Ratio problem- checking my logic

There are 2 ways we can think about this: algebraically or conceptually.

Algebraically:

Target question: c/d = ?
We will only get a sufficient answer if we can get a value for that fraction.

1) The ratio of 3c to 3d is 3 to 4 --> (3c)/(3d) = 3/4
Simplify 3/3 to 1/1 --> c/d = 3/4
Sufficient.

2) The ratio c+3 to d+3 is 4 to 5 --> (c + 3)/(d + 3) = 4/5
We can't simplify the 3's because we have sums in the numerator and denominator. Instead, cross-multiply:
5(c + 3) = 4(d + 3)
5c + 15 = 4d + 12
5c + 3 = 4d
There is no way to manipulate this equation to simply get c/d.
Insufficient.

The answer must be A.

To your question, changing the ratio to 5/5 would change our answer, because a 1:1 ratio means that the terms must be the same:
(c + 3)/(d + 3) = 5/5
Simplify --> (c + 3)/(d + 3) = 1
c + 3 = d + 3
c = d
If the variables are equal, then the ratio c/d = 1
That would have been sufficient.

Conceptually: With ratio problems, it's often faster to think conceptually than to do the algebra.

Target question: What is the ratio of c to d?
If we get a proportional relationship between the two, we'll have sufficient information.

Proportional relationships are always multiplicative (involving multiplication or division):
"one half of something" --> (1/2)n
"5 times something" --> 5y
"25% of something" --> 0.25x
"the ratio of c to d" --> c/d

Additive relationships, on the other hand, give us information about real values, but not about proportions.
"4 more than x" --> x + 4
"10 less than y" --> y - 10

1) The ratio of 3c to 3d is 3 to 4
Multiplying each term by 3 will not change the proportion between them. The ratio of c/d will be the same as the ratio of 3c/3d.

2) The ratio c+3 to d+3 is 4 to 5
Knowing the proportional relationship after we've added real values to c and d wouldn't help. We don't know how much difference the +3 made proportionally.

Try testing numbers:
c + 3 = 4 and d + 3 = 5
c = 1, d = 2
c/d = 1/2
or...
c + 3 = 400 and d + 3 = 500
c = 397, d = 497
c/d = 397/497
That gives us a different ratio, and one that's really close to 4/5. The bigger the numbers get, the less of a difference the +3 made to the overall ratio.

To your question:

My question is if stmt 2 said c+3 to d+3 is 5 to 5 does the answer change?

We can see conceptually that that would work. No matter what numbers we might test, we'll get that c and d are equal, so the ratio must be 1/1.

Does that help?

kdn508 wrote: ↑

Thu Jun 16, 2016 4:25 pm

What is the ratio of c to d?

1. The ratio of 3c to 3d is 3 to 4
2. The ratio c+3 to d+3 is 4 to 5

Solution:

Question Stem Analysis:

We need to determine the ratio of c to d, i.e., the value of c/d.

Statement One Alone:

We are given that (3c) / (3d) = 3/4. Cancelling the 3s on the left hand side, we have c/d = 3/4. Statement one alone is sufficient.

Statement Two Alone:

We are given that (c + 3) / (d + 3) = 4/5. This does not allow us to determine the value of c/d. For example, c could be 1 and d could be 2 so that (c + 3) / (d + 3) = 4/5. However, c could be 5 and d could be 7 so that (c + 3) / (d + 3) = 8/10 = 4/5. In the former case, c/d = 1/2, but in the later case, c/d = 5/7. Statement two alone is not sufficient.

Answer: A