«Golden Treasure Box 2020» Tasks for 6th-grade students Answer the questions by choosing the correct answer option

To solve the given problem, let"s break it down step by step.

Question 1:
We are given that when 30 spectators are seated in the cinema hall, there is always a row with two spectators. And when 26 spectators are seated, at least three rows are empty. We need to determine the number of rows in the cinema hall.

Let"s assume that there are "x" rows in the cinema hall. Since each row has two spectators, the total number of spectators in "x" rows would be 2x. Equating this with 30, we get the equation:

2x = 30

To find the minimum number of rows needed for 26 spectators, we subtract 3 (number of empty rows) from the total number of rows "x" and multiply it by 2. This can be expressed as (x - 3) * 2. Equating this with 26, we get the equation:

(x - 3) * 2 = 26

Now, let"s solve both equations to find the value of "x" (number of rows).

Solving the first equation:
2x = 30
Dividing both sides of the equation by 2:
x = 15

Solving the second equation:
(x - 3) * 2 = 26
Expanding the expression:
2x - 6 = 26
Adding 6 to both sides of the equation:
2x = 32
Dividing both sides of the equation by 2:
x = 16

By solving the equations, we get two different values for the number of rows in the cinema hall: 15 and 16. Since we cannot have a fraction of a row, the correct answer is option D. It is impossible to determine the exact number of rows in the cinema hall based on the information given.

Question 2:
Lena invited 7 classmates to her birthday party. The first guest gave Lena 8 souvenirs. Each subsequent guest gave Lena 1 more souvenir than the previous guest.

To find the total number of souvenirs Lena received, we can use arithmetic progression. The formula to find the sum of the first "n" terms in an arithmetic progression is:

Sum = (n/2) * (2 * first term + (n - 1) * common difference)

In this case, the first term is 8, and the common difference (number of souvenirs given by each subsequent guest) is 1. The value of "n" is 7 (since Lena invited 7 classmates).

Using the formula, we can calculate the total number of souvenirs Lena received:

Sum = (7/2) * (2 * 8 + (7 - 1) * 1)
= (7/2) * (16 + 6)
= (7/2) * 22
= 77

Therefore, Lena received a total of 77 souvenirs (answer to question 2).

I hope this explanation is clear and helps you understand the solution to the given tasks. If you have any further questions, feel free to ask!