TEAS_V_Math

*Perfect Squares*:

1² = 1 2² = 4 3² = 9 4² = 16 5² = 25 6² = 36 7² = 49 8² = 64 9² = 81 10² = 100 11² = 121 12² = 144 13² = 169 14² = 196 15² = 225 16² = 256 17² = 289 18² = 324 19² = 361 20² = 400 25² = 625

*U.S. Measurements*:

*Fluid* +8 ounces = 1 cup +2 cups = 1 pint +2 pints = 1 quart +4 quarts = 1 gallon *Weight* +16 ounces = 1 pound +2000 pounds = 1 ton *Length* +12 inches = 1 foot +3 feet = 1 yard +5280 feet = 1 mile

*Metric Measurements*:

*Mass* 1 gram = 1,000 milligrams 100 centigrams 10 decigrams 1 decagram = 10 grams 1 kilogram = 1,000 grams *Length* +The scale for length is the same as for mass, except the base unit is the meter instead of the gram. So -- 1 meter = 1,000 millimeters 1 meter = 100 centimeters 1 kilometer = 1,000 meters

*Slope Formula*:

+"What direction it's pointed" is known more formally as *slope*. +*rise-over-run* +Starting at one point, you just count how many units *up or down to the other point (the rise)* and write it down as the top of your fraction. (Remember: up means positive, and down means negative). Then count how many units *left or right to get to the other point* (again, left is negative, and right is positive). Write that down as the bottom of your fraction. Now you have rise/run.

*Circles*: +360 degrees total +the measurement from the center to any part of the outside edge is the *radius*. +Double the radius and you get the *diameter* which is the longest measurement across the circle. +Multiply the radius by 2pi and you get the *circumference* which is kind of like a perimeter -- the measurement around the outside.

+*Diameter of a circle*: d = 2r +*Circumference of a circle*: C = 2πr +*Area of a circle*: A = πr²

*Exponents*:

+*Multiplication* -- If a problem has exponents of the *same base* being multiplied, such as 3² x 3⁵, you combine or *ADD* the exponents to get 3⁷. +*Division* -- If a problem has exponents of the same base being divided, such as 4⁷ ÷ 4⁵, you *SUBTRACT* the exponents to get 4². +*Exponents* -- If an exponent is raised to another exponent, such as (3²)⁴, you *MULTIPLY* the exponents to get 3⁸. +Warning - you need to be aware that if you are dealing *with two different bases, you CANNOT combine the exponents in any way*. They cannot be combined because their bases are not the same.

*Multiplying and Dividing Fractions*:

+*Multiplying Fractions*: you just multiply straight across. Try to reduce beforehand if possible. Divide top and bottom to reduce. +Example for *multiplying fractions*: 3/2 x 2/5 = 6/10 = but can be reduced to 3/5. Sometimes you HAVE to wait until after to reduce like this example. +*Dividing Fractions*: you MUST flip the second fraction first and then you can just multiply straight across as you did in multiplying fractions. +Example for *dividing fractions*: 3/10 ÷ 5/4 = Flip the second fraction so it becomes: 3/10 x 4/5. Now you multiply straight across = 12/50 which reduces to 6/25.

*Calculating Hypotenuse and Diagonals*:

+*Pathagorean theorem*: the two legs of a the triangle (or the two sides of a rectangle) are a and b, and the diagonal/hypotenuse is c. +The formula is *a² + b² = c²* +Example: What is the length of a diagonal of a rectangle with sides 4 and 6? → 4² + 6² = c² → 16 + 36 = c² → 52 = c² → √52 = √c² → c = √52 +Another example: Find the length of the base of a triangle that has a height of 4 and a hypotenuse of 5. → 4² + b² = 5² → 16 + b² = 25 →16 - 16 + b² = 25 - 16 → b² = 9 → √b² = √9 → b = 3

*Percentages*:

+*Percent* means *out of one hundred*.

*Rectangles*: +Remember that there are always two pairs of sides, so if you know the length of one side, you know the length of the side directly opposite.

+*Perimeter*: P = 2l + 2w or l + l + w + w +*Area*: A = lw

*Squares*: +All sides are the same length.

+*Perimeter*: P = 4s +*Area*: A = s²

*Triangles*: +b =(base) +h = (height)

+*Perimeter*: P = s₁ + s₂ + s₃ (just add up all the sides) +*Area*: A = 1/2bh (not the hypotenuse). Example: if b x h = 12 the area would be 6 because you take half of 12.

*Prime Numbers*:

+A *prime number* is a number that has exactly two different factors: itself and one. So *prime numbers* CANNOT be negative numbers, 0, or 1. +Some of the smallest *prime numbers* are -- 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. +No prime numbers except 2 are even. So if a number is greater than 2 and even, it's not prime. +Check whether the number is divisible by 3. +A number ending in 0 or 5 cannot be prime.

*Median*:

+A median is simply the number in the *middle* in an ordered list. +For example, if you are asked to find the median of the list 8, 11, 4, 17, and 5, you would start by putting them in numerical order: 4, 5, 8, 11, 17. And then you choose the number in the middle. When you are left with just one number in the middle that is the median. If you have two in the middle, you just take the mean of those two numbers, and that is your median.

*Calculating Percentages Using Fractions or Decimals*:

+A percent is only significant when it is a percent of something else. That means that you are always multilpying the percent times another number. So, if you are calculating 7 percent sales tax on a $20 purchase, you multiply 7% x 20. You do that by converting the percentage to either a fraction or a decimal, whichever you prefer, and multiplying out: 0.07 x 20 = 1.4 You end up with 1.4 or $1.40 in sales tax. +Another example: You find a shirt on clearance and it is marked at 75% off the original price of $40. How do you find 75% of $40? With decimals - that is 0.75 x 40 = 30. So 75% of $40 is $30.

*Roots*:

+Are the opposite of exponents. +You know that 5² = 25, and therefore √25 = 5. We call 25 a *perfect square* because it is the result of squaring a single integer.

*From decimals to percents*:

+Do the opposite; move the decimal two places to the right and add a percent sign. 0.345 becomes 34.5% 0.6 becomes 60% 2.6 becomes 260%

*Adding and Subtracting Fractions*:

+Example for *adding fractions*: 1/2 + 1/3 +Try to reduce first, if possible. Have to divide top and bottom to reduce. +The first step is to multiply 3 x 1 = 3 and multiply 2 x 1 = 2. Then you add them together which = 5. That will be the numerator. For the denominator you take the denominators 2 and 3 and multiply them. = 5/6. +Example for *subtracting fractions*: 5/8 - 3/5 +Multiply 5 x 5 = 25 and 8 x3 = 24. Then you minus those which = 1 and that becomes the numerator. Then you take the denominators 8 and 5 and multiply them which = 40. That becomes the denominator = 1/40

*Translating Word Problems Involving Percents*:

+Example: 5 is what percent of 4? 5 = x% x 4 which then will be 5 = x/100 x 4 which is the same as 5 = 4x/100. Now you can just solve the equation 5 = 4x/100. +Multiply each side by 100, which gives you 500 = 4x. Then divide each side by 4, which gives you 500/4 = 4x/4 And solve: 125 = x +Try another: 80 percent of what number is 4? 80 percent × x = 4 80/100 × x = 4 which is the same as 80x/100 = 4. Multiply each side by 100 to get 80x = 400. Then divide each side by 80 to get 80x/80 = 400/80. And solve: x = 5.

*Factors*:

+Factors are sets of numbers that other numbers can be divided by. +For example, if you are asked to find the factors of 12, you draw a factor T, like this: 12 ---- | +The first pair of factors in the factor T should be 1 and the number itself: 12 ---- 1|12 +Then continue counting and fill in a pair every time you find a number that the original number is divisible by. 12 ---- 1|12 2|6 3|4

*Algebra* and *solving for x*:

+Get x alone. +You want to end with a statement that says x equals this. +Whatever you do to one side of the equation you have to do the same thing to the other side. +If a variable has a number added to it, you subtract to get rid of the number. +If a variable is being divided by a number, multiply to get rid of the number. +Example: 2 + x = 6 2 - 2 + x = 6 - 2 0 + x = 4 which is x = 4 +Another example: 4x = 12 4x/4 = 12/4. So x = 12/4 which means x = 3.

*Another Algebra Word Problem*:

+If 2 less than 3 times a certain number is 13, what is the value of that number? +Translation: 3× - 2 = 13 3× - 2 + 2 = 13 + 2 3× = 15 3×/3 = 15/3 × = 5

*From a ratio to a decimal or percent*:

+If a problem asks you to convert from a ratio to a decimal or a percent , it is the exact same as converting from a fraction; just take the ratio and change it into a fraction. 3:2 becomes 3/2 and then divide to find the decimal and the percent.

*Fahrenheit/Celcius Conversions*:

+To convert Fahrenheit to Celcius: subtract 32 and then multiply by 5/9. +Example: to convert 100 degrees Fahrenheit to Celcius: (100 - 32) x 5/9 68 x 5/9 68 x 5 = 340 /9 = approximately 37.8 Celcius. +To convert Celcius to Fahrenheit: multiply by 9/5 and then add 32. +Example: to convert 20 degrees Celcius to Fahrenheit: 20 x 9/5 + 32 20 x 9 = 180/5 + 32 = 36 + 32 = 68 Fahrenheit.

*Decimals*:

+In *addition* and *subtraction* you just line the decimals up and add. +In *multiplying* you do not have to deal with the decimals until afterwards when you need to remember to add them back on to the answer at the end. *Multiplying example*: 14.6 x 2.1. 146 x 21 ------ 3,066 --- to add back the decimals. Go back to your original numbers 14.6 and 2.1 and count the number of digits after the decimal. There is one in each number, so a total of two. That means your answer has to have two digits after the decimal. 3,066 becomes 30.66. +In *dividing decimals, you can move them ahead of time so you don't have to worry about them afterwards. +*Dividing example*: .10 ÷ .5 So move the decimal in .5 one over, so it becomes just 5. Then you have to change the 10. The decimal is at the end of the 10, so move that one, which would equal 100. So it comes out to be 100 ÷ 5.

*Calculating Roots*:

+Make a prime factor tree. +Example: Start with 200. Try to think of a perfect square that would go into the number (200). 200 2 100 +100 is a perfect square = 10. So 10 goes outside because it can stand alone and 2 goes inside = 10√2

*Mean*:

+Mean is basically another word for *average*. +You are generally given a list of numbers -- such as 8, 11, 4, 17, and 5 -- and asked to find their mean. +Add up the list of numbers: 8 + 11 + 4 + 17 + 5 = 45. Then divide by the number of numbers in the original list, which in this case is 5. The expression 45/5 = 9.

*From decimals to fractions*:

+Move the decimal two places to the right and make the number the numerator of a fraction, with 100 on the bottom. Reduce and you are done. 0.75 becomes 75/100 which reduces to 3/4 0.60 becomes 60/100 which reduces to 3/5

*Percentages, like the bill at a restaurant*:

+One more example: The bill at a restaurant is $80 and you want to leave a 15 percent gratuity. What is 15% of 80? With decimals, that is 0.15 x 80 = 12 which means you are leaving a $12 tip. +By the way, what if the problem asked you to calculate the entire bill, including gratuity? you would just add $12 to the $80 you already had, giving you a $92 total.

*Order of Operations*:

+PEMDAS 1) Parantheses 2) Exponents 3) Multiplication & Division (from left to right) 4) Addition & Subtraction (from left to right)

*More about Circles and Pie Charts*:

+Pie charts are circles, and like all circles, they have 360 degrees. They are usually divided into "slices," with the size of each slice dependent on what percent of the whole the slice represents. So each slice has a certain degree measure, based on that percentage. +Some questions ask you to calculate how many degrees a certain slice encompasses (although the questions may refer to a slice as a "central angle" instead). +For example: If a family's monthly expenses are shown in a pie chart and their mortgage encompasses 40 percent of their monthly expenses, how many degrees is the central angle representing their mortgage payment? +*To solve this question, just take the percentage that the slice represents and multiply it by 360*. → 40% x 360 = 40/100 x 360 = 40 x 360/1 = 144

*From percents to fractions*:

+Remove the percent sign and make the number the numerator of a fraction, with 100 on the bottom. Reduce and you are done. 65% becomes 65/100 which reduces to 13/20 120% becomes 120/100 which reduces to 6/5

*Translating Algebra Word Problems*:

+Some algebra problems really just involve translating from English into math, like with percentages. +*Of* translates into *multiplication x* +*Is* ( as well as has, have, do, does, equals means *=* +*Number* translates to *×* +*More than* means you need to *add +* +*Less than* means you need to *subtract the number BEFORE the less than*. +*Divided by* or *multiply by* means you need to *divide* or *multiply*.

*Combining Terms in Algebra*:

+Some algebra problems will not ask you to solve for x at all; in fact, solving for x will not even be possible. These problems just ask yuo to simplify as much as possible. For example: 4× - 2× + 3× = First take 4 - 2 = 2 + 3 = 5 which means the answer is 5×. +All you have to do is combine terms that are ALIKE. If there are terms that are not alike, you cannot combine them. +Example: 4× - 2× + 3× + 6 which would equal 5× = 6. +Even when a problem requires solving for ×, you may still need to combine terms to do it: For example: If 2× + 3× = 15. Combine 2 + 3 which equals 5× = 15. Then you divide both sides by 5. 15/5 which means × = 3.

*Prime Factors*:

+Sometimes you will need to calculate the *prime factors* of a number, which are all of the prime numbers that are multiplied together to create that number. +To find *prime factors*, it helps to make what is called a factor tree. +Start with 60. +Then you need to determine whether it is divisible by the first prime number, which is 2. If it is, draw two branches leading from the number, one for 2 and one for the result of 60/2 which is 30. 60 2 30 +The two is there to stay, and now you are going to do the same for 30. It is divisible by 2, so you draw a branch for 2 and a branch for 30/2 which is 15. 60 2 30 2 15 +The two is not divisible by 2, so you move on to the next prime number, which is 3. But the 15 is divisible by 3, so you draw a branch for 3 and a branch for 15/3 which is 5. 60 2 30 2 15 3 5 +Now you are done, because the tips of the branches are all prime numbers: 2, 3, and 5. So the prime factors of 60 are 2, 2, 3, and 5, and therefore, 2 x 2 x 3 x 5 = 60.

*From percents to decimals*:

+Take off the percent sign and move the decimal two places to the left. 75% becomes 0.75 500% becomes 5.0

*Place Value*:

+Take the number 36,827. +There is a 7 in the ones place. +A 2 in the tens place. +An 8 in the hundreds place. +A 6 in the thousands place. +And a 3 in the ten thousands place.

*Mode*:

+The mode is the number that appears *most frequently* in the list. +Let us take this list: 11, 8, -4, 11, 4, 17, and 5. First, carefully put the numbers in order. Remember, a negative number never has the same value as a positive. +Your ordered list looks like: -4, 4, 5, 8, 11, 11, 17. So the mode is 11. +If you have a list such as -4, 4, 5, 5, 8, 11, 11, 17 then 5 and 11 are the modes.

*Mixed Numbers*:

+To add, subtract, or otherwise manipulate them, you must convert them to improper fractions first. +Example: 3 5/6 -- keep 6 as the denominator. Then you multiply 3 x 6 = 18. Then add on the numerator (5) to the 18 = 23. +To do the opposite and make an improper fraction into a mixed number, take the example of 8/5. Then you divide the 8 by the 5. 8 ÷ 5. 5 goes into 8 once so bring down the 5 under the 8 which equals 3. Keep the denominator 5 in the example as the denominator in the mixed number as well. So = 1 3/5. (I am using the radical here as a dividing sign.) 1 5 √ 8 - 5 --------- 3

*From fractions to decimals*:

+Treat the fraction as a division problem; actually move the numbers in a division sign and work the math. This time you do not have to convert to percents, so you are done! 1/4 becomes 0.25 11/20 becomes 0.55

*From fractions to percents*:

+Treat the fraction as a division problem; actually move the numbers into the division sign and work the math. Then move the decimal two places to the right and add a percent sign at the end. 4/5 becomes 0.8 or 80% 3/8 becomes 0.375 or 37.5%

*Absolute Value*:

+Whenever you see a number with vertical lines on either side of it, such as |-3|, you are being asked to figure out the *absolute value* of that number. +*Absolute values* are *positive numbers* only, so the absolute value of -3 is 3. Or, to put it in math terms, |-3| = 3.

*Comparing Fractions*:

+to find out which fraction is larger or smaller you can put them side by side = 3/8 and 1/4. Then you multiply 3x4 = 12 and 8 x 1 = 8. So 3/8 is the larger fraction.

*Percent, Fraction & Decimal*:

5% - 1/20 - 0.05 10% - 1/10 - 0.1 or 0.10 15% - 3/20 - 0.15 20% - 1/5 - 0.2 or 0.20 25% - 1/4 - 0.25 30% - 3/10 - 0.3 or 0.30 35% - 7/20 - 0.35 40% - 2/5 - 0.4 or 0.40 45% - 9/20 - 0.45 50% - 1/2 - 0.5 or 0.50 55% - 11/20 - 0.55 60% - 3/5 - 0.6 or 0.60 65% - 13/20 - 0.65 70% - 7/10 - 0.7 or 0.70 75% - 3/4 - 0.75 80% - 4/5 - 0.8 or 0.80 85% - 17/20 - 0.85 90% - 9/10 - 0.9 or 0.90 95% - 19/20 - 0.95 100% - 1/1 or 1 - 1 or 1.0 or 1.00 12.5% - 1/8 - 0.125 37.5% - 3/8 - 0.375 62.5% - 5/8 - 0.625 87.5% - 7/8 - 0.875

*Roman Numerals*

I - 1 V - 5 X - 10 L - 50 C - 100 D - 500 M - 1,000

*Algebra Word Problems*:

If 8 more than a certain number is the same as three times that number, what is the number? 8 + × = 3× And then solve. The most direct way to get × alone is to remove the × from the left side. 8 + × - × = 3× - × 8 = 2× 8/2 = 2×/2 4 = ×