The sum of the two numbers is 407, and the first term is 10 times greater than the second. Find these numbers?

Let us denote the required numbers by x and y. According to the condition of the problem, the sum of these two numbers is 407, therefore, the following relationship is true: x + y = 407. It is also known that one of these numbers is 10 times larger than the other, therefore, the following relationship is true: x = 10 * y. We solve the resulting system of equations. Substituting into the first equation the value x = 10 * y from the second equation, we get: 10 * y + y = 407. We solve the resulting equation: 11 * y = 407; y = 407/11; y = 37. Knowing y, we find x: x = 10 * y = 10 * 37 = 370.
Answer: the required numbers are 370 and 37.