Math Notes

We’ll discuss many of the concepts in this chapter in depth later. But for now, we need a brief review of these concepts for many of the problems that follow.

1. To compare two fractions, cross-multiply. The larger product will be on the same side as the larger fraction.

Example: Given $\frac{5}{6}$ vs. $\frac{6}{7}$ . Cross-multiplying gives 5 × 7 vs. 6 × 6 , or 35 vs. 36. Now 36 is larger than 35, so $\frac{6}{7}$ is larger than $\frac{5}{6}$ .

2. Taking the square root of a fraction between 0 and 1 makes it larger.

Example: $\sqrt{\frac{1}{4}}$ =  $\frac{1}{2}$ and $\frac{1}{2}$ is greater than $\frac{1}{4}$ .

Caution: This is not true for fractions greater than 1. For example, $\sqrt{\frac{9}{4}}$ = $\frac{3}{2}$ . But $\frac{3}{2}$ < $\frac{9}{4}$

3. Squaring a fraction between 0 and 1 makes it smaller.

Example: ( $\frac{1}{2}$)2 = $\frac{1}{4}$ and $\frac{1}{4}$ is less than $\frac{1}{2}$

4. ax2 ≠ (ax)2. In fact, a2x2 = (ax)2 .

Example: 3 × 22 = 3 × 4 = 12 But (3 × 2)2 = 62 = 36 This mistake is often seen in the following form: x2 = (−x)2 To see more clearly why this is wrong, write x2 = (−1)x2 , which is negative. But (−x)2 = (−x)(−x) = x2 , which is positive.

Example: −52 = (−1)52 = (−1)25 = −25 But (−5)2 = (−5)(−5) = 5 • 5 = 25

5. $\frac{1/a}{b}$  ≠  $\frac{1}{a/b}$ In fact, $\frac{1/a}{b}$  =  $\frac{1}{ab}$ and $\frac{1}{a/b}$  =  $\frac{b}{a}$

Example: $\frac{1/2}{3}$ =  $\frac{1}{2}$ ×  $\frac{1}{3}$ = $\frac{1}{6}$ But $\frac{1}{2/3}$ = 1 ×  $\frac{3}{2}$ =  $\frac{3}{2}$

6. –(a + b) ≠ –a + b. In fact, –(a + b) = –a – b.

Example: –(2 + 3) = –5. But –2 + 3 = 1.

Example: –(2 + x) = –2 – x.

7. Memorize the following factoring formulas—they occur frequently on the GRE.

A. x2 − y2 = (x + y)(x - y)

B. x2 ± 2xy + y2 = (x ± y)2

C. a(b + c) = ab + ac

8. Know these rules for radicals:

A.  $\sqrt{x}\sqrt{y}$ = $\sqrt{xy}$
B.
$\sqrt{\frac{x}{y}}$ = $\frac{\sqrt{x}}{\sqrt{y}}$

9. Pythagorean Theorem (For right triangles only):

c2 = a2 + b2

Example: Which is greater column A or column B?

Column A = 10

Column B = The area of the triangle

Solution:

Since the triangle is a right triangle, the Pythagorean Theorem applies: h2 + 32 = 52 , where h is the height of the triangle. Solving for h yields h = 4.  Hence, the area of the triangle is
$\frac{1}{2}$(base)(height) = $\frac{1}{2}$(3)(4) = 6. Because 6 is less than 10,The answer is (A).

10. When parallel lines are cut by a transversal, three important angle relationships are formed:

11. In a triangle, an exterior angle is equal to the sum of its remote interior angles and therefore greater than either of them.

e = a + b  and e > a  and e > b

12. A central angle has by definition the same measure as its intercepted arc.

13. An inscribed angle has one-half the measure of its intercepted arc.

14. There are 180° in a straight angle.

15. The angle sum of a triangle is 180°.

Example: What is greater column A or column B ?

Column A = 30

Column B = The degree measure of angle c

Solution:

Since a triangle has 180˚, we get 100 + 50 + c = 180.  Solving for c yields c = 30. Hence, the columns are equal.

16. To find the percentage increase, find the absolute increase and divide by the original amount.

Example:  If a shirt selling for $18 is marked up to$20, then the absolute increase is 20 – 18 = 2.  Thus, the percentage increase is  = $\frac{2}{18}$ = $\frac{1}{9}$ ≈ 11%

17. Systems of simultaneous equations can most often be solved by merely adding or subtracting the equations.

Example:  If 4x + y = 14 and 3x + 2y = 13, then x – y =

Solution:

Merely subtract the second equation from the first:

18. Rounding Off: The convention used for rounding numbers is “if the following digit is less than five, then the preceding digit is not changed.  But if the following digit is greater than or equal to five, then the preceding digit is increased by one.”

Example: 65,439 —> 65,000 (following digit is 4)
5.5671 —> 5.5700 (dropping the unnecessary zeros gives 5.57)