Bunuel 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
Kudos for a correct solution.
VERITAS PREP OFFICIAL SOLUTION:The way most people would look at this problem is that it’s an algebra problem. The ratio of two numbers is 7:3, and after an exchange of 15 cards, the ratio is now 3:2. I can set up two equations and solve for the two unknowns in this equation, which will give me the number of cards Bill has and the number of cards John has. After that, it’s simply a question of subtracting the two in order to answer the question. Let’s run through the algebra because it’s somewhat time-consuming but otherwise fairly basic (the white-and-gold approach).
The initial ratio, before the gift, can be describes as J / B = 7 / 3.
The final ratio, after the gift, would then be J – 15 / B + 15 = 3 / 2.
Note that we are defining J and B to be the initial values of John and Bill, so we’ll have to keep that in mind for the final calculation.
Cross-multiplying the first equation gives us 3 J = 7 B. This should make sense as John has many more cards than Bill.
Cross-multiplying the second equation gives 2 (J – 15) = 3 (B + 15),
We can expand this to 2J – 30 = 3B + 45.
Finally we can move the constants to one side and get 2 J = 3 B + 75
You can use either the elimination method or the substitution method to solve for the two variables. I prefer the elimination method so I’d multiply the first equation by 2 and the second equation by 3 to isolate J.
6 J = 14 B
6 J = 9 B + 225
Since the left hand sides are the same, we can simplify to 14 B = 9 B + 225.
Subtracting 9 B from both sides gives 5 B = 225.
Dividing 225 by 5 gives 45.
If B is 45, and 3 J = 7 B, then 3 J must be 315, and so J is equal to 105.
We’re still not done, because these are the initial values: 105 and 45. If John gave Bill 15 cards, then the new totals would be 90 for John and 60 for Bill, which is where the 3/2 ratio comes in. The difference in cards is 30 after the gift, so the answer is B.
Other people see this ratio problem and don’t even think about the algebra, they solve it using the underlying concept (the blue-and-black approach). To illustrate this concept, suppose I had 199 cards and you had 101 cards. Since no simplification is possible, the ratio of our cards would be 199:101. But if you then gave me one card, our ratio would suddenly be 2:1. This reduced fraction does not change the fact that I still have 200 cards and you have 100. Simply because the fraction can be simplified, that does not mean that the totals have changed in any way.
Let’s apply that same logic here. The ratio was 7:3. After the gift, the new ratio is 3:2, but the total number of cards has stayed the same. This means that if I can get a new ratio that’s in the same proportions as the old ratio, the problem will seem much simpler. The ratio 7:3 has 10 total parts. The ratio of 3:2 has only 5 total parts, so they are not in the same proportions. However, if I recognize that I can simply multiply 3:2 by 2 to get a ratio of 6:4, I discover a shortcut that can help on ratio problems.
If the ratio used to be 7:3 then became 6:4 after a transfer of 15 cards, then each unit of the ratio must represent 15 cards. This would mean that 7 would drop to 6 and 3 would increase to 4 because of the same 15 card transfer. Thus the old ratio was (7×15): (3×15), or 105:45. The new ratio is similarly (6×15): (4×15), or 90:60. The difference in cards after the gift is still 30, answer choice B, but for some it’s much easier to see using a little logic than a lot of algebra.
On the GMAT, similar to the chameleon dress, your perspective is what’s going to dictate how you approach problems. Not every question will have a shortcut or an instant solution, but every problem can be approached in multiple ways. The only limit is your understanding of the concepts and your skill at analyzing the presented problem. Hopefully, on test day, these strategies will help you avoid feeling blue (and black).
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