Question

Гном хранит у себя в чулане 4 слитка из золота. Он взвесил каждые два слитка и получил такие результаты: 24, 27, 29, 29, 31, 34 грамма (у двух РАЗНЫХ пар получился вес 29 грамм). Сколько весит каждый слиток в отдельности? В ответе напишите объяснение и решение. Заранее спасибо!

Answer 1

Обозначим эти слитки, как x, y, z, w, тогда условие можно переписать в виде:
x + y = 24      (1)
x + z = 27      (2)
x + w = 29     (3)
y + z = 29      (4)
y + w = 31     (5)
z + w  = 34    (6)

Вычитаем первое из второго и прибавляем четвертое
_ x + z = 27  (2)
   x + y = 24  (1)
   -------------
  + z  - y = 3   
     y + z = 29 (4)
  ----------------
    2*z = 32

z = 16

Из (2) получаем x = 27 - 16 = 11
Из (1) получаем y = 24 - 11 = 13
Из (3) получаем w = 29 - 11 = 18

Проверяем
x + y = 11 + 13 = 24
x + z = 11 + 16 = 27
x + w = 11 + 18 = 29
y + z  = 13 + 16 = 29
y + w = 13 + 18 = 31
z + w = 16 + 18 = 34

Все совпало, т.о. искомые веса слитков найдены.

Answer 2

Problem Analysis

The gnome has 4 gold ingots in his pantry. He weighed each pair of ingots and obtained the following results: 24, 27, 29, 29, 31, and 34 grams. Two different pairs of ingots had a combined weight of 29 grams. We need to determine the weight of each individual ingot.

Solution

Let's assume the weights of the four ingots are A, B, C, and D. We can set up the following equations based on the given information:

Equation 1: A + B = 29 (since two different pairs of ingots have a combined weight of 29 grams) Equation 2: A + C = 31 Equation 3: A + D = 34 Equation 4: B + C = 27 Equation 5: B + D = 24 Equation 6: C + D = 29

To solve this system of equations, we can use substitution or elimination. Let's use substitution:

From Equation 1, we can express B in terms of A: B = 29 - A Substituting this value of B into Equation 4, we get: 29 - A + C = 27 Rearranging the equation, we have: C = A - 2

Substituting the values of C and B into Equation 6, we get: A - 2 + 29 - A = 29 Simplifying the equation, we have: 27 = 29 This equation is not possible, which means there is no solution that satisfies all the given conditions.

Therefore, there is no unique solution to this problem.

Conclusion

Based on the given information, there is no unique solution to determine the weight of each individual gold ingot. The provided data leads to an inconsistent system of equations, which means there is no combination of weights that satisfies all the given conditions.