Tasks solved using the equation. Math problem solving

In the course of school mathematics, tasks are sure to meet. Some tame in a few actions, others require some puzzle.

The problems solved using the equation are difficult at first glance. If you practice, this process will come to automatism.

Geometric figures

In order to understand the issue, you need to understand the essence. Carefully read the condition, it is better to reread several times. Equation problems are only seemingly difficult. Consider the simplest example to get you started.

Given a rectangle, you need to find its area. Given: the width is 48% less than the length, the perimeter of the rectangle is 7.6 centimeters.

Solving problems in mathematics requires careful reading, logic. Let's deal with her. What should be considered first? Denote the length by x. Therefore, in our equation, the width will be 0.52x. We are given a perimeter of 7.6 centimeters. We find a half-perimeter, for this we divide 7.6 centimeters by 2, it is 3.8 centimeters. We got an equation with which we will find the length and width:

0.52x + x = 3.8.

When we get x (length), it will not be difficult to find 0.52x (width). If we know these two quantities, then we find the answer to the main question.

The problems solved with the help of the equation are not as complicated as they seem, we could understand this from the first example. We found the length x = 2.5 centimeters, the width (let’s note y) 0.52x = 1.3 centimeters. We pass to the square. It is found by the simple formula S = x * y (for rectangles). In our problem, S = 3.25. This will be the answer.

Consider more examples of solving problems with finding the area. And this time take a rectangle. Solving problems in mathematics to find the perimeter, the area of ​​different figures is quite common. We read the condition of the problem: given a rectangle, its length is 3.6 centimeters more than the width, which is 1/7 of the perimeter of the figure. Find the area of ​​this rectangle.

It will be convenient to indicate the width for the variable x, and the length for ( x + 3.6) centimeters. Find the perimeter:

P = 2x + 3.6 .

We cannot solve the equation, because we have two variables in it. Therefore, we look again at the condition. It says that the width is 1/7 of the perimeter. We get the equation:

1/7 (2x + 3.6) = x .

For the convenience of the solution, we multiply each part of the equation by 7, so we get rid of the fraction:

2x + 3.6 = 7x.

After solving, we get x (width) = 0.72 centimeters. Knowing the width, we find the length:

0.72 + 3.6 = 4.32 cm.

Now we know the length and width, we answer the main question about what is the area of ​​the rectangle.

S = x * y , S = 3.1104 cm.

Milk cans

Solving problems using equations causes many difficulties for students, despite the fact that this topic begins in the fourth grade. There are many examples, we looked at finding the area of ​​the figures, now let's digress a bit from geometry. Let's look at simple tasks with compiling tables, they help visually: the data that helps in solving is better visible.

Invite the children to read the condition of the problem and make a table to help prepare the equation. Here is the condition: there are two cans, in the first three times more milk than in the second. If from the first one pour five liters into the second, then the milk will be equally divided. Question: how much milk was in each can?

To assist in the decision, you must create a table. What should she look like?

How does this help in making the equation? We know that as a result of milk it became evenly divided, which means the equation will look like this:

3x - 5 = x + 5;

2x = 10;

x = 5.

We found the initial amount of milk in the second can, which means the first was: 5 * 3 = 15 liters of milk.

Now a little explanation on the compilation of the table.

Why did we designate the first can for 3: the condition stipulates that in the second can there is three times less milk. Then we read that 5 liters were drained from the first can, therefore it became 3x - 5 , and in the second they poured: x + 5 . Why did we put an equal sign between these conditions? In the condition of the task it is said that the milk has become equally divided.

So we get the answer: the first can is 15 liters, the second is 5 liters of milk.

Depth determination

According to the condition of the task: the depth of the first well is 3.4 meters greater than the second. The first well was increased by 21.6 meters, and the second - three times, after these actions the wells have the same depth. It is necessary to calculate what depth each well originally had.

There are numerous methods for solving problems , you can do by actions, make equations or their system, but the second option is most convenient. To get to the solution, let's create a table, as in the previous example.

We proceed to the preparation of the equation. Since the wells have become the same depth, it has the following form:

x + 3.4 + 21.6 = 3x;

x - 3x = -25;

-2x = -25;

x = -25 / -2;

x = 12.5

We found the initial depth of the second well, now we can find the first:

12.5 + 3.4 = 15.9 m.

After the actions taken, write down the answer: 15.9 m, 12.5 m.

Two brothers

Please note that this task is different from all the previous ones, since according to the condition there were initially the same number of items. Based on this, the auxiliary table is compiled in the reverse order, that is, from "became" to "was".

Condition: two brothers were given equally nuts, but the eldest gave 10 to his brother, after which the younger got five times more nuts. How many nuts does each boy have now?

We compose the equation:

x + 10 = 5x - 10;

-4x = -20;

x = 5 - became the older brother’s nuts;

5 * 5 = 25 - with the younger brother.

Now you can write down the answer: 5 nuts; 25 nuts.

Purchases

At school, you need to buy books and notebooks, the former more expensive than the latter by 4.8 rubles. It is necessary to calculate how much one notebook and one book are worth if the purchase of five books and twenty-one notebooks paid the same amount of money.

Before proceeding to the solution, it is worth answering the following questions:

• What is the problem in question?
• How much did they pay?
• What did you buy?
• What values ​​can be leveled among themselves?
• What do you need to know?
• What value to take for x ?

If you answered all the questions, then go to the solution. In this example, the value of x can be taken as the price of one notebook, and the cost of the book. Let's consider two possible options:

1. x - the cost of one notebook, then x + 4.8 - the price of the book. Based on this, we obtain the equation: 21x = 5 (x + 4.8).
2. x - the cost of the book, then x - 4.8 - the price of the notebook. The equation has the form: 21 (x - 4.8) = 5x.

You can choose a more convenient option for yourself, then we will solve two equations and compare the answers; as a result, they should coincide.

First way

The solution to the first equation:

21x = 5 (x + 4.8);

4.2x = x + 4.8;

4.2x - x = 4.8;

3.2x = 4.8;

x = 1.5 (rubles) - the cost of one notebook;

4.8 + 1.5 = 6.3 (rubles) - the cost of one book.

Another way to solve this equation (opening brackets):

21x = 5 (x + 4.8);

21x = 5x + 24;

16x = 24;

x = 1.5 (rubles) - the cost of one notebook ;

1.5 + 4.8 = 6.3 (rubles) - the cost of one book.

Second way

5x = 21 (x - 4.8);

5x = 21x - 100.8;

16x = 100.8;

x = 6.3 (rubles) - the cost of 1 book;

6.3 - 4.8 = 1.5 (rubles) - the cost of one notebook.

As can be seen from the examples, the answers are identical, therefore, the problem is solved correctly. Make sure the solution is correct; in our example, the answers should not be negative.

There are other problems that can be solved with the help of an equation, for example, on motion. We will consider them in more detail in the following examples.

Two cars

This section will focus on movement tasks. To be able to solve them, you need to know the following rule:

S = V * T,

S - distance, V - speed, T - time.

Let's try to consider an example.

Two cars drove simultaneously from point A to point B. The first drove the entire distance at the same speed, the second half of the way traveled at a speed of 24 km / h, and the second - 16 km / h. It is necessary to determine the speed of the first motorist, if they came to point B simultaneously.

What we need to draw up the equation: the main variable V ₁ (speed of the first car), secondary: S - the path, T ₁ - the travel time of the first car. Equation: S = V ₁ * T _{1 .}

Next: the second car drove the first half of the way (S / 2) at a speed of V ₂ = 24 km / h. We get the expression: S / 2 = 24 * T _{2 .}

He drove the next part of the way with a speed of V ₃ = 16 km / h. Get S / 2 = 16 * T ₃ .

Further from the condition it is seen that the cars arrived at the same time, therefore T ₁ = T ₂ + T ₃ . Now we have to express the variables T ₁ , T ₂ , T ₃ from our previous conditions. We get the equation: S / V ₁ = (S / 48) + (S / 32).

S is taken as unity and we solve the equation:

1 / V ₁ = 1/48 + 1/32;

1 / V ₁ = (2/96) + (3/96);

1 / V ₁ = 5/96;

V ₁ = 96/5;

V ₁ = 19.2 km / h.

This is the answer. The problems solved with the help of the equation are complicated only at first glance. In addition to the above, there may be tasks for work, what it is, we will consider in the next section.

Task to work

To solve this type of task, you need to know the formula:

A = VT ,

where A is work, V is productivity.

For a more detailed description, you need to give an example. The topic "Solving problems by equation" (Grade 6) may not contain such problems, since this is a more complex level, but nevertheless we give an example for familiarization.

We carefully read the condition: two workers work together and complete the plan in twelve days. It is necessary to determine how much time the first employee will need to fulfill the same norm independently. It is known that he performs a volume of work in two days, like a second worker in three days.

Solving problems in the preparation of equations requires careful reading of the conditions. The first thing we understood from the task is that the work is not defined, which means we take it as a unit, that is, A = 1 . If the task refers to a certain number of parts or liters, then the work should be taken according to these data.

We denote the productivity of the first and second workers through V ₁ and V _2, respectively, at this stage it is possible to draw up the following equation:

1 = 12 (V ₁ + V ₂ ) .

What does this equation tell us? That all the work is done by two people in twelve hours.

Further we can state: 2V ₁ = 3V ₂ . Because the first in two days does as much as the second in three. We got a system of equations:

1 = 12 (V1 + V2);

2V ₁ = 3V _2.

As a result of solving the system, we got an equation with one variable:

1 - 8V ₁ = 12V _1;

V ₁ = 1/20 = 0.05.

This is the productivity of the first worker. Now we can find the time for which the first person will cope with all the work:

A = V ₁ * T _1;

1 = 0.05 * T ₁ ;

T ₁ = 20.

Since the day was taken as a unit of time, the answer is 20 days.

Reformulation of the problem

If you have mastered the skill to solve tasks on the move, and you have some difficulties with tasks for work, then it’s possible to get movement out of work. How? If we take the last example, then the condition is as follows: Oleg and Dima move towards each other, they meet in 12 hours. How much Oleg will overcome the path on his own, if it is known that in two hours he travels a path equal to the path of Dima in three hours.