ASVAB- Arithmetic Knowledge

Divisibility Rules

2: All even numbers are divisible by 2 3: Add up the individual digits of the number. If the total is divisible by 3, then the number itself is divisible by 3 Ex: 243 is divisible by 3 because the sum of its digit is 2+4+3= 9, but 367 is not because the sum of it's digits is 3+6+7= 16 and 16 is not a multiple of 3 4: Take the last two digits and divide them by 2. If the result is even, the number is divisible by 4. If the result is odd, then the number is not divisible by 4. 5: All numbers ending in 5 and 0 are divisible by 5 6: All even numbers that meet the test for divisibility by 3 are divisible by 6 8: Divide the number by 2 twice; if the result is even, then the number is divisible by 8 9: Add up the digits of the number; if the total is divisible by 9, then the number is divisible by 9

Factor

A factor is a positive integer that divides evenly into a given number. Ex: The complete list of factors of 12: 1, 2, 3, 4, 6, 12

Factorials

A factorial is the product of the integer before the factorial sign and and all the positive integers below it. You'll know that you are dealing with a factorial when you see an integer followed by an exclamation point. Ex: 7!= 7x6x5x4x3x2x1= 5,040

Fractions

A fraction is a number that is written in the form A/B, where A is the numerator and B is the denominator. An improper fraction has a numerator with a greater absolute value than that of the denominator. A mixed number consists of a whole number and a fraction. AN improper fraction can be converted to a mixed number and vice versa. Ex: -5/6, -3/17, 1/2, 899/901, -65/64, 9/8, 57/10, -1 1/64, 1 1/8, 5 7/10, 2 3/5= 13/5

Scientific Notation

A method of writing very large and very small numbers that also involve moving decimal points. The first part of a number in scientific notation will be equal to or greater 1 and less than 10. The second part of the number will be a power of 10. For powers of 10, the exponent is the number of zeros the number has when written out. Ex: 10^4= 10,000 For numbers written in scientific notation, where the first number always has exactly one digit to the left of the decimal point, the long version can be written by moving the decimal to the right by the same number of places as the value in the exponent of the power of 10. Ex: 1.23 x10^4= 12,300 Scientific notation also uses negative exponents to indicate the proper placement of the decimal point in a very small number. A negative exponent in scientific notation means that you move the decimal point to the left. Once the decimal point has been moved as far left as possible, start adding zeros to the right of the decimal. Ex: 4.321 x 10^-2= 0.04321

Multiple

A multiple of a number is the product of that number and an integer Ex: Some multiples of 12: 0, 12 ,24, 60

Sequences

A sequence is merely a group of numbers placed in order. You are most likely to encounter a special type of sequence on the ASVAB called an arithmetic sequence. The property that creates an arithmetic sequence is that each number is equal to the number before it plus a constant number. The sequence 1, 3, 5, 7 is an arithmetic sequence because each number is 2 more than the previous number in the sequence. One property that makes arithmetic sequences so special is the median. Ex: What is the mean of a sequence of multiples of 3 that begins with -3 and ends with 15. -3,0,3,6,9,12,15 Median is equal to 6

Integers

All whole numbers, including zero, and their negative counterparts Ex: -900, -3, 0, 1, 54

Even/Odd

An even number is an integer that is a multiple of 2. An odd number is an integer that is not a multiple of 2. Fractions and mixed numbers are neither even nor odd. Ex: Even numbers: -8, -2, 0, 12, 188 Odd numbers: -17, -1, 3, 9, 457

Prime Number

An integer greater than 1 that has no factors other than 1 and itself. 2 is the only even prime number. Ex: 2, 3, 5, 7, 11, 59, 83

Perfect Squares

Are integers that are the result of squaring (multiplying by itself) another integer. Ex: 25 is a perfect square because it is 5x5. You can save time and trouble on the ASVAB if you memorize the perfect squares up through 12x12= 144

Percents

Are ratios of an amount to 100, so the techniques used to work with ratios and proportions are valuable tools to work with percents. Ex: A bag contains 8 marbles and 6 of them are green. What percent of the marbles in the bag are green? 6/8= g/100 600= 8g 75=g So, 75% of the marbles are green. Conversion Table %: /100 (or use decimal or fractional equivalent) of: x (times) what x( or n, or any variable you like) is: = (equals) Increase or Decrease a Number by a Given Percent To calculate such increases or decreases, take that percent of the original number and add it to or subtract it from the original number Ex: To increase 25 by 60% , first find 60% of 25 25x0.6= 15 Then add the result to the original number. 25.15= 40 To decrease 25 by the same percent, subtract the 15 25-15= 10 Percent Change New Value- Original Value/Original Value If the percent change is an increase, the result will be positive. If the change is a decrease, the result will be negative.

Exponents

Are the small raised numbers written above and to the right of a variable of a number (the base). They indicate the number of times that the variable or number is multiplied by itself. There are two important things to remember about exponents. First, any number or variable with an exponent of 1 is equal to the base itself. Second, any number or variable with an exponent of 0 equals 1. Ex: x^1= x 5^1= 5 x^0= 1 5^0= 1 To add or subtract terms involving variables and exponents, both the variables and the exponents must be the same. Ex: 2x^2+x^2= 3x^2 3x^4-2x^4= x^4 To multiply terms with the same base, merely add the exponents. This can be done because exponents represent how many times the base is multiplied by itself. Ex 2^3x2^2= (2x2x2) x(2x2) x (2x2)= 2^3+^2= 2^5 Similarly, to divide terms with the same base, subtract the exponent that is in the denominator from the exponent in the numerator. Ex: 3^4/3^2= 3^4-^2= 3^2 To raise a a term involving an exponent to another exponent, multiply the exponent. Ex: (x^2)^4= x^2x^4= x^8

Coefficient

Is a number multiplied by a variable in a term. For instance, the term 2x^2, the coefficient is 2. to multiply terms consisting of coefficients and exponents that have the same variable in the base, multiply the coefficients and add the exponents. Ex: 6x^7x2x^5= (6x2) (x^7+^5)= 12x^12 To divide terms consisting of coefficients and exponents that have the same variable in the base, divide the coefficients and subtract the exponents. Ex: 2^-3= 1/2^3= 1/8

Square Root

Is a number that , when multiplied by itself, produces the given quantity. The radical sign √ is used to represent the positive square root of a number, so √25= 5, since 5x5= 25 (Even though (-5) x (-5) is also 25, when you ask for the square root, the answer will be a positive number). To add or subtract radicals, make sure the numbers under the radical sign are the same. If they are, you can add or subtract the coefficients outside the radical signs. Ex: 2√2+3√2= 5√2 To simplify radicals, factor out the perfect squares under the radical, calculate the square roots of the perfect squares, and put the results in front of the radical sign. Ex: √32= √16x2= 4√2 Note that when you have simplified to the point where the number under the radical does not contain any perfect squares, your result will be sufficient for the ASVAB. To multiply or divide radicals, multiply or divide the coefficients outside and inside the radical separately. Ex: √xx√y=√xy 3√2x4√5= 12√10 √x/√y=√x/y 12√10/3√2= 4√5 To take the square root of a fraction, you can break the fraction into two separate roots and take the square root of the numerator and the denominator separately. Ex: √16/25= √16/√25= 4/5

Prime Factorization

Is the number expressed using multiplication containing only primes, even of those primes repeat. Ex: Factors of 168: 2, 2, 2, 3, 7

Probability

Is the numerical likelihood that particular outcome will occur. To find the probability that something is going to happen, use this formula: Probability= Number of Outcomes of Interest/Number of Possible Outcomes Ex: If there are 12 books on a shelf and 9 of them are mysteries, what is the probability of picking a mystery at random? Probability= 9/12=3/4 This probability can be expressed as 0.75 or 75% To find the probability that both of two events will occur, find the probability that first event occurs and multiply this by the probability that the second event occurs. The probability of two independent events both occurring will be less than the probability of either occurring by itself. When you see a probability question that deals with one event and another event both occurring, you will multiply the probabilities. Ex:If there are 12 book on a shelf and 9 of them are mysteries, what is the probability of picking out a mystery first AND a non-mystery book second if exactly two books are selected and neither of them is replaced on the shelf? Probability of picking a mystery book: 9/12=3/4 (75%) Probability of picking a mystery book first and then a non-mystery book second: 3/4x3/11= 9/44

Averages

Like ratios, proportions, and rates, use fractions to find the answers to a problem. Here is the average formula: Average= Sum of the Terms/Number of Terms Ex: What is the average of 3, 4, and 8? Average= Sum of Terms/Number of Terms= 3+4+8/3= 15/3= 5 Mean is the same as the average and is calculated the same way. The range is how "wide" the group of numbers is and can be calculated by subtracting the smallest number from the largest. The mode is the number that appears most frequently The median is the middle value of a group of numbers

Positive/Negative

Numbers greater than 0 are positive numbers; numbers less than 0 are negative numbers. 0 is neither positive nor negative. Ex: Positive: 7/8, 1, 5, 6, 900 Negative: -64, -40, -1.11, - 6/13

Consecutive Numbers

Numbers that follow one another, in order, without skipping any. In a series of consecutive numbers, the differences between an consecutive numbers are equal. Ex: Consecutive integers: 3, 4, 5, 6 Consecutive even integers: 2, 4, 6, 8, 10 Consecutive multiples of -9: -9, -18, -27, -36,

Order of Operations

Parentheses Exponents Multiplication Division Addition Subtraction Ex: 3^3-8(4-2)+60/4 = 3^3-8(2)+60/4 = 27-8(2)+60/4 = 27-16+15 =26

Word Problems w/ Formulas

Rate= Distance/Time Average= Sum of the Terms/Number of Terms Probability= Number of Outcomes of Interest/Number of Possible Outcomes Ex: If a truck travels at 50 miles per hour for 6.5 hours, how far will the truck travel? Rate= Distance x Time = 50x6.5 = 325 miles

Ratios, Proportions and Rates

Ratios are either part to part or part to whole relationships, depending upon what quantities are being compared. Ex: A class contains 12 male students and 21 female students The part to part ratio of male students to female students is 12/21= 4/7 The part to whole ratio of male students to all the students in the class is 1/12+21= 12/33= 4/11 Always simplify ratios to their lowest terms, as was done in the above example. A proportion is an equation of two ratios that shows the comparative relationship between parts, things, or elements with respect to size, amount, or degree. When working with proportions, you can use a helpful technique called cross multiplying to help solve the equation. To cross-multiply an equation that consists of two fractions, you multiply the numerator of the first fraction by the denominator of the second and vice versa. Ex: 2x/5= 3/4 2x(4)= 3(5) A rate is simply a ratio that compares two different but related quantities, such as distance divided by time (speed), or amount divided by time, or cost per mile. In other words: Rate= Distance/Time or Rate= Amount/Time or Rate = cost/Units Another way to look at rates is think of them as changes in the numerator per changes in the denominator. For instance, if your rate of pay is $15/hour, you know that if you work one more hour you will earn an additional $15. The key to solving rate problems is to set them up as propotions. Ex: 1) Set up rate as proportion 3 diners/ 5 minutes= x diners/1 hour 2) Convert units 3 diners/5 minutes= x diners/60 minutes 3) Cross-multiply and solve 5x= 180 x= 36

Distributive Property

Says that the result of multiplying one number by the sum or difference of two numbers can be obtained by multiplying each number individually and then totaling the results. Ex: 5x5(20+2)= 5x20+5x2= 100+10=110 5x(20-2)= 5x20-5x2=100-10=90 Similarly, a fraction with multiple terms in the numerator and only one term in the denominator can be subdivided. Ex: x^3+4x^2/x^2= x^3/x^2+4x^2/x^2= x+4 Do not attempt to use this property if the denominator has more than one term Ex: x^2/x^3+4x^2 cannot be split up. Not only is the distributive property useful in solving problems on the ASVAB, it can also help you quickly and accurately multiply in the absence of a calculator.

Commutative Property

Says, in a nutshell, that order does not matter when adding or multiplying numbers Ex: 3x4= 4x3 3=4= 4+3 This property does not change PEMDAS; it merely gives you some flexibility when you get to multipy or add part of the order of operations. The commutative property doe not apply to subtraction or division. Ex: 4-3 does not equal 3-4 4/3 does not equal 3/4

Translating Word Problems

Sum, plus, more than, added to, combined, total: + Minus, less than, difference between, decreased by: - Is, was, equals, is equivalent to, is the same as, adds: = Times, product, multiplied by, of: x Divided by, over, quotient, per, out of, into: / What, how much, how many, a number: x, n, etc.

Absolute Value

The distance a number is from zero on a number line. The absolute value of what is between the vertical lines is the positive magnitude of the number, regardless of whether it is positive or negative. Ex: | x-2 | = 4 x= 6 x-2= -4 x= -2