Joe and Anna collect football cards.The greatest common factor of the numbers of cards in their collections is 15.Altogether Joe and Anna have 75 cards.If Joe has more cards than Anna,how many cards do they each have?
since 15 is a common factor the number of cards each has must be a multiple of 15
let the number of cards that Joe has be 15x, let Anna's number of cards be 15y
then 15x + 15y = 75
x + y = 5
also x > y , and of course both x and y must be whole numbers.
there are only 2 possibilities for x and y
(4,1), (3,2)
(4,1) : Joe has 60, Anna has 15
(3,2) : Joe has 45, Anna has 30
both cases satisfy your 3 conditions.
To find out how many cards Joe and Anna have, we need to set up a system of equations based on the given information.
Let's say Joe has x cards, and Anna has y cards.
From the problem, we know that:
1) The greatest common factor of the numbers of cards in their collections is 15. This means that both x and y are divisible by 15.
2) The total number of cards they have together is 75. This can be expressed as the equation: x + y = 75.
Since x and y are divisible by 15, we can rewrite them as 15a and 15b, where a and b are integers:
15a + 15b = 75
Dividing both sides of the equation by 15, we have:
a + b = 5
Now, we have a system of equations:
1) x + y = 75
2) a + b = 5
We need to find the values of x and y, which will give us Joe and Anna's card collections.
To solve this system, we can substitute 15a for x and 15b for y in equation 1:
15a + 15b = 75
Now we can divide both sides by 15 to simplify the equation:
a + b = 5
Since a + b = 5, we know that a = 5 - b.
Substituting a = 5 - b back into x = 15a and y = 15b, we get:
x = 15(5 - b) = 75 - 15b
y = 15b
Now, let’s use the fact that Joe has more cards than Anna:
x > y
75 - 15b > 15b
Combining like terms, we have:
75 > 30b
Dividing both sides by 30:
75/30 > b
5/2 > b
Since b is an integer, b can be 1 or 0.
If b = 1, then a = 5 - b = 4.
So, x = 15a = 15(4) = 60 and y = 15b = 15(1) = 15.
If b = 0, then a = 5 - b = 5.
So, x = 15a = 15(5) = 75 and y = 15b = 15(0) = 0.
Therefore, Joe and Anna could each have either 60 and 15 cards, or 75 and 0 cards.