A group of $4$ friends is playing cards. The deck has $70$ cards. To start the game, the dealer makes a pile of $15$ cards in the center. Then she deals the remaining cards one at a time to each player until all the cards are gone. $\newline$ What is the greatest number of cards any player will have after all the cards are dealt?

Full solution

Q. A group of

4

friends is playing cards. The deck has

70

cards. To start the game, the dealer makes a pile of

15

cards in the center. Then she deals the remaining cards one at a time to each player until all the cards are gone.

\newline

What is the greatest number of cards any player will have after all the cards are dealt?

Determine Total Number of Cards: First, we need to determine the total number of cards that will be dealt to the players. We start with a deck of $70$ cards and remove $15$ cards for the center pile. $\newline$ Calculation: $70$ cards $- 15$ cards $= 55$ cards remaining to be dealt to the players.
Divide Cards Among Players: Next, we need to divide the remaining $55$ cards among the $4$ players. Since the cards are dealt one at a time to each player in rotation, we can divide the total number of cards by the number of players to find out how many full rounds of dealing there will be. $\newline$ Calculation: $55$ cards $\div$ $4$ players = $13$ full rounds with $3$ cards remaining.
Deal Remaining Cards: Each player will receive $13$ cards during the full rounds of dealing. The remaining $3$ cards will be dealt to the first $3$ players, as there are not enough cards for another full round. $\newline$ Calculation: Each player gets $13$ cards, and the first $3$ players get an additional card each.
Calculate Maximum Cards Per Player: The greatest number of cards any player can have is the number of cards the first $3$ players will have after the additional card is dealt to them. $\newline$ Calculation: $13$ cards $+$ $1$ card $=$ $14$ cards for the first $3$ players.