Question

Число 51,2 дважды увеличивали на одно и то же число процентов, а затем дважды уменьшали на то же самое число процентов. В результате получилось число 28,8. На сколько процентов увеличивали, а затем уменьшали это число?

Answer 1

Problem Analysis

We are given that the number 51.2 was increased twice by the same percentage and then decreased twice by the same percentage, resulting in the number 28.8. We need to find the percentage by which the number was increased and then decreased.

Solution

Let's assume the percentage increase and decrease is represented by x.

To find the percentage increase, we can use the formula: 51.2 + (x% of 51.2) + (x% of (51.2 + (x% of 51.2))) = 28.8

To find the percentage decrease, we can use the formula: 51.2 - (x% of 51.2) - (x% of (51.2 - (x% of 51.2))) = 28.8

We can solve these equations to find the value of x.

Let's solve the first equation: 51.2 + (x/100) * 51.2 + (x/100) * (51.2 + (x/100) * 51.2) = 28.8

Simplifying the equation: 51.2 + (x/100) * 51.2 + (x/100) * (51.2 + (x/100) * 51.2) = 28.8 51.2 + (x/100) * 51.2 + (x/100) * 51.2 + (x/100)^2 * 51.2 = 28.8

Combining like terms: 51.2 + 2 * (x/100) * 51.2 + (x/100)^2 * 51.2 = 28.8

Multiplying both sides by 100 to remove the fraction: 5120 + 2 * x * 51.2 + x^2 * 51.2 = 2880

Rearranging the equation: x^2 * 51.2 + 2 * x * 51.2 + 5120 - 2880 = 0 x^2 * 51.2 + 2 * x * 51.2 + 2240 = 0

Now we can solve this quadratic equation to find the value of x.

Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)

where: a = 51.2 b = 2 * 51.2 c = 2240

Substituting the values into the quadratic formula: x = (-2 * 51.2 ± √((2 * 51.2)^2 - 4 * 51.2 * 2240)) / (2 * 51.2)

Simplifying the equation: x = (-102.4 ± √(10485.76 - 457728)) / 102.4 x = (-102.4 ± √(-447242.24)) / 102.4

Since the discriminant is negative, the equation has no real solutions. This means that there is no percentage by which the number was increased and then decreased to result in 28.8.

Therefore, the answer is that the number 51.2 cannot be increased and then decreased by the same percentage to result in 28.8.

Answer

The number 51.2 cannot be increased and then decreased by the same percentage to result in 28.8.