Problem Solving

swerve wrote:The number of baseball cards that John and Bill had was in the ratio of 7:3. After John gave Bill 15 of his baseball cards, the ratio of the number of baseball cards that John had to the number that Bill had was 3:2. After the gift, John had how many more baseball cards than Bill?

A. 15
B. 30
C. 45
D. 60
E. 90

One option is to solve the question using 2 variables

Let J = # of cards that John INITIALLY had
Let B = # of cards that Bill INITIALLY had

The number of baseball cards that John and Bill had was in the ratio of 7:3
So, we can write: J/B = 7/3
Cross multiply to get: 3J = 7B
Rearrange to get: 3J - 7B = 0

After John gave Bill 15 of his baseball cards, the ratio of the number of baseball cards that John had to the number that Bill had was 3:2.
So, John now has J - 15 cards and Bill now has B + 15 cards
We can write: J/B = (J - 15)/(B + 15) = 3/2
Cross multiply to get: (2)(J - 15) = (3)(B + 15)
Expand: 2J - 30 = 3B + 45
Rearrange to get: 2J - 3B = 75

So, we now have the following system:
3J - 7B = 0
2J - 3B = 45

Solve to get: J = 105 and B = 45
NOTE: these are the number of cards they INITIALLY had

After the gift, John had how many more baseball cards than Bill?
AFTER THE GIFT, John had 90 cards and Bill had 60 cards
So, John had 30 more cards than John.

Answer: B

Cheers,
Brent

swerve wrote:The number of baseball cards that John and Bill had was in the ratio of 7:3. After John gave Bill 15 of his baseball cards, the ratio of the number of baseball cards that John had to the number that Bill had was 3:2. After the gift, John had how many more baseball cards than Bill?

A. 15
B. 30
C. 45
D. 60
E. 90

We can let the original number of baseball cards John and Bill had be 7x and 3x, respectively. So we can create the equation:

(7x - 15)/(3x + 15) = 3/2

2(7x - 15) = 3(3x + 15)

14x - 30 = 9x + 45

5x = 75

x = 15

Therefore, John and Bill originally had 105 and 45 cards, respectively. After John gave Bill 15 cards, John now had 90 cards and Bill had 60 cards. So John had 30 more cards than Bill.

Answer: B