Math ARITHMETIC

A particular stock is valued at $40 per share. If the value increases by 20 percent and then decreases by 25 percent, what will be the value of the stock per share after the decrease?

$36 per share

factor

( 2 )(3)( 10) = 60

What is the prime factorization of 585 ?

(3^2 ) (5)( 13)

Find the following. (a) 40% of 15 (b) 150% of 48 (c) 0.6% of 800 (d) 15 is 30% of which number? (e) 11 is what percent of 55 ?

(a) 40٪x = 15 => 15*40 / 100 = 6 (b) 72 (c) 4.8 (d) 50 (e) 20%

Here are some important rules regarding operations with square roots, where a > 0 and b > 0.

(√a^2) = (√a)^2 = a ; (√3)^2 = 3 ( √a √b)=√ab) (√a)/(√b)= (√a/b)

A real number x is less than a real number y if x is to the left of y on the number line, which is written as x < y. A real number y is greater than x if y is to the right of x on the number line, which is written as y > x.

- √5 < -2 1/2 >0 1 < √2 < 2

irrational numbers اعداد گنگ

. Since these two decimals do not terminate or repeat, they are not rational numbers.

The positive integers are greater than 0, the negative integers are less than 0

0 is neither positive nor negative

The monthly enrollment at a preschool decreased by 8% during one month and increased by 6% during the next month. What was the cumulative percent change for the two months?

100% - 8% = 92% ; 0.92 E ( 1.06)(0.92) E = 0.9752E The percent equivalent of 0.9752 is 97.52%, which is 2.48% less than 100%. Thus, the cumulative percent change in the enrollment for the two months is a 2.48% decrease.

When the positive integer n is divided by 3, the remainder is 2 and when n is divided by 5, the remainder is 1. What is the least possible value of n ?

11

What percent of 150 is 12.9 ?

12.9/150 = x /100 => x = 8.6 %

Given the percent and the part, you can calculate the whole . To do this you can either use the decimal equivalent of the percent or you can set up a proportion and solve it.

15 is 60% of what number? 15 = 60٪ X => x = (15/1)/(60/100) = 25

If an athlete's weight decreased from 160 pounds to 152 pounds, what was the percent decrease in the athlete's weight?

160/100=152/X => x=95, so, 100-95=5%

If a person's salary increased from $200 per week to $234 per week, what was the percent increase in the person's salary?

200/100= 234/x => x = 117, 117-100 = 17 %

Which of the integers 312, 98, 112, and 144 are divisible by 8 ?

312, 112, and 144

If the ratio of the number of men to the number of women on a committee of 20 members is 3 to 2, how many members of the committee are women?

8 women 2/3*20=13 20-13=8

fraction کسر

A fraction is a number of the form a/b where a and b are integers and b ≠ 0. The integer a is called the numerator of the fraction, and b is called the denominator. ex: 5/8

proportion

A proportion is an equation relating two ratios;

composite number -عدد مرکب

An integer greater than 1 that is not a prime number is called a composite number. The first ten composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, and 18.

Exponents

Exponents are used to denote the repeated multiplication of a number by itself;

Exponents can also be negative or zero; such exponents are defined as follows.

For all nonzero numbers a, a^0 = 1. The expression 0^0 is undefined. For all nonzero numbers a, a^-1=1/a ; a^-2=1/a^2 ; Note that (a ) (a^-1 ) =( a)(1/a)= 1

Then the remainder is equal to a minus that multiple of b, or r = a - qb, where r is the remainder. The remainder is always greater than or equal to 0 and less than b.

Here are examples that illustrate a few different cases of division resulting in a quotient and remainder. • 100 divided by 45 is 2 remainder 10, since the greatest multiple of 45 that's less than or equal to 100 = (45 * 2 )+ 10 • 24 divided by 4 is 6 remainder 0, since the greatest multiple of 4 that's less than or equal to 24 is 24 itself, which is 0 less than 24. In general, the remainder is 0 if and only if a is divisible by b. • 6 divided by 24 is 0 remainder 6, since the greatest multiple of 24 that's less than or equal to 6 is (0)( 24 ) , or 0, which is 6 less than 6.

even integer

If an integer is divisible by 2, otherwise it is an odd integer. The set of even integers is { . . . , -6, -4, -2, 0, 2, 4, 6, . . . }, and the set of odd integers is { . . . , -5, -3, -1, 1, 3, 5, . . . } .

base

In general, the whole is called the base of the percent

quotient خارج قسمت

The answer after you divide one number by another. dividend ÷ divisor = quotient Example: in 12 ÷ 3 = 4, 4 is the quotient

absolute value

The distance between a number x and 0 on the number line is called the absolute value of x, written as |x|. Therefore, |3| = 3 and |- 3| = 3 because each of the numbers 3 and -3 is a distance of 3 from 0. Note that if x is positive, then |x| = x ; if x is negative, then |x| = - x ; and lastly, |0| = 0. It follows that the absolute value of any nonzero number is positive.

Integers

The integers are the numbers 1, 2, 3, and so on, together with their negatives, -1, - 2, -3, . . . , and 0. Thus, the set of integers is { . . . , - 3, - 2, -1, 0, 1, 2, 3, . . . }.

An investment in a mutual fund increased by 12% in a single day. If the value of the investment before the increase was $1,300, what was the value after the increase?

The percent increase is 12%. Therefore, the value of the increase is 12% of $1,300, or, using the decimal equivalent, the increase is (0.12)( $1,300) = $156. Thus, the value of the investment after the change is $1,300 + $156 = $1,456

real numbers

The set of real numbers consists of all rational numbers and all irrational numbers. The real numbers include all integers, fractions, and decimals. The set of real numbers can be represented by a number line called the real number line.

Percent

The term percent means per hundred, or hundredths. Percents are ratios that are often used to represent parts of a whole, where the whole is considered as having 100 parts. - 1 percent means 1 part out of 100 parts , 1/100 -32 percent means 32 parts out of 100 parts , 32/100 - 50 percent means 50 parts out of 100 parts, 1/2

cross multiplication example:

To find a number x so that the ratio of x to 49 is the same as the ratio of 3 to 21, you can write x/49 = 3/21 => x = 3 * 49 / 21 = 7

cross multiplication

To solve a problem involving ratios, you can often write a proportion and solve it by cross multiplication.

squaring

When the exponent is 2, we call the process squaring. Thus, 6 squared is 36

The integer a is even and the integer b is odd. For each of the following integers, indicate whether even or odd.

a + 2b Even 2a + b ab Even a^b Even ( a + b)^2 a2 - b^2

square root ریشه دوم

a nonnegative number n is a number r such that r^2 = n . 4 is a square root of 16

a- What are the prime divisors of 100 ? b- What are the prime divisors of 144 ?

a- 2 , 5 b- 2 , 3

(a) What is the prime factorization of 372 ? (b) What are the positive divisors of 372 ?

a- 372 = (2^2 ) (3 )(31 ) b-The positive divisors of 372 are 1, 2, 3, 4, 6, 12, 31, 62, 93, 124, 186, and 372.

prime number -عدد اول

an integer greater than 1 that has only two positive divisors: 1 and itself. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29

common denominator- مخرج مشترک

find a common multiple of the two denominators. Then convert both fractions to equivalent fractions with the same denominator. Finally, add the numerators and keep the common denominator.

If the whole is 20 and the part is 13, you can find the percent as follows.

part / whole = 13 / 20 = 0.65 = 65 %

greatest common divisor (or greatest common factor)ب م م

the greatest common divisor of 30 and 75 is 15.

least common multiple کوچکترین مضرب مشترک ک م م

two nonzero integers a and b is the least positive integer that is a multiple of both a and b. ex: the least common multiple of 30 and 75 is 150.

To find 30% of 350, multiply 350 by the decimal equivalent of 30%, or 0.3, as follows.

x = (350)(0.3) = 105

Here are some other examples.

• 100 divided by 3 is 33 remainder 1, since 100 = (33)(3) + 1. • 100 divided by 25 is 4 remainder 0, since 100 = (4)(25) + 0. • 80 divided by 100 is 0 remainder 80, since 80 = (0)( 100) + 80. • When you divide 100 by 2, the remainder is 0. • When you divide 99 by 2, the remainder is 1.

There are some notable differences between odd-order roots and even-order roots (in the real number system):

• For odd-order roots, there is exactly one root for every number n, even when n is negative. • For even-order roots, there are exactly two roots for every positive number n and no roots for any negative number n.

divisor

• The positive factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100. • 25 is a multiple of only six integers: 1, 5, 25, and their negatives. • The list of positive multiples of 25 has no end: 25, 50, 75, 100, 125, 150, etc.; likewise, every nonzero integer has infinitely many multiples. • 1 is a factor of every integer; 1 is not a multiple of any integer except 1 and -1. • 0 is a multiple of every integer; 0 is not a factor of any integer except 0.

Here are some general facts regarding multiplication of integers.

• The product of two positive integers is a positive integer. • The product of two negative integers is a positive integer. • The product of a positive integer and a negative integer is a negative integer. - * - = + * + = + + * - = - * + = -

several useful facts regarding the sum and product of even and odd integers.

• The sum of two even integers is an even integer. • The sum of two odd integers is an even integer. • The sum of an even integer and an odd integer is an odd integer. • The product of two even integers is an even integer. • The product of two odd integers is an odd integer. • The product of an even integer and an odd integer is an even integer.

There are several general properties of real numbers that are used frequently. If a, b, and c are real numbers, then

• a + b = b + a and ab = ba. For example, 8 + 2 = 2 + 8 = 10 and ( -3)( 17 ) = ( 17 )( -3 ) = -51. • ( a + b ) + c = a + (b + c ) and (ab) c = a (bc ) . For example, (7 + 3) + 8 = 7 + (3 + 8) = 18 • a (b + c ) = ab + ac For example, 5 (3 + 16) = (5)(3) + (5)( 16) = 95. • a + 0 = a , ( a)(0) = 0, and ( a )( 1) = a . • If ab = 0, then a = 0 or b = 0, or both. For example, if -2b = 0, then b = 0. • Division by 0 is not defined 5/0 , 0/0 are undefined. • If both a and b are positive, then both a + b and ab are positive. • If both a and b are negative, then a + b is negative and ab is positive. • If a is positive and b is negative, then ab is negative. |a|| b| =| ab |, for ex |5|| - 2| = |(5)(-2)| = |- 10 | = 10 - If a > 1, then a^2 > a. If 0 < b < 1, then b^2 < b. For example, 5^2 = 25 > 5, but (1/5)^2 = 1/25 < 1/5