Question

There are 64 players in a chess tournament. After each round, half of the players are eliminated.How many rounds must be played until the players reach the semifinals with four players left?Write an equation to represent the situation. Then solve the equation.

Answer 1

To find the number of rounds needed until the players reach the semifinals with four players left, we can set up an equation using exponential growth. After solving the equation, we find that 4 rounds must be played.

Step-by-step explanation:

To find the number of rounds needed until the players reach the semifinals with four players left, we can set up an equation using exponential growth.

Initially, there are 64 players.

After each round, half of the players are eliminated, which means the number of players is halved.

Let x represent the number of rounds needed.

So the equation is: 64 / 2ˣ = 4.

We can solve this equation by equating the exponent of 2 on both sides:

2ˣ = 64 / 4 = 16.

Taking the logarithm base 2 of both sides, we get:

x = log2(16) = 4.

Therefore, 4 rounds must be played until the players reach the semifinals with four players left.

Answer 2

4 rounds must be played to reduce the number of players from 64 to 4.

If there are 64 players and half are eliminated in each round, we are essentially dividing the number of players by 2 after each round until we reach 4 players for the semifinals.

We can represent this situation with the equation:

`\[ (64)/(2^r) = 4 \]`

where
`\( r \)` is the number of rounds played.

To solve for
`\( r \)`, we need to determine how many times we must divide 64 by 2 to get 4:

`\[ 2^r = (64)/(4) \]`

`\[ 2^r = 16 \]`

Now, since
`\( 16 = 2^4 \)`, we can equate the exponents:

`\[ r = 4 \]`

This means that 4 rounds must be played to reduce the number of players from 64 to 4.