Algebra

Remember this from high school? Not a popular one. I'm afraid we have to recap algebra for the test**, but** we'll try to keep it short and sweet!

Terminology

A term is a number, a variable (e.g. x, y, z**), or a combination of both** (e.g.. 3z). Terms come in many different formats, such as: 7x, 21y, and 3(x/y).

Expressions**, are combinations of a variety of** terms**, separated by + or – signs,** for example:

Substitution is very important in algebra. It is a method used to evaluate the expression.

For example:

Evaluate 7x + 21y if x = 2 and y = 1

This means the numbers will be substituted into the expression as:

(7 × 2) + (21 × 1)

= 14 + 21

= 35

We previously discussed the various laws of operations, and those same laws apply to expressions in algebra including the commutative law, the associative law, and, at times, both laws. The distributive law is also found in algebraic expressions. (See the Arithmetic section for a recap of these laws)

Factoring Expressions

When you factor a polynomial expression, you are expressing it as the product of two or more simpler expressions.

The common monomial factor is common to every term in the polynomial and can be factored out using the distributive law.

The difference of two perfect squares can be factored into the product of:

b2 – c2 = (b – c)(b + c)

Polynomials using the expression a2 + 2ab + b2 are equivalent to the square of a binomial.

(a + b)2 = a2 + 2ab + b2

Polynomials of the form a2 – 2ab + b2 are equivalent to the square of a binomial, where the binomial is the difference between the two terms:

(a – b)2 = a2– 2ab + b2

If a question isn’t testing any one of the three common expressions above, you can always use the FOIL method. FOIL stands for First, Outer, Inner, Last. Multiply the first two terms of each factor, then multiply the two outer terms, the two inner terms, and finally the two last terms. This will yield:

(x + m)(x + n) = x**2 + xn + xm + mn

This may seem like a lot to take in, but don’t worry, you will get the hang of it – and algebra is only a small portion of the test.

Ø  More Practice from the GMAT**®** Review 13****th Edition: Questions 37, 45, 209

Word Problems with Algebra

Word problems will test what you learned in algebra, arithmetic, and even geometry from your school years. All they do is ask the same questions in a different way. Look at this example to understand how:

In Algebra Form: 3x + 2 = y

In Word Problem Form*:** If the number of cars on the lot tripled, the car lot would have only 2 fewer than the lot next door.*

In this case, the number of cars on the lot is x, and the lot next door is y.

If x = 23 cars then: 3(23) + 2 = y

The number of cars (y) on the other lot is 71.

The key to understanding word problems is to convert what they are saying in English into the universal language of mathematics. This is sometimes easier said than done.

Let’s do some simple math to ease back into the concept of word problems. We will cover addition, multiplication, subtraction, and division in the next four examples.

If Jim bought eight apples for $.29 each and four pears for $.79 each, how much did he pay in total?

(8 × .29) + (4 × .79)

= 2.32 + 3.16

= 5.48

If there are 72 people in the room and 27 are men and 33 are children, how many women are there in the room?

72 – 27 – 33

= 45 – 33

= 12

If you buy one television for $3,300, then how many will 9 cost you?

3,300 × 9

= $29,700

If Jason bought eight apples for $3.72, how much is each apple?

3.72 / 8

= 47 cents

Right there, we can see quickly and easily what we are dealing with – all based on the information in the word problems. Most simple mathematical word problems follow this method.

Since we just covered algebra, it’s now time to put algebra into word problems. This may help you understand the problem much better. Let’s look at an example to see how algebra fits into word problems.

Frank weighs the sum of Jim’s weight multiplied by two, and Mary’s weight multiplied by three. This can be shown as:

F = 2J + 3M

Of course, we need more information to answer this question. If we know that J = 120 and M = 97, then we can do more math:

F = 2(120) + 3(97)

F = 240 + 291

F = 531

So, Frank weights 531 pounds and should probably start looking into joining a gym ;)

That’s it for the basics of word problems. Turn English into math, use the algebra we learned previously and you are all set! In the next section, we will move on to more complex problems.

Ø  More Practice from the GMAT**®** Review 13****th Edition: Questions 6, 64, 135, 153, 140