# Arithmetic

GMAT Study Guide - a prep wikibook

## Operations

Addition is defined as the mathematical operation of adding two or more numbers in order to get a total. This total equals the sum of the numbers added. Addition is written with the plus sign $\Huge +$. For example,

$3+2=5$

$7+20=27$

$10+\frac{3}{5}=10\frac{3}{5}$

### Multiplication

Multiplication can be defined as the mathematical operation of repeated addition. The operation is written with either of the two signs: $\Huge *$ or $\Huge \times$. For example,

$3*7=3+3+3+3+3+3+3=7+7+7=21$

$2 \times 5=2+2+2+2+2=5+5=10$

### Subtraction

Subtraction is defined as the mathematical operation inverse to addition. The inversion can be explained as follows. If we have a number $X$, and add some number $Y$ to it, and then subtract the same number $Y$, we will get the original number $X$. The operation is written with the minus sign $\Huge -$.

$5-2=3$

$5-2+2=5$

### Division

Division is the mathematical operation inverse to multiplication. Division can be expressed with three most common signs: $a/b=\frac{a}{b}=a \div b$.

$\frac{12}{3}=4$, $3*4=12$

$20/5=4$, $5*4=20$

### Exponentiation

Exponentiation might be explained as a repeated multiplication. Exponentiation involves two numbers, i.e. the base and the exponent:

$\Large a^b = \underbrace{\text{a * a * ... * a}}_{\text{b times}}$

$a$ is the base, $b$ is the exponent. The expression above can be read as "$a$ raised to the $b^{th}$ power" or "$a$ to the $b^{th}$ power."

### PEMDAS order

PEMDAS stands for "Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction." Basically, this is the order of operations performed in algebraic expressions. This is what is meant by PEMDAS:

1. Do operations inside the parentheses
2. Calculate the exponents
3. Do multiplication and division from left to right
4. Do addition and subtraction from left to right

Here is an example:

$5^2 - 3*5 + 121^{\frac{1}{2}} - (3 + 4 - 10)$

First, we have to perform the operations inside the parentheses:

$5^2 - 3*5 + 121^{\frac{1}{2}} + 3$

Second, we calculate the exponents:

$25 - 3*5 + 11 + 3$

Third, we do multiplication (as we don't have division here):

$25 - 15 + 11 + 3$

Fourth, we perform addition and subtraction from left to right:

$25 - 15 + 11 + 3$

$10 + 11 + 3$

$21 + 3 = 24$

### Absolute value

The absolute value of a number is its numeric value no matter what the sign is. The absolute value is sometimes referred to as "magnitude". The expression $|x|$ reads "the absolute value of $x$." For example,

$|-5| = 5$

$|11| = 11$

$|-25| = 25$

Read more on solving absolute value equations and inequalities here.

{{#x:box| TODO: This section requires work. When done, remove the {{TODO}} code. All TODOs: Category:TODO.

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## Divisibility

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### Divisibility rules

Integer is divisible by:

2 - Even integer

3 - Sum of digits are divisible by 3

4 - Integer is divisible by 2 twice or Last 2 digits are divisible by 4

5 - Last digit is 0 or 5

6 - Integer is divisbile by 2 AND 3

8 - Integer is divisible by 2 three times

9 - Sum of digits is divisible by 9

10 - Last digit is 0

11 - Take the 1st digit, subtract 2nd, add 3rd, subtract 4th, add 5th... If the sum is divisible by 11 (the sum may equal 0), the number is divisible by 11

• Any two even numbers in a multiplication will ensure the product be divisible by 4
• If 2 numbers have the same factor, then the sum or difference of the two numbers will have the same factor. (e.g. 4 is a factor of 20, 4 is also a factor of 80, then 4 will be a factor of 60 (difference) and also 100 (sum))
• Remember to include '1' if you're asked to count the number of factors of a number

### Factorization

Factorization is the process of decomposing a number into a product of other numbers (or factors). Multiplying the factors gives the original number. Note that only positive factors are taken into account (e.g. 12 factors into $\normal 2*2*3$, not $\normal (-2)*(-2)*3$ ). Factorization usually implies prime factorization, i.e. presenting a number as a product of its prime factors.

#### How to do prime factorization

This is very easy. The best way to do it is like this:

264 | 2
132 | 2
66 | 2
33 | 3
11 | 11
1 | 1

$264 = 2*2*2*3*11$


The advantage of this system is that you can see all the prime factors organized in one line. Also, if you need to find LCM or GCD, you can easily scratch off or circle the factors you need, making sure your numbers are organized and you don't mess up.

### Prime testing

How to determine whether $x$ is a prime number? Basically we need to try dividing it by other primes up to $\sqrt{x}$. Assume that the number in question is greater than 30, for the primes under 30 should be memorized: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

However, before doing that, do the following divisibility tests:

• (%2): Is the number even?
• (%5): Is the last digit 5 or 0?
• (%3): If the sum of the number's digits divisible by 3?
• (%11): Take the 1st digit, subtract 2nd, add 3rd, subtract 4th, add 5th... Is the sum divisible by 11?

If the answer to any of the questions is Yes, the number is not a prime.

Otherwise, you'll need to try dividing by 7, 13, 17, and so on, until $\sqrt{x}$. Dividing by 29 is sufficient for prime tests of numbers up to $29^2 = 841$.

### LCM & GCD

LCM -> Lowest Common Multiple -> Lowest possible common multiple between numbers

GCD -> Greatest Common Divisor -> Largest possible common factor between numbers

GCD is also known as GCF (Greatest Common Factor).

Two integers are relatively prime if there is no integer greater than 1 that divides them both (that is, their greatest common divisor is one).

To find the GCD/LCM, you will need to do prime factorization. This means reducing a number to its prime-factor form.

Example 1

GCD and LCM of 4,18

$4 = 2*2$
$18= 2*3*3$

To find the GCD, take the multiplication of the common factors (pick the lowest power of the common factors). In this case, $GCD= 2$.

To find the LCM, take the multiplication of all the factors (pick the higest power of the common factors). In this case, $LCM=2*2*3*3=36$

Example 2

GCD and LCM of 4,24

$4 = 2*2$
$24 = 2*2*2*3$

$GCD= 2*2 = 4$
$LCM= 2*2*2*3 = 24$

Useful facts:

• $GCD(a,b)*LCM(a,b) = ab$;
• $LCM(a,b) = ab$ if and only if $a$ and $b$ are relatively prime.

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## Fractions

A fraction is a number represented as $\huge \frac{a}{b}$, where $a$ is called the numerator, $b$ - denominator. Numerator and denominator of a fraction are usually expressed with integers.

If two fractions represent the same number, they are equivalent. For example, the two fractions $\frac{3}{7}$ and $\frac{9}{21}$ are equivalent. Dividing 3 by 7 gives the same number as dividing 9 by 21. As we can see, multiplying the numerator and denominator of a fraction by the same number doesn't change a number the fraction represents. This property of a fraction allows us to perform addition and subtraction over the fractions.

In order to add or subtract two fractions you need to have the same denominators in both fractions. If the denominators are different, you have to find the Least Common Denominator, which equals the Lowest Common Multiple (LCM) of the two denominators. For example, we need to add the two fractions $\frac{17}{38}$ and $\frac{7}{38}$. That's easy, because the denominators of the fractions are equal. We have to add the numerators and leave the denominator as it is. We'll do it as follows:

$\frac{17}{38} + \frac{7}{38} = \frac{17+7}{38} = \frac{24}{38} = \frac{\cancel{2} * 12}{\cancel{2} * 19} = \frac{12}{19}$

Let's add another pair of fractions:

$\frac{7}{13} + \frac{5}{7}$

Fisrt, we see that the fractions have different denominators. We have to find the Least Common Denominator and rewrite the fractions to have the same denominator. The LCD for 7 and 13 is equal to their product, as these two numbers are prime. So, the $LCD = 7 * 13 = 91$. Now we have to rewrite the fractions to have the same denominator:

$\frac{7}{13} + \frac{5}{7} = \frac{7*7}{13*7} + \frac{5*13}{7*13} = \frac{49}{91} + \frac{65}{91}$

Now as we have the two fractions with the same denominators, we can perform addition:

$\frac{49}{91} + \frac{65}{91} = \frac{49 + 65}{91} = \frac{114}{91} = 1\frac{23}{91}$

One more example with subtraction:

$\frac{8}{9} - \frac{3}{4} = \frac{8 * 4}{9 * 4} - \frac{3 * 9}{4 * 9} = \frac{32}{36} - \frac{27}{36} = \frac{32 - 27}{36} = \frac{5}{36}$

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### Multiplication and division

To multiply two fractions, you need to multiply both their numerators and denominators. For example:

$\frac{3}{4} * \frac{2}{3} = \frac{3 * 2}{4 * 3} = \frac{6}{12} = \frac{1}{2}$

In order to divide one fraction by the other, you need to multiply the first one by the reciprocal (find reciprocal by switching numerator and denominator) of the second one. For example:

$\frac{5}{8} \div \frac{4}{9} = \frac{5}{8} * \frac{9}{4} = \frac{5 * 9}{8 * 4} = \frac{45}{32} = 1\frac{13}{32}$

As you might have already noticed, multiplying the numerator and the denominator of a fraction by the same number doesn't change the number a fraction represents. For example:

$\frac{3}{5} = \frac{3 * 4}{5 * 4} = \frac{12}{20}$

In fact, we're just multiplying the fraction by 1, which can be expressed with $\frac{4}{4}$ or any other $\frac{x}{x}$. That is why the value of the fraction doesn't change.

### Proper fractions

A proper fraction is a fraction with its numerator being less than denominator. This means that a proper fraction is always less than 1. For example, $\frac{3}{5}, \frac{10}{13}, \frac{99}{997}$ are proper fractions.

### Reducing

Reducing or simplifying a fraction is a process of finding an equivalent fraction with as small numerator and denominator as possible. This is usually done by dividing both numerator and denominator by their Greatest Common Divisor (GCD). After cancelling out the common divisors we get an equivalent fraction to the original one.

$\frac{240}{768}=\frac{2^4*3*5}{2^8*3}=\frac{\cancel{2^4}*\cancel{3}*5}{2^4*\cancel{2^4}*\cancel{3}}=\frac{5}{16}$

$GCD (240,768)=2^4*3=48$.

### Percentages

Percentage is an indication of a ratio of the part to the whole, which is represented by 100 units. That is why any percentage can be rewritten as a fraction. For example,

$30%=\frac{30}{100}=\frac{3}{10}$

$45%=\frac{45}{100}=\frac{9}{20}$

$80%=\frac{80}{100}=\frac{4}{5}$

GMAT is full of problems with percentage increases/decreases. When one is told about a certain number $X$ getting an increase by 30%, it is understood that a number after this increase is to be found by the formula:

Increased number $= X * 1.3$ ($X$ - the original number)

We multiplied the original number by 1.3 because the new number must equal 100% + 30% = 130% of the original number.

We should use the same logic when a number is increased by a percent, greatly exceeding 100. For example, if the original number $Y$ is increased by 345%:

Increased number $= Y * 4.45$

Note that we multiply by 445%, not 345%. If we multiply the original number by 3.45, we'll get only the 245% increase.

### Ratios

A ratio is a representation of an amount of one quantity relative to another. Ratios are usually written as two numbers separated by a colon (e.g. 2:3, 4:7, 86:99, etc.) Note that fractions and percentages are special cases of ratios. Fractions indicate the ratio of a part (numerator) to the whole (denominator), while the percentages indicate the relation of a part to the whole (represented by 100 units). Ratios are frequently used in GMAT word problems.

Ratio questions are very easy to solve if you master the way of thinking.

Basically, if you have

$\frac{a}{b}=\frac{c}{d}$ (or $a:b=c:d$)

then you can immediately derive a variaty of correlated ratios, such as:

$\frac{a}{a+b}=\frac{c}{c+d}$

$\frac{a}{a-b}=\frac{c}{c-d}$

$\frac{a+b}{a-b}=\frac{c+d}{c-d}$

$\frac{a+c}{b+d}=\frac{c}{d}$

$\frac{a-c}{b-d}=\frac{c}{d}$

etc.

Basically, you can do all kinds of additions and subtractions.

Example:

• $\frac{a}{b}=\frac{3}{5}$ (1)
• $2a-b=4$ (2)

What is $a$?

From (1) we get $\frac{a}{2a-b}=\frac{3}{1}$, so $a=3*4=12$

Explanation: $a$ is 3 share, $b$ is 5 share. Two $a$ is 6 share, $2a-b$ is one share. If one share is 4, then 3 share is 12.

Of course, this question can be solved using the more traditional algebra approach:

$b=\frac{5a}{3}$

substitute in (2)

$2a-\frac{5a}{3}=4$

$\frac{1}{3}*a=4$

$a=12$

You can see the two approaches are really the same in nature. However, the first approach is very straight forward and does not involve calculation in fractions. Sometimes it can save you lots of time, especially when using this method with word problems such as mixture problems.

#### Mixture Problems

Example:

A fruit mixture is made up by 25% fruit A and 75% fruit B. Now if the amount of fruit A is doubled, what is their relative share in the new mixture?

$A:B=25:75$

$2A:B=50:75=2:3$

The new mixture total quantity is $2A+B$

$2A:(2A+B)=2:5$

$B:(2A+B)=3:5$

Therefore the new shares are fruit A 40%, fruit B 60%.

Example 2:

In a picnic 60% people ate two hotdogs, 30% people ate one hamburger, and 10% people ate one hotdog. The total number of hotdog and hamburgers consumed is 80. How many hamburgers and hotdogs were consumed?

People:

$T:H:O=6:3:1$ (1)

Food:

$2T+H+O=80$

From (1)

$T:H:O:(2T+H+O)=6:3:1:16$

Therefore,

$T:(2T+H+O)=3:8=30:80$ 30 people ate two hotdogs

$H:T=3:6=1:2=15:30$ 15 people ate one hamburger

$O:T=1:6=5:30$ 5 people ate one hotdogs

Total people 50, total hotdogs 65, total hamburgers 15.

Verify, total hotdogs and hamburgers equal $65+15=80$.

## Exponents and Roots

A square root, also called a radical or surd, of $x$ is a number $r$ such that $r^2=x$.

1. For the math and physics majors, $\sqrt{16} = 4$ only; THE NEGATIVE ROOT IS NOT USED IN ARITHMETIC (this thread in the forum might be of some help)
2. $\sqrt{5}$ + $\sqrt{3}$ = $\sqrt{5}$ + $\sqrt{3}$; don't add/subtract roots with different bases; you can only multiply and divide them: $\sqrt{5} * \sqrt{3} = \sqrt{15}$
3. A power of a number is a collection of factors: $21^4 = 21*21*21*21 = 7^4*3^4 = 7*7*7*7*3*3*3*3 = 194,481$.

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## Comparisons

### Fractions

From fractions with equal denominators, the one with greater numerator is greater.

$\frac{237}{435}>\frac{145}{435}>\frac{59}{435}$ because $237>145>59$.

From fractions with equal numerators, the one with smaller denominator is greater.

$\frac{13}{47}<\frac{13}{35}<\frac{13}{27}$ because $27<35<47$.

If the fractions to be compared have different numerators and denominators, they are usually transformed to have the common denominator. The least common denominator equals the Lowest Common Multiple (LCM) of the denominators.

What fraction is greater?

• $\frac{5}{7}$
• $\frac{6}{10}$
• $\frac{2}{3}$
• $\frac{1}{2}$
• $\frac{4}{5}$

In order to understand which fraction is greater we have to transform all of them to have the common denominator. The $LCM (7,10,3,2,5) = 210$. Transformations of the fractions follow:

• $\frac{5}{7}=\frac{5*30}{7*30}=\frac{150}{210}$
• $\frac{6}{10}=\frac{6*21}{10*21}=\frac{126}{210}$
• $\frac{2}{3}=\frac{2*70}{3*70}=\frac{140}{210}$
• $\frac{1}{2}=\frac{1*105}{2*105}=\frac{105}{210}$
• $\frac{4}{5}=\frac{4*42}{5*42}=\frac{168}{210}$

Now we have to choose E, as it has the greatest numerator. So, the answer is E.

Note that it works the other way round for negative fractions.

$-\frac{5}{7}>-\frac{6}{7}$ but $\frac{5}{7}<\frac{6}{7}$.

### Expressions with exponents

From expressions with equal base (which is a positive integer), the one with greater exponent is greater.

$2^{99}<2^{102}$

$35^{\frac{2}{3}}>35^{\frac{1}{4}}$

From expressions with equal base (which is a positive fraction), the one with the least exponent is greater.

• $(\frac{1}{4})^{\frac{1}{2}}$
• $(\frac{1}{4})^{-\frac{1}{2}}$

$(\frac{1}{4})^{\frac{1}{2}} = \sqrt{\frac{1}{4}}=\frac{1}{2}$

$(\frac{1}{4})^{-\frac{1}{2}} = 4^{\frac{1}{2}} = \sqrt{4}= 2$

$(\frac{1}{4})^{\frac{1}{2}} < (\frac{1}{4})^{-\frac{1}{2}}$

{{#x:box| TODO: This section requires work. When done, remove the {{TODO}} code. All TODOs: Category:TODO.

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• How to compare negative things

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## Sequences

{{#x:box| TODO: This section requires work. When done, remove the {{TODO}} code. All TODOs: Category:TODO.

Introduce sums, averages and link to Statistics }}

### Consecutive numbers

For example:

• 1, 2, 3, 4 are the first 4 consecutive natural numbers
• 4, 6, 8, 10 are consecutive even integers
• 1, 3, 5, 7 are consecutive odd integers
• 2, 3, 5, 7 are consecutive primes
• 7, 14, 21, 28 are consecutive multiples of 7

Useful facts:

For any set of consecutive integers with an odd number of terms, the sum of the integers is always a multiple of the number of terms. For example, the sum of 1, 2, and 3 (three consecutives - an odd number) is 6, which is a multiple of 3.

For any set of consecutive integers with an even number of terms, the sum of the integers is never a multiple of the number of terms. For example, the sum of 1, 2, 3, and 4 (four consecutives - an even number) is 10, which is not a multiple of 4.

### Sum of a Series

A quick way to find the sum of a series where each preceding term is incremented by the same number would be to find the middle term and multiply it by the number of terms.

The middle term can be found by taking the average of the first and last term. In fact, the average of two equidistant from the first and the last terms will also do, as it equals the middle term.

Example:

Find the sum of 4,8,12,16,20

Middle term: 12

Number of terms: 5

Sum $= 12*5=60$.

### Arithmetic progression

Arithmetic progression or series is a sequence

$a_0,\ a_1,\ ..\ a_n\ ..$

such that

$a_n-a_{n-1}=C$

where C is a constant.

For example, positive integers (1, 2, 3, 4 ..) form an arithmetic progression with C = 1, and positive odd integers (1, 3, 5, 6, ..) form an arithmetic progression with C = 2. Note that C can be negative.

#### Common formulas

$\begin{eqnarray*} a_n &=& a_0 + nC = a_m + (n-m)C \\ \sum_{i=0}^{n} a_i &=& (n+1) a_0 + \frac{n(n+1)}{2} C = \frac{n+1}{2} (a_0 + a_n) \\ \sum_{i=m}^{n} a_i &=& (n-m) a_0 + \frac{n(n+1)-m(m+1)}{2} C \end{eqnarray*}$

### Geometric progression

Geometric progression or series is a sequence

$a_0,\ a_1,\ ..\ a_n\ ..$

such that

$a_{n+1}=Ca_n$

where C is a constant.

For example, the series (1, 2, 4, 8, ..) are a geometric progression with C = 2. Note that C can be negative.

#### Common formulas

$\begin{eqnarray*} a_n &=& C^n a_0 = C^{n-m} a_m \\ \sum_{i=0}^{n} a_i &=& a_0 \frac{1-C^{n+1}}{1-C}\ (C \ne 1) \\ \sum_{i=m}^{n} a_i &=& a_0 \frac{C^m-C^{n+1}}{1-C}\ (C \ne 1) \\ \sum_{i=0}^{\infty} a_i &=& \frac{a_0}{1-C}\ (C \ne 1) \end{eqnarray*}$

### Mean, Median, Mode, Range and Standard Deviation

Mean is the arithmetic mean (the sum of all values divided by the number of terms), median is the middle number (or the mean of two middle numbers in case the number of terms is even) of a sorted in ascending order set, mode is the one that appears the most (there can be more than one mode). Most likely they are not equal to each other. For two sets of numbers, if one set of the three Ms are equal, it means nothing about the other three.

Range is the difference between the smallest and the largest values of a set.

Example:

Set 1 contains {1,1,1,1} ; Set 2 contains {-1,1,-1,1}; Set 3 is the union of Set 1 and Set 2 along with the number 2

Set 1 = {1,1,1,1}; Mean=Median=Mode=1, Range=0

Set 2 = {-1,-1,1,1}; Mean=Median=0, Mode=-1,1 (A set of data can have more than one mode), Range = 2

Set 3 = {-1,-1,1,1,1,1,2}; Mean= $\frac{4}{7}$, Median=1, Mode=1, Range=3

Standard deviation measures how much all the numbers vary from one another, basically. You should understand the calculation of it in order to answer some questions on GMAT. So, standard deviation of a set equals square root of the arithmetic mean of all squared distances from the mean of a set to each of the numbers in this set. The greater the difference between the numbers of a set and its mean, the greater the standard deviation of a set. For example, we have to answer the following question:

{{#x:box| There are two sets of numbers: A={2,3,5,6}, B={-1,2,3,4}. What is the product of standard deviations of the two sets?

• $\sqrt{3.75}$
• $\sqrt{5.75}$
• $\sqrt{6.63}$
• $\sqrt{8.75}$
• $\sqrt{9.72}$

}} In order to find the standard deviation, we should do the following:

1. Calculate the arithmetic mean of each of the sets:
• Set A: $\frac{2+3+5+6}{4}=4$
• Set B: $\frac{-1+2+3+4}{4}=2$
2. Find the squared distances between each of the numbers and the mean of a set:
• Set A: $(2-4)^2$, $(3-4)^2$, $(5-4)^2$, $(6-4)^2$
• Set B: $(-1-2)^2$, $(2-2)^2$, $(3-2)^2$, $(4-2)^2$
3. Find the arithmetic mean of the squared distances:
• Set A: $\frac{4+1+1+4}{4}=2.5$
• Set B: $\frac{9+0+1+4}{4}=3.5$
4. Find the square root (the positive value) of the arithmetic mean of the squared distances:
• Set A: $\sqrt{2.5}$
• Set B: $\sqrt{3.5}$

To answer the above question, we should find the product of the two standard deviations:

$\sqrt{2.5}*\sqrt{3.5}=\sqrt{2.5*3.5}=\sqrt{8.75}$