Math 1321-Arithmetic Test

Decimals-Read and Write Decimal Numbers

"And" gets used to say the decimal point. Ex.) 4.73 = 4 and 73 hundredths. 4.603 = 4 and 603 thousandths.

2*5

10

5*2

10

12*9

108

9*12

108

10*11

110

2*6

12

3*4

12

4*3

12

6*2

12

12*10

120

11*11

121

12*11

132

2*7

14

7*2

14

12*12

144

3*5

15

5*3

15

2*8

16

4*4

16

8*2

16

2*9

18

3*6

18

6*3

18

9*2

18

2*1

2

Whole Numbers-First 20 Prime Numbers

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71.

2*10

20

4*5

20

5*4

20

3*7

21

7*3

21

2*11

22

12*2

24

2*12

24

3*8

24

4*6

24

6*4

24

8*3

24

5*5

25

3*9

27

9*3

27

4*7

28

7*4

28

3*10

30

5*6

30

6*5

30

4*8

32

8*4

32

3*11

33

5*7

35

7*5

35

12*3

36

3*12

36

4*9

36

6*6

36

9*4

36

Whole Numbers-First 20 Composite Numbers

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32.

4*10

40

5*8

40

8*5

40

6*7

42

7*6

42

4*11

44

5*9

45

9*5

45

12*4

48

4*12

48

6*8

48

8*6

48

5*10

50

6*9

54

9*6

54

5*11

55

7*8

56

2*3

6

3*2

6

12*5

60

5*12

60

6*10

60

7*9

63

9*7

63

6*11

66

7*10

70

12*6

72

6*12

72

8*9

72

9*8

72

7*11

77

2*4

8

4*2

8

8*10

80

9*9

81

7*12

84

8*11

88

3*3

9

9*10

90

12*8

96

8*12

96

9*11

99

Fractions-Add/Subtract Fractions including Unlike Denominators

ADD with LIKE denominators- The denominators stay the same, only the numerators are being added. Remember-Denominators the SAME-KEEP in the game! ADD up the TOP, simplify, and stop! Ex.) 1/5 + 3/5 = 4/5. SUBTRACT with LIKE denominators-The denominators stay the same, only the numerators are being subtracted. Ex.) 8/9 - 7/9 = 1/9. Add/Subtract with UNLIKE denominators- In order to either of these operations the denominators must be the same, so if they aren't we must make them. In order to do this, we must find the LCM of the denominators. Addition Ex.) 2/5 + 1/4 We know that the LCM of 5 and 4 is 20 because 5*4 is 20. So, to solve this problem we have to multiply each part of each fraction by either 4 or 5 in order to get the denominators to 20. 2/5= 2*4/5*4= 8/20 1/4= 1*5/4*5= 5/20 So, now as our new problem to solve, we have 8/20 + 5/20. 8/20 + 5/20 = 13/20. 13/20 is our solution. Subtraction Ex.) 4/7 - 1/3 4*3/7*3= 12/21 1*7/3*7= 7/21 12/21 - 7/21 = 5/21

Fractions-Divide Mixed Numbers

Change the mixed number into an improper fraction and then multiply the first number by the reciprocal of the second and then change the answer back into a mixed number if necessary. Ex.) 5 1/2 / 3 2/3 = 11/2 / 11/3 = 11/2 * 3/11 = 3/2 = 1 1/2

Fractions-Multiply Mixed Numbers

Change the mixed number to an improper fraction, cross cancel if possible, and then multiply straight across per usual. If the answer is an improper fraction then change it into a mixed number. Ex.) 1 1/2 * 2 1/5 = 3/2 * 11/5 = 33/10 = 3 3/10

Fractions-Divide Improper Fractions

Change the second number of the equation into the reciprocal and cross cancel if possible and then change into a mixed number. Ex.) 4/5 / 2/1 = 4/5 * 1/2 = 2/5 * 1/1 = 2/5

Applied-Calculate Taxes as Percents of a Number

Cost of the item * percentage of tax as a decimal = Sales tax

Whole Numbers-Division Tricks

Divide by: 1-Answer will always be the dividend. Ex.) 1/8=8 2-If the last digit of the number is even, then the entire number is divisible by 2. Ex.) 2/16=8. 3-Add up the sum of all the digits in the number, if the sum is divisible by 3, then the entire number is divisible by 3. Ex.) 12-1+2=3, 12/3=4. 4-If the last two digits of a number are divisible by 4, then the whole number is divisible by 4. Ex.) 14237732 can be divided by 4 because 32/4=8. 5-If the last digit is a 0 or a 5. Ex.) 100/5=20. 6-If the rules for 2 and 3 are true, then the number can also be divided by 6. Ex.) 402 can be divided by 6 because the rules for 2 (Ends in even) and the rules for 3 (Adds up to sum that's divisible by 3) apply. 7-NOTHING, need to know. 8-If the last three digits are divisible by 8. Ex.) 1792 is divisible by 8 because 792/8=99. 9-Add up the sum of all the digits in the number, if the sum is divisible by 9, then the entire number is divisible by 9. Ex.) 18332145=27 and 27/9=3. 10-If the number ends with a 0 then it is divisible by 10. Ex.) 100/10=10 11-11 must divide the difference of the sum of the digits whose place values are odd powers of 10 and the sum of the digits whose place values are even powers of 10. Ex.) 385627- 8+6+7=21 and 3+5+2=10 so 21-10=11, therefore 385627 is divisible by 11.

Fractions-Add/Subtract Mixed Numbers (May Require Regrouping)

Ex.) 2 3/4 + 3 1/2 ~First, you need to convert the mixed numbers into improper fractions. 2 3/4 = 11/4 3 1/2 = 7/2 ~Next, you need to find the LCM between the denominators and multiply it out to the numerator and denominator. 11/4 = 11/4 7/2= 7*2/2*2= 14/4 ~Now, add both of the fractions since they have a common denominator. 11/4 + 14/4 = 25/4 ~Finally, convert your solution back to a mixed number in order to get your final solution. Divide out the fraction, the quotient is the whole number and the remainder is the numerator, the denominator stays the same. 6 1/4 is the solution. These SAME STEPS are done for SUBTRACTION. ~Another way to do this is to just write them one over the other, solve for the fractions, then add the whole numbers.

Percent's-Given Two of Part, Whole, Percent, Find the Other

First, look at the information and see which numbers you're given and which numbers you're trying to find. Ex.) 18 is 20% of what number? *Trying to find the whole* Next, set up your proportion. The format is: Percent/100 = Part/Whole or Is/Of = %/100. Ex.) 20/100 = 18/X Then, cross multiply. Ex.) 20*X and 100*18 so you get 20x = 1800 Finally, divide by the number next to x into both sides to get x by itself and get your answer. Ex.) 20x/20 and 1800/20 so you get x = 90 So, this means that the whole number is 90, or 18 is 20% of 90.

Applied-Calculate Discounts and Sale Prices

First, look at the original price of the item. Ex.) $90 shirt. Next, know the discount percentage. Ex.) The shirt is 20% off. Then, calculate 20% of $90. Ex.) Set up a proportion so 20/100 = X/90 then cross multiply so 20 * 90 = 180 and 100 * x = 100x. Then, divide out 100 to get x alone and get $18. *To do this you can also do 100-20 = 80 then multiply 90 by 0.8 and get your answer* Finally, subtract the $18 from the $90 and get $72 as the total that you would pay. *discount price=original price−original price⋅discount *dis. = orig. * (1-discount)

Percents-Convert Percent to Decimal

First, put the decimal over 100 as the word percent literally means "per hundred". Ex.) 59.2% = 59.2/100 Next, essentially divide the decimal by 100, or in simpler terms just move the decimal point over 2 spaces and get your final answer. Ex.) 59.2/100 = 0.592 Tip~Decimal goes to the left

Percents-Convert Percent to Fraction

First, put the percentage over 100. Ex.) 75% = 75/100 Next, reduce the fraction if possible. Ex.) 75/100 = 75/25 = 3 and 100/25 =4 The reduced fraction will be your answer. Ex.) 3/4 *If it is not possible to reduce like 7% which would be 7/100, then your answer would just be 7/100.*

Ratios & Proportions-Solve a Proportion

First, state the ratio as a fraction. Ex.) Solve for X. 6/15 and X/20 Then, set them equal to each other. Ex.) 6/15 = X/20 Next, cross multiply. Ex.) 6*20 = 120 and 15*x = 15x so 15x = 120 Finally, divide out the number next to the X and get your final answer. Ex.) 15x/15 = 120/15 X = 8

Decimals-Convert Decimal to Percent

First, turn the decimal into a fraction by putting the number over 1. Ex.) 0.601 = 0.601/1 Then, multiply the numerator and the denominator by 100 in order to get a fraction with the denominator of 100 and a changed decimal number. (Shift the decimal point over 2 spaces to the right every time since 100 is to the right 2 places. Ex.) 0.601/1 = 0.601*100 = 60.1 / 1*100 = 100 so 60.1/100 Finally, write the numerator as a percentage to get your final answer. Ex.) 60.1% Tip~Decimal goes to the right.

Decimals-Convert Decimal to Fraction

First, write the decimal point over one. Ex.) 0.75 = 0.75/1 Next, count how many numbers come after the decimal point and multiply that number by 10. The top and bottom both get multiplied by this number. Ex.) 0.75-2 numbers after the decimal point, so, 2*10 = 20. 0.75/1 = 75/100 Then, simplify and reduce if possible. Ex.) 75/100 = 75/5 = 15/5 =3 100/5 = 20/5 = 4 3/4 is the final answer.

Fractions-Convert Fraction or Mixed Number to Percent

Fraction-Divide the fraction, denominator on the outside, numerator on the inside and get a decimal, then multiply that decimal by 100 and that will give you your percent. Ex.) 4/8 = 4 divided by 8 = 0.5 0.5*100 = 50%. Mixed Number-Convert it into an improper fraction, then divide and multiply as you would a normal fraction. Ex.) 2 1/4 = 9/4 = 2.25 = 2.25*100 = 225% OR- 9*100/4 = 900/4 = 225%

Fractions-Convert Fraction or Mixed Number to Decimal-Includes Repeating Decimal and Terminating Decimals

Fraction-Divide the numerator and the denominator. Ex.) 4/5 would be 4 divided by 5. The denominator (5) would be on the outside of the division bar and the numerator (4) would be on the inside. Since 5 is bigger than 4, it cannot be divided by 4, therefore, 4 needs to be turned into 4.0, and then you would figure out how many times 5 can go into 40. 5*8 = 40, so 0.8 would be the answer. Mixed Number-Change the mixed number to an improper fraction and then divide. The beginning of the decimal number will always be the whole number from the original mixed number. Ex.) 3 4/10= 34/10 = long division so 10 is on the outside and 3 is on the inside = 3.4.

Fractions-Put in Lowest Terms

In order to do this you must divide the numerator and the denominator by their greatest common factor. If you are unsure what that is just keep dividing out a number that you know works for both place values until you cant no more, so for example if both the numbers are even but you don't actually know for sure what their GCF is, continue to divide the numbers by 2 until you no longer can. Another trick is to break down both of the numbers into their prime factorization using number trees and then cross off what they have in common, their remainders will give you the answer. Ex.) 20/8 20/4=5 and 8/4=2 so 20/8 becomes 5/2. Prime factorization: 20=5*2*2 and 8=2*2*2 so the common twos cancel and that leave 5/2 as the solution.

Applied-Find Simple Interest

Interest = Rate/100 * Principal Interest = Principal * Rate * Time (Years) Interest = Amount - Principle Ex.) You borrowed $500 and have a 10% interest rate. Find how much you owe after 1 year. I= 10/100 * 500 = 50. So, after 1 year you would end up owing $550 because you need to pay back the $500 you borrowed plus an additional $50 because of interest. Ex.) You borrowed $500 and have a 10% interest rate. Find out how much you owe after 2 and 3 years. I=10/100 * 500 = 50 So after 2 years you would owe $600. (500 + 50 + 50 = 600) and after 3 years you would owe $650. (500 + 50 + 50 + 50 = 650) *You can also set up proportions to solve: (100 is the whole or for every 100 borrowed you owe X.) Principal/100 = Intrest/Rate So like for $6052 borrowed with a 10% interest rate you would set it up as 6025/100 = X/10 and cross multiply so 6025*10 = 100*X which would be 60,250 = 100x. Next, divide by 100 into both sides and get 602.5 as your interest rate.

Decimals-Multiply Decimals

Just line up the two numbers, completely ignoring the decimal points, and multiply per usual. Then count the number of digits after each factor and put the same number of digits behind the decimal for your final answer. Tip: Count back to the decimal point for each number and then add the totals up to see how many places you need to go back for your final answer. Ex.) 9.57(2 palces) * 1.3(1 place) = 12.441 (Go back 3 palces)

Fractions-Multiply Improper Fractions

Multiply straight across per usual and cross cancel if possible. If the answer is an improper fraction then change it into a mixed number. Ex.) 2/3 * 9/4 = 18/12 = 1 6/12

Whole Numbers-Use Order of Operations

P-Parenthesis E-Exponents M-Multiplication D-Division A-Addition S-Subtraction or G-Groupings E-Exponents M-Multiplication OR Division S-Subtraction OR Addition Done Left to Right, which ever comes first between M or D and A or S.

Applied Problems-Multistep Problems

Problems that contain two or more operations that need to be solved. These would be problems that would require using PEMDAS or GEMS to solve. They would look something like this: 12-5+3*2 In word problems you would have to pick out the numbers to use and use the context of the problem to figure out what operation to do.

Whole Numbers-Divide Whole Numbers (Long Division)

Scaffold Method-Subtract out the Divisor until you can't anymore, add up how many times you've subtracted out the divisor to give you the quotient and look at what's left to give you your remainder. Regular-Figure out what number multiplies with the divisor to get to or close to the first numbers in the dividend and then subtract and carry down the next number in the dividend and continue this process till you get a remainder.

Decimals-Add/Subtract Decimals

The first and most important step to doing this is to always line up the decimal points first before doing any solving. If the numbers are not going to line up, like 3.5 + 36.45, just add 0's in the missing place values in order to have the decimals line up. Also when lining up the numbers, make sure to line up the same place values, so the tenths would be lines up the thousandths would be lines up and so on and so forth. Ex.) 13.492 + 7.8 = 13.492 + 7.800 = 21.292 13.492 - 7.8 = 13.492 - 7.800 = 5.692

Decimals-Identify Place Values

The order BEFORE the decimal point goes: Millions Hundred Thousand Ten Thousand Thousands Hundreds Tens Ones The order After the decimal point: Tenths Hundredths Thousandths Ten Thousandths Hundred Thousandths Millionths Tip: Goes in reverse order of the whole number values and each place value ends with "ths"-No "Oneths"

Whole Numbers-Work with the Average of a Set of Numbers

The sum of the numbers provided, divided by the total number of values in the set. Ex.) Find the average of the numbers 24, 55, 17, 87, and 100. 24+55+17+87+100=283 283/5=56.6 56.6 is the average of the set.

Fractions-Change Mixed Numbers to Improper Fractions

To change a mixed number to an improper fraction, you will first multiply, then add. So you multiply the whole number to the denominator and then add the solution to the numerator. The denominator remains the same and the numerator is the solution that you just found. Ex.) Change 3 1/5 to an improper fraction. First, multiply 3*5=15. Then add 1 to the 15 and get 16. So the improper fraction (or answer) is 16/5.

Fractions-Change Improper Fractions to Mixed Numbers

To change an improper fraction to a mixed number, you divide the numerator by the denominator. "How many times does the denominator go into the numerator and what remainder do I have leftover?" Ex.) Change 23/8 to a mixed number. First, set up the division problem, num/denom, do its 23/8. Then solve. 23/8= 2 R7. So the mixed number becomes 2 7/8. The denominator stays the same, the quotient is the whole number of the mixed number and the remainder is the new numerator for the mixed number.

Decimals-Divide Decimals-Includes Repeating Decimals

To start move the decimal point to the right as many times as needed to make the divisor a whole number or multiply by 10 to do so. Next move the decimal point the same number of times for the dividend and do your long division per usual. Leave the decimal point in the same spot as the dividend and just bring it upwards for the quotient. Ex.) 0.25/1.03075 = 25/103.075 = 4.123