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Keeping Things In Proportion

The concepts of direct and inverse proportion are uncommon, but can seriously ruin your day. Geva Stern, Master GMAT Instructor Extraordinaire, explains the ideas behind the words. This article is based on two GMATPREP® questions that make use of the concept.

Question 1:

The rate of a certain chemical reaction is directly proportional to the square of the concentration of chemical A present and inversely proportional to the concentration of chemical B present. If the concentration of chemical B is increased by 100 percent, which of the following is closest to the percent change in the concentration of chemical A required to keep the reaction rate unchanged?
(a) 100% decreaseӬ(b) 50% decreaseӬ(c) 40% decreaseӬ(d) 40% increaseӬ(e) 50% increase

Question 2:

The cost of a square slab of concrete is proportional to the thickness and also proportional to the square of the length. What is the cost of square slab that is 3 meters long and 0.1 meters thick?

(1) 2m long by 0.2m thick is $160 more than 2m long by 0.1m thick

(2) 3m long by 0.1m thick is $200 more than 2m long by 0.1m thick”¨Both have been hashed in the forums, but the explanations are usually a technical jumble of equations, with no thought as to what the concepts mean – which is what I’d like to clarify today.

The main thing to remember about direct and inverse proportion is that it involves a change in ratio. If x is directly proportional to y, then whenever y changes, x changes by the same ratio. For example, if y is multiplied times 2, then x is also multiplied times 2. If y is divided by 5, then x is also divided by 5, and so on.

The important thing to understand here is that the change is always by multiplication, never addition/subtraction. In the Q1 above, the sentence “chemical B is increased by 100 percent” is intentionally confusing – start by translating it into the equivalent “chemical B is increased times 2”.
If you have this down, then you can understand inverse proportion.

If x is inversely proportional to y, then the opposite of direct proportion happens: If y is multiplied by 2, x changes in the opposite direction – x is divided by 2, or halved.

Remember these three things, and you’ll be fine:

1. Always translate the change to multiplication/division, never addition/subtraction
2. Directly proportional: y is multiplied by 2 → x is also multiplied times 2.
3. Inversely proportional: y is multiplied times 2  â†’ x does the opposite: divided by 2.

For “proportional to the square of A” type phrasings, apply the same idea, only for the square of the change. x is directly proportional to the square of y? Fine – If y is multiplied times 2, then x is multiplied times the square of 2, or times 4. If y is halved (=divided by 2), then x is also divided times the square of 2 = divided by 4, or quartered.
So, how does this apply to the above questions?

Answer to Question 1:

Rate is directly prop. to the square of A, and inversely prop. to B.

First, B is increased 100%. Translate into multiplication: B is multiplied times 2, so the rate is divided by 2.
Now, what changes do we need to make to A to return to the original rate? If the rate is half of what it used to be, then we need to increase A , in order to increase the directly prop. rate – a great chance to eliminate answer choices A, B, and C and limit yourself to a 50/50 guess.
Between D and E, if we increase A by 50% (translate to multiplication – times 3/2), then the rate will increase times the square of the change – or times 9/4, or more than twice. That’s too much – we just want to double the rate back to its original level.”¨On the other hand, if we increase A by 40% (again, translate – times 1.4), then the rate will increase times (1.4)2 = ~2. D is the answer.

Answer to Question 2:

”¢    Cost is proportional to thickness
”¢    Cost is proportional to square of length.
”¢    We want the cost of a slab of 3 meters length and 0.1 thickness.

(1) Note that the length of the slabs is the same (2m), but the thickness changes: 0.2 cm and 0.1 cm. In words: when we increase the thickness of a slab times 2 (not “add 0.1”), the cost (which should also increase times 2, because of direct prop.) increases by $160. 2C = C+160, and we get the original cost of a 2m, 0.1 inch slab at $160.

Now, we want the super-sized version of 3 meters, 0.1 cm, but that’s fine – apply the same direct proportion to the change from 2 m to 3m: if the length is multiplied times 3/2, the cost of $160 is multiplied times 9/4 to get the cost of our required slab of 3m, 0.1 cm.

Stat. (2) is used in exactly the same manner: the statement provides info about the difference in cost between two slabs of the same thickness, but one of length 2m and one of length 3 m. In the move from a 2m, 0.1 inch slab to a 3m, 0.1 inch slab, we add $200; according to the proportion, the cost of such a “move” should multiply times (3/2)2  = 9/4, so 9/4*C = C+200 – from which it is possible to find the cost of a 2m slab, and the 3m slab required by the question stem.

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