How to Solve a Math Problem in Your Head
One fascinating and beneficial way to sharpen your math skills is by practicing mental arithmetic. Solving problems solely with your mind not only enhances logic and memory but also turns a mundane task into an enjoyable intellectual exercise. An example of such a problem is found in A.P. Chekhov’s story “The Tutor.”
The crux of the problem is this: a merchant purchases 138 arshins of cloth in two colors, black and blue, for 540 rubles. The blue cloth costs 5 rubles per arshin, while the black cloth is priced at 3 rubles per arshin. The task is to determine how many arshins of each color the merchant bought.
The protagonist of the story, Petya Udodov’s father, solves this problem without using paper or a calculator, relying solely on his knowledge and intuition. He employs visualization and associative thinking, picturing the 138 arshins as two distinct colors of cloth.
Creating associations can significantly simplify problem-solving. Imagine the merchant buying blue cloth for his daughter’s birthday outfit (since 5 is a lucky number) and black cloth to augment his nephew’s wardrobe (as 3 is considered unlucky). By using emotional ties, you can quickly break down and compare the numbers.
If an analytical approach suits you better, break down the total arshins into two groups based on their respective costs. For instance, divide the 540 rubles by the price of blue cloth (5 rubles per arshin) and black cloth (3 rubles per arshin). This helps you understand how much of the total sum goes to each color.
The multiplication table can also be an invaluable tool. Knowing that 5 x 4 = 20, you can easily calculate that 5 x 28 = 140. Now, if you multiply the price of the blue cloth (5 rubles) by the supposed quantity (28 arshins), you get 140 rubles, which is close in value to the 134 arshins of cloth. Similarly, for the black cloth, divide the total cost, 540 rubles, by 3 (the price per arshin of black material), getting 180, then distribute this among the number of arshins (180 / 3 = 60).
By regularly practicing mental arithmetic, visualization techniques, associative thinking, and an analytical approach, and by confidently knowing your multiplication tables, you can greatly enhance your math skills and solve problems in your head with ease.
Solving a System of Equations Using the Traditional School Method
In the classic school approach to solving systems of equations, we typically deal with two variables—x and y. The process often feels like an engaging puzzle, requiring the correct placement of elements to find the desired variable values. Let’s explore how this works with some examples!
Consider the following system of equations:
2x + 3y = 10
x – y = 2
The first step is to choose a variable to eliminate. In this case, it’s practical to eliminate x. We’ll multiply the second equation by 2 to align the coefficients of x:
2(x – y) = 2 * 2
This simplifies to:
2x – 2y = 4
Next, we add this equation to the first one:
2x + 3y = 10
+ 2x – 2y = 4
for free
The result is:
4x + y = 14
To solve for x, we simplify the equation:
4x = 14 – y
Thus,
x = 3.5
Now, we substitute the value of x back into one of the original equations, such as the second one:
3.5 – y = 2
Performing simple subtraction, we get:
y = 1.5
Therefore, the solution to the system is x = 3.5 and y = 1.5.
Let’s look at another example with a practical application:
Imagine this problem: “In a white box, there is a certain amount of black and blue cloth. The total length of the cloth is 138 arshins, with the black cloth being 12 arshins longer than the blue cloth. How many arshins of each type of cloth are in the box?”
We can set up a system of equations to solve this:
x + y = 138
x = y + 12
Substitute the value from the second equation into the first:
(y + 12) + y = 138
Simplify and solve:
2y + 12 = 138
2y = 126
y = 63
Finally, substituting the value of y back into one of the equations gives us x:
x = 63 + 12 = 75
Thus, the box contains 75 arshins of black cloth and 63 arshins of blue cloth. That’s the entire solving process!
Thus, the system of equations not only solves abstract mathematical problems but also aids in finding solutions to real-life issues.
Solving the Problem of Calculating the Amount of Black and Blue Cloth
One of the most fascinating and engaging mathematical problems is determining the amount of black and blue cloth a merchant could purchase, given the total cost of the transaction and the price per yard of both types of cloth.
There are several ways to approach this problem, each not only providing the correct answer but also showcasing the elegance and beauty of mathematical calculations. Let’s explore two primary methods.
The first method is based on finding the difference between the cost of the entire purchase and the cost if the cloth were all black. Imagine that all the yards purchased are black cloth; calculating the cost would be straightforward. Then, determine the difference between the actual purchase cost and the hypothetical cost if all the cloth were black. Divide this difference by the difference in cost between one yard of black and blue cloth. This will give the number of yards of blue cloth. To complete the calculations, subtract the obtained amount of blue cloth from the total number of yards to get the quantity of black cloth.
For clarity, consider an example: A merchant paid 200 rubles for 40 yards of cloth. One yard of black cloth costs 5 rubles, and one yard of blue cloth costs 3 rubles. If all 40 yards were black, the cost would be 40 * 5 = 200 rubles. Since the total cost remained 200 rubles, the difference is zero, indicating there is no blue cloth. This is a simple yet illustrative example.
The second approach assumes the opposite – that the merchant intended to purchase only blue cloth. Similarly, we calculate the cost of an imaginary purchase of blue cloth, subtract it from the total cost, then divide the difference by the difference in cost between one yard of black and blue cloth, and find the number of yards of black cloth.
Let’s consider another example. The same conditions apply: a merchant pays 200 rubles for 40 arshins of cloth, with one arshin of black cloth costing 5 rubles and one arshin of blue cloth costing 3 rubles. If we assume that all the arshins are blue, then 40 arshins would cost 40 * 3 = 120 rubles. The difference between 200 rubles and 120 rubles is 80 rubles. Since one arshin of black cloth is 2 rubles more expensive, the amount of black cloth would be 80 / 2 = 40 arshins. However, this leads to a conflict because the total number of black cloth arshins cannot exceed the total number of arshins, indicating a mistake in the algorithm that needs correction by recalculating based on the remaining amount.
It’s interesting to note that there are actually two methods of solving these types of problems, highlighting that the creators aimed to make them as intuitive and practical as possible. Remember, in the 19th century, a merchant burdened with cumbersome calculations was always looking for ways to reduce the time spent on computations. Algorithms that required only a few theoretical steps were perfect for quickly arriving at the correct answer.
Thus, what initially appears to be a simple mathematical problem actually offers us a glimpse into the past, illustrating the daily lives and mindsets of 19th-century merchants. It serves as an example of how history and culture influenced the development of mathematics and practical calculations in Russia.