Mathematics

Front

What is the sum of two numbers if their product is unknown?

Back

The sum can be known, but the product remains unknown. For example, if Rex and Slinky Dog's numbers sum to 2020, we cannot determine their product without additional information.

Front

How many balls must be taken from a bag of 1000 balls labeled 1-1000 to guarantee a pair sums to 1000?

Back

502 balls must be taken to guarantee that at least one pair sums to 1000.

Front

What is the formula for the area of a rectangle?

Back

Area = length × width. For example, if the sides are (7x + 9) and (51 - 7z), the area is (7x + 9)(51 - 7z).

Front

How do you find the number of cubes with exactly 2 faces painted in a rectangular box?

Back

Count the cubes on the edges of the box, excluding the corners. For an 8x9x10 box, there are 84 cubes with exactly 2 faces painted.

Front

What is the relationship between the sum and product of two numbers when their reciprocals are taken?

Back

If the difference between the sum and product of the reciprocals is 1, it can help find the product of the original numbers.

Front

What is a reciprocal of a number?

Back

The reciprocal of a number x is 1/x. For example, the reciprocal of 5 is 1/5.

Front

What is the maximum area of a rectangle given its dimensions?

Back

To find the maximum area, use the dimensions provided and apply the area formula. For (7x + 9) and (51 - 7z), the maximum area is 900.

Front

How do you calculate the product of two numbers if their sum is known?

Back

You cannot directly calculate the product from the sum alone without additional information about the numbers.

Front

What is the significance of the number 1000 in the context of pairs of balls?

Back

The number 1000 represents the total number of balls, and pairs that sum to 1000 are formed by taking complementary pairs like (1,999), (2,998), etc.

Front

What does it mean for a cube to have exactly 2 faces painted?

Back

It means the cube is located on an edge of the box but not at a corner, exposing only two of its faces to paint.