Drawing up a system of equations. Posts tagged "solving proportion problems"

From a mathematical point of view, 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 create 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 by directly solving proportions can you quickly and easily learn these skills. And this article will help the reader with this.

Properties of proportion and formula

Reversal of proportion. In the case when the given equality looks like 1a: 2b = 3c: 4d, write 2b: 1a = 4d: 3c. (And 1a, 2b, 3c and 4d are prime numbers other than 0).
Multiplying the given terms of the proportion crosswise. In literal expression it looks like this: 1a: 2b = 3c: 4d, and writing 1a4d = 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).
When drawing up a proportion, its property of rearranging the extreme and middle terms can also be useful. The formula of equality 1a: 2b = 3c: 4d can be displayed in the following ways:
- 1a: 3c = 2b: 4d (when the middle terms of the proportion are rearranged).
- 4d: 2b = 3c: 1a (when the extreme terms of the proportion are rearranged).
Its property of increasing and decreasing helps perfectly in solving proportions. When 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).
You can create a proportion 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 made by addition).
- (1a – 3c) : (2b – 4d) = 1a: 2b = 3c: 4d (the proportion is calculated by subtraction).
Also, when solving a proportion containing fractional or large numbers, you can divide or multiply both of its terms by the same number. For example, the components of the proportion 70:40=320:60 can be written as follows: 10*(7:4=32:6).
An option for solving proportions 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 the following equality (formula): a/b = c/d, in which a, b, c and d are four non-zero numbers.

The ability to calculate a percentage of a number when you need to find out a late fee, the amount of an overpayment on a loan, or a company’s profit if its turnover and markup are known.

How to find a number by its percentage?

Rule. To find a number by its specified percentage, you need to divide the given number by the given percentage value, and multiply the result by 100.

With this calculation, we first determine how many units of this number are contained in 1%, and then in the whole number (100%).

For example:
A number whose 23% is 52 is found like this:
52: 23 * 100 = 226.1

This means that if the number 226.1 is equal to 100%, then the number 52 is equal to 23% of this number.

We find a number whose 125% is 240 as follows:
240: 125 * 100 = 192.

When determining a number by its percentage, remember that:

- if the percentage is less than 100%, then the number obtained as a result of calculations is greater than the specified number (if 23%< 100%, то 226,1 > 52);
— if the percentage is greater than 100%, then the number obtained as a result of the calculation is less than the specified number (if 125% > 100%, then 192< 240).

Therefore, when calculating a number by its percentage, for self-control you need to check:

— the percentage specified in the condition is greater or less than 100%;
— the result of a calculation is greater or less than a given number.

How to find out the percentage of the amount in the general case?

After this there are two options:

If you want to find out what percentage another amount is from the original, you just need to divide it by the 1% amount obtained earlier.
If you need an amount that is, say, 27.5% of the original, you need to multiply the amount of 1% by the required amount of interest.

How to calculate a percentage of an amount using a proportion?

To do this, you will have to use knowledge about the method of proportions, which is taught as part of the school mathematics course. It will look like this:

Let A be the principal amount equal to 100%, and B be the amount whose relationship with A as a percentage we need to know. We write down the proportion:

(X in this case is the number of percent).

According to the rules for calculating proportions, we obtain the following formula:

X = 100 * V / A

If you need to find out how much the amount B will be if the number of percentages of the amount A is already known, the formula will look different:

B = 100 * X / A

Now all that remains is to substitute known numbers into the formula - and you can make the calculation.

How to calculate the percentage of an amount using known ratios?

Finally, you can use a simpler method. To do this, just remember that 1% as a decimal is 0.01. Accordingly, 20% is 0.2; 48% - 0.48; 37.5% is 0.375, etc. It is enough to multiply the original amount by the corresponding number - and the result will indicate the amount of interest.

In addition, sometimes you can use simple fractions. For example, 10% is 0.1, that is, 1/10; therefore, finding out how much 10% is is simple: you just need to divide the original amount by 10.

Other examples of such relationships would be:

12.5% - 1/8, that is, you need to divide by 8;
20% - 1/5, that is, you need to divide by 5;
25% - 1/4, that is, divide by 4;
50% - 1/2, that is, it needs to be divided in half;
75% is 3/4, that is, you need to divide by 4 and multiply by 3.

True, not all simple fractions are convenient for calculating percentages. For example, 1/3 is close in size to 33%, but not exactly equal: 1/3 is 33.(3)% (that is, a fraction with infinite threes after the decimal point).

How to subtract a percentage from an amount without using a calculator?

If you need to subtract an unknown number, which is a certain amount of percent, from an already known amount, you can use the following methods:

Calculate the unknown number using one of the above methods, and then subtract it from the original one.
Immediately calculate the remaining amount. To do this, subtract from 100% the number of percentages that need to be subtracted, and convert the resulting result from percentage to number using any of the methods described above.

The second example is more convenient, so let’s illustrate it. Let's say we need to find out how much will remain if we subtract 16% from 4779. The calculation will be like this:

We subtract 16 from 100 (the total number of percent). We get 84.
We calculate how much 84% of 4779 is. We get 4014.36.

How to calculate (subtract) a percentage from an amount with a calculator in hand?

All of the above calculations are easier to do using a calculator. It can be either in the form of a separate device or in the form of a special program on a computer, smartphone or regular mobile phone (even the oldest devices currently in use usually have this function). With their help, the question how to calculate percentage from amount, The solution is very simple:

The initial amount is collected.
The “-” sign is pressed.
Enter the number of percentages you want to subtract.
The “%” sign is pressed.
The “=” sign is pressed.

As a result, the required number is displayed on the screen.

How to subtract a percentage from an amount using an online calculator?

Finally, there are now quite a few sites on the Internet that offer an online calculator function. In this case, you don’t even need to know how to calculate percentage of amount: all user operations are reduced to entering the required numbers into the windows (or moving the sliders to obtain them), after which the result is immediately displayed on the screen.

This function is especially convenient for those who calculate not just an abstract percentage, but a specific amount of tax deduction or the amount of state duty. The fact is that in this case the calculations are more complicated: you not only need to find the percentages, but also add a constant part of the amount to them. An online calculator allows you to avoid such additional calculations. The main thing is to choose a site that uses data that complies with the current law.

Online interest calculator:

calculator.ru - allows you to perform various calculations when working with percentages;

mirurokov.ru - interest calculator;

A source of information:

nsovetnik.ru - article on how to calculate the percentage of the amount;

Problem 1. The thickness of 300 sheets of printer paper is 3.3 cm. What thickness will a pack of 500 sheets of the same paper be?

Solution. Let x cm be the thickness of a stack of paper of 500 sheets. There are two ways to 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. 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 have a thickness 5.5 cm.

This is a classic reasoning and design of a solution to a problem. Such problems are often included in test tasks for graduates, who usually write the solution in the following form:

or they decide orally, reasoning like this: if 300 sheets have a thickness of 3.3 cm, then 100 sheets have a thickness 3 times less. Divide 3.3 by 3, we get 1.1 cm. This is the thickness of a 100-sheet pack of paper. Therefore, 500 sheets will have a thickness 5 times greater, therefore, we multiply 1.1 cm by 5 and 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 down 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?

Solution.

The entire mass of the watermelon (5 kg) is 100%. Water will be x kg or 98%. There are two ways to find how many kg are in 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: 5kg watermelon contains 4.9 kg water.

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

Solution.

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

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 average term ( 21 ). Let's reduce the fraction by 7 .

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

Answer: 35 liters of oil have mass 28 kg.

After 82% of the entire field had been plowed, there was still 9 hectares left to plow. What is the area of the entire field?

Solution.

Let the area of the entire field be x hectares, which is 100%. There are 9 hectares left to plow, which is 100% - 82% = 18% of the entire field. We can express 1% of the field area in two ways. This:

X : 100 or 9 : 18. We make up the proportion:

X : 100 = 9: 18.

We find the unknown extreme term of the proportion. To do this, 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 entire field 50 hectares.

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In the last video lesson we looked at solving problems involving percentages using proportions. Then, according to the conditions of the problem, we needed to find the value of one or another quantity.

This time the initial and final values have already been given to us. Therefore, the problems will require you to find percentages. More precisely, by how many percent has this or that value changed. Let's try.

Task. The sneakers cost 3,200 rubles. After the price increase, they began to cost 4,000 rubles. By what percentage was the price of sneakers increased?

So, we solve through proportion. The first step - the original price was 3,200 rubles. Therefore, 3200 rubles is 100%.

In addition, we were given the final price - 4000 rubles. This is an unknown percentage, so let's call it x. We get the following construction:

3200 — 100%
4000 - x%

Well, the condition of the problem is written down. Let's make a proportion:

The fraction on the left cancels perfectly by 100: 3200: 100 = 32; 4000: 100 = 40. Alternatively, you can shorten it 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 terms. We get:

8 x = 100 10;
8x = 1000.

This is an ordinary linear equation. From here we find x:

x = 1000: 8 = 125

So, we got the final percentage x = 125. But is the number 125 a solution to the problem? No way! Because the task requires finding out by how many percent the price of sneakers was increased.

By what percentage - this means that we need to find the change:

∆ = 125 − 100 = 25

We received 25% - that’s how much the original price was increased. This is the answer: 25.

Problem B2 on percentages No. 2

Let's move on to the second task.

Task. The shirt cost 1800 rubles. After the price was reduced, it began to cost 1,530 rubles. By what percentage was the price of the shirt reduced?

Let's translate the condition into mathematical language. The original price is 1800 rubles - this is 100%. And the final price is 1,530 rubles - we know it, but we don’t know what percentage it is of the original value. Therefore, we denote it by x. We get the following construction:

1800 — 100%
1530 - x%

Based on the received record, we create a proportion:

To simplify further calculations, let's divide both sides of this equation by 100. In other words, we will cross out two zeros from 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 middle terms.

18 x = 1530 1;
18x = 1530.

All that remains is 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. Now we know that the new price obtained after the reduction is 85% of the old one. And in order to find 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! The problem is solved.

Attentive students will probably ask: why in the first problem, when finding the difference, did we subtract the initial number from the final number, and in the second problem did exactly the opposite: from the initial 100% we subtracted the final 85%?

Let's be clear on this point. Formally, in mathematics, a change in a quantity is always the difference between the final value and the initial value. In other words, in the second problem we should have gotten not 15, but −15.

However, this minus should under no circumstances be included in the answer, because it is already taken into account in the conditions of the original problem. It says directly about the price reduction. And a price reduction of 15% is the same as a price increase of −15%. That is why in the solution and answer to the problem it is enough to simply write 15 - without any minuses.

That's it, I hope we have sorted this out. This concludes our lesson for today. See you again!