Each duck weights the same, and each duckling weighs the same. If the total weight of 3 ducks and 2 ducklings is 32 kilograms, the total weight of 4 ducks and 3 ducklings is 44 kilograms, what is the total weight, in kilograms, of 2 ducks and 1 duckling?

Solution:

Let the weight of a duck as and weight of a duckling as

Answer:

A wooden rectangular block, 4 cm x 5 cm x 6 cm, is painted red and then cut into several 1 cm x 1 cm x 1 cm cubes. What is the ratio of the number of cubes with two red faces to the number of cubes with three red faces?Solution:

Only the cubes at the 8 corners of the block have three red faces.

The cubes with two red faces lie along the 12 edges of the block (excluding the corners): - 4 of them have 4 cubes = 16 cubes; - 4 of them have 3 cubes = 12 cubes;- 4 of them have 2 cubes = 8 cubes total of cubes with two faces = 36

The ratio is = 36 : 8 = __ : __

Answer: 9 : 2

What is the smallest amount of numbers in the product

that should be removed so that the product of the remaining numbers is a perfect square?Solution:

Rewrite the equation after prime factorization: =

To get a square number of an integer, the power of the factors must be even.Therefore, we need to

The smallest amount of numbers to be removed is ___

Answer: 5

We need to find

From calculation, we know that

Solution:

In the figure below, the positive numbers are arranged in the grid follow by the arrows’ direction. For example, “8” is placed in Row 2, Column 3 and “9” is placed in Row 3, Column 2.Which Row and which Column that “2008” is placed?

In Row 1, the 3rd number is 6 = 1+2+3, the 5th number is 15 = 1+2+3+4+5Therefore, where the number of term is odd, the number in Row n is: 1+2+3+…+n = In Column 1, the 2nd number is 3 = 1+2, the 4th number is 10 = 1+2+3+4Therefore, where the number of term is even, the number in Column n is: 1+2+3+…+n =

Answer: Row 9, Column 55

2008 is located at Column (2008 – 1954) + 1 = 55, and Row (63 – 55) + 1 = 9