Three pouches (each represented by a filled circle) are kept in each of the nine slots in a 3× 3 grid, as shown in the figure. Every pouch has a certain number of one-rupee coins. The minimum and maximum amounts of money (in rupees) among the three pouches in each of the nine slots are given in the table. For example, we know that among the three pouches kept in the second column of the first row, the minimum amount in a pouch is Rs. 6 and the maximum amount is Rs. 8.
There are nine pouches in any of the three columns, as well as in any of the three rows. It is known that the average amount of money (in rupees) kept in the nine pouches in any column or in any row is an integer. It is also known that the total amount of money kept in the three pouches in the first column of the third row is Rs. 4.
What is the total amount of money (in rupees) in the three pouches kept in the first column of the second row?
Correct Answer: 13
Solution and Explanation
The total number of coins in each slot can vary between a minimum and maximum value. Specifically, the sum of coins in the three slots in the first column can be either 8 + 11 + 4 = 23 (minimum) or 10 + 13 + 4 = 27 (maximum). It is given that the average amount of money in any column or row of the nine pouches is an integer, which is a multiple of nine.
In the first column, the total amount of money must be either 18 or 27. Consequently, the number of coins in the first column of the three rows are 10 (2 + 4 + 4), 13 (3 + 5 + 5), and 4 (1 + 2 + 1). Similarly, in the third row, the sum must be 18, and in the second column, the sum must be 27. Therefore, the number of coins in the second column is 20 (6 + 6 + 8) and 3 (1 + 1 + 1), and in the third column of the first row, it is 6 (1 + 2 + 3), while in the third row, it is 10 (2 + 3 + 5).
In the final column, the value in the second row is calculated as 54 - 16 - 38 = 6 + 12 + 20. This can be expressed as 54 - 16 - 38 = (6 + 12 + 20).
The distribution of coins in the pouches in each slot is represented by the following figure.
The sum of money in the three pouches in the first column of the second row is indeed 13.
How many pouches contain exactly one coin?
Correct Answer: 8
Solution and Explanation
The total number of coins in each slot can vary between a minimum and maximum value. Specifically, the sum of coins in the three slots in the first column can be either 8 + 11 + 4 = 23 (minimum) or 10 + 13 + 4 = 27 (maximum). It is given that the average amount of money in any column or row of the nine pouches is an integer, which is a multiple of nine.
In the first column, the total amount of money must be either 18 or 27. Consequently, the number of coins in the first column of the three rows are 10 (2 + 4 + 4), 13 (3 + 5 + 5), and 4 (1 + 2 + 1). Similarly, in the third row, the sum must be 18, and in the second column, the sum must be 27. Therefore, the number of coins in the second column is 20 (6 + 6 + 8) and 3 (1 + 1 + 1), and in the third column of the first row, it is 6 (1 + 2 + 3), while in the third row, it is 10 (2 + 3 + 5).
In the final column, the value in the second row is calculated as 54 - 16 - 38 = 6 + 12 + 20. This can be expressed as 54 - 16 - 38 = (6 + 12 + 20).
The distribution of coins in the pouches in each slot is represented by the following figure.
Eight pouches contain exactly one coin. Ans: (8)
What is the number of slots for which the average amount (in rupees) of itsthree pouches is an integer?
Correct Answer: 2
Solution and Explanation
The total number of coins in each slot can vary between a minimum and maximum value. Specifically, the sum of coins in the three slots in the first column can be either 8 + 11 + 4 = 23 (minimum) or 10 + 13 + 4 = 27 (maximum). It is given that the average amount of money in any column or row of the nine pouches is an integer, which is a multiple of nine.
In the first column, the total amount of money must be either 18 or 27. Consequently, the number of coins in the first column of the three rows are 10 (2 + 4 + 4), 13 (3 + 5 + 5), and 4 (1 + 2 + 1). Similarly, in the third row, the sum must be 18, and in the second column, the sum must be 27. Therefore, the number of coins in the second column is 20 (6 + 6 + 8) and 3 (1 + 1 + 1), and in the third column of the first row, it is 6 (1 + 2 + 3), while in the third row, it is 10 (2 + 3 + 5).
In the final column, the value in the second row is calculated as 54 - 16 - 38 = 6 + 12 + 20. This can be expressed as 54 - 16 - 38 = (6 + 12 + 20).
The distribution of coins in the pouches in each slot is represented by the following figure.
In only two slots, specifically in (row 2, column 2) and (row 1, column 3), is the average amount in the three pouches an integer. This is reflected in the answer (2).
In contrast, in three slots, namely (row 2, column 1), (row 1, column 2), and (row 2, column 3), the sum of money in the three pouches exceeds 10.
The number of slots for which the total amount in its three pouches strictly exceeds Rs. 10 is
Correct Answer: 3
Solution and Explanation
The total number of coins in each slot can vary between a minimum and maximum value. Specifically, the sum of coins in the three slots in the first column can be either 8 + 11 + 4 = 23 (minimum) or 10 + 13 + 4 = 27 (maximum). It is given that the average amount of money in any column or row of the nine pouches is an integer, which is a multiple of nine.
In the first column, the total amount of money must be either 18 or 27. Consequently, the number of coins in the first column of the three rows are 10 (2 + 4 + 4), 13 (3 + 5 + 5), and 4 (1 + 2 + 1). Similarly, in the third row, the sum must be 18, and in the second column, the sum must be 27. Therefore, the number of coins in the second column is 20 (6 + 6 + 8) and 3 (1 + 1 + 1), and in the third column of the first row, it is 6 (1 + 2 + 3), while in the third row, it is 10 (2 + 3 + 5).
In the final column, the value in the second row is calculated as 54 - 16 - 38 = 6 + 12 + 20. This can be expressed as 54 - 16 - 38 = (6 + 12 + 20).
The distribution of coins in the pouches in each slot is represented by the following figure.
The distribution of revenue and cost percentages in different years is outlined as follows.
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