Question

Ваня во столько раз старше своей сестры Ани, во сколько раз он моложе бабушки. Сколько лет Ване, если Ане нет ещё и 6 лет, а Ване вместе с бабушкой 112 лет? Возраст каждого - целое число лет.

Answer 1

6/Задание № 5:

Ваня во столько раз старше своей сестры Ани, во сколько раз он моложе бабушки. Сколько лет Ване, если Ане нет ещё и 6 лет, а Ване вместе с бабушкой 112 лет? Возраст каждого - целое число лет.

РЕШЕНИЕ: Пусть Ане x<6 лет. Тогда Ване kх лет, а бабушке - k^2x лет.

По условию Ване вместе с бабушкой 112 лет:

kx+k^2x=112

k(1+k)x=112

Если k=1, то k(1+k)=112, однако последовательных целых чисел, дающих в произведении 112 нет.

Если k=2, то k(1+k)=56, тогда k=7, kx=7*2=14 лет Ване

Если k=3, то k(1+k)=112/3 - не целое число

Если k=4, то k(1+k)=28, однако последовательных целых чисел, дающих в произведении 28 нет.

Если k=5, то k(1+k)=112/5 - не целое число

Единственный вариант, когда Ане 2 года и Ване 14 лет.

ОТВЕТ: 14 лет

Answer 2

Problem Analysis

We are given the following information: - Vanya is a certain number of times older than his sister Anya. - Vanya is a certain number of times younger than his grandmother. - Anya is 6 years old. - The combined age of Vanya and his grandmother is 112 years.

We need to determine Vanya's age.

Solution

Let's assume that Vanya is x times older than Anya and y times younger than his grandmother.

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

Equation 1: Vanya's age = x * Anya's age Equation 2: Vanya's age + Grandmother's age = 112 years Equation 3: Grandmother's age = y * Vanya's age

Substituting the values from the given information into the equations, we get:

Equation 1: Vanya's age = x * 6 Equation 2: Vanya's age + Grandmother's age = 112 years Equation 3: Grandmother's age = y * Vanya's age

To solve this system of equations, we need to find the values of x and y.

Let's solve the equations step by step:

From Equation 1, we have Vanya's age = 6x.

Substituting this value into Equation 3, we get Grandmother's age = y * (6x).

Substituting the values of Vanya's age and Grandmother's age into Equation 2, we get:

6x + y * (6x) = 112.

Simplifying the equation, we have:

6x + 6xy = 112.

Factoring out 6x, we get:

6x(1 + y) = 112.

Dividing both sides by 6(1 + y), we get:

x = 112 / (6 * (1 + y)).

Now, we need to find the values of x and y that satisfy the given conditions.

Since the age of each person is a whole number, we can try different values of y and check if x is a whole number.

Let's start with y = 1:

x = 112 / (6 * (1 + 1)) = 112 / (6 * 2) = 112 / 12 = 9.33.

Since x is not a whole number, y = 1 is not a valid solution.

Let's try y = 2:

x = 112 / (6 * (1 + 2)) = 112 / (6 * 3) = 112 / 18 = 6.22.

Since x is not a whole number, y = 2 is not a valid solution.

Let's try y = 3:

x = 112 / (6 * (1 + 3)) = 112 / (6 * 4) = 112 / 24 = 4.67.

Since x is not a whole number, y = 3 is not a valid solution.

Let's try y = 4:

x = 112 / (6 * (1 + 4)) = 112 / (6 * 5) = 112 / 30 = 3.73.

Since x is not a whole number, y = 4 is not a valid solution.

Let's try y = 5:

x = 112 / (6 * (1 + 5)) = 112 / (6 * 6) = 112 / 36 = 3.11.

Since x is not a whole number, y = 5 is not a valid solution.

Let's try y = 6:

x = 112 / (6 * (1 + 6)) = 112 / (6 * 7) = 112 / 42 = 2.67.

Since x is not a whole number, y = 6 is not a valid solution.

Let's try y = 7:

x = 112 / (6 * (1 + 7)) = 112 / (6 * 8) = 112 / 48 = 2.33.

Since x is not a whole number, y = 7 is not a valid solution.

Let's try y = 8:

x = 112 / (6 * (1 + 8)) = 112 / (6 * 9) = 112 / 54 = 2.07.

Since x is not a whole number, y = 8 is not a valid solution.

Let's try y = 9:

x = 112 / (6 * (1 + 9)) = 112 / (6 * 10) = 112 / 60 = 1.87.

Since x is not a whole number, y = 9 is not a valid solution.

Let's try y = 10:

x = 112 / (6 * (1 + 10)) = 112 / (6 * 11) = 112 / 66 = 1.70.

Since x is not a whole number, y = 10 is not a valid solution.

Let's try y = 11:

x = 112 / (6 * (1 + 11)) = 112 / (6 * 12) = 112 / 72 = 1.55.

Since x is not a whole number, y = 11 is not a valid solution.

Let's try y = 12:

x = 112 / (6 * (1 + 12)) = 112 / (6 * 13) = 112 / 78 = 1.43.

Since x is not a whole number, y = 12 is not a valid solution.

Let's try y = 13:

x = 112 / (6 * (1 + 13)) = 112 / (6 * 14) = 112 / 84 = 1.33.

Since x is not a whole number, y = 13 is not a valid solution.

Let's try y = 14:

x = 112 / (6 * (1 + 14)) = 112 / (6 * 15) = 112 / 90 = 1.24.

Since x is not a whole number, y = 14 is not a valid solution.

Let's try y = 15:

x = 112 / (6 * (1 + 15)) = 112 / (6 * 16) = 112 / 96 = 1.17.

Since x is not a whole number, y = 15 is not a valid solution.

Let's try y = 16:

x = 112 / (6 * (1 + 16)) = 112 / (6 * 17) = 112 / 102 = 1.10.

Since x is not a whole number, y = 16 is not a valid solution.

Let's try y = 17:

x = 112 / (6 * (1 + 17)) = 112 / (6 * 18) = 112 / 108 = 1.04.

Since x is not a whole number, y = 17 is not a valid solution.

Let's try y = 18:

x = 112 / (6 * (1 + 18)) = 112 / (6 * 19) = 112 / 114 = 0.98.

Since x is not a whole number, y = 18 is not a valid solution.

Let's try y = 19:

x = 112 / (6 * (1 + 19)) = 112 / (6 * 20) = 112 / 120 = 0.93.

Since x is not a whole number, y = 19 is not a valid solution.

Let's try y = 20:

x = 112 / (6 * (1 + 20)) = 112 / (6 * 21) = 112 / 126 = 0.89.

Since x is not a whole number, y = 20 is not a valid solution.

From the above calculations, we can see that there is no valid solution for the given conditions.

Therefore, it is not possible to determine Vanya's age based on the given information.

Please let me know if I can help you with anything else.