We continue to study elementary problems in mathematics. This lesson is about percentage problems. We will look at several problems, and also touch on those points that were not mentioned earlier when studying percentages, considering that at first they create difficulties for learning.
In most cases, tasks for percentages come down to finding a percentage of a number, finding a number by percentage, expressing any part as a percentage, or expressing the relationship between several objects, numbers, and quantities as a percentage.
Preliminary Skills Lesson contentWays to find a percentage
The percentage can be found in various ways. The most popular way is to divide the number by 100 and multiply the result by the desired percentage.
For example, to find 60% of 200 rubles, you must first divide these 200 rubles into one hundred equal parts:
200 rubles: 100 = 2 rubles.
When we divide a number by 100, we find one percent of that number. So, dividing 200 rubles into 100 parts, we automatically found 1% of two hundred rubles, that is, we found out how many rubles fall into one part. As can be seen from the example, one part (one percent) accounts for 2 rubles.
1% from 200 rubles - 2 rubles
Knowing how many rubles fall on one part (per 1%), we can find out how many rubles fall on two parts, three, four, five, etc. That is, we can find any number of percent. To do this, it is enough to multiply these 2 rubles by the desired number of parts (percentage). Let's find sixty pieces (60%)
2 × 60 = 120 rubles.
2 × 5 = 10 rubles
We will find 90%
2 × 90 = 180 rubles.
We will find 100%
2 × 100 = 200 rubles
100% is all one hundred parts and they make up all 200 rubles.
The second way is to represent percentages as an ordinary fraction and find this fraction from the number from which you want to find the percentage.
For example, let's find the same 60% of 200 rubles. First, let's represent 60% as a fraction. 60% is sixty parts out of a hundred, that is, sixty hundredths:
Now the task can be understood as "find from 200rubles" . This is the one we studied earlier. Recall that to find a fraction of a number, you need to divide this number by the denominator of the fraction and multiply the result by the numerator of the fraction
200: 100 = 2
2 x 60 = 120
Or multiply the number by a fraction ():
The third way is to represent the percentage as a decimal fraction and multiply the number by this decimal fraction.
For example, let's find the same 60% of 200 rubles. Let's start by representing 60% as a fraction. 60% percent is sixty parts out of a hundred
Let's do the division in this fraction. Move the comma in 60 two digits to the left:
Now we find 0.60 from 200 rubles. To find the decimal fraction of a number, you need to multiply this number by the decimal fraction:
200 × 0.60 = 120 rubles
The given method of finding a percentage is the most convenient, especially if a person is used to using a calculator. This method allows you to find the percentage in one step.
As a rule, expressing a percentage in decimal fractions is not difficult. It suffices to assign "zero integers" before the percentage if the percentage is a two-digit number, or to attribute "zero integers" and another zero if the percentage is a single-digit number. Examples:
60% \u003d 0.60 - assigned zero integers before the number 60, since the number 60 is two-digit
6% \u003d 0.06 - assigned zero integers and another zero before the number 6, since the number 6 is single-digit.
When dividing by 100, we used the method of moving the decimal point two digits to the left. In the answer 0.60, the zero after the number 6 was preserved. But if you perform this division by a corner, zero disappears - you get the answer 0.6
It must be remembered that decimal fractions 0.60 and 0.6 are equal and carry the same value.
0,60 = 0,6
In the same “corner”, you can continue dividing indefinitely, each time adding zero to the remainder, but this will be a meaningless action.
You can express percentages as a decimal not only by dividing by 100, but also by multiplying. The percent sign (%) by itself replaces the 0.01 multiplier. And if we take into account that the number of percent and the percent sign are written together, then there is an “invisible” multiplication sign (×) between them.
So, for example, 45% actually look like this
Replace the percent sign with a factor of 0.01
This multiplication by 0.01 is done by moving the decimal point two digits to the left
Task 1. The family budget is 75 thousand rubles a month. Of these, 70% is money earned by dad. How much did mom earn?
Decision
Total 100 percent. If dad earned 70% of the money, then the remaining 30% of the money was earned by mom.
Task 2. The family budget is 75 thousand rubles a month. Of these, 70% is money earned by dad, and 30% is money earned by mom. How much money did each earn?
Decision
We will find 70 and 30 percent of 75 thousand rubles. So we will determine how much money each earned. For convenience, 70% and 30% will be written as decimal fractions
75 × 0.70 \u003d 52.5 (dad earned a thousand rubles)
75 × 0.30 = 22.5 (mother earned a thousand rubles)
Examination
52,5 + 22,5 = 75
75 = 75
Answer: 52.5 thousand rubles Papa earned 22.5 rubles. mother earned.
Task 3. When cooling, the bread loses up to 4% of its mass as a result of water evaporation. How many kilograms will evaporate when 12 tons of bread cools.
Decision
Convert 12 tons to kilograms. There are 1000 kilograms in one ton, 12 times more in 12 tons
1000 × 12 = 12,000 kg
Now let's find 4% of 12000. The result will be the answer to the problem:
12,000 × 0.04 = 480 kg
Answer: when cooling 12 tons of bread, 480 kilograms will evaporate.
Task 4. When dried, apples lose 84% of their weight. How many dried apples will be obtained from 300 kg of fresh ones?
Find 84% of 300 kg
300: 100 × 84 = 252 kg
300 kg of fresh apples will lose 252 kg of their mass as a result of drying. To answer the question how many dried apples will turn out, you need to subtract 252 from 300
300 - 252 = 48 kg
Answer: from 300 kg of fresh apples you get 48 kg of dried ones.
Task 5. Soybean seeds contain 20% oil. How much oil is in 700 kg of soybeans?
Decision
Find 20% of 700 kg
700 × 0.20 = 140 kg
Answer: 700 kg of soy contains 140 kg of oil
Task 6. Buckwheat contains 10% proteins, 2.5% fats and 60% carbohydrates. How many of these products are contained in 14.4 centners of buckwheat?
Decision
Let's translate 14.4 centners into kilograms. There are 100 kilograms in one centner, 14.4 times more in 14.4 centners
100 × 14.4 = 1440 kg
Find 10%, 2.5% and 60% of 1440 kg
1440 × 0.10 = 144 (kg of proteins)
1440 × 0.025 = 36 (kg fat)
1440 x 0.60 = 864 (kg carbs)
Answer: 14.4 kg of buckwheat contains 144 kg of proteins, 36 kg of fat, 864 kg of carbohydrates.
Task 7. Schoolchildren collected 60 kg of oak, acacia, linden and maple seeds for the forest nursery. Acorns made up 60%, maple seeds 15%, linden seeds 20% of all seeds, and the rest was acacia seeds. How many kilograms of acacia seeds were collected by schoolchildren?
Decision
We will take for 100% the seeds of oak, acacia, linden and maple. Let us subtract from these 100% the percentages expressing the seeds of oak, linden and maple. So we find out how many percent are acacia seeds:
100% − (60% + 15% + 20%) = 100% − 95% = 5%
Now we find acacia seeds:
60 × 0.05 = 3 kg
Answer: schoolchildren collected 3 kg of acacia seeds.
Examination:
60 x 0.60 = 36
60 x 0.15 = 9
60 x 0.20 = 12
60 x 0.05 = 3
36 + 9 + 12 + 3 = 60
60 = 60
Task 8. The man bought food. Milk costs 60 rubles, which is 48% of the cost of all purchases. Determine the total amount of money spent on products.
Decision
This is a problem of finding a number by its percentage, that is, by its known part. This problem can be solved in two ways. The first is to express the known number of percent as a decimal fraction and find the unknown number from this fraction
Express 48% as a decimal
48% : 100 = 0,48
Knowing that 0.48 is 60 rubles, we can determine the amount of all purchases. To do this, you need to find an unknown number in decimal fraction:
60: 0.48 = 125 rubles
So the total amount of money spent on food is 125 rubles.
The second way is to first find out how much money falls on one percent, then multiply the result by 100
48% is 60 rubles. If we divide 60 rubles by 48, then we will find out how many rubles fall on 1%
60: 48% = 1.25 rubles
1% accounts for 1.25 rubles. Total 100 percent. If we multiply 1.25 rubles by 100, we get the total amount of money spent on products
1.25 × 100 = 125 rubles
Task 9. 35% of dried plums come out of fresh plums. How many fresh plums do you need to take to get 140 kg of dried ones? How many dried plums will be obtained from 600 kg of fresh ones?
Decision
Let's express 35% as a decimal fraction and find the unknown number from this fraction:
35% = 0,35
140: 0.35 = 400 kg
To get 140 kg of dried plums, you need to take 400 kg of fresh ones.
Let's answer the second question of the problem - how many dried plums will turn out from 600 kg of fresh ones? If 35% of dried plums come out of fresh plums, then it is enough to find these 35% from 600 kg of fresh plums
600 × 0.35 = 210 kg
Answer: to get 140 kg of dried plums, you need to take 400 kg of fresh ones. From 600 kg of fresh plums, 210 kg of dried plums will be obtained.
Task 10. The assimilation of fats by the human body is 95%. For a month, the student consumed 1.2 kg of fat. How much fat can be absorbed by his body?
Decision
Convert 1.2 kg to grams
1.2 x 1000 = 1200 g
Find 95% of 1200 g
1200 × 0.95 = 1140 g
Answer: 1140 g of fat can be absorbed by the student's body.
Expressing numbers as a percentage
Percentage, as mentioned earlier, can be represented as a decimal fraction. To do this, it is enough to divide the number of these percentages by 100. For example, let's represent 12% as a decimal fraction:
Comment. We are not currently finding a percentage of something, but simply writing it as a decimal fraction.
But the reverse process is also possible. A decimal fraction can be represented as a percentage. To do this, multiply this fraction by 100 and put a percent sign (%)
Let's represent the decimal fraction 0.12 as a percentage
0.12 x 100 = 12%
This action is called expressed as a percentage or expressing numbers in hundredths.
Multiplication and division are inverse operations. For example, if 2 × 5 = 10, then 10: 5 = 2
Similarly, division can be written in reverse. If 10:5 = 2, then 2 × 5 = 10:
The same thing happens when we express a decimal as a percentage. So, 12% was expressed as a decimal as follows: 12: 100 = 0.12 but then the same 12% was “returned” using multiplication, writing the expression 0.12 × 100 = 12%.
Similarly, you can express as a percentage any other numbers, including integers. For example, let's express the number 3 as a percentage. Multiply this number by 100 and add the percent sign to the result:
3 x 100 = 300%
Large percentages like 300% can be confusing at first, because people are used to counting 100% as the maximum share. From additional information about fractions, we know that one whole object can be denoted by unity. For example, if there is a whole uncut cake, then it can be denoted by 1
The same cake can be designated as 100% cake. In this case, both unit and 100% will denote the same whole cake:
Let's cut the cake in half. In this case, the one will turn into the decimal number 0.5 (because it's half a unit), and 100% will turn into 50% (because 50 is half a hundred)
We will return the whole cake, one unit and 100%
Let's draw two more such cakes with the same notation:
If one cake is a unit, then three cakes are three units. Each cake is 100% intact. If you add these three hundred, you get 300%.
Therefore, when converting integers to percentages, we multiply these numbers by 100.
Task 2. Express as a percentage the number 5
5 x 100 = 500%
Task 3. Express as a percentage the number 7
7 x 100 = 700%
Task 4. Express as a percentage the number 7.5
7.5 x 100 = 750%
Task 5. Express as a percentage the number 0.5
0.5 x 100 = 50%
Task 6. Express as a percentage the number 0.9
0.9 x 100 = 90%
Example 7. Express as a percentage the number 1.5
1.5 x 100 = 150%
Example 8. Express as a percentage the number 2.8
2.8 x 100 = 280%
Task 9. George is walking home from school. For the first fifteen minutes he walked 0.75 of the way. The rest of the time he went the remaining 0.25 of the way. Express as a percentage the parts of the path traveled by George.
Decision
0.75 x 100 = 75%
0.25 x 100 = 25%
Task 10. John was treated to half an apple. Express this half as a percentage.
Decision
Half an apple is written as a fraction of 0.5. To express this fraction as a percentage, multiply it by 100 and add the percent sign to the result.
0.5 x 100 = 50%
Analogues in the form of fractions
A value expressed as a percentage has its counterpart in the form of an ordinary fraction. So, the analogue for 50% is a fraction. Fifty percent can also be called the word "half".
The analogue for 25% is a fraction. Twenty-five percent can also be called the word "quarter".
The analogue for 20% is a fraction. Twenty percent can also be called the words "fifth".
The analogue for 40% is a fraction.
The analogue for 60% is a fraction
Example 1. Five centimeters is 50% of a decimeter, or just half. In all cases, we are talking about the same value - five centimeters out of ten
Example 2. Two and a half centimeters is 25% of a decimeter or just a quarter
Example 3. Two centimeters is 20% of a decimeter or
Example 4. Four centimeters is 40% of a decimeter or
Example 5. Six centimeters is 60% of a decimeter or
Decrease and increase in interest
When increasing or decreasing a value expressed as a percentage, the preposition "on" is used.
Examples:
- Increase by 50% - means to increase the value by 1.5 times;
- Increase by 100% - means to increase the value by 2 times;
- To increase by 200% means to increase by 3 times;
- Decrease by 50% - means to reduce the value by 2 times;
- To reduce by 80% means to reduce by 5 times.
Example 1. Ten centimeters increased by 50%. How many centimeters did you get?
To solve such problems, you need to take the initial value as 100%. The original value is 10 cm. 50% of them are 5 cm
The original 10 cm was increased by 50% (by 5 cm), so it turned out 10 + 5 cm, that is, 15 cm
An analogue of increasing ten centimeters by 50% is a multiplier of 1.5. If you multiply 10 cm by it, you get 15 cm
10 × 1.5 = 15 cm
Therefore, the expressions "increase by 50%" and "increase by 1.5 times" mean the same thing.
Example 2. Five centimeters increased by 100%. How many centimeters did you get?
Let's take the original five centimeters as 100%. One hundred percent of these five centimeters will themselves be 5 cm. If you increase 5 cm by the same 5 cm, you get 10 cm
An analogue of increasing five centimeters by 100% is a factor of 2. If you multiply 5 cm by it, you get 10 cm
5×2=10cm
Therefore, the expressions "increase by 100%" and "increase by 2 times" mean the same thing.
Example 3. Five centimeters increased by 200%. How many centimeters did you get?
Let's take the original five centimeters as 100%. Two hundred percent is two times one hundred percent. That is, 200% of 5 cm will be 10 cm (5 cm for every 100%). If you increase 5 cm by these 10 cm, you get 15 cm
An analogue of an increase in five centimeters by 200% is a factor of 3. If you multiply 5 cm by it, you get 15 cm
5×3=15cm
Therefore, the expressions "increase by 200%" and "increase by 3 times" mean the same thing.
Example 4. Ten centimeters have been reduced by 50%. How many centimeters are left?
Let's take the original 10 cm as 100%. Fifty percent of 10 cm is 5 cm. If you reduce 10 cm by these 5 cm, there will be 5 cm
The analogue of reducing ten centimeters by 50% is the divisor 2. If you divide 10 cm by it, you get 5 cm
10:2=5cm
Therefore, the expressions "reduce by 50%" and "reduce by 2 times" mean the same thing.
Example 5. Ten centimeters have been reduced by 80%. How many centimeters are left?
Let's take the original 10 cm as 100%. Eighty percent of 10 cm is 8 cm. If you reduce 10 cm by these 8 cm, there will be 2 cm
The analogue of reducing ten centimeters by 80% is the divisor 5. If you divide 10 cm by it, you get 2 cm
10:5=2cm
Therefore, the expressions "reduce by 80%" and "reduce by 5 times" mean the same thing.
When solving problems for decreasing and increasing interest, you can multiply / divide the value by the multiplier specified in the task.
Task 1. By what percentage did the value change if it increased by 1.5 times?
The value referred to in the problem can be designated as 100%. Then multiply these 100% by a factor of 1.5
100% × 1.5 = 150%
Now, from the obtained 150%, subtract the original 100% and get the answer to the problem:
150% − 100% = 50%
Task 2. By what percentage did the value change if it decreased by 4 times?
This time there will be a decrease in the value, so we will perform the division. The value of which is mentioned in the problem is denoted as 100%. Next, we divide these 100% by a divisor of 4
From the original 100%, subtract the resulting 25% and get the answer to the problem:
100% − 25% = 75%
This means that with a decrease in the value by 4 times, it decreased by 75%.
Task 3. By what percent did the value change if it decreased by 5 times?
The value of which is mentioned in the problem is denoted as 100%. Next, we divide these 100% by a divisor of 5
From the original 100%, subtract the resulting 20% and get the answer to the problem:
100% − 20% = 80%
This means that when the value decreases by 5 times, it decreases by 80%.
Task 4. By what percent did the value change if it decreased by 10 times?
The value of which is mentioned in the problem is denoted as 100%. Next, divide these 100% by a divisor of 10
From the original 100%, subtract the resulting 10% and get the answer to the problem:
100% − 10% = 90%
This means that when the value decreases by 10 times, it decreases by 90%.
The task of finding the percentage
To express something as a percentage, you first need to write down a fraction showing what part the first number is from the second, then divide in this fraction and express the result as a percentage.
For example, suppose there are five apples. Two apples are red and three are green. Express red and green apples as a percentage.
First you need to find out what part red apples make up. There are five apples in total, and two red ones. So two out of five or two-fifths are red apples:
There are three green apples. So three out of five or three-fifths are green apples:
We have two fractions and . Let's do the division in these fractions
We got decimal fractions 0.4 and 0.6. Now let's express these decimal fractions as a percentage:
0.4 x 100 = 40%
0.6 x 100 = 60%
So 40% are red apples, 60% are green.
And all five apples make up 40% + 60%, that is, 100%
Task 2. The mother gave 200 rubles to two sons. Mom gave 80 rubles to the younger brother, and 120 rubles to the older brother. Express as a percentage the money given to each brother.
Decision
The younger brother received 80 rubles out of 200 rubles. We write the fraction eighty-two hundredths:
The elder brother received 120 rubles out of 200 rubles. We write the fraction one hundred and twenty-two hundredths:
We have fractions and . Let's do the division in these fractions
Let's express the results as a percentage:
0.4 x 100 = 40%
0.6 x 100 = 60%
This means that the younger brother received 40% of the money, and the elder brother received 60%.
Some fractions, showing what part the first number is from the second, can be reduced.
So fractions could be reduced. From this, the answer to the problem would not change:
Task 3. The family budget is 75 thousand rubles a month. Of these, 52.5 thousand rubles. - money earned by dad. 22.5 thousand rubles - money earned by mom. Express as a percentage the money earned by dad and mom.
Decision
This problem, like the previous one, is a problem of finding a percentage.
Let's express as a percentage the money earned by dad. He earned 52.5 thousand rubles out of 75 thousand rubles
Let's do the division in this fraction:
0.7 x 100 = 70%
So dad earned 70% of the money. Further, it is not difficult to guess that the remaining 30% of the money was earned by my mother. After all, 75 thousand rubles is all 100% of the money. Let's check to be sure. Mom earned 22.5 thousand rubles. out of 75 thousand rubles. We write down the fraction, perform the division and express the result as a percentage:
Task 4. The student is training to do pull-ups on the crossbar. Last month, he could do 8 pull-ups per set. This month he can do 10 pull-ups per set. By what percent did he increase his pull-ups?
Decision
Find out how many more pull-ups a student does this month than last
Find out what part two pull-ups are from eight pull-ups. To do this, we find the ratio 2 to 8
Let's do the division in this fraction
Let's express the result as a percentage:
0.25 x 100 = 25%
So the student increased the number of pull-ups by 25%.
This problem can also be solved by the second, faster method - find out how many times 10 pull-ups are more than 8 pull-ups and express the result as a percentage.
To find out how many times ten pull-ups are more than eight pull-ups, you need to find the ratio of 10 to 8
Perform division in the resulting fraction
Let's express the result as a percentage:
1.25 x 100 = 125%
The pull-up rate for the current month is 125%. This statement must be understood as "is 125%", not how “the indicator increased by 125%”. These are two different statements expressing different quantities.
The statement "is 125%" should be understood as "eight pull-ups that are 100% plus two pull-ups that are 25% of eight pull-ups." Graphically it looks like this:
And the statement “increased by 125%” should be understood as “to the current eight pull-ups, which were 100%, another 100% was added (8 more pull-ups) plus another 25% (2 pull-ups)”. There are 18 pull-ups in total.
100% + 100% + 25% = 8 + 8 + 2 = 18 pull-ups
Graphically, this statement looks like this:
In total, it turns out 225%. If we find 225% of eight pull-ups, we get 18 pull-ups
8 × 2.25 = 18
Task 5. Last month, the salary was 19.2 thousand rubles. In the current month, it amounted to 20.16 thousand rubles. By what percent did the salary increase?
This problem, like the previous one, can be solved in two ways. The first is to first find out by how many rubles the salary has increased. Next, find out how much this increase is from last month's salary
Find out by how much the salary increased:
20.16 - 19.2 \u003d 0.96 thousand rubles.
We will find out what part of 0.96 thousand rubles. is from 19.2. To do this, we find the ratio of 0.96 to 19.2
Perform division in the resulting fraction. Along the way, remember:
Let's express the result as a percentage:
0.05 x 100 = 5%
So the salary has increased by 5%.
Let's solve the problem in the second way. We will find out how many times 20.16 thousand rubles. more than 19.2 thousand rubles. To do this, we find the ratio of 20.16 to 19.2
Let's perform the division in the resulting fraction:
Let's express the result as a percentage:
1.05 x 100 = 105%
The salary is 105%. That is, this includes 100%, which amounted to 19.2 thousand rubles, plus 5%, which amounted to 0.96 thousand rubles.
100% + 5% = 19,2 + 0,96
Task 6. The price of a laptop has increased by 5% this month. What is its price if last month it cost 18.3 thousand rubles?
Decision
Find 5% of 18.3:
18.3 x 0.05 = 0.915
Let's add these 5% to 18.3:
18.3 + 0.915 = 19.215 thousand rubles
Answer: the price of a laptop is 19.215 thousand rubles.
Task 7. The price of a laptop has decreased by 10% this month. What is its price if last month it cost 16.3 thousand rubles?
Decision
Find 10% of 16.3:
16.3 x 0.10 = 1.63
Subtract these 10% from 16.3:
16.3 − 1.63 = 14.67 (thousand rubles)
Such tasks can be written briefly:
16.3 - (16.3 × 0.10) = 14.67 (thousand rubles)
Answer: the price of a laptop is 14.67 thousand rubles.
Task 8. Last month, the price of a laptop was 21 thousand rubles. This month the price has risen to 22.05 thousand rubles. By what percentage has the price increased?
Decision
Determine how much rubles the price increased
22.05 − 21 = 1.50 (thousand rubles)
We will find out what part of 1.05 thousand rubles. is from 21 thousand rubles.
Express the result as a percentage
0.05 x 100 = 5%
Answer: laptop price increased by 5%
Task 8. The worker was supposed to make 600 parts according to the plan, and he made 900 parts. By what percentage did he complete the plan?
Decision
Let's find out how many times 900 parts are more than 600 parts. To do this, we find the ratio of 900 to 600
The value of this fraction is 1.5. Let's express this value as a percentage:
1.5 x 100 = 150%
So the worker fulfilled the plan by 150%. That is, he completed it 100%, having made 600 parts. Then he made another 300 parts, which is 50% of the original plan.
Answer: the worker completed the plan by 150%.
Percent comparison
We have already compared values many times in various ways. Our first tool was the difference. So, for example, to compare 5 rubles and 3 rubles, we wrote down the difference 5−3. Having received the answer 2, one could say that "five rubles is more than three rubles by two rubles."
The answer obtained as a result of subtraction in everyday life is called not “difference”, but “difference”.
So, the difference between five and three rubles is two rubles.
The next tool we used to compare quantities was the ratio. The ratio allowed us to find out how many times the first number is greater than the second (or how many times the first number contains the second).
So, for example, ten apples are five times more than two apples. Or put another way, ten apples contains two apples five times. This comparison can be written using the relation
But the values can also be compared in percentages. For example, to compare the price of two goods not in rubles, but to evaluate how much the price of one goods is more or less than the price of the other as a percentage.
To compare values in percent, one of them must be designated as 100%, and the second based on the conditions of the problem.
For example, let's find out how many percent ten apples are more than eight apples.
For 100%, you need to designate the value with which we compare something. We are comparing 10 apples to 8 apples. So for 100% we denote 8 apples:
Now our task is to compare how many percent of 10 apples are more than these 8 apples. 10 apples is 8+2 apples. This means that by adding two more apples to eight apples, we will increase 100% by another number of percent. To find out which one, let's determine how many percent of eight apples are two apples
Adding this 25% to eight apples, we get 10 apples. And 10 apples is 8 + 2, that is, 100% and another 25%. Total we get 125%
So ten apples are more than eight apples by 25%.
Now let's solve the inverse problem. Find out how much percent eight apples are less than ten apples. The answer immediately suggests itself that eight apples are 25% less. However, it is not.
We are comparing eight apples to ten apples. We agreed that for 100% we will take what we compare with. Therefore, this time we take 10 apples for 100%:
Eight apples is 10−2, that is, by reducing 10 apples by 2 apples, we will reduce them by some percentage. To find out which one, let's determine how many percent of ten apples are two apples
Subtracting these 20% from ten apples, we get 8 apples. And 8 apples is 10−2, that is, 100% and minus 20%. Total we get 80%
So eight apples are less than ten apples by 20%.
Task 2. How many percent is 5,000 rubles more than 4,000 rubles?
Decision
Let's take 4000 rubles for 100%. 5,000 is more than 4,000 per 1,000. So by increasing four thousand by one thousand, we will increase four thousand by a certain percentage. Let's find out which one. To do this, we determine what part one thousand is from four thousand:
Let's express the result as a percentage:
0.25 x 100 = 25%
1000 rubles from 4000 rubles is 25%. If you add these 25% to 4000, you get 5000 rubles. So 5000 rubles is 25% more than 4000 rubles
Task 3. How many percent is 4,000 rubles less than 5,000 rubles?
This time we compare 4000 with 5000. Let's take 5000 as 100%. Five thousand more than four thousand for one thousand rubles. Find out what part one thousand is from five thousand
One thousand out of five thousand is 20%. If we subtract these 20% from 5,000 rubles, we get 4,000 rubles.
So 4000 rubles is less than 5000 rubles by 20%
Tasks for concentration, alloys and mixtures
Suppose there was a desire to prepare some kind of juice. We have water and raspberry syrup available.
Pour 200 ml of water into a glass:
Add 50 ml of raspberry syrup and stir the resulting liquid. As a result, we get 250 ml of raspberry juice (200 ml water + 50 ml syrup = 250 ml juice)
What part of the resulting juice is raspberry syrup?
Raspberry syrup makes up juice. We calculate this ratio, we get the number 0.20. This number indicates the amount of dissolved syrup in the resulting juice. Let's call this number syrup concentration.
The concentration of a solute is the ratio of the amount of a solute or its mass to the volume of a solution.
The concentration is usually expressed as a percentage. Let's express the syrup concentration as a percentage:
0.20 x 100 = 20%
Thus, the concentration of syrup in raspberry juice is 20%.
Substances in solution can be heterogeneous. For example, let's mix 3 liters of water and 200 grams of salt.
The mass of 1 liter of water is 1 kg. Then the mass of 3 liters of water will be 3 kg. Converting 3 kg to grams, we get 3 kg = 3000 g.
Now, in 3000 g of water, we lower 200 g of salt and mix the resulting liquid. The result will be a saline solution, the total mass of which will be 3000 + 200, that is, 3200 g. Let's find the concentration of salt in the resulting solution. To do this, we find the ratio of the mass of dissolved salt to the mass of the solution
This means that when mixing 3 liters of water and 200 g of salt, a 6.25% salt solution will be obtained.
Similarly, the amount of a substance in an alloy or in a mixture can be determined. For example, an alloy contains tin with a mass of 210 g, and silver with a mass of 90 g. Then the mass of the alloy will be 210 + 90, that is, 300 g. The alloy will contain tin and silver. The percentage of tin will be 70% and silver 30%
When two solutions are mixed, a new solution is obtained, consisting of the first and second solutions. The new solution may have a different concentration of the substance. A useful skill is the ability to solve concentration problems, alloys and mixtures. In general, the meaning of such tasks is to track the changes that occur when mixing solutions of different concentrations.
Mix two raspberry juices. The first juice of 250 ml contains 12.8% raspberry syrup. And the second juice with a volume of 300 ml contains 15% raspberry syrup. Pour these two juices into a large glass and mix. As a result, we get a new juice with a volume of 550 ml.
Now we will determine the concentration of syrup in the resulting juice. The first 250 ml drained juice contained 12.8% syrup. And 12.8% of 250 ml is 32 ml. So the first juice contained 32 ml of syrup.
The second drained juice of 300 ml contained 15% syrup. And 15% of 300 ml is 45 ml. So the second juice contained 45 ml of syrup.
Add up the amount of syrups:
32 ml + 45 ml = 77 ml
These 77 ml of syrup are contained in the new juice, which has a volume of 550 ml. Determine the concentration of syrup in this juice. To do this, we find the ratio of 77 ml of dissolved syrup to the volume of juice 550 ml:
So when mixing 12.8% raspberry juice with a volume of 250 ml and 15% raspberry juice with a volume of 300 ml, 14% raspberry juice with a volume of 550 ml is obtained.
Task 1. There are 3 solutions of sea salt in water: the first solution contains 10% salt, the second contains 15% salt and the third contains 20% salt. Mixed 130 ml of the first solution, 200 ml of the second solution and 170 ml of the third solution. Determine the percentage of sea salt in the resulting solution.
Decision
Determine the volume of the resulting solution:
130 ml + 200 ml + 170 ml = 500 ml
Since in the first solution there were 130 × 0.10 = 13 ml of sea salt, in the second solution 200 × 0.15 = 30 ml of sea salt, and in the third - 170 × 0.20 = 34 ml of sea salt, then in the resulting solution will be contain 13 + 30 + 34 = 77 ml of sea salt.
Let us determine the concentration of sea salt in the resulting solution. To do this, we find the ratio of 77 ml of sea salt to the volume of a solution of 500 ml
This means that the resulting solution contains 15.4% sea salt.
Task 2. How many grams of water must be added to 50 g of a solution containing 8% salt to obtain a 5% solution?
Decision
Note that if water is added to the existing solution, then the amount of salt in it will not change. Only its percentage will change, since adding water to the solution will lead to a change in its mass.
We need to add such an amount of water that eight percent of the salt would become five percent.
Determine how many grams of salt are contained in 50 g of solution. To do this, we find 8% of 50
50 g × 0.08 = 4 g
8% of 50 grams is 4 grams. In other words, there are 4 grams of salt for eight parts in a hundred. Let's make sure that these 4 grams are not eight parts, but five parts, that is, 5%
4 grams - 5%
Now knowing that there are 4 grams per 5% solution, we can find the mass of the entire solution. For this you need:
4 g: 5 = 0.8 g
0.8 g × 100 = 80 g
80 grams of solution is the mass at which 4 grams of salt will fall on 5% of the solution. And to get these 80 grams, you need to add 30 grams of water to the original 50 grams.
This means that to obtain a 5% salt solution, you need to add 30 g of water to the existing solution.
Task 2. Grapes contain 91% moisture, and raisins - 7%. How many kilograms of grapes are required to produce 21 kilograms of raisins?
Decision
Grapes consist of moisture and pure substance. If fresh grapes contain 91% moisture, then the remaining 9% will account for the pure substance of this grape:
Raisins contain 93% pure substance and 7% moisture:
Note that in the process of turning grapes into raisins, only the moisture of this grape disappears. The pure substance remains unchanged. After the grapes turn into raisins, the resulting raisins will be 7% moisture and 93% pure matter.
Let's determine how much pure substance is contained in 21 kg of raisins. To do this, we find 93% of 21 kg
21 kg × 0.93 = 19.53 kg
Now back to the first picture. Our task was to determine how many grapes you need to take to get 21 kg of raisins. A pure substance weighing 19.53 kg will fall on 9% of the grapes:
Now knowing that 9% of the pure substance is 19.53 kg, we can determine how many grapes are required to produce 21 kg of raisins. To do this, you need to find the number by its percentage:
19.53 kg: 9 = 2.17 kg
2.17 kg × 100 = 217 kg
So to get 21 kg of raisins, you need to take 217 kg of grapes.
Task 3. In an alloy of tin and copper, copper is 85%. How much alloy should be taken to contain 4.5 kg of tin?
Decision
If copper is 85% in the alloy, then the remaining 15% will be tin:
The question is how much alloy should be taken so that it contains 4.5 tin. Since the alloy contains 15% tin, then 4.5 kg of tin will fall on these 15%.
And knowing that 4.5 kg of the alloy is 15%, we can determine the mass of the entire alloy. To do this, you need to find the number by its percentage:
4.5 kg: 15 = 0.3 kg
0.3 kg × 100 = 30 kg
So the alloy needs to be taken 30 kg so that it contains 4.5 kg of tin.
Task 4. A certain amount of a 12% hydrochloric acid solution was mixed with the same amount of a 20% solution of the same acid. Find the concentration of the resulting hydrochloric acid.
Decision
Let us draw the first solution in the form of a straight line in the figure and select 12%
Since the number of solutions is the same, the same figure can be drawn side by side, illustrating a second solution with a hydrochloric acid content of 20%
We got two hundred parts of a solution (100% + 100%), thirty-two parts of which are hydrochloric acid (12% + 20%)
Determine what part 32 parts are from 200 parts
This means that when mixing a 12% solution of hydrochloric acid with the same amount of a 20% solution of the same acid, a 16% solution of hydrochloric acid will be obtained.
To check, imagine that the mass of the first solution was 2 kg. The mass of the second solution will also be 2 kg. Then when mixing these solutions, 4 kg of solution will be obtained. In the first solution of hydrochloric acid, there were 2 × 0.12 = 0.24 kg, and in the second - 2 × 0.20 = 0.40 kg. Then in the new hydrochloric acid solution there will be 0.24 + 0.40 \u003d 0.64 kg. The concentration of hydrochloric acid will be 16%
Tasks for independent solution
on , we will find 60% of the number
Now let's increase the number by the found 60%, i.e. per number
Answer: the new value is
Task 12. Answer the following questions:
1) Spent 80% of the amount. What percentage of this amount is left?
2) Men make up 75% of all factory workers. What percentage of the plant's employees are women?
3) Girls make up 40% of the class. What percentage of the class are boys?
BUT Decision
Let's use a variable. Let be P This is the original number that is mentioned in the problem. Let's take this original number P for 100%
Decrease this original number P by 50%
The new number is now 50% of the original number. Find out how many times the original number P more than the new number. To do this, we find the ratio of 100% to 50%
The original number is twice the new one. This can be seen even in the picture. And to make the new number equal to the original, it must be doubled. And doubling the number means increasing it by 100%.
This means that the new number, which is half of the original number, must be increased by 100%.
Considering the new number, it is also taken as 100%. So, in the figure above, the new number is half of the original number and is signed as 50%. In relation to the original number, the new number is half. But if we consider it separately from the original, it must be taken as 100%.
Therefore, in the figure, the new number, which is represented by a line, was first designated as 50%. But then we designated this number as 100%.
Answer: to get the original number, the new number must be increased by 100%.
Problem 16. Last month there were 15 accidents in the city.
This month, this figure dropped to 6. By what percent did the number of road accidents decrease?
Decision
There were 15 accidents last month. This month, 6. So the number of accidents decreased by 9.
Let's take 15 accidents as 100%. By reducing 15 accidents by 9, we will reduce them by a certain number of percent. To find out which one, we find out what part of 9 accidents is from 15 accidents
Answer: the concentration of the resulting solution is 12%.
Problem 18. A certain amount of a 11% solution of a certain substance was mixed with the same amount of a 19% solution of the same substance. Find the concentration of the resulting solution.
Decision
The mass of both solutions is the same. Each solution can be taken as 100%. After adding the solutions, a 200% solution will be obtained. In the first solution there was 11% of the substance, and in the second 19% of the substance. Then in the resulting 200% solution there will be 11% + 19% = 30% of the substance.
Determine the concentration of the resulting solution. To do this, we find out what part thirty parts of a substance make up from two hundred parts of a substance:
1,10. So the price for the first month will be 1.10.
For the second month, the price also increased by 10%. We add ten percent of this price to the current price of 1.10, we get 1.10 + 0.10 × 1.10 . This sum is equal to the expression 1.21 . So the price for the second month will be 1.21.
For the third month, the price also increased by 10%. Let's add ten percent of this price to the current price of 1.21, we get 1.21 + 0.10 × 1.21. This sum is equal to the expression 1.331 . Then the price for the third month will be 1.331.
Calculate the difference between the new and old price. If the original price was equal to 1, then it increased by 1.331 − 1 = 0.331. Expressing this result as a percentage, we get 0.331 × 100 = 33.1%
Answer: for 3 months, food prices increased by 33.1%.
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In the last video tutorial, we considered solving percentage problems using proportions. Then, according to the condition of the problem, we needed to find the value of one or another quantity.
This time, the initial and final values are already given to us. Therefore, in tasks it will be required to find percentages. More precisely, by what percentage has this or that value changed. Let's try.
Task. Sneakers cost 3200 rubles. After the price increase, they began to cost 4000 rubles. By what percentage was the price of the sneakers increased?
So, we solve through proportion. The first step - the original price was equal to 3200 rubles. Therefore, 3200 rubles is 100%.
In addition, we were given the final price - 4000 rubles. This is an unknown percentage, so let's denote it as x . We get the following construction:
3200 — 100%
4000 - x%
Well, the condition of the problem is written down. We make a proportion:
The fraction on the left is perfectly reduced by 100: 3200: 100 = 32; 4000: 100 = 40. In addition, you can reduce by 4: 32: 4 = 8; 40: 4 = 10. We get the following proportion:
Let's use the basic property of proportion: the product of the extreme terms is equal to the product of the middle ones. We get:
8 x = 100 10;
8x = 1000.
This is the usual linear equation. From here we find x :
x=1000:8=125
So, we got the final percentage x = 125. But is the number 125 the solution to the problem? No way! Because the task requires you to find out by what percentage the price of sneakers was increased.
By how many percent - this means that we need to find the change:
∆ = 125 − 100 = 25
We got 25% - that's how much the original price was increased. This is the answer: 25.
Problem B2 for interest #2
Let's move on to the second task.
Task. The shirt cost 1800 rubles. After the price reduction, it began to cost 1530 rubles. By what percentage was the price of the shirt reduced?
We translate the condition into mathematical language. The initial price of 1800 rubles is 100%. And the final price is 1530 rubles - we know it, but it is not known how many percent it is of the original value. Therefore, we denote it by x. We get the following construction:
1800 — 100%
1530 - x%
Based on the resulting record, we make up the proportion:
To simplify further calculations, let's divide both parts of this equation by 100. In other words, we will cross out two zeros at the numerator of the left and right fractions. We get:
Now let's use the basic property of proportion again: the product of the extreme terms is equal to the product of the average ones.
18 x = 1530 1;
18x = 1530.
It remains to find x :
x = 1530: 18 = (765 2) : (9 2) = 765: 9 = (720 + 45) : 9 = 720: 9 + 45: 9 = 80 + 5 = 85
We got that x = 85. But, as in the previous problem, this number in itself is not the answer. Let's go back to our condition. We now know that the new price after the cut is 85% of the old price. And in order to find the changes, you need from the old price, i.e. 100%, subtract the new price, i.e. 85%. We get:
∆ = 100 − 85 = 15
This number will be the answer: Please note: exactly 15, and in no case 85. That's all! Problem solved.
Attentive students will probably ask: why in the first task, when finding the difference, we subtracted the initial number from the final number, and in the second task we did exactly the opposite: from the initial 100% we subtracted the final 85%?
Let's clear this up. Formally, in mathematics, the change in value is always the difference between the final value and the initial one. In other words, in the second problem, we should have got not 15, but -15.
However, in no case should this minus be included in the answer, because it has already been taken into account in the condition of the original problem. It says right there about the price reduction. A 15% price decrease is the same as a -15% price increase. That is why in the solution and answer of the problem it is enough to write just 15 - without any minuses.
All, I hope, with this moment we have understood. This concludes our lesson for today. See you soon!
Today, in the modern world, it is impossible to do without interest. Even at school, starting from the 5th grade, children learn this concept and solve problems with this value. Interest is found in every area of modern structures. Take, for example, banks: the amount of overpayment of the loan depends on the amount specified in the contract; the dimension of profit is also affected. Therefore, it is vital to know what a percentage is.
The concept of interest
According to one legend, the percentage appeared due to a silly typo. The compositor was supposed to set the number 100, but mixed it up and put it like this: 010. This caused the first zero to rise slightly, and the second to fall. The unit has become a backslash. Such manipulations led to the appearance of the percent sign. Of course, there are other legends about the origin of this value.
The Hindus knew about percentages as early as the 5th century. In Europe, with which our concept is closely interconnected, appeared after a millennium. For the first time in the Old World, the judgment of what a percentage is was introduced by a scientist from Belgium, Simon Stevin. In 1584, a table of magnitudes was first published by the same scientist.
The word "percent" originates in Latin as pro centum. If you translate the phrase, you get "from a hundred." So, a percentage is understood as one hundredth of a value, a number. This value is denoted by the sign%.
Thanks to percentages, it became possible to compare parts of one whole without much difficulty. The appearance of shares greatly simplified the calculations, which is why they have become so common.
Converting fractions to percentages
To convert a decimal fraction to a percentage, you may need the so-called percent formula: the fraction is multiplied by 100,% is added to the result.
If you need to convert an ordinary fraction to a percentage, first you need to make it a decimal, and then use the above formula.
Converting percentages to fractions
As such, the percentage formula is rather conventional. But you need to know how to convert this value into a fractional expression. To convert shares (percentages) to decimal fractions, you need to remove the% sign and divide the indicator by 100.
The formula for calculating the percentage of a number
1) 40 x 30 = 1200.
2) 1200: 100 = 12 (students).
Answer: control work on "5" was written by 12 students.
You can use the ready-made table, which shows some fractions and percentages that correspond to them.
It turns out that the percentage formula looks like this: C \u003d (A ∙ B) / 100, where A is the original number (in a specific example, equal to 40); B - the number of percent (in this problem, B = 30%); C is the desired result.
Formula for calculating a number from a percentage
The following task will demonstrate what a percentage is and how to find a number from a percentage.
The garment factory produced 1,200 dresses, of which 32% are new-style dresses. How many new-style dresses did the clothing factory make?
1. 1200: 100 = 12 (dresses) - 1% of all manufactured items.
2. 12 x 32 = 384 (dresses).
Answer: The factory made 384 new style dresses.
If you need to find a number by its percentage, you can use the following formula: C \u003d (A ∙ 100) / B, where A is the total number of items (in this case, A \u003d 1200); B - the number of percent (in a specific task B = 32%); C is the desired value.
Increase, decrease a number by a given percentage
Students must learn what percentages are, how to count them and solve various problems. To do this, you need to understand how the number increases or decreases by N%.
Often tasks are given, and in life you need to find out what the number increased by a given percentage will be equal to. For example, given the number X. You need to find out what the value of X will be if it is increased, say, by 40%. First you need to convert 40% to a fractional number (40/100). So, the result of increasing the number X will be: X + 40% ∙ X \u003d (1 + 40 / 100) ∙ X \u003d 1.4 ∙ X. If we substitute any number instead of X, take, for example, 100, then the whole expression will be equal to : 1.4 ∙ X \u003d 1.4 ∙ 100 \u003d 140.
Approximately the same principle is used when decreasing a number by a given percentage. It is necessary to carry out calculations: X - X ∙ 40% \u003d X ∙ (1-40 / 100) \u003d 0.6 ∙ X. If the value is 100, then 0.6 ∙ X \u003d 0.6. 100 = 60.
There are tasks where you need to find out by what percentage the number has increased.
For example, given the task: The driver was driving along one section of the track at a speed of 80 km/h. On another section, the speed of the train increased to 100 km/h. By what percent did the speed of the train increase?
Let's say 80 km/h is 100%. Then we make calculations: (100% ∙ 100 km / h) / 80 km / h = 1000: 8 = 125%. It turns out that 100 km / h is 125%. To find out how much the speed has increased, you need to calculate: 125% - 100% = 25%.
Answer: the speed of the train on the second section increased by 25%.
Proportion
There are often cases when it is necessary to solve problems for percentages using a proportion. In fact, this method of finding the result greatly facilitates the task for students, teachers and not only.
So what is proportion? This term refers to the equality of two relations, which can be expressed as follows: A / B \u003d C / D.
In mathematics textbooks, there is such a rule: the product of the extreme terms is equal to the product of the average. This is expressed by the following formula: A x D = B x C.
Thanks to this formulation, any number can be calculated if the other three terms of the proportion are known. For example, A is an unknown number. To find it, you need
When solving problems by the method of proportion, it is necessary to understand from what number to take percentages. There are times when shares need to be taken from different values. Compare:
1. After the end of the sale in the store, the cost of the T-shirt increased by 25% and amounted to 200 rubles. What was the price during the sale.
In this case, the value of 200 rubles corresponds to 125% of the original (sales) price of the T-shirt. Then, to find out its value during the sale, you need (200 x 100): 125. You get 160 rubles.
2. There are 200,000 inhabitants on the planet Vitsencia: people and representatives of the humanoid race Naavi. Naavi make up 80% of the total population of Vicencia. Of the people, 40% are employed in the maintenance of the mine, the rest are mined for tetanium. How many people mine tetanium?
First of all, you need to find in numerical form the number of people and the number of Naavi. So, 80% of 200,000 will equal 160,000. So many representatives of the humanoid race live on Vicencia. The number of people, respectively, is 40,000. Of these, 40%, that is, 16,000, serve the mine. So, 24,000 people are engaged in the extraction of tetanium.
Multiple change of a number by a certain percentage
When it is already clear what a percentage is, you need to study the concept of absolute and relative change. An absolute transformation is understood as an increase in a number by a specific number. So, X has increased by 100. Whatever one substitutes for X, this number will still increase by 100: 15 + 100; 99.9 + 100; a + 100, etc.
A relative change is understood as an increase in a value by a certain number of percent. Let's say X has increased by 20%. This means that X will be equal to: X + X ∙ 20%. Relative change is implied whenever we talk about a half or third increase, a quarter decrease, a 15% increase, etc.
There is another important point: if the value of X is increased by 20%, and then by another 20%, then the total increase will be 44%, but not 40%. This can be seen from the following calculations:
1. X + 20% ∙ X = 1.2 ∙ X
2. 1.2 ∙ X + 20% ∙ 1.2 ∙ X = 1.2 ∙ X + 0.24 ∙ X = 1.44 ∙ X
This shows that X has increased by 44%.
Examples of tasks for percentages
1. What percentage of the number 36 is the number 9?
According to the formula for finding a percentage of a number, you need to multiply 9 by 100 and divide by 36.
Answer: The number 9 is 25% of 36.
2. Calculate the number C, which is 10% of 40.
According to the formula for finding a number by its percentage, you need to multiply 40 by 10 and divide the result by 100.
Answer: The number 4 is 10% of 40.
3. The first partner invested 4,500 rubles in the business, the second - 3,500 rubles, the third - 2,000 rubles. They made a profit of 2400 rubles. They shared the profits equally. How much in rubles did the first partner lose compared to how much he would have received if they divided the income according to the percentage of invested funds?
So, together they invested 10,000 rubles. The income for each amounted to an equal share of 800 rubles. To find out how much the first partner should have received and how much he lost, respectively, you need to find out the percentage of invested funds. Then you need to find out how much profit this contribution makes in rubles. And the last thing is to subtract 800 rubles from the result.
Answer: the first partner lost 280 rubles when sharing profits.
A bit of economy
Today, a rather popular question is the issue of a loan for a certain period. But how to choose a profitable loan so as not to overpay? First, you need to look at the interest rate. It is desirable that this indicator be as low as possible. Then you should apply for a loan.
As a rule, the size of the overpayment is affected by the amount of debt, the interest rate and the method of repayment. There are annuity and In the first case, the loan is repaid in equal installments every month. Immediately, the amount that covers the main loan grows, and the cost of interest gradually decreases. In the second case, the borrower pays constant amounts to repay the loan, to which interest is added on the balance of the principal debt. Monthly, the total amount of payments will decrease.
Now you need to consider both methods. So, with the annuity option, the amount of the overpayment will be higher, and with the differential option, the amount of the first payments. Naturally, the terms of the loan are the same for both cases.
Conclusion
So, interest. How to count them? Simple enough. However, sometimes they can be problematic. This topic begins to be studied at school, but it catches up with everyone in the field of loans, deposits, taxes, etc. Therefore, it is advisable to delve into the essence of this issue. If you still can’t make calculations, there are a lot of online calculators that will help you cope with the task.
The concept of percentage occurs in our lives too often, so it is very important to know how to solve problems with percentages. In principle, this is not a difficult matter, the main thing is to understand the principle of working with interest.
What is a percentage
We operate with the concept of 100 percent, and accordingly, one percent is a hundredth of a certain number. And all the calculations are already based on this ratio.
For example, 1% of 50 is 0.5, 15 of 700 is 7.
How to decide
- Knowing that one percent is one hundredth of the number presented, you can find any number of required percentages. In order to make it clearer, let's try to find 6 percent of the number 800. This is done simply.
- First we find one percent. To do this, divide 800 by 100. It turns out 8.
- Now we multiply this very one percent, that is, 8, by the number of percent we need, that is, by 6. It turns out 48.
- Fix the result by repetition.
15% of 150. Solution: 150/100*15=22.
28% of 1582. Solution: 1582/100*28=442.
- There are other problems when you are given values, and you need to find percentages. For example, you know that there are 5 scarlet roses out of 75 white roses in the store, and you need to know what percentage of scarlet roses. If we do not know this percentage, then we will denote it as x.
There is a formula for this: 75 - 100%
In this formula, the numbers are multiplied cross by cross, that is, x \u003d 5 * 100/75. It turns out that x \u003d 6% So the percentage of scarlet roses is 6%.
- There is another type of problem for percentages, when you need to find by what percentage one number is greater or less than another. How to solve problems with percentages in this case?
There are 30 students in the class, 16 of them are boys. The question is how many percent of boys are more than girls. First you need to calculate what percentage of students are boys, then you need to find out what percentage of girls. And finally find the difference.
So let's get started. We make a proportion of 30 accounts. - 100%
16 accounts -X %
Now we count. X=16*100/30, x=53.4% of all students in the class are boys.
Now find the percentage of girls in the same class. 100-53.4=46.6%
It remains now only to find the difference. 53.4-46.6=6.8%. Answer: there are more boys than girls by 6.8%.
Key Points in Interest Solving
So, so that you do not have problems with how to solve problems for percentages, remember a few basic rules:
- In order not to get confused in problems with percentages, always be vigilant: go from specific values to percentages and vice versa if necessary. The main thing is never to confuse one with the other.
- Be careful when calculating percentages. It is important to know from what specific value you need to count. For successive changes in values, the percentage is calculated from the last value.
- Before writing down the answer, read the entire problem again, because it may be that you have found only an intermediate answer, and you need to perform one or two more actions.
Thus, solving problems with percentages is not such a difficult matter, the main thing in it is attentiveness and accuracy, as, indeed, in all mathematics. And don't forget that practice is required to improve any skill. So decide more, and everything will be fine or even excellent for you.
1% is a hundredth of a number.1% = 0,01.
Finding percentages of a number.
To find a percentage of a number, you can express the percentage as a decimal fraction and multiply the number by the resulting decimal fraction.
Finding a number by its percentage.
To find a number by its percentage, you can represent the percentage as a decimal fraction and divide this number by the resulting decimal fraction.
To find how many percent one number is from another, you can divide one number by another and multiply the resulting product by 100.
How to solve percentage problems. Examples.
Finding a percentage of a number is related to finding a fraction of a number. Interest is a special way of writing an ordinary fraction, so you should begin to reveal the meaning of the concept of interest from understanding the concept of an ordinary fraction.
Let's take a few common fractions, for example. What is the meaning of each such entry?
These are examples of regular fractions. The denominator of each of them shows how many equal parts you need to divide some real or abstract object, the numerator shows how many such parts you need to take. Let's take a regular fraction as an example. For example. The meaning of this expression can be revealed as follows. Some real object was divided into 3 equal parts and 2 parts were taken from them.
As a real object, you can take, for example, a rectangle.
This expression is the quotient of a and b, where b is not equal to 0.
This is the ratio of the numbers a and b, where b is not equal to 0.
This is an ordinary fraction. a is the numerator, b is the denominator (b is not equal to 0).
Example 1 The capacity of the barrel was 200 liters. The barrels were filled with water. What is the meaning of this proposal?
- this fraction means that a certain object was divided into 5 equal parts and 2 parts were taken from them. The object in this problem is the volume of the barrel equal to 200 liters, therefore,
200:5 = 40,
402 = 80.
80 liters of water were poured into a barrel.
The above example is a typical example of finding a fraction of a number.
To find a fraction of a number, you need to multiply the number by that fraction.
Now we can move on to percentages.
The concept of percentage is defined as follows: 1% of a number is a hundredth of a number, i.e. 1% \u003d 0.01.
Then the meaning of the sentence a% of number b can be explained like this. Some object (the value of which is equal to b units) divided into 100 equal parts and taken from them a parts.
Example 2 Masha had 400 rubles. She spent 24% of this amount. What is the meaning of this saying?
Since 24% \u003d 0.24, and 0.24 means that a certain object was divided into 100 equal parts and 24 parts were taken from them. In this case, the object is the amount of money equal to 400 rubles, therefore,
400: 100 =4,
424 = 96.
Masha spent 96 rubles.
The above example is a typical example of finding percentages of a number.
Example 3 Need to find R%
from number b
.
Let x be the number we need to find.
p%
= 0,01p,
x = b
0,01p
To find percentages of a number, you need to represent the number of percents as a decimal fraction and multiply the given number by this decimal fraction.
Another approach to this problem. You can use the concept and properties of proportion. If we recall that proportion is the equality of two ratios, and the ratio of two numbers is an ordinary fraction, then this method is also associated with the concept of an ordinary fraction.
b - 100%,
x - p%,
We have a proportion:
b: 100 = x: p, (b is to 100 as x is to p) whence,
Example 4 Let there be numbers a and b , moreover, a >b Then the number a more number b on the %.
Let's approach this problem a little differently. We will consider a simple special case, for example, this: "How many percent is the number 10 greater than the number 2?".
1. Subtract the smaller number from the larger number. 10 - 2 = 8. Then 10 is greater than 2 by 8.
2. Find the ratio of the found number to a smaller number. 8:2=4 is the ratio of two numbers!
3 We express the ratio as a percentage 4100 = 400%.
The number 10 is greater than the number 2 by 400%.
If we divide 8 by 10, we will find a ratio showing how much of 10 2 is less than 10 (here the comparison is with the number 10.
The number 2 is less than the number 10 by 80%.
Example 5 The tractor driver plowed 6 hectares, which is from the entire field. What is the area of the entire field.
This is a typical problem of finding a number by its fraction. Let the area of the entire field be x,
then we have the equation x= 6. Whence x = 6:; x = 26. Field area is 26 ha.
To find a number by its fraction, you need to divide the number corresponding to the given fraction by the fraction.
Example 6 . Given a number b, which is p% from number a. Find a number a.
p%
= 0,01p
b
= 0,01pa
a = b: (0.01p)
Given a number b , which is p% from number a .
Find a number a .
a - 100%
b-p%
a:100 = b:p
Compound interest formula.
If the deposit has an amount a monetary units, and the bank charges R%
per annum, then through n
years, the amount on the deposit will be monetary units, or
a(1+0.01p)n
monetary units.
Example 7 The construction of the house cost 9,800 rubles, of which 35% was paid for the work, and the rest was paid for the material. How much did the materials cost?
Paid for work:
0,359800 = 3430.
Therefore, the materials cost: 9800 - 3430 = 6370.
Answer: 6370 rubles.
Example 8 37.4 tons of gasoline were poured into the tank, after which 6.5% of the tank's capacity remained unfilled. How much gasoline must be added to the tank to fill it?
If the unfilled part of the tank is 6.5% of the capacity, then the filled part is: 100% - 6.5% = 93.5%. Then, if x is the mass of gasoline that remains to be added to the tank, then we have the proportion
where 
.
Answer: 2.6 tons.
Example 9 Find a number knowing that 25% of it is 45% of 640.
Let x be the desired number. We have
0.25x = 0.45640.
Answer: 1152.
Example 10 The number a is 92% of the number b. If the number b is increased by 700, then the new number will be 9% greater than the number a. Find numbers a and b.
From the condition of the problem we have a system of equations:
Solving the resulting system, we find, a = 230000, b = 250000.
Answer: 230000; 250000.
Example 11. The first number is 50% of the second. What percentage of the first is the second?
Let's denote the second number by x, then the first number is equal to 0.5x. To find out what percentage is the number x of the number 0.5x; Let's make a proportion:
from which we find
Answer: 200%.
Example 12. There are 260 students in the lyceum, of which 10% fail. After the expulsion of a certain number of poor performers, their percentage dropped to 6.4%. How many students dropped out?
Before expulsion, the number of underachievers before expulsion was solo
Let x people be expelled. Then, in total, 260 students remained in the lyceum, of which 26 were unsuccessful. We have a proportion
260 - x - 100%,
(260 - x)0.064=(26 - x)100,
Solving the resulting equation, we find x = 10.
Example 13 By what percentage is 250 greater than 200?
Let's do two things.
1) We find out how many percent is the number 250 tons of the number 200:
2) Since the number 200 in this example is 100%, then the number 250 is greater than the number 200 by 125% -100% = 25%.
Answer: 25%.
Example 14 What percentage is 200 less than 250?
1) Find out how many percent is the number 200 of the number 250 (unlike the previous example, here you need to take the number 250 as 100%!):
2) The number 200 is less than the number 250 by 100% - 80% = 20%.
Answer: 20%.
Example 15 The length of the brick was increased by 30%, the width by 20%, and the height was reduced by 40%. Did the volume of bricks increase or decrease from this and by what percentage?
Let the original length of the brick be x, width - y, height - z. Then the initial volume of the brick: V 1 = xyz. New brick sizes: 1.3x; 1.2y; 0.6z and new volume: V 2 \u003d 1.3x1.2y0.6z \u003d 0.936xyz. Since V 2< V 1 , объем кирпича уменьшился. Уменьшение V 2 - V 1 = 0,064xyz и составляет 6,4% от V 1.
Answer: decreased by 6.4%.
Example 16 The price of a commodity went down by 40%, then another 25%. By what percentage has the price of the product decreased from its original price?
Let x be the original price of the product. After the first decrease, the price will be equal to
x - 0, 4x = 0.6x.
The second price decrease is 25% of the new price of 0.6x, so after the second decrease we will have the price
0.6x - 0.250.6x = 0.45x;.
After two declines, the total price change is:
x - 0.45x = 0.55x.
Since the value is 0.55x; is 55% of x, then the price of the good has decreased by 55%.
Answer: 55%.
Example 17. The initial cost of a unit of production was 75 rubles. During the first year of production, it increased by a certain number of percent, and during the second year it decreased (in relation to the increased value) by the same number of percent, as a result of which it became equal to 72 rubles. Determine the percentage increase and decrease in the cost of a unit of production.
Let x% be the percentage increase (and decrease) in the cost of a unit of output. By definition, x% of 75 is 750.01x. Then after the first increase the price will be equal to 75 + 0.75x.
During the second year, the price will decrease by
0.01x(75+0.75x) = 0.75x + 0.0075x2.
Now we can write the equation for the final price
(75 + 0.75x) - (0.75x + 0.0075x 2) = 72;
x 2 \u003d 400; hence x 1 = - 20, x 2 = 20.
Only one root of this equation is suitable: x 2 \u003d 20.
Answer: 20%.
Example 18. 10 thousand rubles were deposited in the bank account. After the money lay for one year, 1 thousand rubles were withdrawn from the account. A year later, the account was 11 thousand rubles. Determine what percentage per annum the bank charges.
Let the bank charge p% per annum.
1) The amount of 10,000 rubles, deposited in a bank account at p% per annum, in a year will increase to the value
10000 + 0.01p10000 = 10000 + 100 rub.
When 1000 rubles are withdrawn from the account, 9000 + 100 rubles will remain there.
2) In another year, the latter value will increase to 9000 + 100r + 0.01p (9000 + 100r) = r 2 + 190r + 9000 rubles due to the accrual of interest.
By condition, this value is equal to 11,000 rubles, so we have a quadratic equation.
p 2 + 190r + 9000 = 11000;
r 2 + 190r - 2000 = 0
, we solve this quadratic equation using Viette's theorem, p 1 \u003d 10, p 2 \u003d -200.
The negative root is not suitable.
Answer: 10%.
Example 19. The city currently has 48,400 inhabitants. It is known that the population of this city increases annually by 10%. How many inhabitants were there in the city two years ago?
Suppose that two years ago the number of inhabitants of the city was x people, then the number of inhabitants is currently expressed through x using the compound interest formula:
x(1+0.1) 2 = 1.21x.
From the problem statement:
Answer: 40,000 people.
