More Math Problem Solving

Logistic Growth Problem
On a certain scale of intensity, each increment of 10 in magnitude represents a tenfold increase in intensity. On this scale, an intensity corresponding to a magnitude of 165 is how many times an intensity corresponding to a magnitude of 125?
a. 40
b. 100
c. 400
d. 1,000
e. 10,000

Solution:
The increase form 125 to 165 is 40, which is 4 increments of 10.
So there are four tenfold increases, that is 10 to the power 4 = 10,000
Answer choice (e).

This kind of problem also appears in the Official Guide 10th Edition. Incidentally this is the concept behind logarithms, but don’t worry – logs are not on the GMAT.

Readers may be familiar with the Richter Scale for earthquake intensity. When the measurement increases by 1, the intensity increases by 10. For example, an earthquake of 7 on the Richter Scale is 100 times more intense than an earthquake of 5 on the Richter Scale.



Algebra and Arithmetic Problem in Geometry Disguise
The volume of a sphere with radius r is (4πr^3) /3 and the surface area is 4πr^2. If a spherical balloon has a volume of 972π cubic centimeters, what is the surface area of the balloon in square centimeters?
324
729
243π
324π
729π

At first glance this appears to be a geometry problem but at closer inspection it is a straightforward algebra problem.

To find the surface area SA, we need the radius r.
We can find r because we know what V is, and how it is related to r.
V=972π = (4πr^3) /3

It might appear that you’ll have to do a lot of arithmetic either mentally or on paper, but cancelling shouldn' make it too hard.
Find r^3in terms of V:
r^3=3V/4π
Here r^3 = 3*972 π/4 π
Cancelling, we obtain r^3 = 3*243 = 729

The GMAT Hero method recommends that you learn by heart the list of squares and cubes of numbers up to 12, jsut as you (should) have learned multiplication tables at primary school.
You should recognize that 9^3=729.
So r = 9.

Plug that into the given formula for surface area,
SA = 4π(9^2) = 4(81) π
SA = 324 π
Answer choice (d).

The main difficulty with this problem is finding the cube root of 729.


Percentage Problem
On a Saturday night, each of the rooms at a certain motel was rented for either $40 or $60. If 10 of the rooms that were rented for $60 had instead been rented for $40, then the total rent the motel charged for that night would have been reduced by 25 percent. What was the total rent the motel actually charged for that night?
$600
$800
$1,000
$1,600
$2,400

Initially it may appear that not enough information is given, but keep calm – you can do this!

The difference between $60 and $40 is $20, so the 10 rooms would have $20x10 = $200 value.
From the info given, $200 is the 25% reduction from the actual total (call it T)
Algebraically, 200 = 0.25T
So T = 200/0.25 = 800
Answer choice (b).