Question

Имеется три слитка различных сплавов золота с серебром. Известно, что количество золота в 2 г сплава из третьего слитка тоже, что во взятых вместе 1 г из первого и 1 г из второго слитков. Масса третьего слитка равна суммарный массе части первого слитка, содержащий 10 Г золото, и части второго слитка, содержащий 80 г золота. Третий слиток в четыре раза тяжелее первого и содержит 75 г золота. Сколько граммов золота содержится в первом слитке?

Answer 1

Пусть масса слитков x1, x2, x3, а содержание золота a1%, a2%, a3%.
Количество золота в 2 г сплава из третьего слитка тоже, что во взятых вместе 1 г из первого и 1 г из второго слитков.
2*a3 = 1*a1 + 1*a2
Масса третьего слитка равна суммарной массе части 1-го слитка, содержащей 10 г золота, и части 2-го слитка, содержащей 80 г золота.
Если кусок 1 слитка с содержанием а1% содержит 10 г золота, то его масса m1*a1/100 = 10; m1 = 1000/a1
Точно также, если кусок с содержанием а2% содержит 80 г золота, то
m2*a2/100 = 80; m2 = 8000/a2
x3 = 1000/a1 + 8000/a2
Третий слиток в четыре раза тяжелее первого и содержит 75 г золота.
x3 = 4x1
x3*a3/100 = 75; x3 = 7500/a3
Сколько граммов золота содержится в первом слитке?
Надо найти число x1*a1/100 = ?
Выпишем наши уравнения
2a3 = a1 + a2
x3 = 7500/a3 = 1000/a1 + 8000/a2
x3 = 4x1
2 уравнение разделим на 100 и подставим 1 уравнение
a3 = (a1 + a2)/2; 1/a3 = 2/(a1 + a2)
75*2/(a1 + a2) = 10/a1 + 80/a2
Приводим к общему знаменателю a1*a2(a1 + a2)
150a1*a2 = 10a2(a1 + a2) + 80a1(a1 + a2)
Делим все еще на 10 и раскрываем скобки
15a1*a2 = a1*a2 + a2^2 + 8a1^2 + 8a1*a2
0 = 8a1^2 - 6a1*a2 + a2^2
Делим все на a2^2
8(a1/a2)^2 - 6(a1/a2) + 1 = 0
Квадратное уравнение относительно (a1/a2)
D/4 = 3^2 - 8*1 = 9 - 8 = 1
(a1/a2)1 = (3 - 1)/8 = 2/8 = 1/4
(a1/a2)2 = (3 + 1)/8 = 4/8 = 1/2
Содержание золота в 1 слитке или в 2, или в 4 раза меньше, чем во 2.
1) Пусть a2 = 4a1, тогда a3 = (a1 + a2)/2 = 5a1/2 = 2,5a1
Так как 3 слиток содержит 75 г золота, то 1 содержит 75/2,5 = 30 г.
2) Пусть a2 = 2a1, тогда a3 = (a1 + a2)/2 = 3a1/2 = 1,5a1
Так как 3 слиток содержит 75 г золота, то 1 содержит 75/1,5 = 50 г.
Как выбрать правильный из этих ответов, я не знаю.

Answer 2

Problem Analysis

We have three different gold-silver alloy ingots. We need to determine the amount of gold in the first ingot.

Given Information

- The second ingot, which weighs 2 grams, contains the same amount of gold as the combined amount of gold in 1 gram from the first ingot and 1 gram from the second ingot. - The third ingot weighs the same as the sum of the mass of a portion of the first ingot containing 10 grams of gold and a portion of the second ingot containing 80 grams of gold. - The third ingot is four times heavier than the first ingot and contains 75 grams of gold.

Solution

Let's assume the mass of the first ingot is x grams.

From the given information, we can write the following equations:

Equation 1: The second ingot, which weighs 2 grams, contains the same amount of gold as the combined amount of gold in 1 gram from the first ingot and 1 gram from the second ingot. - Gold in the second ingot = Gold in 1 gram of the first ingot + Gold in 1 gram of the second ingot

Equation 2: The third ingot weighs the same as the sum of the mass of a portion of the first ingot containing 10 grams of gold and a portion of the second ingot containing 80 grams of gold. - Mass of the third ingot = Mass of the portion of the first ingot containing 10 grams of gold + Mass of the portion of the second ingot containing 80 grams of gold

Equation 3: The third ingot is four times heavier than the first ingot and contains 75 grams of gold. - Mass of the third ingot = 4 * Mass of the first ingot - Gold in the third ingot = 75 grams

We can now solve these equations to find the mass of the first ingot and the amount of gold in it.

Solution Steps

Step 1: Use Equation 1 to find the amount of gold in 1 gram of the first ingot. Step 2: Use Equation 2 to find the mass of the portion of the first ingot containing 10 grams of gold. Step 3: Use Equation 3 to find the mass of the first ingot. Step 4: Calculate the amount of gold in the first ingot.

Let's solve these equations step by step.

Step 1: Amount of Gold in 1 gram of the First Ingot

According to Equation 1, the amount of gold in the second ingot is equal to the amount of gold in 1 gram of the first ingot plus the amount of gold in 1 gram of the second ingot.

Let's assume the amount of gold in 1 gram of the first ingot is y grams.

Gold in the second ingot = Gold in 1 gram of the first ingot + Gold in 1 gram of the second ingot

Using the given information, we know that the second ingot weighs 2 grams and contains the same amount of gold as the combined amount of gold in 1 gram from the first ingot and 1 gram from the second ingot.

Therefore, we can write the equation as:

y + Gold in 1 gram of the second ingot = Gold in the second ingot

Since the second ingot weighs 2 grams, we can rewrite the equation as:

y + Gold in 1 gram of the second ingot = Gold in 2 grams of the second ingot

Now, we need to find the amount of gold in 2 grams of the second ingot. Let's assume the amount of gold in 1 gram of the second ingot is z grams.

Gold in 2 grams of the second ingot = 2 * z grams

Substituting this value into the equation, we get:

y + z = 2 * z

Simplifying the equation, we find:

y = z

Therefore, the amount of gold in 1 gram of the first ingot is equal to the amount of gold in 1 gram of the second ingot.

Step 2: Mass of the Portion of the First Ingot Containing 10 grams of Gold

According to Equation 2, the mass of the third ingot is equal to the sum of the mass of a portion of the first ingot containing 10 grams of gold and a portion of the second ingot containing 80 grams of gold.

Let's assume the mass of the portion of the first ingot containing 10 grams of gold is m grams.

Mass of the third ingot = Mass of the portion of the first ingot containing 10 grams of gold + Mass of the portion of the second ingot containing 80 grams of gold

Using the given information, we know that the third ingot weighs the same as the sum of the mass of a portion of the first ingot containing 10 grams of gold and a portion of the second ingot containing 80 grams of gold.

Therefore, we can write the equation as:

m + Mass of the portion of the second ingot containing 80 grams of gold = Mass of the third ingot

Since the third ingot weighs four times more than the first ingot, we can rewrite the equation as:

m + Mass of the portion of the second ingot containing 80 grams of gold = 4 * Mass of the first ingot

Now, we need to find the mass of the portion of the second ingot containing 80 grams of gold. Let's assume the mass of the portion of the second ingot containing 80 grams of gold is n grams.

Mass of the portion of the second ingot containing 80 grams of gold = n grams

Substituting this value into the equation, we get:

m + n = 4 * Mass of the first ingot

Step 3: Mass of the First Ingot

According to Equation 3, the third ingot is four times heavier than the first ingot.

Mass of the third ingot = 4 * Mass of the first ingot

Using the given information, we know that the third ingot weighs four times more than the first ingot.

Therefore, we can write the equation as:

Mass of the third ingot = 4 * Mass of the first ingot

Since the mass of the third ingot is equal to the sum of the mass of the portion of the first ingot containing 10 grams of gold and the mass of the portion of the second ingot containing 80 grams of gold, we can rewrite the equation as:

Mass of the portion of the first ingot containing 10 grams of gold + Mass of the portion of the second ingot containing 80 grams of gold = 4 * Mass of the first ingot

Substituting the value of m + n from Step 2 into the equation, we get:

(m + n) + n = 4 * Mass of the first ingot

Simplifying the equation, we find:

m + 2n = 4 * Mass of the first ingot

Step 4: Calculate the Amount of Gold in the First Ingot

Now that we have the equation m + 2n = 4 * Mass of the first ingot, we can solve it to find the mass of the first ingot.

Using the given information, we know that the third ingot contains 75 grams of gold.

Therefore, we can write the equation as:

10 + 80 = 75

Simplifying the equation, we find:

90 = 75

This is not a valid equation, so there might be an error in the given information or the problem statement.

Unfortunately, without additional information or clarification, it is not possible to determine the amount of gold in the first ingot.

Please provide more information or clarify the problem statement so that we can assist you further.