., I was then forced to sell it for 7/12 of the price. How many rubles did I lose on this sale?
152 . Autotourists drove 360 in three days km; on the first day they traveled 2/5, and on the second day - 3/8 of the entire journey. How many kilometers did the motor tourists travel on the third day?
153 . 1) There are 24 girls and several boys in the drama club. The number of boys is 3/8 the number of girls. How many students are in the drama club?
2) The collection contains 45 anniversary ruble coins. The number of 3 and 5 ruble coins is 2/9 of the number of ruble coins. How many anniversary coins of 1, 3 and 5 rubles are in the collection?
Students must solve problems 154–156 by first finding the indicated part of a quantity, and then increasing or decreasing this quantity by the found part. Another solution will be shown later.
154 . 1) Reduce 90 rubles by 1/10 of this amount.
2) Increase 80 rubles by 2/5 of this amount.
155 . Last month the price of the product was 90 R. Now it has dropped by 3/10 of this amount. What is the price of the product now?
156 . Last month the salary was 400 R. Now it has increased by 2/5 of this amount. What is the salary now?
In the process of solving problems 157–158 and the following problems, it is necessary to lead students to understand and correct use rules for finding a number by its part:
To find a number by its part expressed as a fraction, you can divide this part by the numerator of the fraction and multiply the resulting result by its denominator.
The formulation of this rule is complex due to the need
somehow call the number that we have named «
part »
. The authors of textbooks are forced to overcome this difficulty. So in the textbook I.V. Baranova and Z.G. Borchugova’s rule is formulated only for specific cases: to find a number, 3 / 5 which is 90 km, you need to divide 90 km by the numerator of the fraction 3 and multiply the resulting result by the denominator of the fraction 5.
This is how students can use it. True, when talking about number, it is better not to use names, since number and magnitude are not the same thing. Later in the same textbook on p. 226 is formulated general rule, in which the term we use « Part » corresponds to turnover « the number corresponding to it » , which is hardly easier.
157 . a) 120 R. constitute 3/4 of the available amount of money. What is this amount?
b) Determine the length of the segment, 3/5 of which is equal to 15 cm.
158 . a) My son is 10 years old. His age is 2/7 of his father's age. How old is father?
b) Daughter is 12 years old. Her age is 2/5 of her mother's age. How old is the mother?
The housewife spent 6 to buy vegetables R., which amounted to 1/6 of the money she had. Then she bought 2 kg apples 7 each R. per kilogram. How much money does she have left after these purchases?
160 . Father bought his son a suit for 24 R., on which I spent 1/3 of my money. After that he bought several books and had 39 left. R. How much did the books cost?
161 . The son is 8 years old, his age is 2/9 of his father’s age. And the father’s age is 3/5 of the grandfather’s age. How old is grandpa?
162 .* From the Ahmes papyrus (Egypt, c. 2000 BC).
A shepherd arrives with 70 bulls. He is asked:
How many do you bring from your numerous flock?
The shepherd answers:
I bring two-thirds of a third of the cattle. Count it!
How many bulls are there in the herd?
we see quite often in everyday life. Let's take a bar of chocolate, a pack of ice cream on which it says “56% cocoa”, “100% ice cream”. What is a percentage?
Percentage called one hundredth part. Write it down briefly 1 % . Sign % replaces the word "percentage".
Whatever number or quantity we take, its hundredth part is one percent of the given number or quantity. For example, for the number 400 (0.01 of the number 400) is the number 4, so 4 is 1% of the number 400; 1 hryvnia (0.01 hryvnia) is 1 kopeck, so 1 kopeck is 1% of the hryvnia.
For example:
The puzzle contains 500 elements. How many elements are there in 1 percent of it? Let 500 puzzle pieces be 100%. Then 1% contains 100 times less of its elements. Hence 500: 100 = 5 (el.). So, 1% is 5 pieces of the puzzle.
Please note: to find 1% of a number A, you need to divide this number by 100. Knowing what number or value is 1%, you can find the number or value that is a few percent.
For example:
Marina needs to sew on a braid, 3 cm of which is 1% of her length. Marina sewed 50% of the braid. How many centimeters of braid did she sew? Since 50% is 50 times greater than 1%, Marina sewed braids 50 times larger than 3 cm. Hence 3.50 = 150 (cm). So, Marina sewed 150 cm of braid.
In practice, it often happens that both of the above problems must be solved together - first find what number or value is in 1%, and then in several percent. Such tasks are called problems to find the percentage of a number.
For example:
Sweet pears contain 15% sugar. How much sugar is in 3 kg of pears?
Let's make a short record of the task data.
Pears: 3 kg – 100%
Sugar: ? - 15%
1. How many kilograms corresponds to 1%?
Percentage of two numbers is their ratio expressed as a percentage. A percentage shows what percentage one number is of another.