Question

Сумма трёх чисел равна 125. Первое число составляет 54% этой суммы. Второе число в три раза меньше первого. Найдите разность между наибольшим и наименьшим числами. Запишите решение и ответ.

Answer 1

Problem Analysis

To solve this problem, we need to find three numbers that add up to 125. The first number is 54% of the sum, and the second number is three times smaller than the first number. We are asked to find the difference between the largest and smallest numbers.

Solution

Let's break down the problem step by step:

1. We are given that the sum of the three numbers is 125. Let's call the three numbers x, y, and z. So we have the equation: x + y + z = 125

2. The first number (x) is 54% of the sum. We can express this as: x = 0.54 * (x + y + z)

3. The second number (y) is three times smaller than the first number. We can express this as: y = (1/3) * x

Now, let's solve these equations to find the values of x, y, and z.

Solving the Equations

To solve the equations, we can use substitution. Let's start with the equation for x:

x = 0.54 * (x + y + z)

Substituting the value of y from the second equation:

x = 0.54 * (x + (1/3) * x + z)

Simplifying the equation:

x = 0.54 * (4/3) * x + 0.54 * z

Multiplying both sides by (3/4) to eliminate the fraction:

(3/4) * x = 0.54 * z

Simplifying further:

x = (4/3) * (0.54 * z)

Now, let's substitute this value of x into the equation for y:

y = (1/3) * x

Substituting the value of x:

y = (1/3) * (4/3) * (0.54 * z)

Simplifying:

y = (4/9) * (0.54 * z)

Finally, let's substitute the values of x and y into the equation for the sum:

x + y + z = 125

(4/3) * (0.54 * z) + (4/9) * (0.54 * z) + z = 125

Simplifying:

(16/27) * (0.54 * z) + (4/9) * (0.54 * z) + z = 125

Now, we can solve this equation to find the value of z.

Calculating the Value of z

Let's solve the equation:

(16/27) * (0.54 * z) + (4/9) * (0.54 * z) + z = 125

Multiplying both sides by 27 to eliminate the fraction:

16 * (0.54 * z) + 12 * (0.54 * z) + 27 * z = 3375

Simplifying:

(8.64 * z) + (6.48 * z) + 27 * z = 3375

Combining like terms:

15.12 * z + 27 * z = 3375

Simplifying further:

42.12 * z = 3375

Dividing both sides by 42.12 to solve for z:

z = 3375 / 42.12

Calculating the value of z:

z ≈ 80.07

Now that we have the value of z, we can substitute it back into the equation for x and y to find their values.

Calculating the Values of x and y

Let's calculate the values of x and y using the equations we derived earlier:

x = (4/3) * (0.54 * z)

Substituting the value of z:

x = (4/3) * (0.54 * 80.07)

Calculating the value of x:

x ≈ 57.68

Now, let's calculate the value of y:

y = (4/9) * (0.54 * z)

Substituting the value of z:

y = (4/9) * (0.54 * 80.07)

Calculating the value of y:

y ≈ 30.46

Finding the Difference between the Largest and Smallest Numbers

Now that we have the values of x, y, and z, we can find the difference between the largest and smallest numbers.

The largest number is x, which is approximately 57.68, and the smallest number is y, which is approximately 30.46.

The difference between the largest and smallest numbers is:

57.68 - 30.46 ≈ 27.22

Solution and Answer

The solution to the problem is as follows: - The first number (x) is approximately 57.68. - The second number (y) is approximately 30.46. - The third number (z) is approximately 80.07. - The difference between the largest and smallest numbers is approximately 27.22.

Therefore, the difference between the largest and smallest numbers is approximately 27.22.

Note: The values of x, y, and z are approximate due to rounding in the calculations.