Math Symbols

The SAT likes to include a math few problems with unusual symbols, such as ◊, or ♠. For some exam takers, this is a signal to panic, since they have no idea what ◊ or ♠ mean, but really, these are simple replacement questions, similar to the f(x) questions I addressed in an earlier post.

Really, no one knows what what ◊ or ♠ mean.  The College Board (the writers of the exam) make up a meaning for the symbols as they write the question.  In each case they give you the definition of the symbol, then ask you to apply that definition.

For example:

a ♣ b = a + (b x 2)

3 ♣ 7 = ?

The first line defines the symbol ♣.  We'll use that definition as a template for answering the question on the second line.

Note that in the definition, a appears before the ♣. And in the question, 3 appears before the ♣. The test makers have replaced a with 3 in the left half of the equation; we just need to make the same replacements in the right side of the equation:

a ♣ b = a + (b x 2)

3 ♣ 7 = 3 + (b x 2)

We still have to account for b, though. In the left side of the equation, the test maker replaced b with 7, so now we need to make the same replacement on the right:

a ♣ b = a + (b x 2)

3 ♣ 7 = 3 + (7 x 2)

At this point, we have an equation that we can easily solve. Following the usual order of operations (PEMDAS), we get:

3 ♣ 7 = 3 + (7 x 2)
= 3 + 14
= 17

MCAT Math

The MCAT does not allow calculators, (source: MCAT Essentials (will download as pdf)) a fact that leaves many test-takers in something of a panic. With proper preparation, however, there should be no difficulty in completing MCAT math.

First, recognize that since the test takers aren't allowed to use calculators, the test makers can't include math that requires a calculator. This means that the math on the exam has to be relatively basic, and the numbers used have to be easy to manipulate. Note that sometimes manipulation includes simplification: the acceleration due to gravity on earth is 9.8 m/s^2, but on exam day we typically simplify that number to 10; a sample with a mass of 5.1 grams may be simplified to 5 grams; etc.

Second, keep in mind that as a multiple-choice test, the MCAT has to actually give you the answer - all you have to do is separate out the correct answer from the incorrect answers. It doesn't matter how you do it. Eliminating all of the wrong answers gives you just as much credit as crunching the numbers to arrive at the correct answer. Skilled test-takers work at questions from both ends, eliminating patently absurd answer choices and only then bothering to really grapple with the problem. If they're given answer choices such as 51 J, 51 V, 53 J, and 53 V, their first step will probably be to determine what units the correct answer will have, since doing so will eliminate 67% of the wrong answers. Only then will they bother with the math.

Third, realize that since you're not taking the exam today, you have a chance to improve your math skills. And the best way to do that is to use those skills: between now and exam day, every time you need to do math, do it by hand. Check your work with a calculator if you want, but do it by hand first. And make use of every opportunity, too; when you buy items at a store, estimate or calculate the pre-tax total before you get to the register. Then do the total with tax. Calculate fifteen and twenty percent tips at restaurants. When you're at the gas station, estimate how many gallons you'd get for $20, then crunch the numbers and see how close you were. Balance your check book every night.

Keep these points in mind, and make use of every opportunity to practice, and MCAT math will be a breeze.

If f (x) = 3x + 4

Function problems, like f (x), g (x), h (x), etc, leave a lot of students confused. This is unfortunate, because function problems are nothing more than substitution problems. I'll give you an example:

f (x) = 3x + 4
f (2) = ?

We note that in left half of the question, 2 has been plugged in where x was. To solve the problem, then, all we need to do is substitute (plug in) 2 for x in the right half of the equation:

f (2) = 3(2) + 4 = 6 + 4 = 10

Similarly,

f (3) = 3(3) + 4 = 9 + 4 = 13
f (4) = 3(4) + 4 = 12 + 4 = 16
f (5) = 3(5) + 4 = 15 + 4 = 19
f (6) = 3(6) + 4 = 18 + 4 = 22
f (7) = 3(7) + 4 = 21 + 4 = 25
f (8) = 3(8) + 4 = 24 + 4 = 28
f (9) = 3(9) + 4 = 27 + 4 = 31
f (10) = 3(10) + 4 = 30 + 4 = 34
f (y) = 3(y) + 4 = 3y + 4
f (z) = 3(z) + 4 = 3z + 4
etc.

Sometimes, we might find multiple functions used together. When this is the case, we just follow the usual rules of math to untangle the question:

if f (x) = 3x + 4, and g(x) = 5x - 7
f (4) - g (2) = ?

As before, we merely substitute in. Let's work with each function separately, then put them together, being sure to keep straight that 4 was given to us for the f function and 2 was given to us for the g function:

f (4) = 3(4) + 4 = 12 + 4 = 16
g (2) = 5(2) - 7 = 10 - 7 = 3

Taking the original equation and then substituting in these values, we have:

f (4) - g (2) = ?
16 - 3 = ?

and of course that equals 13.

We may also find cases where functions are nested within parentheses:

if f (x) = 3x + 4, and g(x) = 5x - 7
f (g (10)) = ?

Note that I've defined a new function for g. These are solved in the same way: by following the usual rules of math. g (10) is found inside a set of parentheses, so we start with that:

g (10) = 5(10) - 7 = 50 - 7 = 43

We then substitute this value in for g (10):

f (g (10)) = f (43)

And then we solve f (43):

f (43) = 3(43) + 4 = 129 + 4 = 134

PS: If you had to reach for your calculator to do any of that math, then you're relying on your calculator too much.