Recognizing Blocks

Today, let’s discuss a math strategy I like to call “Recognizing Blocks.” This is a strategy that can sometimes be used on standardized tests such as the ACT, SAT, GRE, and AP Calculus exams.

First, let me give my definition of a block: A block is an algebraic expression that appears more than once in a given problem.

Very often in math problems on standardized tests a block can be treated just like a variable. In particular, blocks should usually not be manipulated—treat them as a single unit.

This is simple in theory. Let’s first try it with a basic algebra problem:

Example 1

If 3x – 5y = 7 , what is the value of 6(3x − 5y) ?

(A) 50
(B) 42
(C) 36
(D) 30
(E) 4

In this example there is a block of 3x – 5y. Let’s look at the following seemingly easier problem:

If a = 7, what is the value of 6a?

The answer to this problem is (6)(7) = 42.

But this is essentially the same problem as the one we were given. We just replaced the block by the variable a. So the answer to the original question is also 42, choice (B).

You can be really sharp by doing this problem in your head in just a few seconds: (6)(7) = 42. choice (B).

Easy, right?

Here is a more difficult problem for you:

Example 2

If b = 7(u + v + x + y +z)/5, then in terms of b what is the average (arithmetic mean) of u, v, x, y, and z?

(A) b/35
(B) b/7
(C) b/5
(D) 5b
(E) 7b

In this example there is a block of a = (u + v + x + y + z)/5 which is precisely the average of u, v, x, y, and z. So solving this problem is the same as solving the following equation for a.

b = 7a

To solve this equation for a we will simply divide both of the sides by 7 to get a = b/7, choice (B).

These problems as relatively straightforward to solve as long as you treat all blocks as variables and do not try to manipulate them.