**Miscellaneous**

- To compare two fractions, cross-multiply. The larger product will be on the same side as the larger fraction.
- Taking the square root of a fraction between 0 and 1 makes it larger. Caution: This is not true for fractions greater than 1. For example, $\sqrt{{\displaystyle \frac{9}{4}}}$ = $\frac{3}{2}$ . But $\frac{3}{2}$ < $\frac{9}{4.}$
- Squaring a fraction between 0 and 1 makes it smaller.
*a**x*^{2}â‰ (*ax*)^{2}. In fact,*a*^{2}*x*^{2}= (*ax*)^{2}.- $\frac{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$a$}\right.}}{b}\xe2\u2030\frac{1}{{\displaystyle \raisebox{1ex}{$a$}\!\left/ \!\raisebox{-1ex}{$b$}\right.}}$ . In fact, $\frac{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$a$}\right.}}{b}=\frac{1}{ab}$ and $\frac{1}{{\displaystyle \raisebox{1ex}{$a$}\!\left/ \!\raisebox{-1ex}{$b$}\right.}}=\frac{b}{a}$.
- â€“(
*a*+*b*) â‰ â€“*a*+*b*. In fact, â€“(*a*+*b*) = â€“*a*â€“*b*. *percentage increase*= $\frac{increase}{originalamount}$- Often you can solve a system of two equations in two unknowns by merely adding or subtracting the equations.
- 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.
- The number of integers between two integers
__inclusive__is one more than their difference. - Principles for solving
**quantitative comparisons**:- You can add or subtract the same term (number) from both sides of a quantitative comparison problem.
- You can multiply or divide both sides of a quantitative comparison problem by the same positive term (number). (Caution: this cannot be done if the term can ever be negative or zero.)
- When using substitution on quantitative comparison problems, you must plug in all five major types of numbers: positives, negatives, fractions, 0, and 1. Test 0, 1, 2, â€“2, and 1/2, in that order.
- If there are only numbers (i.e., no variables) in a quantitative comparison problem, then "not-enough-information" cannot be the answer.

- â€‹Substitution (Special Cases):
- A. In a problem with two variables, say,
*x*and*y*, you must check the case in which*x*=*y*. (This often gives a double case.) - B. When you are given that
*x*< 0, you must plug in negative whole numbers, negative fractions, and â€“1. (Choose the numbers â€“1, â€“2, and â€“1/2, in that order.) - C. Sometimes you have to plug in the first three numbers (but never more than three) from a class of numbers.

- A. In a problem with two variables, say,
**Elimination strategies**:- A. On hard problems, if you are asked to find the least (or greatest) number, then eliminate the least (or greatest) answer-choice.
- B. On hard problems, eliminate the answer-choice "not enough information."
- C. On hard problems, eliminate answer-choices that
__merely__repeat numbers from the problem. - D. On hard problems, eliminate answer-choices that can be derived from elementary operations.
- E. After you have eliminated as many answer-choices as you can, choose from the more complicated or more unusual answer-choices remaining.

- To solve a fractional equation, multiply both sides by the LCD (lowest common denominator) to clear fractions.
- You can cancel only over multiplication, not over addition or subtraction. For example, the câ€™s in the expression $\frac{c+x}{c}$ cannot be canceled.
- The average of N numbers is their sum divided by N, that is, Average = Sum/N.
*Weighted average:*The average between two sets of numbers is closer to the set with more numbers.*Average Speed*= $\frac{TotalDistanc\mathrm{e\; /}}{TotalTime}$- Distance=Rate * Time
- Work = Rate*Time, or W=R * T. The amount of work done is usually 1 unit. Hence, the formula becomes 1 =
*R***T*. Solving this for*R*gives*R*= 1T. - Interest =
*Amount***Time***Rate*