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GMAT Quantatative > Arithmetic & Fractions > Flashcards

Arithmetic & Fractions Flashcards

1
Q

Real Number

A

ANY number on the number line (positive, negative, fractions, decimals, INCLUDING zero)

Remember that zero is not positive nor negative

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2
Q

Integer

A

All positive and negative whole numbers, including zero.

Does NOT include fractions or decimals

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3
Q

Positive Integer

A

All positive whole numbers

Does NOT include fractions, decimals, or zero

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4
Q

Sum

A

Result of addition

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5
Q

Difference

A

Result of subtraction

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6
Q

Product

A

Result of multiplication

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7
Q

Quotient

A

Result of division

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8
Q

Sign Rules for Multiplication

A

positive x positive = positive

negative x negative = positive

negative x positive = negative

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9
Q

Sign Rules for Division

A

positive / positive = positive

negative / negative = positive

negative / positive = negative

positive / negative = negative

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10
Q

Absolute Value

A

Gives the distance of the number from the origin

|x| = the distance of x from the origin

|x - 5| = the distance of x from +5

|x + 3| = the distance of x from -3

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11
Q

Order of Operations

A

G - Grouping Symbols (parentheses, brackets, square root signs, long fraction bar, exponent slot)

E - Exponents

M/D - Multiplication & Division

A/S - Addition & Subtraction

Goblins Eat Many Doughnuts A Second

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12
Q

Decimals as Powers

A
  1. 1 = 10^ -1
  2. 01 = 10^ -2
  3. 001 = 10^ -3
  4. 0001 = 10^ -4

For negative powers of 10, the power equals how many decimal places to the right of the decimal point.

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13
Q

Adding and Subtracting Decimals

A

Line up decimal points and add or subtract vertically

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14
Q

Multiplying Decimals

A

1) Count the number of digits to the right of the decimal point for each.

For example: 6.25 x 0.048

  • 6.25 has 2 decimal places to the right of the decimal point
  • 0.048 has 3 decimal places to right of the decimal point

2) Add those two numbers together.
- 2 + 3 = 5 SO the product will have 5 decimal places
3) Ignore the decimal points completely and multiply as if they were two positive integers.
- 625 x 48 = 30,000
4) with 5 decimal places to the left, the answer is 0.3

Note that powers are a special case of multiplying

(0.03)^3 = 0.03 x 0.03 x 0.03
2 + 2 + 2 = 6 SO the product will have 6 decimal places to the right of the decimal point.

3^3 = 27

The 7 must fall 6 places to the right of the decimal point

The answer is 0.000027

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15
Q

Dividing Decimals

A

1) Move the decimals of the numerator and denominator/divisor to the right until they are both whole numbers.

For example: 0.56/0.0007

0.56/0.0007 = 5.6/0.007 = 56/0.07 = 560/0.7 = 5600/7 = 800

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16
Q

Numerator

A

The top number of a fraction

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17
Q

Denominator

A

The bottom number of a fraction

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18
Q

Multiples of 10 greater than 1

A

For multiples of 10 greater than 1, the number of zeroes after the one equals the factors of 10 the multiple contains

For example:

10,000 = 10^4 = 10x10x10x10

1,000,000 = 10^6 = 10x10x10x10x10x10

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19
Q

Multiples of 10 less than 1

A

For powers of 10 less than 1, the number of factors of ten in the denominator equals the number of places to the right of the decimal point.

For example:

  1. 01 = 10^-2 = 1/10x10
  2. 0001 = 10^-4 = 1/10x10x10x10
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20
Q

Multiplying by Positive Powers of 10

A

When multiplying a number by positive powers of 10, we move the decimal point the number of spaces to the right equal to the power, that is, equal to the number of factors of 10

350x100

100 = 10^2

So we have to move the decimal point two to the right.

= 35,000

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21
Q

Dividing by Positive Powers of 10 OR Multiplying by Negative Powers of 10

A

When we divide any number by any positive power of 10 OR multiply any number by a negative power of 10, we move the decimal point a number of spaces to the left equal to the absolute value of the power, that is, equal to the factors of 10.

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22
Q

Dividing by a Negative Power of 10

A

This is the equivalent of multiplying by a positive power of 10 and will ultimately make the product bigger

2.35/0.01 = 2.35 x 100 = 235

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23
Q

Scientific Notation

A

N = A x 10^p

Where A is greater than or equal to 1 and is less than or equal to 10

AND

Where p is an integer

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24
Q

Multiplying/Dividing in Scientific Notation

A

We can multiply/divide numbers in scientific notation by multiplying/dividing the numbers in front and then adding (for multiplication) or subtracting (for dividing) the exponents of 10.

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25
Q

Adding and Subtracting in Scientific Notation

A

We can add or subtract two numbers in scientific notation, but FIRST we have to make the powers of 10 the same.

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26
Q

Equivalent Fractions

A

Fractions are equivalent if they have equal numerical value despite having different numerators and denominators.

We can always find an equivalent fraction by multiplying the numerator and denominator by the sane integer.

For example:
3/8 = 3x4/8x4 = 12/32

We can also factor out the same positive integer from both the numerator and denominator.

For example:
6/42 = 6x1/6x7
The sixes cancel and we’re left with 1/7

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27
Q

Lowest Term Fraction/Simplest Form

A

An equivalent fraction is in its simplest form when the lowest possible integer values are in the numerator and denominator

28
Q

Improper Fraction

A

When a fraction is greater than one and expressed by a numerator that is greater than the denominator

Ex: 5/3

29
Q

Mixed Numeral

A

A fraction greater than one that is expressed by an integer part plus a fraction part

Ex: 1 2/3

Note that this is addition! 1 + 2/3

30
Q

1/2

A

0.5

31
Q

1/4

A

0.25

32
Q

3/4

A

0.75

33
Q

2/5

A

0.4

34
Q

4/5

A

0.8

35
Q

1/3

A

0.33333 repeating

36
Q

2/3

A

0.666666 repeating ~ 0.667

37
Q

1/5

A

0.2

38
Q

3/5

A

0.6

39
Q

1/6

A

0.1666666 repeating ~ 0.1667

40
Q

5/6

A

0.833333 repeating

41
Q

1/7

A

0.142857 repeating ~ 0.143

42
Q

1/8

A

0.125

43
Q

3/8

A

0.375

44
Q

5/8

A

0.625

45
Q

7/8

A

0.875

46
Q

1/9

A

0.11111 repeating

This means that any numerator on top of 9 will just be that number repeating as a decimal.

2/9 = 0.22222 repeating

3/9 = 0.33333 repeating

4/9 = 0.44444 repeating

Etc.

47
Q

Fractions with denominators that are multiples of 10

A

Break up the fraction to simplify to a fraction you already know as a decimal and then move the decimal place based on the factor of 10.

1/40 = 1/4 x 1/10 = (0.25)(10^-1) = 0.025

1/600 = 1/6 x 1/100 = (0.1667)(10^-2)
= 0.001667

48
Q

n/1

A

n/1 = n

Fractions with 0 in the denominator are UNDEFINED

Fractions with 0 in the numerator always equal 0

Note: 0/0 is also illegal

49
Q

n/n

A

n/n = 1

Anything divided by itself equals 1 and we can always multiply a fraction by n/n because it doesn’t change the value of the product.

50
Q

Reciprocal Fraction

A

For a/b, the reciprocal fraction is flipped over and equals b/a.

The product of a fraction and its reciprocal always equals 1.

Ex: 2/3 x 3/2 = 1

1 divided by a fraction will always equal its reciprocal.

Ex: 1/(3/7) = 7/3

51
Q

Changing the numerator but keeping the same denominator

A

If a > b, then a/c > b/c

Ex: 4/13 < 6/14

52
Q

Changing the denominator but keeping the same numerator

A

If p>q, then s/p < s/q

If we’re dividing by more, the quotient is less…bigger denominators make smaller fractions.

Ex: 2/5 > 2/7

53
Q

Adding the same number to the numerator and denominator

A

The fraction will move closer to 1.

If the original fraction is LESS THAN 1 to begin with, adding the same number to the numerator and denominator will result in a BIGGER fraction

Ex: 2/5 (less than 1)
If you add 6 to numerator and denominator, the resulting fraction is 8/11, which is closer to 1 on the number line and GREATER THAN 2/5

8/11 > 2/5

If the original fraction is MORE THAN 1 to begin with, adding the same number to the numerator and denominator will result in a SMALLER fraction

Ex: 7/4 (greater than 1)
If you add 2 to the numerator and denominator, the resulting fraction is 9/6, which is closer to 1 on the number line and LESS THAN 7/4.

54
Q

Adding different numbers to the numerator and denominator

A

Suppose we add 2 to the numerator and 5 to the denominator…the resulting fraction will be closer to 2/5 on the number line than the original fraction was.

SO if the original fraction was LESS THAN 2/5, adding 2 to the numerator and 5 to the denominator would result in a BIGGER fraction.

BUT if the original fraction was MORE THAN 2/5, adding 2 to the numerator and 5 to the denominator would result in a SMALLER fraction.

Ex: 1/8 (smaller than 2/5)

(1+2)/(8+5) = 3/13

1/8 < 3/13

Ex: 3/4 (greater than 2/5)

(3+2)/(4+5) = 5/9

3/4 > 5/9

55
Q

Add to numerator and subtract from denominator (or vice versa!)

A

Whichever fraction has a larger numerator AND a smaller denominator is bigger

56
Q

Cross Multiplying Fractions

A

Used when we have an equation in the form fraction = fraction.

For a/b = c/d, we can multiply both sides by the denominator of the opposite side to eliminate all fractions. The numerator stays on the appropriate side!

a x d = c x b

This also works if we want to quickly compare two relatively close fractions.

Instead of an equal sign, out a ? in between the two fractions. This could represent greater than or less than. But if only works if we keep the inequality pointing in the same direction. If we have to multiply or divide by a negative, the inequality sign will flip and this won’t work!

Ex: 7/11 and 5/8

7/11 ?? 5/8

7 x 8 ?? 5 x 11

56 ?? 55

56 > 55 and because we didn’t change the direction of the inequality, it’s true of the original fractions as well and 7/11 > 5/8

57
Q

Adding/Subtracting Fractions

A

We can only perform addition and subtraction on fractions when they share a common denominator — that is, they must have the same denominator.

If you’re given a fraction that does not have a common denominator, find an equivalent fraction with the same denominator. You must multiply fractions to ensure there’s a common denominator and once you have the common denominator you can add/subtract across the numerator.

Ex: 1/3 + 1/7

(1/3 x 7/7) + (1/7 x 3/3) = 7/21 + 3/21

= 10/21

58
Q

Multiplying Fractions

A

You simply multiply across numerators and denominators.

Remember: when multiplying two or more fractions, you can cancel any numerator with any denominator. Always cancel before you multiply!

59
Q

Dividing Fractions

A

To divide by a fraction, we simply multiply by its reciprocal.

Ex: 1/4 / 3/2 = 1/4 x 2/3

= 1/6

To divide a whole number by a fraction, multiply the whole number by the reciprocal of the fraction.

Ex: 6/(3/4) = 6/1 x 4/3 = 8/1

= 8

To divide a fraction by a whole number, multiply the fraction by the reciprocal or the whole number, which will be in the form 1/n.

Ex: (3/5) / 2 = 3/5 x 1/2

= 3/10

60
Q

Addition or Subtraction in Numerator

A

We CAN separate a fraction into two fractions by addition or subtraction in the numerator.

a + b/c = a/c + b/c

61
Q

Addition or Subtraction in the Denominator

A

We CANNOT separate a fraction into two fractions by addition or Subtraction in the denominator.

Illegal move!

62
Q

Addition or Subtraction in Numerator and Denominator

A

If we have addition or Subtraction in both the numerator and denominator, we can split up the numerator but the denominator remains unchanged.

Ex: a + b /c + d = a/c + d + b/c + d

63
Q

Multiplying a Fraction by its Denominator

A

Fraction x Denominator = Numerator

Can be very useful in solving equations.

64
Q

Simplifying complex fractions

A

Strategy: multiply numerator and denominator of the big fraction by each denominator of the inner fractions

65
Q

Cancellation in Proportions

A

For proportion

a/b = c/d

You’re allowed to cancel numerators and denominators vertically (a & b and c & d)

You’re also allowed to cancel horizontally across numerators and across denominators (a & c and b & d)

You are NOT ALLOWED to cancel diagonally - this is only allowed in multiplication, it’s completely illegal to use in proportions.

66
Q

Word Problem Secret

A

In most word problems:

  • is = equals
  • of = multiply

GMAT Quantatative (9 decks)

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