How to find proportion in percentage. Percentage Problems: Standard Calculation Using Proportions

A proportion is a mathematical expression in which two or more numbers are compared to each other. In proportions, absolute values ​​​​and quantities can be compared or parts of a larger whole. Proportions can be written and calculated in several different ways, but the basic principle is the same.

Steps

Part 1

What is proportion

Find out what proportions are for. Proportions are used as in scientific research, as well as in Everyday life to compare different values ​​and quantities. In the simplest case, two numbers are compared, but a proportion can include any number of values. When comparing two or more quantities, you can always apply a proportion. Knowing how quantities relate to each other allows, for example, to write chemical formulas or recipes for various dishes. Proportions will come in handy for a variety of purposes.

1. Learn what proportion means. As noted above, proportions allow you to determine the relationship between two or more quantities. For example, if it takes 2 cups of flour and 1 cup of sugar to make a cookie, we say that there is a 2 to 1 ratio between the amount of flour and sugar.

   • With proportions, you can show how different quantities relate to each other, even if they are not directly related to each other (unlike a recipe). For example, if there are five girls and ten boys in the class, the ratio of the number of girls to the number of boys is 5 to 10. In this case, one number does not depend on the other and is not directly related to it: the proportion can change if someone leaves the class or vice versa , new students will come to it. Proportion simply allows you to compare two quantities.

2. pay attention to various ways proportion expressions. Proportions can be written in words or mathematical symbols can be used.

   • In everyday life, proportions are more often expressed in words (as above). Proportions are used in a wide variety of areas, and if your profession is not related to mathematics or another science, most often you will come across this way of writing proportions.
   • Proportions are often written with a colon. When comparing two numbers using a proportion, they can be written with a colon, such as 7:13. If more than two numbers are being compared, a colon is inserted consecutively between each two numbers, for example 10:2:23. In the class example above, we are comparing the number of girls and boys, with 5 girls: 10 boys. Thus, in this case, the proportion can be written as 5:10.
   • Sometimes when writing proportions, a fraction sign is used. In our class example, the ratio of 5 girls to 10 boys would be written as 5/10. In this case, the “divide” sign should not be read and it must be remembered that this is not a fraction, but the ratio of two different numbers.

   Part 2

   Operations with proportions

   1. Bring the proportion to its simplest form. Proportions can be simplified, like fractions, by reducing their members by a common divisor. To simplify a proportion, divide all the numbers in it by common divisors. However, one should not forget about the initial values ​​\u200b\u200bthat led to this proportion.

      • In the example above with a class of 5 girls and 10 boys (5:10), both sides of the proportion have a common divisor of 5. Dividing both by 5 (greatest common divisor), we get a ratio of 1 girl to 2 boys (i.e. 1:2) . However, when using a simplified proportion, one should remember the initial numbers: there are not 3 students in the class, but 15. The reduced proportion only shows the ratio between the number of girls and boys. There are two boys for every girl, but this does not mean that there are 1 girl and 2 boys in the class.
      • Some proportions are not amenable to simplification. For example, the ratio 3:56 cannot be reduced, since the quantities included in the proportion do not have common divisor: 3 is a prime number and 56 is not divisible by 3.

   2. For "scaling" proportions can be multiplied or divided. Proportions are often used to increase or decrease numbers in proportion to each other. Multiplying or dividing all the quantities in a proportion by the same number keeps the ratio between them unchanged. Thus, the proportions can be multiplied or divided by the “scale” factor.

      • Suppose a baker needs to triple the amount of cookies they bake. If flour and sugar are taken in a ratio of 2 to 1 (2:1), to increase the number of cookies by three times this proportion should be multiplied by 3. The result will be 6 cups of flour for 3 cups of sugar (6:3).
      • You can also do the opposite. If the baker needs to halve the amount of cookies, both parts of the proportion should be divided by 2 (or multiplied by 1/2). The result is 1 cup of flour for half a cup (1/2, or 0.5 cup) of sugar.

   3. Learn how to find an unknown quantity using two equivalent proportions. Another common problem for which proportions are widely used is finding an unknown quantity in one of the proportions, if a second proportion similar to it is given. The multiplication rule for fractions greatly simplifies this task. Write each proportion as a fraction, then equate these fractions to each other and find the desired value.

      • Suppose we have a small group of students of 2 boys and 5 girls. If we want to keep the ratio between boys and girls, how many boys should there be in a class with 20 girls? First, let's make up both proportions, one of which contains an unknown value: 2 boys: 5 girls \u003d x boys: 20 girls. If we write proportions as fractions, we get 2/5 and x/20. After multiplying both sides of the equation by the denominators, we get the equation 5x=40; we divide 40 by 5 and as a result we find x=8.

   Part 3

   Error detection

   1. When dealing with proportions, avoid addition and subtraction. Many proportion problems sound like this: “It takes 4 potatoes and 5 carrots to make a dish. If you want to use 8 potatoes, how many carrots do you need?” Many make the mistake of simply trying to add up the corresponding values. However, to maintain the same proportion, you should multiply, not add. Here is the error and correct solution given task:

      • Wrong method: “8 - 4 = 4, that is, 4 potatoes were added to the recipe. So, you need to take the previous 5 carrots and add 4 to them, so that ... something is not right! Proportions work differently. Let's try again".
      • The correct method is: “8/4 = 2, that is, the number of potatoes has doubled. This means that the number of carrots should also be multiplied by 2. 5 x 2 = 10, that is, 10 carrots must be used in the new recipe.

   2. Convert all values ​​to the same units. Sometimes the problem arises because the values ​​have different units. Before writing down the proportion, convert all quantities to the same units of measurement. For example:

      • The dragon has 500 grams of gold and 10 kilograms of silver. What is the ratio of gold to silver in dragon reserves?
      • Grams and kilograms are different units of measurement, so they should be unified. 1 kilogram = 1,000 grams, so 10 kilograms = 10 kilograms x 1,000 grams/1 kilogram = 10 x 1,000 grams = 10,000 grams.
      • So the dragon has 500 grams of gold and 10,000 grams of silver.
      • The ratio of the mass of gold to the mass of silver is 500 grams of gold / 10,000 grams of silver = 5/100 = 1/20.

   3. Write down units of measurement in the solution of the problem. In problems with proportions, it is much easier to find an error if you write down after each value its unit of measurement. Remember that if the numerator and denominator have the same units of measure, they are reduced. After all possible abbreviations, the correct units of measurement should be obtained in the answer.

      • For example: given 6 boxes, and in every three boxes there are 9 balls; how many balls are there?
      • Wrong method: 6 boxes x 3 boxes / 9 marbles = ... Hmm, nothing is reduced, and the answer is “boxes x boxes / marbles“. This makes no sense.
      • Correct method: 6 boxes x 9 balls / 3 boxes = 6 boxes x 3 balls / 1 box = 6 x 3 balls / 1 = 18 balls.

Often there are cases when you need to find the percentage of a number. Those who are now in school should easily cope with this task. But it happens that the decision flies out of your head, and you need to calculate it urgently. So how do you find the percentage of a number?

How to find the percentage of a number and the number of a percentage on paper

The easiest way to find a percentage of a number is a proportion. Let's say you need to calculate the percentage of defective goods. It is known that a total of 300 parts were produced, 20 of them defective. Here's how to find the percentage of a number using proportion:

Now remember how the proportions were calculated at school: (20 * 100) / 300 \u003d 6.66%. AT reverse side it works like this: you need to find out how much is one percent of the number and multiply by one hundred. Suppose you need to calculate how many cars were produced if 120 cars were delivered to the city, which is 5% of the entire batch. Divide 120 by 5 and get 24. Now it remains to multiply by a hundred, and you will find out how many cars were produced in total. So, knowing how to find the percentage of a number and the number of a percentage, you can solve problems of this kind on paper.

Learn also how to convert a simple fraction to a decimal and vice versa.

What is the difference between compound and simple interest? Read.

Using Third Party Programs

In order to quickly calculate the interest, you can use the usual office tools - a browser or Microsoft Excel. If you have an internet connection, then you can use the services online calculator percent. There are a lot of similar services on the network, so you are sure to find what you are looking for. Or you can just write in the Google search line "5% of 100", for example. He will give you the answer instantly, counting on the built-in calculator.

But the most popular office solution is Microsoft Excel. Most often, interest is needed when compiling tables, and when there is a powerful tool at hand that can calculate interest for you (for example, the amount tax deduction), then it would be a sin not to use it.

How to find percentage of a number in excel? Exactly the same as on paper, with the only difference that you do not have to recalculate manually. Formulas in Excel are written in cells and begin with the "=" sign. Translating the proportion described above into the Excel formula language, you will get the following expression: \u003d B1 / A1, where A1 is the total number of parts, and B1 is the number of defective parts. After that, in the context menu of cell C1, select the “cell format” item and select the percentage numeric format. The answer will automatically be converted into percentages. After you can copy the formula to other cells, the cell addresses will change automatically.

To solve most problems in high school mathematics, knowledge of proportioning is required. This simple skill will help you not only perform complex exercises from the textbook, but also delve into the very essence of mathematical science. How to make a proportion? Now let's figure it out.

by the most simple example is a problem where three parameters are known, and the fourth must be found. The proportions are, of course, different, but often you need to find some number by percentage. For example, the boy had ten apples in total. He gave the fourth part to his mother. How many apples does the boy have left? This is the simplest example that will allow you to make a proportion. The main thing is to do it. There were originally ten apples. Let it be 100%. This we marked all his apples. He gave one-fourth. 1/4=25/100. So, he has left: 100% (it was originally) - 25% (he gave) = 75%. This figure shows the percentage of the amount of fruit left over the amount of fruit that was available first. Now we have three numbers by which we can already solve the proportion. 10 apples - 100%, X apples - 75%, where x is the desired amount of fruit. How to make a proportion? It is necessary to understand what it is. Mathematically it looks like this. The equal sign is for your understanding.

10 apples = 100%;

x apples = 75%.

It turns out that 10/x = 100%/75. This is the main property of proportions. After all, the more x, the more percent is this number from the original. We solve this proportion and get that x=7.5 apples. Why the boy decided to give a non-integer amount, we do not know. Now you know how to make a proportion. The main thing is to find two ratios, one of which contains the desired unknown.

Solving a proportion often comes down to simple multiplication and then division. Children are not taught in schools why this is so. While it is important to understand that proportional relationships are mathematical classics, the very essence of science. To solve proportions, you need to be able to handle fractions. For example, it is often necessary to convert interest to common fractions. That is, a record of 95% will not work. And if you immediately write 95/100, then you can make solid reductions without starting the main count. It’s worth saying right away that if your proportion turned out with two unknowns, then it cannot be solved. No professor can help you here. And your task, most likely, has a more complex algorithm for correct actions.

Consider another example where there are no percentages. The motorist bought 5 liters of gasoline for 150 rubles. He thought about how much he would pay for 30 liters of fuel. To solve this problem, we denote by x the required amount of money. You can solve this problem yourself and then check the answer. If you have not yet figured out how to make a proportion, then look. 5 liters of gasoline is 150 rubles. As in the first example, let's write 5l - 150r. Now let's find the third number. Of course, it's 30 liters. Agree that a pair of 30 l - x rubles is appropriate in this situation. Let's move on to mathematical language.

5 liters - 150 rubles;

30 liters - x rubles;

We solve this proportion:

x = 900 rubles.

That's what we decided. In your task, do not forget to check the adequacy of the answer. It happens that with the wrong decision, cars reach unrealistic speeds of 5000 kilometers per hour and so on. Now you know how to make a proportion. Also you can solve it. As you can see, there is nothing complicated in this.

From the point of view of mathematics, a proportion is the equality of two ratios. Interdependence is characteristic of all parts of the proportion, as well as their unchanging result. You can understand how to make a proportion by familiarizing yourself with the properties and formula of proportion. To understand the principle of solving proportions, it will be sufficient to consider one example. Only directly solving proportions, you can easily and quickly learn these skills. And this article will help the reader in this.

Proportion properties and formula

1. Reversal of proportion. In the case when the given equality looks like 1a: 2b = 3c: 4d, write 2b: 1a = 4d: 3c. (Moreover, 1a, 2b, 3c and 4d are prime numbers, other than 0).
2. Multiplying the given members of the proportion crosswise. In literal terms, this looks like this: 1a: 2b \u003d 3c: 4d, and writing 1a4d \u003d 2b3c will be equivalent to it. Thus, the product of the extreme parts of any proportion (the numbers at the edges of the equality) is always equal to the product of the middle parts (the numbers located in the middle of the equality).
3. When compiling a proportion, such a property of it as a permutation of the extreme and middle terms can also be useful. The equality formula 1a: 2b = 3c: 4d can be displayed in the following ways:
   • 1a: 3c = 2b: 4d (when the middle members of the proportion are rearranged).
   • 4d: 2b = 3c: 1a (when the extreme members of the proportion are rearranged).
4. Perfectly helps in solving the proportion of its property of increase and decrease. With 1a: 2b = 3c: 4d, write:
   • (1a + 2b) : 2b = (3c + 4d) : 4d (equality by increasing proportion).
   • (1a - 2b) : 2b = (3c - 4d) : 4d (equality by decreasing proportion).
5. You can create proportions by adding and subtracting. When the proportion is written as 1a:2b = 3c:4d then:
   • (1a + 3c) : (2b + 4d) = 1a: 2b = 3c: 4d (the proportion is added).
   • (1a - 3c) : (2b - 4d) = 1a: 2b = 3c: 4d (the proportion is subtracted).
6. Also, when solving a proportion containing fractional or large numbers, you can divide or multiply both of its members by the same number. For example, the components of the proportion 70:40=320:60 can be written like this: 10*(7:4=32:6).
7. The variant of solving the proportion with percentages looks like this. For example, write down, 30=100%, 12=x. Now you should multiply the middle terms (12 * 100) and divide by the known extreme (30). Thus, the answer is: x=40%. In a similar way, if necessary, you can multiply the known extreme terms and divide them by a given average number, obtaining the desired result.

If you are interested in a specific proportion formula, then in the simplest and most common version, the proportion is such an equality (formula): a / b \u003d c / d, in which a, b, c and d are four non-zero numbers.

Task 1. The thickness of 300 sheets of printer paper is 3.3 cm. How thick would a stack of 500 sheets of the same paper be?

Decision. Let x cm be the thickness of a 500-sheet paper ream. In two ways we find the thickness of one sheet of paper:

3,3: 300 or x : 500.

Since the sheets of paper are the same, these two ratios are equal to each other. We get the proportion reminder: proportion is the equality of two ratios):

x=(3.3 · 500): 300;

x=5.5. Answer: pack 500 sheets of paper has a thickness 5.5 cm.

This is a classic reasoning and formulation of a solution to a problem. Such problems are often included in graduate tests, which usually write the solution in this form:

or they decide orally, arguing as follows: if 300 sheets have a thickness of 3.3 cm, then 100 sheets have a thickness 3 times smaller. We divide 3.3 by 3, we get 1.1 cm. This is the thickness of a 100 sheet of paper. Therefore, 500 sheets will have a thickness 5 times greater, therefore, we multiply 1.1 cm by 5 and we get the answer: 5.5 cm.

Of course, this is justified, since the time for testing graduates and applicants is limited. However, in this lesson we will reason and write the solution as it should be done in 6 class.

Task 2. How much water is contained in 5 kg of watermelon if it is known that watermelon consists of 98% water?

Decision.

The entire mass of watermelon (5 kg) is 100%. Water will be x kg or 98%. In two ways, you can find how many kg fall on 1% of the mass.

5: 100 or x : 98. We get the proportion:

5: 100 = x : 98.

x=(5 · 98): 100;

x=4.9 Answer: in 5kg watermelon contains 4.9 kg of water.

The mass of 21 liters of oil is 16.8 kg. What is the mass of 35 liters of oil?

Decision.

Let the mass of 35 liters of oil be x kg. Then in two ways you can find the mass of 1 liter of oil:

16,8: 21 or x : 35. We get the proportion:

16,8: 21=x : 35.

Find the middle term of the proportion. To do this, we multiply the extreme terms of the proportion ( 16,8 and 35 ) and divide by the known middle term ( 21 ). Reduce the fraction by 7 .

Multiply the numerator and denominator of the fraction by 10 so that the numerator and denominator contain only integers. We reduce the fraction by 5 (5 and 10) and on 3 (168 and 3).

Answer: 35 liters of oil have a mass 28 kg.

After 82% of the entire field had been plowed, 9 hectares remained to be plowed. What is the area of ​​the entire field?

Decision.

Let the area of ​​the entire field be x ha, which is 100%. It remains to plow 9 hectares, which is 100% - 82% = 18% of the entire field. Let's express 1% of the field area in two ways. This is:

X : 100 or 9 : 18. We make a proportion:

X : 100 = 9: 18.

We find the unknown extreme term of the proportion. To do this, we multiply the average terms of the proportion ( 100 and 9 ) and divide by the known extreme term ( 18 ). We reduce the fraction.

Answer: area of ​​the whole field 50 ha.

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