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General Articles
Official Contest Dates 2017-2018 School Year
Contest Level Registration Deadline* Shipping Date Contest Date Registration Fee (Prices include shipping)
4th Grade January 31, 2018 March April 15th or later $40 per set of 30
5th Grade January 31, 2018 March April 15th or later $40 per set of 30
6th Grade December 31, 2017 January 3rd Tues. in Feb. - February 20, 2018 or 4th Tues. in Feb. - February 27, 2018 $40 per set of 30
7th Grade December 31, 2017 January 3rd Tues. in Feb. - February 20, 2018 or 4th Tues. in Feb. - February 27, 2018 $40 per set of 30
8th Grade December 31, 2017 January 3rd Tues. in Feb. - February 20, 2018 or 4th Tues. in Feb. - February 27, 2018 $40 per set of 30
Algebra Course 1 January 31, 2018 March April 15th or later $40 per set of 30
High School September 30, 2017 October HS Contest 1 - October 17, 2017
HS Contest 2 - November 14, 2017
HS Contest 3 - December 12, 2017
HS Contest 4 - January 9, 2018
HS Contest 5 - February 13, 2018
HS Contest 6 - March 20, 2018

Alternate contest dates may be scheduled one week after the contest dates, in the event of scheduling conflicts.

$90 / 6 contests, 30 of each

* Late registrations will be accepted.

 

Official Contest Dates 2016-2017 School Year
Contest Level Registration Deadline* Shipping Date Contest Date Registration Fee (Prices include shipping)
4th Grade January 31, 2017 March April 15th or later $40 per set of 30
5th Grade January 31, 2017 March April 15th or later $40 per set of 30
6th Grade December 31, 2016 January 3rd Tues. in Feb. - February 21, 2017 or 4th Tues. in Feb. - February 28, 2017 $40 per set of 30
7th Grade December 31, 2016 January 3rd Tues. in Feb. - February 21, 2017 or 4th Tues. in Feb. - February 28, 2017 $40 per set of 30
8th Grade December 31, 2016 January 3rd Tues. in Feb. - February 21, 2017 or 4th Tues. in Feb. - February 28, 2017 $40 per set of 30
Algebra Course 1 January 31, 2017 March April 15th or later $40 per set of 30
High School September 30, 2016 October HS Contest 1 - October 18, 2016 
HS Contest 2 - November 15, 2016
HS Contest 3 - December 13, 2016
HS Contest 4 - January 10, 2017
HS Contest 5 - February 7, 2017
HS Contest 6 - March 14, 2017

Alternate contest dates may be scheduled one week after the contest dates, in the event of scheduling conflicts.

$90 / 6 contests, 30 of each

* Late registrations will be accepted.

Contest Results and Newsletters Previous Years (old site)

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Official Contest Dates 2008-2009 School Year
Contest LevelRegistration
Deadline
 *
Shipping DateContest Date Registration Fee
(Prices include shipping)
4th GradeFebruary 28, 2009MarchApril 15th or later $30 per set of 30
5th GradeFebruary 28, 2009MarchApril 15th or later $30 per set of 30
6th Grade

January 31, 2009

January3rd Tues. in Feb. - February 17, 2009
or 4th Tues. in Feb. - February 24, 2009
$30 per set of 30
7th GradeJanuary 31, 2009January3rd Tues. in Feb. - February 17, 2009
or 4th Tues. in Feb. - February 24, 2009
$30 per set of 30
8th GradeJanuary 31, 2009January3rd Tues. in Feb. - February 17, 2009
or 4th Tues. in Feb. - February 24, 2009
$30 per set of 30
Algebra Course 1February 28, 2009MarchApril 15th or later$30 per set of 30
High School

October 15, 2008

October6 Contests, Oct.-Mar.$75 / 6 contests, 30 of each

 * Late registrations will be accepted.


Algebra Course 1 Contest

Math League's Algebra Course 1 Contests are a great way to motivate students learning algebra for the first time. The questions range from basic algebra skills, to more difficult problems, requiring creative solutions by applying algebra course 1 techniques. A wide variety of word problems are included on each contest, in addition to computational problems, to stress the value of applied algebra techniques. These contests are provided for intraschool competition, and come with certificates of merit for your school's high scoring students, and one of our High School Contest problem books for your school's top scorer. Here's a quote from one of last year's Algebra Course 1 students who participated in our contest: "I hope you keep making those tests. It really gets your brain to work" -Amber L.

Contest Format: Each contest consists of 30 multiple-choice questions that you can do in 30 minutes. On each 3-page contest, the questions on the 1st page are generally straightforward, those on the 2nd page are moderate in difficulty, and those on the 3rd page are more difficult. The questions require no more knowledge than that of a first year high school algebra course.


 Sample Algebra 1 Contest (PDF - 195k) • Solutions (PDF - 208k) 
Dates / Fees • Register Online • Print Order Form
Rules
 • FAQ • Check School Registration


To best view sample contests get Acrobat Reader (free) 

6th, 7th, and 8th Grade Contests

Math League's 6th, 7th, and 8th grade contests challenge students and schools in interschool league competitions. Students in each league compete for the highest scores, while schools compete for the highest team score: the total of the top 5 scores in each school. Each contest's questions cover material appropriate to each grade level. Questions may cover: basic topics, plus exponents, fractions, reciprocals, decimals, rates, ratios, percents, angle measurement, perimeter, area, circumference, basic roots, patterns, sequences, integers, triangles and right triangles, and other topics, depending on the grade level. Detailed solution sheets demonstrate the methods used to solve each problem. These contests encourage a variety of problem-solving skills and methods, to improve students' abilities and understanding of mathematical connections, while having fun!

Contest Format: Each contest consists of 35 multiple-choice questions that you can do in 30 minutes. On each 3-page contest, the questions on the 1st page are generally straightforward, those on the 2nd page are moderate in difficulty, and those on the 3rd page are more difficult. There is a 6th Grade Score Report, and a 7th and 8th Grade Score Report sent to schools in each league after the contest.


 

 Sample 6th Grade Contest (PDF - 240k) • Solutions (PDF - 256k) 
 
Sample 7th Grade Contest (PDF - 232k) • Solutions (PDF - 253k) 
 Sample 8th Grade Contest (PDF - 195k) • Solutions (PDF - 247k) 
Dates / Fees • Register Online • Print Order Form
 Score Reports • Rules • FAQ • Check School Registration

To best view sample contests get Acrobat Reader (free)

High School Contests

Math League's High School Contests are the best in high school mathematics competition. Students in each league compete for the highest scores, while schools compete for the highest team score: the total of the top 5 scores in each school. These contests consist of 6 High School Contests each year, with 6 questions per contest. There are 6 score reports per year for each league, showing each participating school's team scores, high scoring schools and students, and students with a perfect score. Each score report is accompanied by a newsletter, which includes comments and alternate solutions from teachers and students. All high school students in accredited schools are welcome to compete. Problems draw from a wide range of high school topics: geometry, algebra, trigonometry, logarithms, series, sequences, exponents, roots, integers, real numbers, combinations, probability, coordinate geometry, and more. No knowledge of calculus is required to solve any of these problems. Detailed solution sheets demonstrate the methods used to solve each problem, including various approaches where appropriate. Working through these problems and our contest problem books is excellent practice for the SAT and college-bound students.

Contest Format: There are 6 High School Contests each year, with 6 questions per contest. There is a 30 minute time limit for each contest. On each contest, the last two questions are generally more difficult than the first four. The final question on each contest is intended to challenge the very best mathematics students. The problems require no knowledge beyond secondary school mathematics. No knowledge of calculus is required to solve any of these problems. Two to four of the questions on each contest only require a knowledge of elementary algebra. Starting with the 1992-93 school year, students have been permitted to use any calculator on any of our contests.


 Sample 05-06 High School Contest & Solutions (PDF) 
  
Dates / Fees
 • Register Online • Print Order Form
 Score Reports • Rules • FAQ • Check School Registration


To best view sample contests get Acrobat Reader (free) 



 


2010-2011 High School Contest Dates

Contest #Official Date* (Tuesdays) 
HS Contest 1  
HS Contest 2  
HS Contest 3  
HS Contest 4  
HS Contest 5  
HS Contest 6 
October 19, 2010
November 16, 2010
December 14, 2010
Janaury 11, 2011
February 22, 2011
March 22, 2011
 *Alternate contest dates may be scheduled one week before
the contest dates, in the event of scheduling conflicts.

Positive and negative numbers

About positive and negative numbers
The number line
Absolute value of positive and negative numbers
Adding positive and negative numbers
Subtracting positive and negative numbers
Multiplying positive and negative numbers
Dividing positive and negative numbers
Coordinates
Comparing positive and negative numbers
Reciprocals of negative numbers

 


About Positive and Negative Numbers

Positive numbers are any numbers greater than zero, for example: 1, 2.9, 3.14159, 40000, and 0.0005. For each positive number, there is a negative number that is its opposite. We write the opposite of a positive number with a negative or minus sign in front of the number, and call these numbers negative numbers. The opposites of the numbers in the list above would be: -1, -2.9, -3.14159, -40000, and -0.0005. Negative numbers are less than zero (see the number line for a more complete explanation of this). Similarly, the opposite of any negative number is a positive number. For example, the opposite of -12.3 is 12.3.
We do not consider zero to be a positive or negative number. 
The sum of any number and its opposite is 0.
The
sign of a number refers to whether the number is positive or negative, for example, the sign of -3.2 is negative, and the sign of 442 is positive.
We may also write positive and negative numbers as fractions or mixed numbers.

The following fractions are all equal:

(-1)/3, 1/(-3), -(1/3) and - 1/3.

The following mixed numbers are all equal:

-1 1/6, -(1 1/6), (-7)/6, 7/(-6), and - 7/6.

 


The Number Line

The number line is a line labeled with positive and negative numbers in increasing order from left to right, that extends in both directions. The number line shown below is just a small piece of the number line from -4 to 4.

 

For any two different places on the number line, the number on the right is greater than the number on the left.

Examples:

4 > -2, 1 > -0.5, -2 > -4, and 0 > -15


Absolute Value of Positive and Negative Numbers

The number of units a number is from zero on the number line. The absolute value of a number is always a positive number (or zero). We specify the absolute value of a number n by writing n in between two vertical bars: |n|.

Examples:

|6| = 6
|-0.004| = 0.004
|0| = 0
|3.44| = 3.44
|-3.44| = 3.44
|-10000.9| = 10000.9


Adding Positive and Negative Numbers

1) When adding numbers of the same sign, we add their absolute values, and give the result the same sign.

Examples:

2 + 5.7 = 7.7
(-7.3) + (-2.1) = -(7.3 + 2.1) = -9.4 
(-100) + (-0.05) = -(100 + 0.05) = -100.05

2) When adding numbers of the opposite signs, we take their absolute values, subtract the smaller from the larger, and give the result the sign of the number with the larger absolute value.

Example:

7 + (-3.4) = ?
The absolute values of 7 and -3.4 are 7 and 3.4. Subtracting the smaller from the larger gives 7 - 3.4 = 3.6, and since the larger absolute value was 7, we give the result the same sign as 7, so 7 + (-3.4) = 3.6.

Example:

8.5 + (-17) = ? 
The absolute values of 8.5 and -17 are 8.5 and 17. Subtracting the smaller from the larger gives 17 - 8.5 = 8.5, and since the larger absolute value was 17, we give the result the same sign as -17, so 8.5 + (-17) = -8.5.

Example:

-2.2 + 1.1 = ?
The absolute values of -2.2 and 1.1 are 2.2 and 1.1. Subtracting the smaller from the larger gives 2.2 - 1.1 = 1.1, and since the larger absolute value was 2.2, we give the result the same sign as -2.2, so -2.2 + 1.1 = -1.1.

Example:

6.93 + (-6.93) = ?
The absolute values of 6.93 and -6.93 are 6.93 and 6.93. Subtracting the smaller from the larger gives 6.93 - 6.93 = 0. The sign in this case does not matter, since 0 and -0 are the same. Note that 6.93 and -6.93 are opposite numbers. All opposite numbers have this property that their sum is equal to zero. Two numbers that add up to zero are also called additive inverses.


Subtracting Positive and Negative Numbers

Subtracting a number is the same as adding its opposite.

Examples:

In the following examples, we convert the subtracted number to its opposite, and add the two numbers.
7 - 4.4 = 7 + (-4.4) = 2.6 
22.7 - (-5) = 22.7 + (5) = 27.7 
-8.9 - 1.7 = -8.9 + (-1.7) = -10.6 
-6 - (-100.6) = -6 + (100.6) = 94.6

Note that the result of subtracting two numbers can be positive or negative, or 0.


Multiplying Positive and Negative Numbers

To multiply a pair of numbers if both numbers have the same sign, their product is the product of their absolute values (their product is positive). If the numbers have opposite signs, their product is the opposite of the product of their absolute values (their product is negative). If one or both of the numbers is 0, the product is 0.

Examples:

In the product below, both numbers are positive, so we just take their product.
0.5 × 3 = 1.5

In the product below, both numbers are negative, so we take the product of their absolute values.
(-1.1) × (-5) = |-1.1| × |-5| = 1.1 × 5 = 5.5

In the product of (-3) × 0.7, the first number is negative and the second is positive, so we take the product of their absolute values, which is |-3| × |0.7| = 3 × 0.7 = 2.1, and give this result a negative sign: -2.1, so (-3) × 0.7 = -2.1

In the product of 21 × (-3.1), the first number is positive and the second is negative, so we take the product of their absolute values, which is |21| × |-3.1| = 21 × 3.1 = 65.1, and give this result a negative sign: -65.1, so 21 × (-3.1) = -65.1.

To multiply any number of numbers:

1. Count the number of negative numbers in the product. 
2. Take the product of their absolute values.
3. If the number of negative numbers counted in step 1 is even, the product is just the product from step 2, if the number of negative numbers is odd, the product is the opposite of the product in step 2 (give the product in step 2 a negative sign). If any of the numbers in the product is 0, the product is 0.

Example:

2 × (-1.1) × 5 (-1.2) × (-9) = ? 
Counting the number of negative numbers in the product, we see that there are 3 negative numbers: -1.1, -1.2, and -9. Next, we take the product of the absolute values of each number: 2 × |-1.1| × 5 × |-1.2| × |-9| = 2 × 1.1 × 5 × 1.2 × 9 = 118.8 
Since there were an odd number of numbers, the product is the opposite of 118.8, which is -118.8, so 2 × (-1.1) × 5 (-1.2) × (-9) = -118.8.


Dividing Positive and Negative Numbers

To divide a pair of numbers if both numbers have the same sign, divide the absolute value of the first number by the absolute value of the second number.
To divide a pair of numbers if both numbers have different signs, divide the absolute value of the first number by the absolute value of the second number, and give this result a negative sign.

Examples:

In the division below, both numbers are positive, so we just divide as usual.
7 ÷ 2 = 3.5

In the division below, both numbers are negative, so we divide the absolute value of the first by the absolute value of the second. 
(-2.4) ÷ (-3) = |-2.4| ÷ |-3| = 2.4 ÷ 3 = 0.8

In the division (-1) ÷ 2.5, both number have different signs, so we divide the absolute value of the first number by the absolute value of the second, which is |-1| ÷ |2.5| = 1 ÷ 2.5 = 0.4, and give this result a negative sign: -0.4, so (-1) ÷ 2.5 = -0.4.

In the division 9.8 ÷ (-0.7), both number have different signs, so we divide the absolute value of the first number by the absolute value of the second, which is |9.8| ÷ |-0.7| = 9.8 ÷ 0.7 = 14, and give this result a negative sign: -14, so 9.8 ÷ (-0.7) = -14.


Coordinates

Number coordinates are pairs of numbers that are used to determine points in a grid, relative to a special point called the origin. The origin has coordinates (0,0). We can think of the origin as the center of the grid or the starting point for finding all other points. Any other point in the grid has a pair of coordinates (x,y). The x value or x-coordinate tells how many steps left or right the point is from the point (0,0), just like on the number line (negative is left of the origin, positive is right of the origin). The y value or y-coordinate tells how many steps up or down the point is from the point (0,0), (negative is down from the origin, positive is up from the origin). Using coordinates, we may give the location of any point in the grid we like by simply using a pair of numbers.

Example:

The origin below is where the x-axis and the y-axis meet. Point A has coordinates (2.3,3), since it is 2.3 units to the right and 3 units up from the origin. Point B has coordinates (-3,1), since it is 3 units to the left, and 1 unit up from the origin. Point C has coordinates (-4,-2.5), since it is 4 units to the left, and 2.5 units down from the origin. Point D has coordinates (9.2,-8.4); it is 9 units to the right, and 8.4 units down from the origin. Point E has coordinates (-7,6.6); it is 7 units to the left, and 6.6 units up from the origin. Point F has coordinates (8,-5.7); it is 8 units to the right, and 5.7 units down from the origin.


Comparing Positive and Negative Numbers

We can compare two different numbers by looking at their positions on the number line. For any two different places on the number line, the number on the right is greater than the number on the left. Note that every positive number is greater than any negative number.

Examples:

9.1 > 4, 6 > -9.3, -2 > -8, and 0 > -5.5
-2 < -13, -1 < -0.5, -7 < -5, and -10 < 0.1


Reciprocals of Negative Numbers

The reciprocal of a positive or negative fraction is obtained by switching its numerator and denominator, the sign of the new fraction remains the same. To find the reciprocal of a mixed number, first convert the mixed number to an improper fraction, then switch the numerator and denominator of the improper fraction. Notice that when you multiply negative fractions with their reciprocals, the product is always 1 (NOT -1).

Examples:

What is the reciprocal of -2/7? We just switch the numerator and denominator, and keep the same sign: -7/2.

What is the reciprocal of - 5 1/8? First, we convert to a negative improper fraction: -5 1/8 = - 41/8, then we switch the numerator and denominator, and keep the same sign: - 8/41.

 

Introduction to Algebra

Variables 
Expressions 
Equations 
Solution of an equation 
Simplifying equations 
Combining like terms 
Simplifying with addition and subtraction 
Simplifying by multiplication 
Simplifying by division 
Word problems as equations 
Sequences 

Variables

A variable is a symbol that represents a number. Usually we use letters such as nt, or x for variables. For example, we might say that s stands for the side-length of a square. We now treat s as if it were a number we could use. The perimeter of the square is given by 4 × s. The area of the square is given by s× s. When working with variables, it can be helpful to use a letter that will remind you of what the variable stands for: let n be the number of people in a movie theater; let t be the time it takes to travel somewhere; let d be the distance from my house to the park. 

Expressions

An expression is a mathematical statement that may use numbers, variables, or both.

Example:

The following are examples of expressions:

2

x

3 + 7

2 × y + 5

2 + 6 × (4 - 2)

z + 3 × (8 - z)

Example:

Roland weighs 70 kilograms, and Mark weighs k kilograms. Write an expression for their combined weight. The combined weight in kilograms of these two people is the sum of their weights, which is 70 + k.

Example:

A car travels down the freeway at 55 kilometers per hour. Write an expression for the distance the car will have traveled after h hours. Distance equals rate times time, so the distance traveled is equal to 55 × h..

Example:

There are 2000 liters of water in a swimming pool. Water is filling the pool at the rate of 100 liters per minute. Write an expression for the amount of water, in liters, in the swimming pool after m minutes. The amount of water added to the pool after m minutes will be 100 liters per minute times m, or 100 × m. Since we started with 2000 liters of water in the pool, we add this to the amount of water added to the pool to get the expression 100 × + 2000.

To evaluate an expression at some number means we replace a variable in an expression with the number, and simplify the expression.

Example:

Evaluate the expression 4 × z + 12 when z = 15.

We replace each occurrence of z with the number 15, and simplify using the usual rules: parentheses first, then exponents, multiplication and division, then addition and subtraction.

4 × z + 12 becomes

4 × 15 + 12 =

60 + 12 =

72

Example:

Evaluate the expression (1 + z) × 2 + 12 ÷ 3 - z when z = 4.

We replace each occurrence of z with the number 4, and simplify using the usual rules: parentheses first, then exponents, multiplication and division, then addition and subtraction.

(1 + z) × 2 + 12 ÷ 3 - z becomes

(1 + 4) × 2 + 12 ÷ 3 - 4 =

5 × 2 + 12 ÷ 3 - 4 =

10 + 4 - 4 =

10. 


Equations

An equation is a statement that two numbers or expressions are equal. Equations are useful for relating variables and numbers. Many word problems can easily be written down as equations with a little practice. Many simple rules exist for simplifying equations.

Example:

The following are examples of equations:

2 = 2

17 = 2 + 15

x = 7

7 = x

t + 3 = 8

3 × n +12 = 100

w + 4 = 12 - w

y - 1 - 2 - 9.3 = 34

3 × (d + 4) - 11 = 321 - 23

Example:

Translate the following word problem into an equation:

My age in years y plus 20 is equal to four times my age, minus 10.

The first expression stands for "my age in years plus 20", which is y + 20.

This is equal to the second expression for "four times my age, minus 10", which is 4 × y - 10.

Setting these two expressions equal to one another gives us the equation:

y + 20 = 4 × y - 10 


Solution of an Equation

When an equation has a variable, the solution to the equation is the number that makes the equation true when we replace the variable with its value.

Example:

We say y = 3 is a solution to the equation 4 × y + 7 = 19, for replacing each occurrence of y with 3 gives us

4 × 3 + 7 = 19  ==>

12 + 7 = 19  ==>

19 = 19 which is true.

Examples:

x = 100 is a solution to the equation x ÷ 2 - 40 = 10

= 12 is a solution to the equation 5 × (z - 6) = 30

Counterexample:

y = 10 is NOT a solution to the equation 4 × y + 7 = 19. When we replace each y with 10, we get

4 × 10 + 7 = 19  ==>

40 + 7 = 19  ==>

47 = 19 not true!

Counterexamples:

x = 200 is NOT a solution to the equation x ÷ 2 - 40 = 10

z = 20 is NOT a solution to the equation 5 × (z - 6) = 30 


Simplifying Equations

To find a solution for an equation, we can use the basic rules of simplifying equations. These are as follows:

1) You may evaluate any parentheses, exponents, multiplications, divisions, additions, and subtractions in the usual order of operations. When evaluating expressions, be careful to use the associative and distributive properties properly.

2) You may combine like terms. This means adding or subtracting variables of the same kind. The expression 2x + 4x simplifies to 6x. The expression 13 - 7 + 3 simplifies to 9.

3) You may add any value to both sides of the equation.

4) You may subtract any value from both sides of the equation. This is best done by adding a negative value to each side of the equation.

5) You may multiply both sides of the equation by any number except 0.

6) You may divide both sides of the equation by any number except 0.

Hint: Since subtracting any number is the same as adding its negative, it can be helpful to replace subtractions with additions of a negative number.

Example:

This problem illustrates grouping like terms and dealing with subtraction in an equation.

Solve x - 12 + 20 = 37.

Replacing the -12 with a +(-12), we get

x + (-12) + 20 = 37.

Since addition is associative, the two like terms (the integers) may be combined.

(12) + 20 = 8

The left side of the equation becomes

x + 8 = 37.

Now we may subtract 8 from each side of the equation, (we will actually add a -8 to each side).

x + 8 + (-8) = 37 + (-8)

x + 0 = 29

x = 29

We can check this solution in the original equation:

29 - 12 + 20 = 37x + 0 = 29

17 + 20 = 37

37 = 37 so our solution is correct.

Example:

This problem illustrates the proper use of the distributive property.

Solve 2 × (x + 1 + 4) = 20.

Grouping like terms in the parentheses, the left side of the equation becomes

2 × (x + 1 + 4)  ==>  2 × (x + 5).

Using the distributive property,

2 × (x + 5)  ==>  2 × x + 2 × 5.

Carrying out multiplications,

2 × x + 2 × 5  ==>  to 2x + 10.

The equation now becomes

2x + 10 = 20.

Subtracting a 10 (adding a -10) to each side gives us

2x + 10 + (-10) = 20 + (-10)  ==>

2x + (10 + (-10)) = 20 - 10  ==>

2x + 0 = 10  ==>

2x = 10.

Since the x is multiplied by 2, we divide both sides by 2 to solve for x:

2x = 10  ==>

2x ÷ 2 = 10 ÷ 2  ==>

(2x)/2 = 5  ==>

x = 5.

We can check this solution in the original equation:

2 × (5 + 1 + 4) = 20  ==>

2 × 10 = 20  ==>

20 = 20 so our solution is correct. 


Combining like terms

One of the most common ways to simplify an expression is to combine like terms. Numeric terms may be combined, and any terms with the same variable part may be combined.

Example:

Consider the expression 2 + 7x + 12 - 3x - 5. The numeric like terms are the numbers 2, 12, and 5. The variable like terms are 7x and 3x. Combining the numeric like terms, we have 2 + 12 - 5 = 14 - 5 = 9. Combining the variable like terms, we have 7x - 3x = 4x, so the expression 2 + 7x + 12 - 3x - 5 simplifies to 9 + 4x


Simplifying with addition and subtraction

We can use addition and subtraction to get all the terms with variables on one side of an equation, and all the numeric terms on the other.

The equations 3x = 17, 21 = y, and z/12 = 24 each have a variable term on one side of the = sign, and a number on the other.

The equations x + 3 = 12, 21 = 30 - y, and (z + 2) × 4 = 10 do not.

We usually do this after simplifying each side using the distributive rules, eliminating parentheses, and combining like terms. Since addition is associative, it can be helpful to add a negative number to each side instead of subtracting to avoid mistakes.

Examples:

For the equation 3x + 4 = 12, we can isolate the variable term on the left by subtracting a 4 from both sides:

3x + 4 - 4 = 12 - 4  ==>

3x = 8.

For the equation 7y - 200 = 10, subtracting the 200 on the left side is the same as adding a -200:

7y + (-200) = 10.

If we add 200 to both sides of the equation, the 200 and -200 will cancel each other:

7y + (-200) + 200 = 10 + 200  ==>

7y = 210.

For the equation 8 = 20 - z, we can add z to both sides to get 8 + z = 20 - z + z  ==>  8 + z = 20. Now subtracting 8 from both sides,

8 + z - 8 = 20 - 8  ==>

z = 12, so we get a solution for z


Simplfying by multiplication

When solving for a variable, we want to get a solution like x = 3 or z = 2001. When a variable is divided by some number, we can use multiplication on both sides to solve for the variable.

Example:

Solve for x in the equation x ÷ 12 = 5.

Since the x on the left side is being divided by 12, the equation is the same as x × 1/12 = 5. Multiplying both sides by 12 will cancel the 1/12 on the left side:

x × 1/12 × 12 = 5 × 12  ==>

x × 1 = 60  ==>

x = 60. 


Simplifying by division

When solving for a variable, we want to get a solution like x = 3 or z = 2001. When a variable is multiplied by some number, we can use division on both sides to solve for the variable.

Example:

Solve for x in the equation 7x = 133. Since the x on the left side is being multiplied by 7, we can divide both sides by 7 to solve for x:

7x ÷ 7 = 133 ÷ 7  ==>

(7x)/7 = 133 ÷ 7  ==>

x/1 = 19  ==>

x = 19.

Note that dividing by 7 is the same as multiplying both sides by 1/7. 


Word problems as equations

When converting word problems to equations, certain "key" words tell you what kind of operations to use: addition, multiplication, subtraction, and division. The table below shows some common phrases and the operation to use.
WordOperation ExampleAs an equation 
sumadditionThe sum of my age and 10 equals 27. y + 10 = 27
differencesubtractionThe difference between my age and my younger sister's age, who is 11 years old, is 5 years. y - 11 = 5
productmultiplicationThe product of my age and 14 is 168.y × 14 = 168 
timesmultiplicationThree times my age is 60. 3 × y = 60 
less thansubtractionSeven less than my age equals 32.y - 7 = 32 
totaladditionThe total of my pocket change and 20 dollars is $22.43. y + 20 = 22.43
more thanadditionEleven more than my age equals 43. 11 + y = 43

Sequences

A sequence is a list of items. We can specify any item in the list by its place in the list: first, second, third, fourth, and so on. Many useful lists have patterns so we know what items occur in each place in the list. There are 2 kinds of sequences. A finite sequence is a list made up of a finite number of items. An infinite sequence is a list that continues without end.

Examples:

The following are examples of finite sequences.

The sequence 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 is the sequence of the first 10 odd numbers.

The sequence a, e, i, o, u, is the sequence of vowels in the alphabet.

The sequence m, m, m, m, m, m is the sequence of 6 m's.

The sequence 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0 is the sequence of 12 alternating 1's and 0's.

The sequence 1, 2, 3, 4, ..., 9998, 9999, 10000 is the sequence of the first ten thousand integers.

The sequence 0, 1, 4, 9, 16, 25, 36, 49 is the sequence of the squares of the first 8 whole numbers.

Examples:

The following are examples of infinite sequences.

The sequence 2, 4, 6, 8, 10, 12, 14, 16, ... is the sequence of even whole numbers. The 100th place in this sequence is the number 200.

The sequence a, b, c, a, b, c, a, b, c, a, b, ... is the sequence of the letters a, b, c, repeating in this pattern forever.

The 100th place in this sequence is the letter a. The 300th place in this sequence is the letter c.

The sequence -1, 2, -3, 4, -5, 6, -7, 8, -9, ... is the sequence of integers with alternating signs. The 10th place in this sequence is 10. The 100th place in this sequence is 100. The 101st place in this sequence is -101.

The sequence 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, ... is a sequence of 1's separated by 1 zero, then 2 zeros, then 3 zeros, and so on. The 100th place in this sequence is a 0. The 105th place in this sequence is a 1.

The sequence 1, 3, 6, 10, 15, 21, 28, 36, 45, ... is the sequence of places the 1 occurs in the sequence of 1's and 0's above! If this sequence seems strange, note the difference between pairs of numbers next to one another:

3 - 1 = 2

6 - 3 = 3

10 - 6 = 4

15 - 10 = 5

21 - 15 = 6

28 - 21 = 7

Checking these differences makes the pattern clearer.

1, 1, 1, 1, 1, 1, ... is the sequence where every item in the list is the number 1.

1, 2, 3, 4, 5, 6, 7, ... is the sequence of counting numbers. Each item in the list is its place number in the list.

a, b, a, b, a, b, a, b, ... is the sequence of alternating letters a and b. The a's occur in odd-numbered places, and the b's occur in the even-numbered places.

1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, ... is the sequence of reciprocals of the whole numbers.

1, 4, 9, 16, 25, 36, 49, 64, 81, ... is the sequence of squares of the whole numbers.

a, e, i, o, u, a, e, i, o, u, a, e, ... is the repeating sequence of vowels in the alphabet.

4, 7, 10, 13, 16, 19, 22, 25, ... is the sequence of numbers beginning with the number 4, and each number in the list is 3 more than the number before it.

Metric units and Measurement

Length
Volume
Mass
Time
Temperature
Decimals in measurement

 


Length

The standard unit of length in the metric system is the meter. Other units of length and their equivalents in meters are as follows: 
1 millimeter = 0.001 meter
1 centimeter = 0.01 meter
1 decimeter = 0.1 meter
1 kilometer = 1000 meters

We abbreviate these lengths as follows:
1 millimeter = 1
mm
1 centimeter = 1 cm
1 meter = 1 m
1 decimeter = 1 dm
1 kilometer = 1 km

For reference, 1 meter is a little longer than 1 yard or 3 feet. It is about half the height of a very tall adult. A centimeter is nearly the diameter of a dime, a little less than half an inch. A millimeter is about the thickness of a dime.


Volume

The standard unit of volume in the metric system is the liter. One liter is equal to 1000 cubic centimeters in volume. Other units of volume and their equivalents in liters are as follows: 
1 milliliter = 0.001 liter
1 centiliter = 0.01 liter
1 deciliter = 0.1 liter
1 kiloliter = 1000 liters

From these units, we see that 1000 milliliters equal 1 liter; so 1 milliliter equals 1 cubic centimeter in volume. We abbreviate these volumes as follows:
1 milliliter = 1
ml
1 centiliter = 1 cl
1 deciliter = 1 dl
1 liter = 1 l
1 kiloliter = 1 kl

For reference, 1 liter is a little more than 1 quart. One teaspoon equals about 5 milliliters.


Mass

The standard unit of mass in the metric system is the gram. Other units of mass and their equivalents in grams are as follows:
1 milligram = 0.001 gram
1 centigram = 0.01 gram
1 decigram = 0.1 gram
1 kilogram = 1000 grams

We abbreviate these masses as follows:
1 milligram = 1
mg
1 centigram = 1 cg
1 decigram = 1 dg
1 gram = 1 g
1 kilogram = 1 kg

For reference, 1 gram is about the mass of a paper clip. One kilogram is about the mass of a liter of water.


Time

The following conversions are useful when working with time:
1 minute = 60 seconds
1 hour = 60 minutes = 3600 seconds
1 day = 24 hours
1 week = 7 days
1 year = 365 1/4 days (for the Earth to travel once around the sun)

In practice, every three calendar years will have 365 days, and every fourth year is a "leap year", which has 366 days, to make up for the extra quarter day over four years. The years 1992, 1996, 2000, and 2004 are all leap years. This gives us a total of 52 complete 7 day weeks in each calendar year, with 1 day left over (or 2 in a leap year).

The year is divided into 12 months, each of which has 30 or 31 days, except for February, which has 28 days (or 29 days in a leap year).


Temperature

Temperature is expressed in degrees Celsius in the metric system. The boiling point of water (at sea level) is 100°Celsius, or 100°C. The freezing point of water (at sea level) is 0° Celsius. A hot day is about 30° Celsius.


Decimals in measurement

We use decimals to specify units of measurement when we need more precision about length, volume, mass, or time. For example, when specifying the height of a person 1.63 meters tall, to say that person is 1 or 2 meters tall doesn't give us a very good idea of how tall that person really is. 
The prefixes for the different units of length, volume, and mass in the metric system obey the following rules:

 

Prefix Multiply by
milli- 0.001
centi- 0.01
deci- 0.1
deka- 10
hecto- 100
kilo- 1000

 

So for example:
1
hectometer = 100 meters
1
centigram = 0.01 gram
3
milliliters = 3 × (0.001 liters) = 0.003 liters
0.9
kilometers = 0.9 × (1000 meters) = 900 meters

Integers

Positive and negative integers
The number line
Absolute value of an integer
Adding integers
Subtracting integers
Multiplying integers
Dividing integers
Integer coordinates
Comparing integers


Positive and Negative Integers

Positive integers are all the whole numbers greater than zero: 1, 2, 3, 4, 5, ... . Negative integers are all the opposites of these whole numbers: -1, -2, -3, -4, -5, …. We do not consider zero to be a positive or negative number. For each positive integer, there is a negative integer, and these integers are called opposites. For example, -3 is the opposite of 3, -21 is the opposite of 21, and 8 is the opposite of -8. If an integer is greater than zero, we say that its sign is positive. If an integer is less than zero, we say that its sign is negative. 

Example:

Integers are useful in comparing a direction associated with certain events. Suppose I take five steps forwards: this could be viewed as a positive 5. If instead, I take 8 steps backwards, we might consider this a -8. Temperature is another way negative numbers are used. On a cold day, the temperature might be 10 degrees below zero Celsius, or -10°C.


The Number Line

The number line is a line labeled with the integers in increasing order from left to right, that extends in both directions:

 

For any two different places on the number line, the integer on the right is greater than the integer on the left.

Examples:

9 > 4, 6 > -9, -2 > -8, and 0 > -5


Absolute Value of an Integer

The number of units a number is from zero on the number line. The absolute value of a number is always a positive number (or zero). We specify the absolute value of a number n by writing n in between two vertical bars: |n|.

Examples:

|6| = 6
|-12| = 12
|0| = 0
|1234| = 1234
|-1234| = 1234


Adding Integers

1) When adding integers of the same sign, we add their absolute values, and give the result the same sign.

Examples:

2 + 5 = 7
(-7) + (-2) = -(7 + 2) = -9 
(-80) + (-34) = -(80 + 34) = -114

2) When adding integers of the opposite signs, we take their absolute values, subtract the smaller from the larger, and give the result the sign of the integer with the larger absolute value.

Example:

8 + (-3) = ? 
The absolute values of 8 and -3 are 8 and 3. Subtracting the smaller from the larger gives 8 - 3 = 5, and since the larger absolute value was 8, we give the result the same sign as 8, so 8 + (-3) = 5.

Example:

8 + (-17) = ?
The absolute values of 8 and -17 are 8 and 17. 
Subtracting the smaller from the larger gives 17 - 8 = 9, and since the larger absolute value was 17, we give the result the same sign as -17, so 8 + (-17) = -9.

Example:

-22 + 11 = ?
The absolute values of -22 and 11 are 22 and 11. Subtracting the smaller from the larger gives 22 - 11 = 11, and since the larger absolute value was 22, we give the result the same sign as -22, so -22 + 11 = -11.

Example:

53 + (-53) = ?

The absolute values of 53 and -53 are 53 and 53. Subtracting the smaller from the larger gives 53 - 53 =0. The sign in this case does not matter, since 0 and -0 are the same. Note that 53 and -53 are opposite integers. All opposite integers have this property that their sum is equal to zero. Two integers that add up to zero are also called additive inverses.


Subtracting Integers

Subtracting an integer is the same as adding its opposite.

Examples:

In the following examples, we convert the subtracted integer to its opposite, and add the two integers.
7 - 4 = 7 + (-4) = 3
12 - (-5) = 12 + (5) = 17
-8 - 7 = -8 + (-7) = -15 
-22 - (-40) = -22 + (40) = 18

Note that the result of subtracting two integers could be positive or negative.


Multiplying Integers

To multiply a pair of integers if both numbers have the same sign, their product is the product of their absolute values (their product is positive). If the numbers have opposite signs, their product is the opposite of the product of their absolute values (their product is negative). If one or both of the integers is 0, the product is 0.

Examples:

In the product below, both numbers are positive, so we just take their product.
4 × 3 = 12

In the product below, both numbers are negative, so we take the product of their absolute values.
(-4) × (-5) = |-4| × |-5| = 4 × 5 = 20

In the product of (-7) × 6, the first number is negative and the second is positive, so we take the product of their absolute values, which is |-7| × |6| = 7 × 6 = 42, and give this result a negative sign: -42, so (-7) × 6 = -42.

In the product of 12 × (-2), the first number is positive and the second is negative, so we take the product of their absolute values, which is |12| × |-2| = 12 × 2 = 24, and give this result a negative sign: -24, so 12 × (-2) = -24.

To multiply any number of integers:

1. Count the number of negative numbers in the product. 
2. Take the product of their absolute values.
3. If the number of negative integers counted in step 1 is even, the product is just the product from step 2, if the number of negative integers is odd, the product is the opposite of the product in step 2 (give the product in step 2 a negative sign). If any of the integers in the product is 0, the product is 0.

Example:

4 × (-2) × 3 × (-11) × (-5) = ?

Counting the number of negative integers in the product, we see that there are 3 negative integers: -2, -11, and -5. Next, we take the product of the absolute values of each number:
4 × |-2| × 3 × |-11| × |-5| = 1320. 
Since there were an odd number of integers, the product is the opposite of 1320, which is -1320, so
4 × (-2) × 3 × (-11) × (-5) = -1320.


Dividing Integers

To divide a pair of integers if both integers have the same sign, divide the absolute value of the first integer by the absolute value of the second integer.
To divide a pair of integers if both integers have different signs, divide the absolute value of the first integer by the absolute value of the second integer, and give this result a negative sign.

Examples:

In the division below, both numbers are positive, so we just divide as usual.
4 ÷ 2 = 2.

In the division below, both numbers are negative, so we divide the absolute value of the first by the absolute value of the second. 
(-24) ÷ (-3) = |-24| ÷ |-3| = 24 ÷ 3 = 8.

In the division (-100) ÷ 25, both number have different signs, so we divide the absolute value of the first number by the absolute value of the second, which is |-100| ÷ |25| = 100 ÷ 25 = 4, and give this result a negative sign: -4, so (-100) ÷ 25 = -4.

In the division 98 ÷ (-7), both number have different signs, so we divide the absolute value of the first number by the absolute value of the second, which is |98| ÷ |-7| = 98 ÷ 7 = 14, and give this result a negative sign: -14, so 98 ÷ (-7) = -14.


Integer coordinates

Integer coordinates are pairs of integers that are used to determine points in a grid, relative to a special point called the origin. The origin has coordinates (0,0). We can think of the origin as the center of the grid or the starting point for finding all other points. Any other point in the grid has a pair of coordinates (x,y). The x value or x-coordinate tells how many steps left or right the point is from the point (0,0), just like on the number line (negative is left of the origin, positive is right of the origin). The y value or y-coordinate tells how many steps up or down the point is from the point (0,0), (negative is down from the origin, positive is up from the origin). Using coordinates, we may give the location of any point in the grid we like by simply using a pair of numbers.

Example:

The origin below is where the x-axis and the y-axis meet. Point A has coordinates (2,3), since it is 2 units to the right and 3 units up from the origin. Point B has coordinates (3,1), since it is 3 units to the right, and 1 unit up from the origin. Point C has coordinates (8,-5), since it is 8 units to the right, and 5 units down from the origin. Point D has coordinates (9,-8); it is 9 units to the right, and 8 units down from the origin. Point E has coordinates (-4,-3); it is 4 units to the left, and 3 units down from the origin. Point F has coordinates (-7,6); it is 7 units to the left, and 6 units up from the origin.


Comparing Integers

We can compare two different integers by looking at their positions on the number line. For any two different places on the number line, the integer on the right is greater than the integer on the left. Note that every positive integer is greater than any negative integer.

Examples:

9 > 4, 6 > -9, -2 > -8, and 0 > -5 
-2 < 1, 8 < 11, -7 < -5, and -10 < 0