Arithmetic and Fractions
n/n
n/n always equals 1. Doesn't matter how ugly a. 8/8 = 1 b. 0.045/0.045= 1 c. (3^3)/(3^3)= 1
(-46) - 37 (-22) + (-61) (-25) + (-41) (-61) - 11
(-46) - 37 = -(46 + 37) = -83 (-22) + (-61) = -(22 + 61) = -83 (-25) + (-41) = -(25 + 41) = -66 (-61) - 11 = -(61 + 11) = -72
(54x56x72)/(64x45x60)=
(6x56x72)/(64x5x60)= (6x56x9)/(8x5x60)= (1x56x9)/(8x5x10)= (7x9)/(5x10)= 63/50 (answer) Note: Always cancel before your multiply.
-(1/5): (different looks)
-1/5 = 1/-5 = -(1/5) Note: the negative sign can move about
Zero Product Property
If the product of two number is zero, one of the factors MUST BE zero. If ab = 0, then a = 0 OR b = 0
0.0013/0.025
0.0013/0.025 = 0.013/0.25 = (0.013x4)/(0.25x4) = 0.052/1 = 0.052 Note: If you can easily times the denominator by something to make it 1, do it as soon ass you can (make sure to times numerator and denominator by the same number).
Translating words to math
1. 'is' means equals 2. 'of' means multiply Example: What is 3/5 of 400 = (3/5)x400
AxB/CxD (different combinations. Remember Division and Multiplication is at the same level)
1. = (B/CxD)xA 2. = (A/D)x(B/C) 3 =[(A/C)xB]/D Note: ALWAYS choose to cancel before you multiply
Multiplication of Decimals Procedures
1. Count the number of digits to the right of the decimal point: 6.25x0.048 2. The first factor has 2 dec. places and the second has three. Add those two: the product will have 2+3 = 5 decimal places. 3. Ignore the decimal, and find the product of the two positive integers: 625x48
Multiplying and Dividing positive and negative numbers procedures
1. Determine the sign of the product/quotient 2. Treat both factors as positive, and perform the mult. or division. 3. Give the result the appropriate sign.
GEMDAS (Order of operations)
1. Grouping symbols 2. Exponents 3. Multiplication & Division (same level) 4. Addition & Subtraction (same level) Note: Always work from the inside out if there are multiple layers of parentheses/grouping symbols.
Fractions (properties 1)
1. If a>b, then (a/c) > (b/c) (4/13 > 3/13) 2. Bigger denominators with same numerators make smaller fractions (2/5 > 2/7) 3. If the numerator gets bigger and the denominator gets smaller, the fraction gets bigger (3/8 < 4/7) 4. Cross multiply to decide if two random fractions are bigger.
Possible forms of a fraction that is greater than one.
1. Improper fraction: numerator > denominator 2. Mixed numeral: integer part + fraction part Note: the mixed numeral represents an addition relationship, NOT a multiplication relationship Note 2: Usually better to use improper fractions on the test, but not always.
When to use mixed numeral or improper fractions.
1. Used mixed numerals to locate a number on the number line. 2. For adding and subtracting (doesn't really matter) 3. For multiplication, division, and exponents ALWAYS use improper fractions.
Fraction Properties II (takeaway)
1. We CANNOT separate a fraction into two fractions by addition or subtraction in the denominator (e.g. a/(b+C) IS NOT a/b + a/c) 2. 1 does not hold true for addition and subtraction in the numerator (e.g. (a+b)/c IS (a/c + b/c) 3. If we have addition and subtraction in both the numerator and denominator, the numerator can be split up, but the denominator must stay the same--(a+b)/(c+d)= [a/(c+d)]+[b/(c+d)].
Operations with proportions (takeaway) Note: proportions have an equal sign in the middle
1. We can get rid of proportions through cross multiplication (5/7 = 3/x; 5x=21; x=21/5) 2. For proportions with larger numbers we should try to cancel first. However, the rules for cancelation differ, so be careful. diagonal cancellation IS ILLEGAL...DON'T DO IT. Not the same as multiplication. 3. Horizontal and Vertical Cancelations are okay.
Positive/Negative sign rules for multiplication
1. positive x positive = positive 2. negative x negative = positive 3. postive x negative = negative
Postive/Negative sign rules for division
1. positive/positive = positive 2. negative/negative = positive 3. positive/negative = negative (any order)
"Comparing Fractions II (Advanced)" Takeaway (Review the lesson again to crystalize).
1.. If we start with a proper fraction and add the same number to both the numerator and the denominator, that resultant fraction is closer to 1. 2. If we start with a fraction, and add p to the numerator and q to the denominator, that resultant fraction is closer to p/q
12/5x = 8/15 solve for x
12/5x = 8/15; 3/x = 2/3; 9=2x; x=9/2
256 (Place Values)
2 hundreds 5 tens 6 ones 2x100 + 5x10 + 6x1
39.0625 (Place Values)
3 in the tens place (10) 9 in the ones place (10) 0 in the tenths place (1/10) 6 in the hundredths place (1/100) 2 in the thousandths place (1/1,000) 5 in the ten thousandths place (1/10,000)
4+3/5(mixed numeral to improper example)
4 + 3/5 = 4x5/5 + 3/5 = 20/5 + 3/5 = 23/5
6/42 (factoring out example)
6/42 = (6x1)/(6x7) = 1/7 Note: canceling has been done here. It is a form of division. Note: on the text always write fractions in the simplest form. (The answer choice will almost always be in simplest form in MC)
Real Number
A real number is any number on the number line. This includes round numbers as well as fractions, decimals, and negative numbers.. NOTE: "number" always means "real number" on the test.
Integer
All positive & negative WHOLE numbers, including zero
Absolute Value sample problem: Consider the positive integers from 1-100. If n is a number in that set, then for how many numbers n is it true that |n-30| > 20?
Answer is 59
**Practice Problem QA: 147/200 QB: 150/203
Answer: Quantity B is bigger Principle: If you add the same number to both the num. and dem. of a proper fraction, the result is bigger. However, if you add the same number to an improper fraction, the result is smaller. (Very easy for comparison in some situations).
a. -6 x -7 b. -65/5 c. -30/-12
Answers: a. 42 b. -13 c. 5/2
Equivalent fractions
Fractions that have the same numerical values, but may differ with respect to their numerators and denominators. 2/3 = 10/15
0.56/0.0007 (dividing by decimals examples)
Note: slide decimal until denominator is an integer a. 0.56/0.0007 = 5.6/0.007 = 56/0.07 = 560/0.7 = 5600/7 = 800
Numerator vs Denominator: 3/16
Numerator: top 3 Denominator: bottom 16
Rounding module takeaway
Only look immediately to the right of the place value you are rounding. Look no further (e.g.3.14159 rounded to the nearest thousandths is 3.142 & 59,049 rounded to the nearest hundreds place is 59,000). DO NOT double round.
Absolute value
The absolute value of a number gives the distance of the number from an origin (written as positive). 0 is the exception. So, a. |x| = the distance of x from the origin. b. |x-5| = the distance of x from +5 c. |x+3| = the distance of x from -3
the number zero
The only number that is neither positive nor negative.
Reciprocal of a fraction
The reciprocal of a fraction, a/b, is the flipped over fraction, b/a (a not 0, and b not 0) 1. the product of any fraction with its reciprocal is 1 ((4/17)x(17/4)= 1). 2. The reciprocal of a positive integer is one divided by that integer (6 is 1/6) 3. One divided by any fraction equals the reciprocal of that fraction (1/(3/7)= 7/3 4. If a number is bigger than 1, then its reciprocal is smaller, between 0 and 1. If a number is between 0 and 1, its reciprocal is larger than 1
[27(y+5)(2y-2)]/[2(y-1)]= canceling algebraic expressions
[27(y+5)(2y-2)]/[2(y-1)]= [9(y+5)(2y-2)]/[(2y-2]= [9(y+5)x 2(y-1)]/2(y-1)= 9(y+5)
Distributive Property
a(b+c) = ab + ac Or a(b-c) = ab - ac
a. (1/4)+(2/3) b. (3/5)-(1/10 c. (5/6)+(1/4)
a. (1/4)+(2/3)= (3/12)+(8/12)= 11/12 b. (3/5)-(1/10)= (6/10)-(1/10)= 5/10= 1/2 c. (5/6)+(1/4)= (10/12)+(3/12)= 13/12
a. (1/4)x(8/13)
a. (1/4)x(8/13)= 1x(2/13)= 2/13 Note: Always cancel before you multiply.
a. (7/10)/(7/15)= b. (24/35)/(25/36)= c. (8/9)/6=
a. (7/10)/(7/15)= 3/2 b. (24/35)/(25/36)= 10/21 c. (8/9)/6= 4/27 Note: Always cancel before you multiply
(0.03)^3 (decimal multiplication example)
a. 0.03^3 = .03x.03x.03 b. (2 + 2 + 2 = 6 decimal places) c. 3^3 = 27, so the 7 must land six places to the right of the decimal d. 0.000027 (answer)
a. 0.1 b. 0.01 c. 0.001
a. 0.1 = one tenth = 1/10 = 10^-1 b. 0.01 = one hundredth = 1/100 = 10^-2 c. 0.001 = one thousandth = 1/1000 = 10^-3
a. 1/10 b. 1/100 c. 1/1000 d. 1/20
a. 1/10 = 0.1 b. 1/100= 0.01 c. 1/1000= 0.001 d. 1/20= (1/20)x(5/5)= 5/100= 0.05
Note: Memorize every decimal form of a fraction with a single digit denominator--there are gaps because some fractions reduce down: a. 1/2 b. 1/4 c. 3/4
a. 1/2 = 0.5 b. 1/4 = 0.25 c. 3/4 = 0.75 q. 1/9= 0.11111... (trend for every fraction with 9) r. 2/9= 0.2222...
a. 1/3 b. 2/3 express as decimals
a. 1/3= 0.33333 b. 2/3= 0.66667
a. 1/40 b. 1/600 (write in decimal form)
a. 1/40= (1/4)x(1/10)= (0.25)(0.1)= 0.025 b. 1/600= (1/6)x(1/100)= (0.16666..)(0.01)= 0.00166..) Note: extrapolate for all multiples of 10
a. 1/5 b. 2/5 c. 3/5 d. 4/5 express as decimals
a. 1/5= 0.2 b. 2/5= 0.4 c. 3/5= 0.6 d. 4/5= 0.8
a. 1/6 b. 5/6 c. 1/7 d. 2/7 e. 3/7 f. 4/7 g. 5/7 h. 6/7 express as decimals
a. 1/6= 0.1667 b. 5/6= 0.8333... c. 1/7= 0.143 d. .285 e. 0.428 f. 0.571 g. 0.714 h.0.857
a. 1/8 b. 3/8 c. 5/8 d. 7/8 express as decimals
a. 1/8= 0.125 b. 3/8= 0.375 c. 5/8= 0.625 d. 7/8= 0.875
a. 1/9 b. 2/9 express as decimals
a. 1/9= 0.1111.... b. 2/9= 0.2222 Note: trend for every fraction with 9
a. 11 - 78 b. 47 -65 c. 28 - 43 d. 62 - 74
a. 11 - 78 = -(78 - 11) = -67 b. 47 -65 = -(65 - 47) = -18 c. 28 - 43 = -(43 - 28) = -15 d. 62 - 74 = -(74 - 62) = -12
a. 1235/100 b. 0.064x10^-2 c. 37.5/10000 d. 64,000x0.0001 e. 5.4x 10^-5 f. 20.25/10^-6
a. 1235/100 = 12.35 b. 0.064x10^-2 = 0.00064 c. 37.5/10000 = 0.00375 d. 64,000x0.0001 = 6.4 e. 5.4x 10^-5 = 0.000054 f. 20.25/10^-6 = 0.00002025
a. 24/10 b. 0.02/10 c. 39.85 X 0.1 d. 0.00072 x 0.1
a. 24/10 = 2.4 b. 0.02/10 = 0.002 c. 39.85 X 0.1 = 3.985 d. 0.00072 x 0.1 = 0.000072 Note: When we divide any number by ten, or multiply by .1, we move the decimal point one place to the left.
a. 24/10 b. 0.02/10 c. 39.85 x 0.1 d. .00072 x 0.1
a. 24/10 = 2.4 b. 0.02/10 = 0.002 c. 39.85 x 0.1 = 3.985 d. .00072 x 0.1 = 0.000072
a. 24x10 b. 2.53x10 c. 6400x10 d. 0.00045x10
a. 24x10 = 240 b. 2.53x10 = 25.3 c. 6400x10 = 64,000 d. 0.00045x10 = 0.0045
a. 350x100 b. 0.01728x1000 c. 8.3 x 10^6
a. 350x100 = 35,000 b. 0.01728x1000 = 17.28 c. 8.3 x 10^6 = 8,300,000
47 + 36
a. 40 + 30 = 70 b. 7 + 6 = 13 c. 70 + 13 = 83 (Rule: You can simplify addition of two digit numbers by treating the digits separately).
47 + 36 (mental addition 2 digit example)
a. 40 + 30 = 70 b. 7 + 6 = 13 c. 70 + 13 = 83 (Rule: You can treat the digits separately in addition if two digits)
Reciprocal practice problem: The reciprocal of a positive number times the cube of the same number equals 5. What is the number?
a. 5=(1/x)(x)(x)(x) b. 5 = (1)(x)(x)--one x has canceled out c. 5 = x^2 d. square root of 5 = x (answer)
Practice Problem QA: 6/200 QB: 7/235
a. 6/200= 3/100= 1/33.3 b. we have to add 1/35 to reach quantity B c. 1/35 is smaller than 1/33.3, so adding 1 to the num. and 35 to the denom. will decrease the ratio. Thus, quantity A is larger. Note: when the ratio decreases the original fraction is the largest. When the ration increases, the other option is the largest.
a. 83 -17 b. 40 - 18 c. 52 - 27 d. 71 - 15
a. 83 -17 = 86 - 20 = 66 b. 40 - 18 = 42 - 20 = 22 c. 52 - 27 = 55 - 30 = 25 d. 71 - 15 = 76 - 20 = 56 (Rule: simplify subtraction by adding the same number to both terms)
9/20 ?? 4/9 (use cross multiplication)
a. 9x9 = 81 ?? 4x10=80 b. 81 > 80 c. 9/20 > 4/9 (answer)
Practice word problem with fractions Cathy's salary is 3/7 of Nora's salary and is 5/4 of Teresa's salary. Nora's salary is what fraction of Teresa's salary?
a. C= (3/7)N b. C= (5/4)T c. (3/7)N = (5/4)T d. N= (7/3)x(5/4)T e. N= (35/12)T
**QA: 449/150 QB: 20/7 (number sense)
a. They are both almost three b. (450/150) - (1/150) vs. (21/7) - (1/7) c. If you subtract something smaller you get something bigger d. 1/150 < 1/7 e. A is bigger
**|x-1| > 4 (think number line)
the distance between x and +1 is grater than +4 Or X < -3 OR x > 5 = |x-1| > 4
{[5+(5/8)]/[4+(1/2)]}=
{[5+(5/8)]/[4+(1/2)]}= (45/8)/(9/2)= (45/8)x(2/9)= (5/4)x1= 5/4= 1+(1/4) (answer)