A good deal of success in math is simply about careful detail management. If you are, by nature, someone already comfortable with math, highly organized and detail oriented, probably none of these mistakes will plague your work. This is a post for folks who may be a little rusty in math, and need to be a little more careful with the basic work of detail management.
Dropping the negative sign
Suppose you are solving the equation
5 – 2x = 13
We want to isolate x. One tactic would be to begin by subtracting 5 from both sides. On the right, 13 – 5 = 8. On the left, the 5′s cancel, but with what are we left? It would be a mistake to subtract 5 and wind up with:
I have that in red, with an unequal sign, to emphasize that it is wrong. Of course, the mistake is: when we subtract the 5 and get rid of it, the 2x term does not magically change from negative to positive. It still has a negative sign in front of it. Therefore, the next steps are:
-2x = 8
x = -4
Actually, if you notice any tendency toward making this mistake, I highly recommend: make your first step to add any subtracted variable to other side, to make it positive. If your first step, automatically, is to make the variable positive, then you will be considerably less likely to make this mistake.
Dividing by the numerator
Suppose you have this equation to solve:
Both sides are clearly divisible by 5, so one possible first step would be to divide by sides by 5. On the right, . On the left, the 5′s cancel, but the question is: what is on the left side after dividing by 5? Just x? No! That’s a very tempting mistake to make! In the equation above, x is in the denominator, and “being in the denominator” is not a condition that goes away just because a number in the numerator is cancelled. If we divide by sides by 5, the proper result is
You can multiply by x, and divide by 4 to solve —- or, you simply could take the reciprocal of both sides (always a completely legitimate move when you have fraction = number or fraction = fraction). Either way, the answer is .
Distributing a square
These next three mistakes are part of a broad category. First of all, one of most fundamental laws underlying all arithmetic and algebra is a law called the Distributive Law. Symbolically, it states
When read left-to-right, it is called distributing: we distribute A over (B + C). When this same equation is read right-to-left, it is called “factoring out.” See this post for a more extended panegyric to the Distributive Law, in a more advanced context. That equation is 100% true, 100% of the time. In words, we can say: multiplication distributes over addition and subtraction. It’s one of the most fundamental laws in all of mathematics.
That pattern is very important, and has a wide variety of applications in elementary and advanced mathematics. For some reason, though, this pregnant pattern is ripe for vast over-generalization. The mind seems almost magnetically drawn to distributing all kind of things other than multiplication over addition and subtraction.
One example is: an exponent. Suppose we are asked to expand algebraically the expression:
Be careful here, because unless you are a pro at math, your mind is going to be magnetically attracted to the wrong thing to do. Here’s the wrong thing to do:
If you notice, this mistake involves following the Distributive Law pattern, but with an exponent rather than with multiplication. That’s illegal. What’s the correct procedure? Well, squaring anything means multiplying it by itself, so the first step would be:
From there, you would FOIL out the expression. That’s the step-by-step way to get to the answer. It can be a very handy shortcut to have the following two patterns memorized.
The Square of a Sum:
The Square of a Difference:
Those formulas take into account the proper FOILing. Memorizing these can be a time-saving shortcut and also might help you to remember to avoid Mistake #3 here.
Distributing a fraction
This is another mistake of the “over-extend the Distributive Law to regions that are not valid” variety. Here is the succinct way to express this mistake.
In other words, you can neither combine nor separate fractions by additions in the denominator. This one has far-reaching ramifications. For example, in the following fraction …
… what can you cancel? NOTHING! If the 12 were over just the 3x, or if the 12 were just over the 8, then you would be able to cancel, but because you can’t separate the fraction, you can do absolutely no canceling. BTW, in the fraction ….
… even though some cancellation is possible, you can’t do any while the fraction is still like this. You have to separate it, by the addition in the numerator (a 100% legitimate move) and then you can cancel:
Another related mistake. Suppose we have to solve the following equation.
While it’s true in general that you can take the reciprocal of both sides, unfortunately, you can only take the reciprocal of a single number or a single fraction, NOT a sum or difference of fractions.
The reciprocal of a sum is not the sum of the reciprocals. How do you find the reciprocal of a sum? You would have to add the two fractions, using a common denominator, combining them into a single fraction. Here, by far the easiest solution would be to begin by subtracting 1/48 from both sides, and performing the fraction subtraction on the left side, so that you have a single fraction equals 1/x. Then, you would legitimately be allowed to take the reciprocal of both sides to solve.
Distributing a root
The final mistake, yet another example of illegitimately over-extending the pattern of the Distributive Law, is distributing root signs. Succinctly, this mistake says:
You cannot separate a square-root by addition or subtraction. You can separate a root by multiplication or division: see this post for more on that. You can see more about roots in general here.
If you have the equation…
… it is illegal to try to simplify that by taking a square-root of each term:
Think about it. Mr. Pythagoras was a very intelligent individual. If it were possible to simplify to a + b = c, then that’s how he would have stated the famous theorem. The fact is: he had to state it as , because it is absolutely illegal to simplify that to a + b = c.
In fact, whereas the former is true for the three side of every right triangle, the latter is not true for the three sides of any triangle. In fact, it constitutes a blatant violation of the Triangle Inequality, a law that is true for every possible triangle.
These are very common mistakes, and GMAT questions are often designed to elicit falling into one of these mistakes. If you can simply learn and avoid these five mistakes, then you will avoid the common traps to which so many GMAT takers will readily succumb.