Chapter 1 | Essential GMAT Skills

Percent

- Means "divided by 100" — 5% = 5/100 —> 1/20 — 500% = 500/100 —> 5 — 50% = 50/100 —> 1/2

Non-Terminating & Non-Repeating Decimal

- .123456..., .1010010001..., 3.14159265...

Perfect Squares Units Digit

- 0, 1, 4, 5, 6 or 9

Terminating Decimal

- A decimal which has a finite number of digits - 3.4, .12, .0056

Decimal to Percent

- Move the decimal point 2 places to the right and attach the % sign - .35 = 35% - 2 = 200% - .005 = .5%

Fraction to Percent

- Multiply by 100 and attach the % sign - 3/4 —> 3/4 x 100 = 300/4 = 75% - 7/100 —> 7/100 x 100 = 700/100 = 7%

Multiplying Fractions

- Multiply the numerators and place the product over the product of the denominators - Formula: a/b x c/d = ac/bd - 2/3 x 5/4 = 2 x 5 / 3 x 4 = 10/12

Subtracting from 1,000

- Subtract all but the last number from 9, then subtract the last number from 10

Units Digits Trick

1) Look for the repeating pattern, usually out of 4 2) Figure out where the pattern will be at the desired power

Dividing Decimals

1) Move the decimal point of the divisor (the term in the denominator) to the right until it becomes a whole number 2) Move the decimal point in the dividend (the term in the numerator) to the right the same number of places you have moved the decimal point of the divisor 3) Divide the numbers by long division and leave the decimal points in the same place as the quotient you find

Multiplying Decimals

1. Multiply the numbers as if they were whole numbers 2. Tally the total number of decimal places in the factors and add them: 2.2 and 2.33 has three total decimal places 3. Add the total number of decimal places to the product, counting from left to right: 22 x 233 = 5126 --> 5.126

1/7 - Conversions

1/7 = .143 = 14.3% 2/7 = .286 = 28.6% 3/7 = .429 = 42.9% 4/7 = .571 = 57.1% 5/7 = .714 = 71.4% 6/7 = .857 = 85.7%

1-9 - Conversions

1/9 = .111 = 11.1% 2/9 = .222 = 22.2% 3/9 = .333 = 33.3% 4/9 = .444 = 44.4% 5/9 = .556 = 55.6% - Changes here for last digit to be +1 6/9 = .667 = 66.7% 7/9 = .778 = 77.8% 8/9 = .889 = 88.9%

Divisibility Rules

2 - If the number is even 3 - If the sum of all the digits is divisible by 3 4 - If the last two digits are divisible by 4 5 - If the number ends on 0 or 5 6 - If the number is divisible by 2 and 3 9 - If the sum of all the digits is divisible by 9

Distributive Property of Division over Addition

Formula: a+b/c = a/c + b/c - Most commonly, we use this in Remainder questions

Factoring Factorials

- 11! + 10! = 10!(11 + 1) = 10!(12) - 20! + 19! + 18! —> 20 x 19 x 18! + 19 x 18! + 18! —> 18!(20 x 19 + 19 + 1) —> 18!(400)

Repeating Decimal

- A decimal which has repeating digits or a repeating pattern of digits - 3.444, .121212, .056056056

Time Saving Properties Of Multiplication

- Commutative: a x b = b x a - Associative: (a x b) x c = a x (b x c) - When dealing with just multiplication in a sequence, we can go in whatever order we want

Guessing Large Squares

- Define the range and see where the number falls in the spectrum, then guess accordingly!

Dividing by 5

- Double the numerator and move the decimal point one place to the left

Percent to Decimal

- Drop the percent sign and move the decimal two places to the left - 70% = .7 - 3,000% = 30 - 15% = .15 - 298% = 2.98

Use Foil For Multiplying Two-Digit Integers

- FOIL each to closest multiple of 10 and use FOIL to multiply across - Great for large #'s

Comparing The Size of Fractions: Reference Point Method

- Find a reference point and compare each fraction to that point A: 2/3 and B: 4/5 — A is 1/3 from 1, B is 1/5 from 1 — 1/5 is smaller than 1/3, hence, B is closer to 1 and larger than A

Adding and Subtracting Fractions

- Find the common denominator, add/subtract the numerators and place over the common denominator Common Denominators: a/b + c/b = (a + c)/b Unlike Denominators: a/b + c/d = ad + bc /bd

Comparing The Size of Fractions: Bow Tie Method

- Formula: a/b > c/d if ad > bc - Is 7/9 > 6/8? — 7 x 8 = 56, 9 x 6 = 54 — 56 > 54, so 7/9 is > 6/8

Comparing Decimals

- Give each number the same number of decimal places, ignore the decimal and see which is the largest number - Which is larger — 0.02 or 0.1 —> .02 or .10 —> 10 is greater than 2, so .10 is the larger value - Compare the decimals place by place and whichever has the larger number at the first point of difference is the larger number

Fraction Logic: Adding a Constant to both Numerator and Denominator

- Given a fraction, adding a non-zero constant WILL change the value of the fraction - Simple Fraction: Value will be larger - 2/3, adding 2 to both N and D becomes 4/5, which is larger and closer to 1 than the OG fraction - Complex Fraction: Value will be smaller - 5/3, adding 2 to both N and D becomes 7/5, which is smaller than the OG fraction

Fraction Logic: Subtracting a Constant to both Numerator and Denominator

- Given a fraction, subtracting a non-zero constant WILL change the value of the fraction - Simple Fraction: Value will be smaller - 2/3, subtracting 1 from both N and D becomes 1/2, which is smaller and closer to 0 than the OG - Complex Fraction: value will be larger - 6/4, subtracting 2 from both N and D becomes 4/2, which is larger than the OG fraction

Fraction Logic: Multiplication / Division of a Constant to Both Numerator and Denominator

- Given a fraction, this WILL NOT change the value - 3/5 — 3 x 2 / 5 x 2 = 6/10, which is equal to 3/5

Decimal to Fraction (Terminating)

- If a terminating decimal has n number of decimal places, the numerator is the number without a decimal point over 1 followed by n zeros If a terminating decimal has 1 decimal place: — Write the number with no decimal point over 10 —> .3 = 3/10 and 3.4 = 34/10 If a terminating decimal has 2 decimal places: — Write the number with no decimal point over 100 —> 1.47 = 147/100 and .12 = 12/100 If a terminating decimal has 3 decimal places: — Write the number with no decimal point over 1,000 —> .679 = 679/1,000 and .125 = 125/1,000

Comparing the Size of Fractions: LCD Method

- If asked to arrange a set of fractions in ascending or descending order, find the LCD of the denominators and notice that the larger the numerators, the larger the value - Arrange from largest to smallest: 1/3, 3/5 and 4/6 — The LCD of 3, 5, and 6 is 30 — 10/30, 18/30 and 20/30 — Therefore, 4/6 > 3/5 > 1/3

Comparing the Size of Fractions: LCD of Numerator Method

- If asked to arrange a set of fractions in ascending or descending order, fractions with common numerators share the following properties: - The larger the denominator, the smaller the fraction: 3/2 and 3/4, 3/2 > 3/4 - The smaller the denominator, the larger the fraction: 3/2 and 3/4, 3/2 > 3/4 If the LCD of the denominator is too large to calculate, look to this method to compare a few of the fractions in clusters to see if you can find the answer

Multiplying Large Numbers (2x Trick)

- Important when asked tío multiply 2 large #'s by each other

Dividing Fractions

- Invert the second fraction to its reciprocal and multiply it against the numerator - Formula: a/b / c/d = ad/bc - 2/3 / 5/4 = 2 x 4 / 3 x 5 = 8/15

Adding Decimals

- Line up the decimal points and add normally - 23.62 + 8.369 = 31.989

Subtracting Decimals

- Line up the decimal points and subtract normally - 20.02 - 1.36 = 18.66)

Square Root Rule

- On the GMAT, the square root of a number is always positive - Except for 0, which is neither negative nor positive

PEMDAS

- Parentheses, Exponents, Multiplication, Division, Addition, Subtraction Remember, absolute value bars and radicals are the same as parentheses

11 Times (Trick)

- Take the original number and put a space between the two digits - Add the two numbers together and put them in the middle - If the number in the middle is 2 digits, add 1 to the leading number and leave the second digit in the middle

Least Common Denominator (The LCD) of The Denominator

- The LCD of 2 or more fractions is the smallest non-zero whole number that is divisible by each of the denominators - To find the LCD, list the multiples of all the denominators until you find a common number

Multiplying Decimals: Shortcut

- The number of decimal places a product of two numbers will have is equal to their unique sum of decimal places — 15.534 x 2.8 will have 4 decimal places — We can eliminate answer choices that do not have 4 decimals places off the bat — .76854 x .823 will have 8 decimal places — We do not need to do the ugly math to narrow our choices down * If the product of the two numbers ends in a 0, do not use this method* — .5 x .6 = .3, but we would assume it would equal .03 if we were just too look at answer choices and not use the longer method of multiplying decimals

Factorials

- The product of an integer and all the integers after it - 4! = 4 x 3 x 2 x 1 = 24 - 0! = 1, 1! = 1

When the product of two integers is 1

- Then either both are 1 or both are -1

Rounding Decimals

- To round decimals, first find the place value you want to round to (the rounding digit) and look at the digit directly to the right - If the digit is less than five, do not change the rounding digit and drop all #'s after the rounding digit - If the digit is greater than or equal to five, increase the rounding digit by 1 and drop all #'s after the rounding digit If rounding 5.298 to the nearest hundredth, notice that the 9 would become a 0 — We must keep the 0 to keep that number rounded to the hundredth place —> 5.298 = 5.30, not 5.3 — Do not drop the last 0!

Rounding Whole Numbers

- To round whole numbers, first find the place value you want to round to (the rounding digit) and look at the digit directly to the right - If the digit is less than five, do not change the rounding digit and replace all #'s after the rounding digit with zeros - If the digit is greater than or equal to five, increase the rounding digit by 1 and replace all #'s after the rounding digit with zeros

Equivalent Fraction Rule

- Two fractions are equivalent if A/B = C/D is equal to A x D = B x C - Two fractions, although seemingly different, are of equal value if this holds true - This is an important method for DS questions when looking to manipulate the stem

Canceling Factorials

- When stuck with a factorial over a factorial in fraction format, do not forget to expand the factorial in the numerator to cancel out the factorial in the denominator

Factorial Distribution

- When you have to solve for integers, consider distributing factorials out! To better visualize the scenario at hand

Factorial Greater Than or Equal to 5!

- Will have a 0 as its units digit, because there is a (5 x 2) pair

Finding Square Roots (Trick)

- Works for any square root greater than 10 1) Take number and subtract the units and add the units digit, multiplying the two resulting numbers 2) Square the units digit and add it to the product of the multiplication in step 1

Percent to Fraction

- X% = X/100 (Reduce if possible) - 60% = 60/100, or, 3/5