When some cards are dealt to six people, each receives three more than when the same cards are dealt to eight people. How many cards are there?

Question

When some cards are dealt to six people, each receives three more than when the same cards are dealt to eight people. How many cards are there?

Answer 1

First, we need to reason through this word problem and translate it into a mathematical problem. There are three numbers that we don't know:

The total number of cards

The number of cards each person has if the deck is dealt to six people

The number of cards each person has if the deck is dealt to eight people.

 

We will call these numbers x, y and z:

 

x=The total number of cards
y=The number of cards each person has if the deck is dealt to six people
z=The number of cards each person has if the deck is dealt to eight people.

 

Now, we need to write equations that specify how these numbers are related to each other. 

 

When the deck is dealt to six people, this means that each person has one sixth of the cards available, i.e.

 

y=x/6. 

 

The same holds true when it is dealt to eight people. Each person has one eighth of the deck:

 

z = x/8.

 

Finally, we know that

 

y = z+3, since if the cards are dealt to six people, each receives three more than when the same cards are dealt to eight people. We therefore have three equations and three unknowns in our linear system:

 

(1) y=x/6

(2) z=x/8

(3) y=z+3

 

This can be quickly solved by substitution. Using equation (3), we can replace the y in equation (1) with z+3

 

(1) z+3=x/6

(2) z=x/8

 

Now, we can multiply both sides of equation (1) by 6 and multiply both sides of equation (2) by 8:

 

(1) 6(z+3)=x (after multiplying both sides by 6)

     6z+18=x

 

(2) 8z =x (after multiplying both sides by 8)

 

Since x is equal to both 6z + 18 and 8z, we can use the transitive property of equality to get

 

6z+18=8z

 

Subtracting 6z from both sides, we get

18=2z or

2z=18

 

Now, we can get z by dividing both sides of the equation by 2:

 

z=9

 

To get the other two variables, we use the two equations that we eliminated earlier:

 

y=z+3=9+3=12,

and

 

z=x/8

x=8z=8*9 = 72. Remember that x was the total number of cards, so this is the answer.

 

To check, we should make sure that the conditions of the problem really are satisfied. If 72 cards are divided equally among six people, each person gets 72/6 = 12 cards. If 72 cards are equally divided among eight people, each person gets 72/8 = 9 cards, which is three fewer than 12 as required.