# 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 5/6 vs. 6/7. Cross-multiplying gives 5 x 7 vs. 6 x 6, or 35 vs. 36. Now 36 is larger than 35, so 6/7 is larger than 5/6.

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

Example:
and 1/2 is greater than 1/4.
Caution: This is not true for fractions greater than 1.
For example, . But .

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

Example:
and 1/4 is less than 1/2.

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

Example:
3 x 22 = 3 x 4 = 12. But (3 x 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 x 5 = 25.

5. . In fact, and .

Example:
. But .

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.

1. x2y2 = (x + y)(xy)
2. x2 ± 2xy + y2 = (x ± y)2
3. a(b + c) = ab + ac

8. Know these rules for radicals:

9. Pythagorean Theorem (For right triangles only):
c2 = a2 + b2

Example
What is the area of the triangle?
1. 6
2. 7
3. 8
4. 10
5. 16
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 .

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°.
a + b + c = 180°

Example: In the triangle to the right, what is the degree measure of angle c ?
1. 30
2. 35
3. 40
4. 41
5. 45
Solution: Since a triangle has 180˚, we get 100 + 50 + c = 180. Solving for c yields c = 30. Hence, the answer is (A).

16. Consecutive integers are written x, x + 1, x + 2, . . . . Consecutive even or odd integers are written x, x + 2, x + 4, . . . .

17. 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

18. 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:

19. When counting elements that are in overlapping sets, the total number will equal the number in one group plus the number in the other group minus the number common to both groups. Venn diagrams are very helpful with these problems.

Example: If in a certain school 20 students are taking math and 10 are taking history and 7 are taking both, how many students are taking either math or history?

Solution:
By the principle stated above, we add 10 and 20 and then subtract 7 from the result.
Thus, there are (10 + 20) – 7 = 23 students.

20. The number of integers between two integers inclusive is one more than their difference.

For example: The number of integers between 49 and 101 inclusive is (101 – 49) + 1 = 53. To see this more clearly, choose smaller numbers, say, 9 and 11. The difference between 9 and 11 is 2. But there are three numbers between them inclusive—9, 10, and 11—one more than their difference.

21. 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