Math-36 Basic Math Concepts

Add Fractions with Different Denominators

1/2 + 1/4=2/4 + 1/4

Positive and Negative Numbers

A positive number is any number greater than zero; no sign is written in front of the number, A negative is any number less than zero (-) sign in front of number. Adding positive and negative numbers is that same as subtracting the second number from first. For example: 3+(-2)=3-2=1 Subtracting negative numbers is same as adding positive values: 3-(-2)=3+2=5 When 2 positives are multiplied, product is a positive number: 2x2=4 When two negatives are multiplied, product is a positive number -2 x -2=4 When a positive and negative number is multiplied, product is negative; 2 x -2=-4

Real and Rational Numbers

A real number is any number that can be placed on an infinite number line stretching from negative infinity to positive infinity. Integers, fractions and decimals including infinite decimals (for example .33333.....), are real numbers. ACT math uses only Real numbers so you don't have to worry about imaginary numbers, such as I (the square root of -1) both rational and irrational umbers are used in ACT Math. A rational number can be expressed as a fraction consisting of an integer divided by another integer. Expressed as decimal, a rational number is a finite or repeating decimal (for example, 3, .5, .33333 and .32323232). Most numbers on the ACT are rational, but some are irrational for example 3.412....., square root 2 and e...) You probably wont be asked to distinguish between rational and irrational Remember-that rational numbers can be represented by a single fraction of two integers OR as a finite or repeating decimal, and that irrational numbers CANNOT.

Integers

ACT math questions often ask you to "round your answer to the nearest "integer" or "find all integers between 1 & 20 that"... An integer is a number that can be expressed without a decimal point or fraction. For example: 1,3,849 but NOT 2.4, 1/2 or 7 5/8). Negative numbers can also be integers (for example -2, -3, -87)

Ratio, Proportions, and percentages

All are closely linked. Ratio and proportion relate one part of a whole to another part of a whole, while percentage relates one part to the whole itself. =Example: a+b=c; ratio and proportion relate a and b to one another; while percentage relate a OR b to c

Proportion

Applies a ratio to more than on whole. For ex: may state that ratio of blue to green sprinkles on your cupcake is 5:3 and ask you to calculate the number of green sprinkles if there are 800 sprinkles on the cupcake: If there were eight sprinkles, five of them blue and three green you can write a proportion: 3/8=?/800 then cross multiply 3x800=8x so x =300

To Divide Fractions,

Cross multiply the the numerators and denominators that is multiply the first fraction by the reciprocal of second(flip). EX 2/3 divide by 4/5=2/3 x 5/4= 10/12=5/6

DISTRIBUTION

Distribution involves multiplication when there are two or more terms inside parentheses. fir example you can distribute x(y+z) as (xy + xz): here is another example: 3(2+x)= 3(2) + 3(x) = 6 + 3x

Even and Odd Numbers

Even numbers are those evenly divisible by 2; example 2,4,6...; Odd numbers are those NOT evenly divisible by 2 examples 1,3,5,7.... When two even numbers or two odd numbers are added the sum is an EVEN number. When an Even number and an odd number are added, the sum is an ODD number, When two even numbers are multiplied, the product is an EVEN number; When two ODD numbers are Multiplied the Product is ODD; When and even number and an odd number are multiplied, the Product is an EVEN number. Just test in head by using 2 as your even number or 1 as odd.

Factors and multiples

Factors of a number at all the integers that can evenly divide into the number. Put another way, the factors if a number is the set of all integers that can be multiplied by a single integer. Here are factors of 24=1,2,3,4,6,8,12,24 because 1x24=24, 2x12=24 and so on) Factors can be listed in pairs whose products are the number here are factor pairs of 70: 1x70, 2x35, 5x14, 7x10. The Greatest Common Factor: GCF is Often seen in ACT math. The GCF of two numbers is the largest integer that is a factor of both numbers. Example: GCF of 12 & 8 is 4 because 3x4=12 & 2x4=8 See pictures for fractions:

Order of Operations

Mathematical calculations are performed in specific order remember rule: PEMDAS; Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. Do work in that order:

To Multiply Fractions

Multiply the numerators and multiply denominators: 2/7 x 3/4=2x3/7x4=6/28=3/14 (Remember to reduce to lowest form)

Percentage

Must know this: A percentage is expressed as a fraction f 100 for example 50%=50/100 or is PART/WHOLE The question will give 2 quantities and ask you to figure out 3rd. 5/7=.714 x 100=or 71.4% Or to calculate the Part from the whole and percentage, multiply the whole b the percentage (expressed as a decimal) For ex: the question may say professor reads 89% of 27 term papers and asks the number of papers she read: 27 x .89=24 to calculate the whole from part and the percentage, divide the part by the percentage (expressed as decimal). EX: question may state that birch used 52 photos on a project, which represents 6% of all photos she took last year and ask how many photos she took? 52/0.06= 867 photos When problem involves % increase, be aware that the total percentage is the percent increase PLUS 100%; thus a 50% increase means 150% of the original value. When problem involves a % decrease, subtract the percent decrease from 100%; thus a 20% decrease means 80% of original value.

Pre-algebra 23% of Math questions

Number problems: These are word problems that require you to make calculations (Add, subtract, multiply, divide) with integers, decimals and fractions)

Scientific Notation

Scientific notation appears rarely in ACT math, but occurs regularly in ACT Science. It is used to express VERY LARGE numbers and VERY SMALL numbers. i.e. 6,687,567,000,000=6.7 x 1012 ( power of 12) or 0.00000007.3 =7.3 x 10 -7(power of)

Least Common + Multiple (LCM)

The LCM is often used in ACT Math. The LCM is the smallest integer that is a multiple of both numbers. For example, LCM of 12 and 20 is 60, the LCM of 54 & 9 is 54; The LCM is used when adding and subtracting fractions; to calculate LCM of original denominator: EX: 1/8 + 1/12=3/24 + 2/24 = 5/24

Exponents

The exponents of a number indicates how many time the number is to be multiplied by itself. x2=x times x to multiply two terms with exponents, combine the terms by adding the exponents x2 x x3 = x5 only if terms are same. Cannot do this if is a x & y or xy... For negative exponents x-1 is reciprocal of result i.e. 1/x Remember Exponents are NOT distributive. (See picture)

Square Roots

The square root of a number is the number that, multiplied by itself, yields the first number. The square root of x is y that: Y x Y=x or Y2 =x or written as : (Picture)

Ratio

Written several ways: 3:2 like 3 red bricks to 2 green bricks; can also be expressed 3/2 when doing ratio problems Picture: When working on these problems remember to relate one part to another part, even if question states one part to the whole; you will need to calculate the other part Example: Sally flipped a coin 30 times and got 18 heads, then question asks ratio of heads to tails is ? 18:12 NOT 18:30 ACT exam writers like to TRICK you on this so pay attention to what being asked Ratio is comparing one PART to ANOTHER not to the Whole.