The question "how to calculate percentages" begins to prevail among schoolchildren in grade 5. It is then that such a topic appears in mathematics. It seems that fifth-graders are not offered difficult problems. Then why do many have problems with these tasks? Perhaps everything is hidden in a misunderstanding of the essence of this concept.
The basis of everything is understanding the meaning
This is the key to all problems on this topic. If a person can determine one percent, then he can do thirteen, eighty-nine and one hundred thirty-five. At least four hundred and twenty ...
And this is one hundredth of the total number, which is discussed in the problem. Moreover, it can be set explicitly, but it happens that it is only indirectly referred to.
What situations exist?
Find out the percentage of the number
That is, a certain value is known and it is required to calculate how much x% of it will be. This is the main task in the topic. So how do you calculate the percentage of a number? It is necessary to make a proportion, in the first line the known data will be written, in the second - the required data. Now you need to multiply the known value by the desired percentage and divide by 100%.
If you write it shorter, you get the following proportion:
known number - 100%,
the required number is x%.
From this entry, you can compose the formula:
desired \u003d (known * x%) / 100%.
The result is obtained by multiplying crosswise two known quantities and dividing by the one that was left without a pair.
If the task total number consists of several, then the question arises of how to calculate the percentage of the amount. There are two ways to do this:
Find out the number by a known percentage
In this situation, the part of the number and the corresponding% are known. To find out how to calculate percentages correctly, you will need to use the proportion already recorded. Only the first line will contain the required number, and the second will contain the known one:
desired \u003d (known * 100%) / x%.
Find out the percentage of one number from another
You are given two values, and you need to calculate how many percent will be more or less. Typically, such tasks contain information about an overfulfilled plan or, conversely, about a decrease in the quantity compared to an earlier value.
Again, proportion is required. It is customary to take the value with which it is compared for 100%.
the first value is 100%,
the second value is x%,
x \u003d (second value * 100%) / first value.
Find out the percentage when nothing is known about the total
In such problems, it is reported that a number is a known percentage and another is an unknown percentage. Here it also needs to be calculated. How to calculate interest in this case? Again using proportion:
the first number is a known%,
second number - x%,
x \u003d (second * known%) / first.
Tasks with more difficult questions
Find out how much the numbers differ
There are two possibilities here. The first is when you need to compare more with less. And find, by what percentage the second is less. In this situation, the question of how to calculate the percentage comes down to figuring out what to choose for 100%. The one that's bigger. And then the proportion will look like this:
higher number - 100%,
smaller number - x%,
x \u003d (less * 100%) / more.
But that's not the answer. For it, you need to subtract the found value of x from 100%.
The second option is when the lower number is compared with the one that is larger. In it, the lower value is taken as 100%. The proportion looks like this:
smaller number - 100%,
larger number - x%,
x \u003d (more * 100%) / less.
To calculate the final value, you need to find out how much x% will be - 100%.
Find out the result of increasing the number by a known percentage
In such tasks, you need to find the answer that will turn out after increasing the known percentage of it by some value. In this case, the proportion will look like this:
known number - 100%,
the required number - 100 + x%,
desired \u003d (known * (100 + x%)) / 100%.
Find out the result from reducing the number by a known percentage
known number - 100%,
the required number - 100 - x%,
desired \u003d (known * (100 - x%)) / 100%.
Calculator as an assistant in calculating interest
It can be used in two ways. The first is when all the steps described above are performed step by step on the device screen. Everything is simple here. You just need to not get confused with the procedure. In general, a calculator will simply replace a person at the moment of practical calculation.
In the second method, he will do everything himself. For example, you can select the engineering type of the calculator on a computer and enter the entire formula with brackets and the necessary actions into it at once. After pressing the "\u003d" key, the answer will be displayed in the window.
It happens that the calculation option is simple, when you need to find out the percentage of a known value. Then you can use a special function, which is indicated by the "%" button.
To do this, you need to dial a known value on the calculator. Then press the multiplication sign. Then the number of percent and the "%" button. The answer will immediately appear on the screen.
They will allow you not to get confused. Moreover, it will be able to answer any question about how to calculate the percentage of the amount or difference, you no longer have to think about it - everything will be solved automatically.
- Always go to specific values. Percentage is something impersonal. But the kilograms, students and boxes are quite tangible and understandable. And you need to strive for them.
- Read the condition very carefully. Because there are situations when interest is taken several times and on different values.
- Check the answer. Is it really finite? Or maybe it's just an intermediate value.
Anonymous Number A is 56% less than number B, which is 2.2 times less than number C. What percentage of number C relative to number A? NMitra A \u003d B - 0.56 ⋅ B \u003d B ⋅ (1 - 0.56) \u003d 0.44 ⋅ BB \u003d A: 0.44 C \u003d 2.2 ⋅ B \u003d 2.2 ⋅ A: 0.44 \u003d 5 ⋅ AC 5 times more AC 400% more A Anonymous Help. In 2001, revenue increased by 2 percent compared to 2000, although it was planned to double. By what percentage is the plan under fulfilled? NMitra A - 2000 B - 2001 B \u003d A + 0.02A \u003d A ⋅ (1 + 0.02) \u003d 1.02 ⋅ A B \u003d 2 ⋅ A (plan) 2 - 100% 1.02 - x% х \u003d 1.02 ⋅ 100: 2 \u003d 51% (the plan is fulfilled) 100 - 51 \u003d 49% (the plan is not fulfilled) Anonymous Help answer the question. Watermelon contains 99% moisture, but after drying (put in the sun for several days), its moisture content is 98%. How much will the WEIGHT of a watermelon change after drying? If you calculate mathematically, it turns out that my watermelon has completely dried up. For example: with a weight of 20 kg, water is 99% of the mass, that is, the dry weight is 1% \u003d 0.2 kg. Here the watermelon loses liquid, and is already 98%, therefore, the dry weight is 2%. But the dry weight cannot change due to the loss of water, so it is still 0.2 kg. 2% \u003d 0.2 \u003d\u003e 100% \u003d 10 kg. Anonymous Can you please tell me how to calculate the percentage itself in the range of 2 values? Say, what is the percentage of 37 in the range of 22-63? I need a formula for the application, I used to solve such problems in a couple of minutes, but now my brain has dried up). Help out. NMitra This is how it works out for me: percentage \u003d (number - z0) ⋅ 100: (z1-z0) z0 - the initial value of the range z1 - the final value of the range For example, x \u003d (37-22) ⋅ 100: (63-22) \u003d 1500 : 41 \u003d 37% For the example below converges
| 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| 35 | 50% | 10 | 45 |
| 16 | 23% | 4,6 | 20,6 |
| 18 | 26% | 5,2 | 23,2 |
| 1 | 1% | 0,2 | 1,2 |
| 70 | 100% | 20 | 90 |
| 35 | 50% | 10 | 45 | 67,5 |
| 16 | 23% | 4,6 | 20,6 | 30,9 |
| 18 | 26% | 5,2 | 23,2 | 34,8 |
| 1 | 1% | 0,2 | 1,2 | 1,8 |
| 70 | 100% | 20 | 90 | 135 |
A ratio (in mathematics) is a relationship between two or more numbers of the same kind. Ratios compare absolute values \u200b\u200bor parts of a whole. Ratios are calculated and written in different ways, but the basic principles are the same for all ratios.
Steps
Part 1
Determination of ratios-
Determination of ratios. A ratio is a relationship between two (or more) values \u200b\u200bof the same kind. For example, if you need 2 cups of flour and 1 cup of sugar to make a cake, then the ratio of flour to sugar is 2 to 1.
- The ratios can also be used in cases where the two quantities are not related to each other (as in the example with the cake). For example, if there are 5 girls and 10 boys in a class, then the ratio of girls to boys is 5 to 10. These values \u200b\u200b(the number of boys and the number of girls) are independent of each other, that is, their values \u200b\u200bwill change if someone leaves the class or a new student will come to the class. Ratios simply compare the values \u200b\u200bof quantities.
-
pay attention to different ways representation of relations. Relationships can be expressed in words or in mathematical symbols.
- Very often the ratios are expressed in words (as shown above). Especially this form of representation of ratios is used in everyday life, far from science.
- Also, ratios can be expressed through a colon. When comparing two numbers in a ratio, you will use one colon (for example, 7:13); when comparing three or more values, put a colon between each pair of numbers (for example, 10: 2: 23). In our class example, you can express the ratio of girls to boys like this: 5 girls: 10 boys. Or like this: 5:10.
- Less commonly, ratios are expressed using a slash. In the class example, it can be written like this: 5/10. Nevertheless, this is not a fraction and such a ratio is not read as a fraction; Moreover, remember that in the ratio, the numbers do not represent part of a whole.
Part 2
Using ratios-
Simplify the ratio. The ratio can be simplified (similar to fractions) by dividing each term (number) of the ratio by. However, do not lose sight of the original ratio values \u200b\u200bwhen doing this.
- In our example, there are 5 girls and 10 boys in the class; the ratio is 5:10. The greatest common divisor of the terms of the ratio is 5 (since both 5 and 10 are divisible by 5). Divide each ratio number by 5 to get the ratio of 1 girl to 2 boys (or 1: 2). However, keep the original values \u200b\u200bin mind when simplifying the ratio. In our example, there are not 3 students in the class, but 15. The simplified ratio compares the number of boys and the number of girls. That is, for every girl there are 2 boys, but there are not 2 boys and 1 girl in the class.
- Some relationships are not simplified. For example, the ratio 3:56 is not simplified because these numbers have no common divisors (3 is a prime number, and 56 is not divisible by 3).
-
Use multiplication or division to increase or decrease the ratio. Common tasks in which it is necessary to increase or decrease two values \u200b\u200bproportional to each other. If you are given a ratio and need to find a larger or smaller ratio corresponding to it, multiply or divide the original ratio by some given number.
- For example, a baker needs to triple the amount of ingredients given in a recipe. If the recipe has a flour to sugar ratio of 2 to 1 (2: 1), then the baker will multiply each term in the ratio by 3 to get a 6: 3 ratio (6 cups flour to 3 cups sugar).
- On the other hand, if a baker needs to halve the amount of ingredients given in a recipe, then the baker will divide each term in the ratio by 2 and get a ratio of 1: ½ (1 cup flour to 1/2 cup sugar).
-
Finding an unknown value when given two equivalent relationships. This is a problem in which you need to find an unknown variable in one relation using the second relation, which is equivalent to the first. To solve such problems, use. Write down each ratio as an ordinary fraction, put an equal sign between them and multiply their terms crosswise.
- For example, a group of students is given, in which there are 2 boys and 5 girls. What will be the number of boys if the number of girls is increased to 20 (the proportion remains the same)? First, write down two ratios - 2 boys: 5 girls and x boys: 20 girls. Now write these ratios as fractions: 2/5 and x / 20. Multiply the terms of the fractions crosswise to get 5x \u003d 40; therefore, x \u003d 40/5 \u003d 8.
Part 3
Common mistakes-
Avoid addition and subtraction in ratio word problems. Many word problems look something like this: “In the recipe, you need to use 4 potato tubers and 5 carrot roots. If you want to add 8 potato tubers, how many carrots do you need to keep the ratio unchanged? " When solving such problems, students often make the mistake of adding the same amount of ingredients to the original number. However, to keep the ratio, you need to use multiplication. Here are examples of right and wrong solutions:
- False: “8 - 4 \u003d 4 - so we added 4 potato tubers. So, you need to take 5 carrot root crops and add 4 more to them ... Stop! Relationships are not calculated that way. It is worth trying again. "
- It is true: "8 ÷ 4 \u003d 2 - so we multiplied the amount of potatoes by 2. Accordingly, 5 carrots must be multiplied by 2. 5 x 2 \u003d 10 - 10 carrots must be added to the recipe." Write down the units of measurement after each value. In word problems, it is much easier to recognize an error if you write down the units after each value. Remember that quantities with the same unit in both the numerator and denominator are canceled. By shortening the expression, you get the right answer.
- Example: 6 boxes are given, in every third box there are 9 balls. How many balls are there?
- Incorrect: 6 boxes x 3 boxes / 9 balls \u003d ... Stop, nothing can be cut. The answer would be "boxes x boxes / balls". It doesn't make sense.
- Correct: 6 boxes x 9 balls / 3 boxes \u003d 6 boxes * 3 balls / 1 box \u003d 6 boxes * 3 balls / 1 box \u003d 6 * 3 balls / 1 \u003d 18 balls.
Using ratios. Ratios are used both in science and in everyday life to compare values. The simplest ratios relate only two numbers, but there are ratios that compare three or more values. In any situation in which more than one quantity is present, a ratio can be written. By linking some values, ratios can, for example, suggest how to increase the amount of ingredients in a recipe or substances in a chemical reaction.
In some cases, the user is tasked with not counting the sum of the values \u200b\u200bin a column, but counting their number. That is, in simple terms, you need to count how many cells in a given column are filled with certain numeric or text data. Excel has a number of tools that can solve this problem. Let's consider each of them separately.
Depending on the user's goals, Excel can count all the values \u200b\u200bin a column, only numerical data and those that meet a certain specified condition. Let's look at how to solve the assigned tasks in different ways.
Method 1: indicator in the status bar
This method is the simplest and requires the least amount of steps. It allows you to count the number of cells containing numeric and text data. You can do this simply by looking at the indicator in the status bar.
To complete this task, just hold down the left mouse button and select the entire column in which you want to count the values. As soon as the selection is made, in the status bar, which is located at the bottom of the window, near the parameter "Amount" the number of values \u200b\u200bcontained in the column will be displayed. Cells filled with any data (numeric, text, date, etc.) will participate in the calculation. Empty items will be ignored when counting.
In some cases, the number of values \u200b\u200bindicator may not appear in the status bar. This means that it is most likely disabled. To enable it, right-click on the status bar. A menu appears. In it, you need to check the box next to the item "Amount"... After that, the number of cells filled with data will be displayed in the status bar.
The disadvantages of this method include the fact that the result obtained is not recorded anywhere. That is, as soon as you deselect it, it disappears. Therefore, if it is necessary to fix it, you will have to record the resulting total manually. In addition, using this method, you can count only all the cells filled with values \u200b\u200band you cannot set the counting conditions.
Method 2: COUNT operator
Using the operator COUNTas in the previous case, it is possible to count all the values \u200b\u200blocated in the column. But unlike the option with an indicator in the status bar, this method provides an opportunity to fix the result in a separate element of the sheet.
The main task of the function COUNT, which belongs to the statistical category of operators, is just counting the number of nonblank cells. Therefore, we can easily adapt it to our needs, namely, to count the elements of a column filled with data. The syntax for this function is as follows:
COUNT (value1; value2; ...)
In total, the operator can have up to 255 general group arguments "Value"... The arguments are just references to cells or a range in which you want to count values.
As you can see, unlike the previous method, this option offers to output the result to a specific element of the sheet with its possible saving there. But unfortunately the function COUNT still does not allow specifying the selection conditions for values.
Method 3: COUNT operator
Using the operator SCORE only the numerical values \u200b\u200bin the selected column can be counted. It ignores text values \u200b\u200band does not include them in the grand total. This function also belongs to the category of statistical operators, like the previous one. Its task is to count cells in the selected range, and in our case, in a column that contains numeric values. The syntax for this function is almost identical to the previous statement:
COUNT (value1; value2; ...)
As you can see, the arguments for SCORE and COUNT are exactly the same and represent cell or range references. The difference in syntax is only in the name of the operator itself.
Method 4: COUNTIF operator
Unlike the previous methods, using the operator COUNTIF allows you to set conditions that meet the values \u200b\u200bthat will take part in the calculation. All other cells will be ignored.
Operator COUNTIF also belongs to the statistical group of Excel functions. Its only task is to count non-empty elements in a range, and in our case in a column, that meet a given condition. The syntax for this operator is noticeably different from the previous two functions:
COUNTIF (range, criterion)
Argument "Range" is represented as a link to a specific array of cells, and in our case, to a column.
Argument "Criterion" contains the specified condition. It can be either an exact numeric or text value, or a value specified by signs. "more" (> ), "less" (< ), "not equal" (<> ) etc.
Let's count how many cells with the name "Meat" are located in the first column of the table.
Let's change the task a little. Now let's count the number of cells in the same column that do not contain the word "Meat".
Now let's count all the values \u200b\u200bthat are greater than 150 in the third column of this table.
Thus, we can see that there are a number of ways in Excel to count the number of values \u200b\u200bin a column. The choice of a particular option depends on the specific goals of the user. So, the indicator on the status bar only allows you to see the number of all values \u200b\u200bin the column without fixing the result; function COUNT provides an opportunity to record their number in a separate cell; operator SCORE only counts items containing numeric data; and using the function COUNTIF you can set more complex conditions for counting elements.
Percentage is one hundredth of a whole number. Percentages are used to indicate the relationship of a part to a whole, as well as to compare values.
1% = 1 100 = 0,01
The interest calculator allows you to perform the following operations:
Find the percentage of the number
To find the percentage p from a number, you need to multiply this number by a fraction p 100
Find 12% of 300:
300 12
100
\u003d 300 0.12 \u003d 36
12% of 300 is 36.
For example, a product costs 500 rubles and has a 7% discount. Let's find the absolute value of the discount:
500 7
100
\u003d 500 0.07 \u003d 35
Thus, the discount is 35 rubles.
How many percent is one number from another
To calculate the percentage of numbers, you need to divide one number by another and multiply by 100%.
Let's calculate how many percent is the number 12 of the number 30:
12
30
100 \u003d 0.4100 \u003d 40%
The number 12 is 40% of the number 30.
For example, a book contains 340 pages. Vasya read 200 pages. Let's calculate what percentage of the whole book Vasya read.
200
340
100% \u003d 0.59100 \u003d 59%
Thus, Vasya read 59% of the entire book.
Add percent to number
To add to the number p percent, you need to multiply this number by (1 + p 100)
Add 30% to 200:
200 (1 + 30
100
) \u003d 200 1.3 \u003d 260
200 + 30% equals 260.
For example, a subscription to the pool costs 1000 rubles. From next month they promised to raise the price by 20%. Let's calculate how much the subscription will cost.
1000 (1 + 20
100
) \u003d 1000 1.2 \u003d 1200
Thus, the subscription will cost 1200 rubles.
Subtract percentage from the number
To subtract from the number p percent, you need to multiply this number by (1 - p 100)
Subtract 30% of 200:
200 (1 - 30
100
) \u003d 200 0.7 \u003d 140
200 - 30% equals 140.
For example, a bicycle costs 30,000 rubles. The store has made a 5% discount on it. Let's calculate how much the bike will cost, taking into account the discount.
30,000 (1 - 5
100
) \u003d 30,000 0.95 \u003d 28,500
Thus, the bike will cost 28,500 rubles.
How much percent is one number greater than the other
To calculate how many percent one number is greater than another, you need to divide the first number by the second, multiply the result by 100 and subtract 100.
Let's calculate by what percentage the number 20 is greater than the number 5:
20
5
100 - 100 \u003d 4 100 - 100 \u003d 400 - 100 \u003d 300%
The number 20 is 300% more than the number 5.
For example, a boss's salary is RUB 50,000, and an employee's salary is RUB 30,000. Let's find by what percentage the boss's salary is higher:
50000
35000
100 - 100 \u003d 1.43 * 100 - 100 \u003d 143 - 100 \u003d 43%
Thus, the salary of the boss is 43% higher than the salary of the employee.
What percentage is one number less than the other
To calculate how many percent one number is less than another, you need to subtract the ratio of the first number to the second, multiplied by 100 from 100.
Let's calculate by what percentage the number 5 is less than the number 20:
100 - 5
20
100 \u003d 100 - 0.25 100 \u003d 100 - 25 \u003d 75%
The number 5 is 75% less than the number 20.
For example, freelancer Oleg in January completed orders for 40,000 rubles, and in February for 30,000 rubles. Let's find out by what percentage Oleg earned less in February than in January:
100 - 30000
40000
100 \u003d 100 - 0.75 * 100 \u003d 100 - 75 \u003d 25%
Thus, in February, Oleg earned 25% less than in January.
Find 100 percent
If the number x this is p percent, then you can find 100 percent by multiplying the number x on 100 p
Find 100% if 25% is 7:
7 100
25
\u003d 7 4 \u003d 28
If 25% equals 7, then 100% equals 28.
For example, Katya copies photos from a camera to a computer. In 5 minutes, 20% of the photos were copied. Let's find how long the copying process takes:
five · 100
20
\u003d 5 5 \u003d 25
We get that the process of copying all photos takes 25 minutes.
