Arithmetic Calculations SummaryFor the four basic operations, understand deeply how to apply them to integers, fractions, and decimals. You should focus on how the process works and the role of each component in these operations so that you can deal with abstract, conceptual questions.

Spend extra time preparing for division problems. Make sure that you thoroughly understand the relationship between the dividend, divisor, quotient, and remainder. There are numerous division problems in the homework section that you can use to practice this difficult question type.

For a review of fractions and decimal calculations (and more practice with integer calculations), go back to the Skillbuilder, as those calculations have not been covered in the lesson section.

Become efficient at performing any basic arithmetic calculation in your head or on paper. To accomplish that, try to minimize work as you perform the calculations in the homework problems in this book. It is the rare Arithmetic problem on the GMAT that requires more than a couple of written calculations (if that!). If you find yourself filling up a page with work on a problem, there must be a better way.

It is helpful to memorize the shortcut tools (perfect squares, fraction/decimal conversions, etc.) presented in the Skillbuilder so that you can leverage them to speed up calculations.

Factors, Multiples, and the Number Line Summary• Know the definitions of and be clear about the difference between prime factors, factors, and prime factorization.

• Be extremely proficient with prime factorization and avoid factor trees unless absolutely necessary. You should be able to break down all two-digit numbers into their prime factorization without written work (just use your times tables!)

• Know the divisibility rules for all single digits (except 7, for which you should just use your understanding of universal divisibility).

• Know how to find the total number of unique factors for a given number by leveraging the prime factorization (either with the unique factors trick or simply pairing up the prime factors).

• Know that all perfect squares have an odd number of total unique factors.

• Know how to check whether a number is prime (and, more importantly, not prime).

• Know the definition of a multiple and remember that 0 is a multiple of all numbers.

• Understand deeply the least common multiple and greatest common factor. Be able to find the LCM and GCF by using the prime factorization of each number.

• Understand the repeating patterns of multiples on the number line.

• Recognize that (and understand why) two numbers beside each other on the number line (x and x + 1) can share no factors other than 1. The greatest common factor of x and x + 1 will always be 1, and the least common multiple of x and x + 1 will always be (x) (x + 1).

• Be able to put all these concepts together for broader number line questions like the last one in this section.

Number Properties Summary• Units digit properties are both explicitly tested and important for shortcutting calculations. In either case, units digit problems are simply testing how the four basic operations work. Any time you are presented with a problem that seems to involve tedious calculations or prohibitively large numbers, focus on the units digit instead of the entire calculation.

• The odd/even properties themselves are relatively straightforward: odd +/− odd = even odd • odd = odd

odd +/− even = odd odd • even = even

even +/− even = even even • even = even

• The most difficult odd/even property questions involve division, because it is not as well understood. Remember: Even divided by even can be either odd or even (or a non-integer).

• Don’t forget that odd/even properties can also be used to avoid tedious calculations. For instance, if only one answer choice is odd, then that might be enough to solve the problem without actually executing calculations.

• Like odd/even properties, positive/negative properties relating to arithmetic operations are relatively straightforward: positive • positive = positive positive + positive = positive

negative • positive = negative negative + negative = negative

negative • negative = positive negative + positive = negative or positive

• Positive/negative number property questions become much more complicated when exponents are involved (as you will see in the Algebra lesson).

• Most difficult number property questions involve unusual patterns or properties relating to factors, multiples, and the number line. To unlock these difficult, “create your own” number property questions, focus on two important strategies: Do something, and look for patterns. The key to most of these questions is to start with something easy that you do understand, and then look for patterns as you execute some basic operations.

Ratios Summary• Always consider the total number of parts in any ratio problem. Make sure that you answer the proper question. If you are told, for instance, that the ratio of men to women at a picnic is 1:4, then that represents the ratio of one group to another—not the ratio of one group to the total. In this example, men represent one-fifth of the total, not one-fourth, as many people think. This is the most commonly exploited mistake relating to ratios.

• Understand the multiplier concept well. When you are given a ratio, it only gives you the proportion, but not the actual numbers involved. To determine the multiplier, you must be given an actual number for one of the corresponding parts (or at least enough information to determine that actual number). Once you have determined the multiplier, you can leverage that to solve for any part in the ratio.

• Remember to use the total number of parts when you are asked to solve for the actual total amount in a ratio problem. People tend to add up all the individual amounts instead of just multiplying the total number of parts by the multiplier.

• Remember that most ratio problems involve things that must have a whole-number multiplier. For instance, if the ratio of men to women at a picnic is 5:3, then it is impossible to have 7 men at the picnic. Why? Because then the multiplier would be a non-integer, and you would have fractional number of women at the picnic—clearly nonsensical. Other than problems involving quantities (weight, volume, etc.), it is generally necessary to have a whole-number multiplier.

• Be careful with any problem involving addition and subtraction with ratios. When you add or subtract quantities in a ratio with an unknown multiplier, you do not know how that addition or subtraction will affect the original ratio. This is a very important concept on the GMAT.

• Be flexible on ratio problems. Remember that sometimes it is easier to think of ratios in terms of fractions or fractional parts, whereas other times it is easier to think in terms of the multiplier.

Percents Summary• Percent calculations need to become second nature, as they are so prevalent on the GMAT. While the “by the book” set-ups are given in the Skillbuilder section, you should able to calculate most percent calculations mentally or with very little written work.

• Make sure that you can move seamlessly between percents, decimals, and fractions. This allows you to pick the calculation method that works best on any given problem. Generally speaking, fractions are easier to use than decimals, but sometimes decimals or other tricks (like the one for 15%) will be faster.

• Read percent problems carefully. People tend to get percent problems wrong—not because they calculate the percent incorrectly, but because they are answering the wrong question. Always take the extra time to make sure you are answering the proper question.

• The most common mistake that people make in percents is that they calculate the correct percent of the wrong number. Particularly on discount problems, make sure that you are setting up the problem correctly and taking the proper percent of the proper number.

• With many percent problems, it is easier to pick a number and work with that number than it is to use variables. If no specific number is given in a problem, just use a simple number like 100 and work with that to keep track of the percent changes.

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