General
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Line graphsUsing Data and Statistics
Pie charts
Bar graphs
Mean
Median
Mode
Line Graphs
A line graph is a way to summarize how two pieces of information are related and how they vary depending on one another. The numbers along a side of the line graph are called the scale.
Example 1:
The graph above shows how John's weight varied from the beginning of 1991 to the beginning of 1995. The weight scale runs vertically, while the time scale is on the horizontal axis. Following the gridlines up from the beginning of the years, we see that John's weight was 68 kg in 1991, 70 kg in 1992, 74 kg in 1993, 74 kg in 1994, and 73 kg in 1995. Examining the graph also tells us that John's weight increased during 1991 and 1995, stayed the same during 1991, and fell during 1994. Example 2:
This line graph shows the average value of a pickup truck versus the mileage on the truck. When the truck is new, it costs $14000. The more the truck is driven, the more its value falls according to the curve above. Its value falls $2000 the first 20000 miles it is driven. When the mileage is 80000, the truck's value is about $4000.
Pie Charts
A pie chart is a circle graph divided into pieces, each displaying the size of some related piece of information. Pie charts are used to display the sizes of parts that make up some whole. Example 1: The pie chart below shows the ingredients used to make a sausage and mushroom pizza. The fraction of each ingredient by weight is shown in the pie chart below. We see that half of the pizza's weight comes from the crust. The mushrooms make up the smallest amount of the pizza by weight, since the slice corresponding to the mushrooms is smallest. Note that the sum of the decimal sizes of each slice is equal to 1 (the "whole" pizza"). Example 2: The pie chart below shows the ingredients used to make a sausage and mushroom pizza weighing 1.6 kg. This is the same chart as above, except that the labels no longer tell the fraction of the pizza made up by that ingredient, but the actual weight in kg of the ingredient used. The sum of the numbers shown now equals 1.6 kg, the weight of the pizza. The size of each slice is still the same, and shows us the fraction of the pizza made up from that ingredient. To get the fraction of the pizza made up by any ingredient, divide the weight of the ingredient by the weight of the pizza. What fraction of the pizza does the sausage make up? We divide 0.12 kg by 1.6 kg, to get 0.075. This is the same value as in the pie chart in the previous example.
Example 3: The pie chart below shows the ingredients used to make a sausage and mushroom pizza. The fraction of each ingredient by weight shown in the pie chart below is now given as a percent. Again, we see that half of the pizza's weight, 50%, comes from the crust. Note that the sum of the percent sizes of each slice is equal to 100%. Graphically, the same information is given, but the data labels are different. Always be aware of how any chart or graph is labeled. Example 4: The pie chart below shows the fractions of dogs in a dog competition in seven different groups of dog breeds. We can see from the chart that 4 times as many dogs competed in the sporting group as in the herding group. We can also see that the two most popular groups of dogs accounted for almost half of the dogs in the competition. Suppose 1000 dogs entered the competition in all. We could figure the number of dogs in any group by multiplying the fraction of dogs in any group by 1000. In the toy group, for example, there were 0.12 × 1000 = 120 dogs in the competition.
Bar graphs consist of an axis and a series of labeled horizontal or vertical bars that show different values for each bar. The numbers along a side of the bar graph are called the scale. Example 1: The bar chart below shows the weight in kilograms of some fruit sold one day by a local market. We can see that 52 kg of apples were sold, 40 kg of oranges were sold, and 8 kg of star fruit were sold.
Bar Graphs
Example 2: A double bar graph is similar to a regular bar graph, but gives 2 pieces of information for each item on the vertical axis, rather than just 1. The bar chart below shows the weight in kilograms of some fruit sold on two different days by a local market. This lets us compare the sales of each fruit over a 2 day period, not just the sales of one fruit compared to another. We can see that the sales of star fruit and apples stayed most nearly the same. The sales of oranges increased from day 1 to day 2 by 10 kilograms. The same amount of apples and oranges was sold on the second day.
The mean of a list of numbers is also called the average. It is found by adding all the numbers in the list and dividing by the number of numbers in the list. Example: Find the mean of 3, 6, 11, and 8. We add all the numbers, and divide by the number of numbers in the list, which is 4. (3 + 6 + 11 + 8) ÷ 4 = 7 So the mean of these four numbers is 7. Example: Find the mean of 11, 11, 4, 10, 11, 7, and 8 to the nearest hundredth. (11 + 11 + 4 + 10 + 11 + 7 + 8) ÷ 7 = 8.857… which to the nearest hundredth rounds to 8.86. The median of a list of numbers is found by ordering them from least to greatest. If the list has an odd number of numbers, the middle number in this ordering is the median. If there is an even number of numbers, the median is the sum of the two middle numbers, divided by 2. Note that there are always as many numbers greater than or equal to the median in the list as there are less than or equal to the median in the list. Example: The students in Bjorn's class have the following ages: 4, 29, 4, 3, 4, 11, 16, 14, 17, 3. Find the median of their ages. Placed in order, the ages are 3, 3, 4, 4, 4, 11, 14, 16, 17, 29. The number of ages is 10, so the middle numbers are 4 and 11, which are the 5th and 6th entries on the ordered list. The median is the average of these two numbers: (4 + 11)/2 = 15/2 = 7.5 Example: The tallest 7 trees in a park have heights in meters of 41, 60, 47, 42, 44, 42, and 47. Find the median of their heights. Placed in order, the heights are 41, 42, 42, 44, 47, 47, 60. The number of heights is 7, so the middle number is the 4th number. We see that the median is 44.
Mean
Median
The mode in a list of numbers is the number that occurs most often, if there is one. Example: The students in Bjorn's class have the following ages: 5, 9, 1, 3, 4, 6, 6, 6, 7, 3. Find the mode of their ages. The most common number to appear on the list is 6, which appears three times. No other number appears that many times. The mode of their ages is 6.
Mode
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Decimals, Whole Numbers, and Exponents
Decimal numbers
Whole number portion
Expanded form of a decimal number
Adding decimals
Subtracting decimals
Comparing decimal numbers
Rounding decimal numbers
Estimating sums and differences
Multiplying decimal numbers
Dividing whole numbers, with remainders
Dividing whole numbers, with decimal portions
Dividing decimals by whole numbers
Dividing decimals by decimals
Exponents (powers of 2, 3, 4, ...)
Factorial notation
Square roots
Decimal Numbers
Decimal numbers such as 3.762 are used in situations which call for more precision than whole numbers provide.
As with whole numbers, a digit in a decimal number has a value which depends on the place of the digit. The places to the left of the decimal point are ones, tens, hundreds, and so on, just as with whole numbers. This table shows the decimal place value for various positions:
Note that adding extra zeros to the right of the last decimal digit does not change the value of the decimal number.
Place (underlined)
Name of Position
1.234567
Ones (units) position
1.234567
Tenths
1.234567
Hundredths
1.234567
Thousandths
1.234567
Ten thousandths
1.234567
Hundred Thousandths
1.234567
Millionths
Example:
In the number 3.762, the 3 is in the ones place, the 7 is in the tenths place, the 6 is in the hundredths place, and the 2 is in the thousandths place.
Example:
The number 14.504 is equal to 14.50400, since adding extra zeros to the right of a decimal number does not change its value.
Whole Number Portion
The whole number portion of a decimal number are those digits to the left of the decimal place.
Example:
In the number 23.65, the whole number portion is 23.
In the number 0.024, the whole number portion is 0.
Expanded Form of a Decimal Number
The expanded form of a decimal number is the number written as the sum of its whole number and decimal place values.
Example:
3 + 0.7 + 0.06 + 0.002 is the expanded form of the number 3.762.
100 + 3 + 0.06 is the expanded form of the number 103.06.
Adding Decimals
To add decimals, line up the decimal points and then follow the rules for adding or subtracting whole numbers, placing the decimal point in the same column as above.
When one number has more decimal places than another, use 0's to give them the same number of decimal places.
Example:
76.69 + 51.37
1) Line up the decimal points:
76.69
+51.37
2) Then add.
76.69
+51.37
128.06
Example:
12.924 + 3.6
1) Line up the decimal points:
12.924
+ 3.600
2) Then add.
12.924
+ 3.600
16.524
Subtracting Decimals
To subtract decimals, line up the decimal points and then follow the rules for adding or subtracting whole numbers, placing the decimal point in the same column as above.
When one number has more decimal places than another, use 0's to give them the same number of decimal places.
Example:
18.2 - 6.008
1) Line up the decimal points.
18.2
- 6.008
2) Add extra 0's, using the fact that 18.2 = 18.200
18.200
- 6.008
3) Subtract.
18.200
- 6.008
12.192
Comparing Decimal Numbers
Symbols are used to show how the size of one number compares to another. These symbols are < (less than), > (greater than), and = (equals). To compare the size of decimal numbers, we compare the whole number portions first. The larger decimal number is the one with the lager whole number portion. If the whole number parts are both equal, we compare the decimal portions of the numbers. The leftmost decimal digit is the most significant digit. Compare the pairs of digits in each decimal place, starting with the most significant digit until you find a pair that is different. The number with the larger digit is the larger number. Note that the number with the most digits is not necessarily the largest.
Example:
Compare 1 and 0.002. We begin by comparing the whole number parts: in this case 1>0, 0 being the whole number part of 0.002, and so 1>0.002.
Example:
Compare 0.402 and 0.412. The numbers 0.402 and 0.412 have the same number of digits, and their whole number parts are both 0. We compare the next most significant digit of each number, the digit in the tenths place, 4 in each case. Since they are equal, we go on to the hundredths place, and in this case, 0<1, so 0.402<0.412.
Example:
Compare 120.65 and 34.999. Comparing the whole number parts, 120>34, so 120.65>34.999.
Example:
Compare 12.345 and 12.097. Since the whole number parts are both equal, we compare the decimal portions starting with the tenths digit. Since 3>0, we have 12.345>12.097.
Note:
Remember that adding extra zeros to the right of a decimal does not change its value:
2.4 = 2.40 = 2.400 = 2.4000.
Rounding Decimal Numbers
To round a number to any decimal place value, we want to find the number with zeros in all of the lower places that is closest in value to the original number. As with whole numbers, we look at the digit to the right of the place we wish to round to. Note: When the digit 5, 6, 7, 8, or 9 appears in the ones place, round up; when the digit 0, 1, 2, 3, or 4 appears in the ones place, round down.
Examples:
Rounding 1.19 to the nearest tenth gives 1.2 (1.20).
Rounding 1.545 to the nearest hundredth gives 1.55.
Rounding 0.1024 to the nearest thousandth gives 0.102.
Rounding 1.80 to the nearest one gives 2.
Rounding 150.090 to the nearest hundred gives 200.
Rounding 4499 to the nearest thousand gives 4000.
Estimating Sums and Differences
We can use rounding to get quick estimates on sums and differences of decimal numbers. First round each number to the place value you choose, then add or subtract the rounded numbers to estimate the sum or difference.
Example:
To estimate the sum 119.36 + 0.56 to the nearest whole number, first round each number to the nearest one, giving us 119 + 1, then add to get 120.
Multiplying Decimal Numbers
Multiplying decimals is just like multiplying whole numbers. The only extra step is to decide how many digits to leave to the right of the decimal point. To do that, add the numbers of digits to the right of the decimal point in both factors.
Example:
4.032 × 4
We can multiply 4032 by 4 to get 16128. There are three decimal places in 4.032, so place the decimal three digits from the right:
4.032 × 4 = 16.128
Example:
6.74 × 9.063
We can multiply 674 by 9063 to get 6108462. Then there are 5 decimal places: two in the number 6.74 and three in the number 9.063, so place the decimal five digits from the right:
6.74 × 9.063 = 61.08462.
Dividing Whole Numbers, with Remainders
Example:
1400 ÷ 7..
Since 14 ÷ 7 = 2, and 1400 is 100 times greater than 14, the answer is 2 × 100 = 200.
Many problems are similar to the above example, where the answer is easily obtained by adding on or taking off an appropriate number of 0's. Others are more complicated.
Example:
4934 ÷ 6. Use long division.
So the answer is 822 with a remainder of 2, written 822 R2.
To double-check that the answer is correct, multiply the quotient by the divisor and add the remainder:
(822 × 6) + 2 = 4932 + 2 = 4934.
Dividing Whole Numbers, with Decimal Portions
Example:
Find 32 ÷ 6 to the nearest whole number.
32 ÷ 6 = 5 r2. 6 is the divisor; 2 is the remainder.
2 is closer to 0 than 6, so round down. The answer is 5.
Dividing Decimals by Whole Numbers
To divide a decimal by a whole number, use long division, and just remember to line up the decimal points:
Example:
13.44 ÷ 12.
When rounding an answer, divide one place further than the place you're rounding to, and round the result. Add 0's to the right of the number being divided, if necessary.
Example:
1.0 ÷ 6. Round to the nearest thousandth.
To round 0.16666 . . . to the nearest thousandth, we take 4 places to the right of the decimal point and round to 3 places. Here, we round 0.1666 to 0.167, the answer.
Dividing Decimals by Decimals
To divide by a decimal, multiply that decimal by a power of 10 great enough to obtain a whole number. Multiply the dividend by that same power of 10. Then the problem becomes one involving division by a whole number instead of division by a decimal.
Example:
0.144 ÷ 0.12
Multiplying the divisor (0.12) and the dividend (0.144) by 100, then dividing, gives the same result.
The answer is 1.2.
Be aware that some problems are less difficult and do not require this procedure.
Example:
6 ÷ 2.00
This is the same as 6 ÷ 2! The answer is 3.
Exponents (Powers of 2, 3, 4, ...)
Exponential notation is useful in situations where the same number is multiplied repeatedly. The notation is often shown as "^"
The number being multiplied is called the base, and the exponent tells how many times the base is multiplied by itself.
Example:
4 ×4 ×4 ×4 ×4 ×4 = 46
The base in this example is 4, the exponent is 6.
We refer to this as four to the sixth power, or four to the power of six, written as 4^6.
Examples:
2 ×2 ×2 = 2^3 = 8
1.1"2 = 1.1 × 1.1 = 1.21
0.5^3 = 0.5 × 0.5 × 0.5 = 0.125
10^6 = 10 × 10 × 10 × 10 × 10 × 10 = 1000000
Observe that the base may be a decimal.
Special Cases:
A number with an exponent of two is referred to as the square of a number.
The square of a whole number is known as a perfect square. The numbers 1, 4, 9, 16, and 25 are all perfect squares.
A number with an exponent of three is referred to as the cube of a number.
The cube of a whole number is known as a perfect cube. The numbers 1, 8, 27, 64, and 125 are all perfect cubes.
Note:
A number written with an exponent of 1 is the same as the given number.
23^1 = 23.
Factorial Notation n!
The product of the first n whole numbers is written as n!, and is the product
1 × 2 × 3 × 4 × … × (n - 1) × n.
Examples:
4! = 1 × 2 × 3 × 4 = 24
11! = 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10 × 11 = 39916800
Tricks:
When dividing factorials, note that many of the numbers cancel out!
Note:
The number 0! Is defined to be 1.
Square Roots
The square root of a whole number n is the number r with the property that r × r = n.
We write this as
.
We say that the number n is the square of the number r.
Examples:
The square root of 9 is 3, since 3 × 3 = 9.
The square root of 289 is 17, since 17 × 17 = 289.
The square root of 2 is close to 1.41421. We say close to because the digits to the right of the decimal point in the square root of 2 continue forever, without any repeating pattern. Such a number is called an irrational number, meaning that it cannot be written as a fraction.
Tricks:
Since the square root of a whole number n is the number r with the property that r × r = n, we always have
That is, the square of the square root of any number is just the original number.
We also have, for any number r that the square root of the square of r is the absolute value of r.
We say the absolute value, because the notation actually means the positive square root of n.
Example:
From the example above, we see that each positive number n actually has 2 numbers r that satisfy r × r = n, one is positive, and the other is negative.
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Whole Numbers and Their Basic Properties
Using Whole Numbers
Whole numbers
Place value
Expanded form
Ordering
Rounding whole numbers
Divisibility tests
Operations and Their Properties
Commutative property of addition and multiplication
Associative property
Distributive property
The zero property of addition
The zero property of multiplication
Multiplicative identity
Order of operations
The whole numbers are the counting numbers and 0. The whole numbers are 0, 1, 2, 3, 4, 5, ...
The position, or place, of a digit in a number written in standard form determines the actual value the digit represents. This table shows the place value for various positions:
| Place (underlined) |
Name of Position |
|
1 000 |
Ones (units) position |
|
1 000 |
Tens |
|
1 000 |
Hundreds |
|
1 000 |
Thousands |
|
1 000 000 |
Ten thousands |
|
1 000 000 |
Hundred Thousands |
|
1 000 000 |
Millions |
|
1 000 000 000 |
Ten Millions |
|
1 000 000 000 |
Hundred millions |
|
1 000 000 000 |
Billions |
Example:
The number 721040 has a 7 in the hundred thousands place, a 2 in the ten thousands place, a one in the thousands place, a 4 in the tens place, and a 0 in both the hundreds and ones place.
The expanded form of a number is the sum of its various place values.
Example:
9836 = 9000 + 800 + 30 + 6.
Symbols are used to show how the size of one number compares to another. These symbols are < (less than), > (greater than), and = (equals.) For example, since 2 is smaller than 4 and 4 is larger than 2, we can write: 2 < 4, which says the same as 4 > 2 and of course, 4 = 4.
To compare two whole numbers, first put them in standard form. The one with more digits is greater than the other. If they have the same number of digits, compare the most significant digits (the leftmost digit of each number). The one having the larger significant digit is greater than the other. If the most significant digits are the same, compare the next pair of digits from the left. Repeat this until the pair of digits is different. The number with the larger digit is greater than the other.
Example: 402 has more digits than 42, so 402 > 42.
Example: 402 and 412 have the same number of digits. We compare the leftmost digit of each number: 4 in each case. Moving to the right, we compare the next two numbers: 0 and 1. Since 0 < 1, 402 < 412.
To round to the nearest ten means to find the closest number having all zeros to the right of the tens place. Note: when the digit 5, 6, 7, 8, or 9 appears in the ones place, round up; when the digit 0, 1, 2, 3, or 4 appears in the ones place, round down.
Examples:
Rounding 119 to the nearest ten gives 120.
Rounding 155 to the nearest ten gives 160.
Rounding 102 to the nearest ten gives 100.
Similarly, to round a number to any place value, we find the number with zeros in all of the places to the right of the place value being rounded to that is closest in value to the original number.
Examples:
Rounding 180 to the nearest hundred gives 200.
Rounding 150090 to the nearest hundred thousand gives 200000.
Rounding 1234 to the nearest thousand gives 1000.
Rounding is useful in making estimates of sums, differences, etc.
Example:
To estimate the sum 119360 + 500 to the nearest thousand, first round each number in the sum, resulting in a new sum of 119000 + 1000.. Then add to get the estimate of 120000.
There are many quick ways of telling whether or not a whole number is divisible by certain basic whole numbers. These can be useful tricks, especially for large numbers.
| Divisibility by 2 |
Commutative Property of Addition and Multiplication
Addition and Multiplication are commutative: switching the order of two numbers being added or multiplied does not change the result.
Examples:
100 + 8 = 8 + 100
100 × 8 = 8 × 100
Addition and multiplication are associative: the order that numbers are grouped in addition and multiplication does not affect the result.
Examples:
(2 + 10) + 6 = 2 + (10 + 6) = 18
2 × (10 × 6) = (2 × 10) × 6 =120
The distributive property of multiplication over addition: multiplication may be distributed over addition.
Examples:
10 × (50 + 3) = (10 × 50) + (10 × 3)
3 × (12+99) = (3 × 12) + (3 × 99)
Adding 0 to a number leaves it unchanged. We call 0 the additive identity.
Example:
88 + 0 = 88
The Zero Property of Multiplication
Multiplying any number by 0 gives 0.
Example:
88 × 0 = 0
0 × 1003 = 0
We call 1 the multiplicative identity. Multiplying any number by 1 leaves the number unchanged.
Example:
88 × 1 = 88
The order of operations for complicated calculations is as follows:
1) Perform operations within parentheses.
2) Multiply and divide, whichever comes first, from left to right.
3) Add and subtract, whichever comes first, from left to right.
Example:
1 + 20 × (6 + 2) ÷ 2 =
1 + 20 × 8 ÷ 2 =
1 + 160 ÷ 2 =
1 + 80 =
81.
A whole number is divisible by 2 if the digit in its units position is even, (either 0, 2, 4, 6, or 8).
Examples:
The number 84 is divisible by 2 since the digit in the units position is 4, which is even.
The number 333336 is divisible by 2 since the digit in the units position is 6, which is even.
The number 1297000 is divisible by 2 since the digit in the units position is 0, which is even.
A whole number is divisible by 3 if the sum of all its digits is divisible by 3.
Examples:
The number 177 is divisible by three, since the sum of its digits is 15, which is divisible by 3.
The number 8882151 is divisible by three, since the sum of its digits is 33, which is divisible by 3.
The number 162345 is divisible by three, since the sum of its digits is 21, which is divisible by 3.
If a number is not divisible by 3, the remainder when it is divided by 3 is the same as the remainder when the sum of its digits is divided by 3.
Examples:
The number 3248 is not divisible by 3, since the sum of its digits is 17, which is not divisible by 3. When 3248 is divided by 3, the remainder is 2, since when 17, the sum of its digits, is divided by three, the remainder is 2.
The number 172345 is not divisible by 3, since the sum of its digits is 22, which is not divisible by 3. When 172345 is divided by 3, the remainder is 1, since when 22, the sum of its digits, is divided by three, the remainder is 1.
A whole number is divisible by 4 if the number formed by the last two digits is divisible by 4.
Examples:
The number 3124 is divisible by 4 since the number formed by its last two digits, 24, is divisible by 4.
The number 1333336 is divisible by 4 since the number formed by its last two digits, 36, is divisible by 4.
The number 1297000 is divisible by 4 since the number formed by its last two digits, 0, is divisible by 4.
If a number is not divisible by 4, the remainder when the number is divided by 4 is the same as the remainder when the last two digits are divided by 4.
Example:
The number 172345 is not divisible by 4, since the number formed by its last two digits, 45, is not divisible by 4. When 172345 is divided by 4, the remainder is 1, since when 45 is divided by 4, the remainder is 1.
A whole number is divisible by 5 if the digit in its units position is 0 or 5.
Examples:
The number 95 is divisible by 5 since the last digit is 5.
The number 343370 is divisible by 5 since the last digit is 0.
The number 129700195 is divisible by 5 since the last digit is 5.
If a number is not divisible by 5, the remainder when it is divided by 5 is the same as the remainder when the last digit is divided by 5.
Example:
The number 145632 is not divisible by 5, since the last digit is 2. When 145632 is divided by 5, the remainder is 2, since 2 divided by 5 is 0 with a remainder of 2.
The number 7332899 is not divisible by 5, since the last digit is 9. When 7332899 is divided by 5, the remainder is 4, since 9 divided by 5 is 1 with a remainder of 4.
A number is divisible by 6 if it is divisible by 2 and divisible by 3. We can use each of the divisibility tests to check if a number is divisible by 6: its units digit is even and the sum of its digits is divisible by 3.
Examples:
The number 714558 is divisible by 6, since its units digit is even, and the sum of its digits is 30, which is divisible by 3.
The number 297663 is not divisible by 6, since its units digit is not even.
The number 367942 is not divisible by 6, since it is not divisible by 3. The sum of its digits is 31, which is not divisible by 3, so the number 367942 is not divisible by 3.
A whole number is divisible by 8 if the number formed by the last three digits is divisible by 8.
Examples:
The number 88863024 is divisible by 8 since the number formed by its last three digits, 24, is divisible by 8.
The number 17723000 is divisible by 8 since the number formed by its last three digits, 0, is divisible by 8.
The number 339122483984 is divisible by 8 since the number formed by its last three digits, 984, is divisible by 8.
If a number is not divisible by 8, the remainder when the number is divided by 8 is the same as the remainder when the last three digits are divided by 8.
Example:
The number 172045 is not divisible by 8, since the number formed by its last three digits, 45, is not divisible by 8. When 172345 is divided by 8, the remainder is 5, since when 45 is divided by 8, the remainder is 5.
A whole number is divisible by 9 if the sum of all its digits is divisible by 9.
Examples:
The number 1737 is divisible by nine, since the sum of its digits is 18, which is divisible by 9.
The number 8882451 is divisible by nine, since the sum of its digits is 36, which is divisible by 9.
The number 762345 is divisible by nine, since the sum of its digits is 27, which is divisible by 9.
If a number is not divisible by 9, the remainder when it is divided by 9 is the same as the remainder when the sum of its digits is divided by 9.
Examples:
The number 3248 is not divisible by 9, since the sum of its digits is 17, which is not divisible by 9. When 3248 is divided by 9, the remainder is 8, since when 17, the sum of its digits, is divided by 9, the remainder is 8.
The number 172345 is not divisible by 9, since the sum of its digits is 22, which is not divisible by 9. When 172345 is divided by 9, the remainder is 4, since when 22, the sum of its digits, is divided by 9, the remainder is 4.
A whole number is divisible by 10 if the digit in its units position is 0.
Examples:
The number 1229570 is divisible by 10 since the last digit is 0.
The number 676767000 is divisible by 10 since the last digit is 0.
The number 129700190 is divisible by 10 since the last digit is 0.
If a number is not divisible by 10, the remainder when it is divided by 10 is the same as the units digit.
Examples:
The number 145632 is not divisible by 10, since the last digit is 2. When 145632 is divided by 10, the remainder is 2, since the units digit is 2.
The number 7332899 is not divisible by 10, since the last digit is 9. When 7332899 is divided by 10, the remainder is 4, since the units digit is 9.
Starting with the units digit, add every other digit and remember this number. Form a new number by adding the digits that remain. If the difference between these two numbers is divisible by 11, then the original number is divisible by 11.
Examples:
Is the number 824472 divisible by 11? Starting with the units digit, add every other number:2 + 4 + 2 = 8. Then add the remaining numbers: 7 + 4 + 8 = 19. Since the difference between these two sums is 11, which is divisible by 11, 824472 is divisible by 11.
Is the number 49137 divisible by 11? Starting with the units digit, add every other number:7 + 1 + 4 = 12. Then add the remaining numbers: 3 + 9 = 12. Since the difference between these two sums is 0, which is divisible by 11, 49137 is divisible by 11.
Is the number 16370706 divisible by 11? Starting with the units digit, add every other number:6 + 7 + 7 + 6 = 26. Then add the remaining numbers: 0 + 0 + 3 + 1=4. Since the difference between these two sums is 22, which is divisible by 11, 16370706 is divisible by 11.
A number is divisible by 12 if it is divisible by 4 and divisible by 3. We can use each of the divisibility tests to check if a number is divisible by 12: its last two digits are divisible by 4 and the sum of its digits is divisible by 3.
Examples:
The number 724560 is divisible by 12, since the number formed by its last two digits, 60, is divisible by 4, and the sum of its digits is 30, which is divisible by 3.
The number 36297414 is not divisible by 12, since the number formed by its last two digits, 14, is not divisible by 4.
The number 367744 is not divisible by 12, since it is not divisible by 3. The sum of its digits is 29, which is not divisible by 3, so the number 367942 is not divisible by 3.
A number is divisible by 15 if it is divisible by 3 and divisible by 5. We can use each of the divisibility tests to check if a number is divisible by 15: its units digit is 0 or 5, and the sum of its digits is divisible by 3.
Example:
The number 7145580 is divisible by 15, since its units digit is even, and the sum of its digits is 30, which is divisible by 3.
A whole number is divisible by 16 if the number formed by the last four digits is divisible by 16.
Examples:
The number 898630032 is divisible by 16 since the number formed by its last four digits, 32, is divisible by 16.
The number 1772300000 is divisible by 16 since the number formed by its last four digits, 0, is divisible by 16.
The number 339122481296 is divisible by 16 since the number formed by its last four digits, 1296, is divisible by 16.
If a number is not divisible by 16, the remainder when the number is divided by 16 is the same as the remainder when the last four digits are divided by 16.
Example:
The number 172411045 is not divisible by 16, since the number formed by its last four digits, 1045, is not divisible by 16. When 172411045 is divided by 16, the remainder is 5, since when 1045 is divided by 16, the remainder is 5.
A number is divisible by 18 if it is divisible by 2 and divisible by 9. We can use each of the divisibility tests to check if a number is divisible by 18: its units digit is even and the sum of its digits is divisible by 9.
Examples:
The number 7145586 is divisible by 18, since its units digit is even, and the sum of its digits is 36, which is divisible by 9.
The number 2976633 is not divisible by 18, since its units digit is not even.
The number 367942 is not divisible by 18, since it is not divisible by 9. The sum of its digits is 31, which is not divisible by 9, so the number 367942 is not divisible by 9.
A number is divisible by 20 if its units digit is 0, and its tens digit is even. In other words, the last two digits form one of the numbers 0, 20, 40, 60, or 80.
Examples:
The number 3351002760 is divisible by 20, since the number formed by its last two digits is 60.
The number 802199730000 is divisible by 20, since the number formed by its last two digits is 0.
A number is divisible by 22 if it is divisible by the numbers 2 and 11. We can use each of the divisibility tests to check if a number is divisible by 22: its units digit is even, and the difference between the sums of every other digit is divisible by 11.
Example:
Is the number 117524 divisible by 22? The units digit is even, so it is divisible by 2. The two sums of every other digit are 4 + 5 + 1 = 10 and 2 + 7 + 1 = 10, which have a difference of 0. Since 0 is divisible by 11, 117524 is divisible by 11. Thus, 117524 is divisible by 22, since it is divisible by both 2 and 11.
A number is divisible by 25 if the number formed by the last two digits is any of 0, 25, 50, or 75 (the number formed by its last two digits is divisible by 25).
Examples:
The number 73224050 is divisible by 25, since its last two digits form the number 50.
The number 1008922200 is divisible by 25, since its last two digits form the number 0.
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This is a help resource for 4th through 8th grades. We have just redesigned and reformatted these pages to enable faster loading and display times, but some pages still load slowly. Different browsers display graphics differently, so some math symbols may not appear to line up perfectly in some sentences. Our educational CD-ROMs contain the complete, searchable, indexed help facility shown below, along with the helpful problem-solving tips section. We hope you find this a helpful resource.
Whole numbers and their basic properties
Decimals, whole numbers, and exponents
Using data and statistics
Fractions
Geometry
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Integers
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Introduction to algebra
Positive and negative numbers
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MATHEMATICS LEAGUE
4TH, 5TH, 6TH, 7TH, 8TH GRADE
and ALGEBRA COURSE 1 CONTEST ORGANIZATION
QUESTIONS (TIME LIMITS AND TOPICS) Each contest is a 30-minute multiple-choice test. Questions may involve any topic appropriate to the grade level of the contest. If, for any reason, a question must be dropped, no replacement will be made.
CONTEST COPIES Each school will receive 30 copies of each contest in which they are participating. Schools requiring additional copies of any contest are permitted, on the day of the contest, to make as many additional copies as are required. A separate registration form should be submitted from each participating school.
CONTEST MATERIALS PACKAGE You should receive materials for Algebra Course 1 and Grades 4 & 5 by April 1; for Grade 6 by the last Tuesday in February; for Grades 7 & 8 by the next to the last Tuesday in January. If any contest materials have not been received by these dates, the League should be phoned immediately at 1-201-568-6328. Each contest materials package includes 30 copies of the contest and a solution key for the contest. A school needing additional copies of any contest is permitted, on the day of the contest, to make as many additional copies of the contest as are required. Special arrangements for blind or other handicapped students, or for non-English-speaking students, may be made by any school.
CONTEST AWARDS
GRADES 4 AND 5 and ALGEBRA COURSE 1 In each school, the highest scoring student on each contest receives a book award. Other high scoring students in each school receive certificates of merit.
GRADES 6, 7, AND 8 In each school, a certificate of merit is awarded to the highest scoring student on each contest. For each contest, awards are given to the two schools with the highest total scores in the League and also to the two students with the highest total scores in the League. For each contest, additional regional awards are given to the highest scoring school in each region (which may be a county, province, state or other grouping as determined by the League). Counties/Provinces/States with fewer than fifteen participating schools are grouped together into regions for the purposes of awards. No school may win both a regional award and an overall League award on any one grade level in the same school year. The League reserves the right to break ties based upon performance on selected questions, or, at its option, to issue duplicate awards.
CONTEST LOCATION Each school may administer the contests on its own premises or any other suitable site.
6TH, 7TH, AND 8TH GRADE CONTEST PROCEDURES
(The 4th and 5th Grade and Algebra Course 1 Contests are non-competitive; these procedures do not apply.)
CONTEST DATE Except in unusual circumstances, the contests must be held on the scheduled date. In the event of school closings, special testing days, school trips or other administrative functions, severely inclement weather, or similar disruptions of the normal school day, permission is granted to conduct the contest on a proximate school day.
STARTING TIME Each contest may be held, on the scheduled date, at any time convenient for the school. All students officially participating in a contest within the same school should take that contest at the same time. Scores of students taking the contest at any later time should not be included on the score report filed with the League.
PROCTORING Each contest must be actively proctored at all times by a teacher. Neither the proctor nor anyone else may interpret any question to any student during the contest.
ELIGIBILITY Only students officially registered in the same accredited school of record may participate on that school's team. A student may take only a contest designed for a grade the student has not completed or a higher grade (regardless of the math course in which the student is enrolled.) Students taking the 4th or 5th grade contest may also take the 6th, 7th, or 8th grade contest. Students taking the 6th, 7th, or 8th grade contest may take only one such contest (but they may take the Algebra Course 1 Contest). On each contest, all official participants must take the contest in school at the exact same time. A student taking the contest at a later time or period or on a later day must not be included on the score report. Students absent on the contest day may take the contest but must not be listed on the score report.
MATERIALS ALLOWED Only plain paper, pencil or pen, and any calculator without a QWERTY keyboard, may be used by the participants. No graph paper, compasses, straight edges, rulers, printed mathematical tables, or other devices shall be allowed, except where special arrangements have been made for handicapped students or when dictionaries are made accessible to non-English-speaking students.
START OF THE CONTEST Each contestant should be given a copy of the contest and should complete the information requested on the cover page of the contest. Answers submitted for each question must appear in the appropriate space in the answer column. Answers written elsewhere will receive no credit. After the signal to begin is given by the proctor, the timing of the contest will begin.
TIME WARNINGS Warnings that "fifteen minutes remain," that "five minutes remain," and that "one minute remains" should be given at the appropriate times. No other warnings or announcements (relative to the contest) should be made to any contestant during the contest.
MARKING THE ANSWERS At the end of the contest, the question papers should be collected by the advisor. The advisor should then open the sealed envelope containing the solution key and should mark each paper, awarding 1 point for each correct answer. All papers should be marked exactly according to the solution key. If you wish to appeal an answer, please follow the appeals procedure, but score your students' papers according to the official answer key. The League has the option to disqualify any school that submits a mismarked paper.
SUBMITTING CONTEST RESULTS ONLINE The advisor should score the contests. For each contest, the advisor should submit the scores of the school's participants to the League's Internet Score Report Center. The school score for each contest will be the sum of the scores of the five highest scoring participants from the same school of record. The score report must be submitted to our Internet Score Report Center by Friday of the contest week.Student papers may be returned to the students, except that papers with scores above 30 must be held until June 1.
APPEALS PROCEDURE Appeals will be awarded only on the basis of an incorrect official answer or a correct alternative interpretation of a question. Detailed explanations of alternative interpretations should be made in the comments section when filing the score reports. Appeals filed with the League must include the names of all students listed on the score report for whom an appeal is being filed. If you disagree with an official answer, file an appeal. You must use only the official solution key in grading student papers.
AUTHENTICATION OF RESULTS League policy is to authenticate scores and eligibility of participants from schools winning major awards. The League reserves the right to authenticate scores and/or to reexamine students or validate student solutions before granting official status to any score. Schools must keep all papers with scores above 30 until June 1. The League has the option to disqualify any school that submits a mismarked paper. A school disqualified for cause on any contest is ineligible for awards in any League contest.
Directors:
Dan Flegler / Phone: 1-201-568-6328, Fax: 1-201-816-0125
Steve Conrad / Phone: 1-516-365-5656, Fax: 1-516-365-5657
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How to get your school involved in Math League Contests
Check the Math League Registration Center for the Math League Contests in your state to see if your school is participating, and in which contests. Schools are listed by County in the U.S., and by Province in Canada. If your school is not yet registered, and you would like to see them compete in Math League Contests, you can encourage them to participate. Here's how:
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The Math League was formed in 1977 by Steven R. Conrad and Daniel Flegler.
In October 1985, Steven R. Conrad and Daniel Flegler were both honored by President Ronald Reagan as recipients of Presidential awards for "Excellence in Mathematics Teaching." Mr. Conrad was the winner from New York, and Mr. Flegler was the winner from New Jersey. Mr. Flegler was the 1977 recipient of Princeton University's award for "Distinguished Secondary School Teaching." Mr. Conrad and Mr. Flegler have been preparing contests for math students across North America since 1977. They have co-authored 18 books.
Steven R. Conrad taught at Roslyn High School, Roslyn, New York, from 1980 to 1998. Prior to that, he taught at Benjamin Cardozo High School in Bayside, New York and Francis Lewis High School in Flushing, New York.
He began his undergraduate education at Washington University, St. Louis. He received a B.S. from Queens College and an M.S. from Yeshiva University. He has done additional graduate study at the University of San Francisco, Fordham University, and St. John's University, where he earned a certificate in School Administration.
More than 60 of Mr. Conrad's students have been named to the honors group of the Intel National Science Talent Search for mathematics papers they have written. Six of them have finished in the top 10 nationwide. His sons are both math professors.
He was Problem Editor for The Mathematics Student Journal (an official journal of the National Council of Teachers of Mathematics) from 1972 to 1978 and was an associate editor of The New York State Mathematics Teachers' Journal for 3 years. He has contributed problems to many journals, and has had articles published in Mathematics Magazine, The Mathematics Teacher, , American Mathematical Monthly, and Crux Mathematicorum. He is a reviewer for The Mathematics Teacher. For 4 years, Mr. Conrad was the author for the American Regions Mathematics League. He has also been the contest author for the Association of Mathematics Teachers of New Jersey, and director of contests for New York City. Currently, he is the author for contests sponsored by Bergen County, New Jersey, Suffolk County, New York, and Fairfax County, Virginia. He has also authored contests for Montgomery County, Maryland and Nassau County, New York. County, New York. Mr. Conrad served for six years as a member of the committee which developed the SAT II for the College Board.
Daniel Flegler taught at Waldwick High School, Waldwick, New Jersey, from 1965 to 1991, where he served as department chairman for 11 years. Mr. Flegler received his B.A. from Brown University and his M.A. from Columbia University. He has done additional graduate work at Fairleigh Dickinson University, the University of Iowa, and Rutgers University, from which he received his certificate in educational administration.
More than 15 of Mr. Flegler's students have been named to the honors group of the Intel National Science Talent Search for mathematics papers they have written. Three of them have finished in the top 40 nationwide.
From 1972-78, Mr. Flegler served on the Executive Committee of the Association of Mathematics Teachers of New Jersey, and was assistant editor of the New Jersey Mathematics Teacher. In addition, he has written contest problems for both New York City and Nassau County, New York. He has also directed the contests for Bergen County, New Jersey.
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Frequently Asked Questions
Books and Copying
Math Help
- I have a math question. Can I e-mail you for help on my math homework from Math League?
- Can you mail or e-mail me the solutions for contest xxx?
Fees & Billing
- How much does the High School League cost for the school year?
- I didn't order the high school contests but I received a set anyway. Must I pay for these?
Contests
- How many contests are in a set?
- How do I check if my school is registered for contests?
- I received an invoice for my order but I haven't received the materials yet. Do I have to pay in advance?
- How do I obtain additional Certificates of Merit for my top students?
- How much does the High School League cost for the school year?
- My school is closed or on a special schedule the day of the contest. May we still participate?
- My package of 4th Grade Contests (or 5th Grade or Algebra Course 1) did not include a score report form. How do I report the scores?
- One of my best students was absent from school on the day of the contest. May this student take the contest later?
- I'm scheduled for both the 7th and 8th grade contest. Must both groups take the test at the same time or on the same day? Must all my students take the contest at the same time?
- I didn't order the high school contests but I received a set anyway. Must I pay for these?
- I'm a new high school advisor this year, although my school was in the League last year. I have not yet received my high school package. What should I do?
- Some of my 8th grade students are studying algebra (or other high school math classes). Are they permitted to participate in the 8th grade contest?
- Should my 8th grade students enroll in the algebra course 1 contest or the 8th grade contest?
- I teach a 6th grade student who is doing 7th grade math. In which contest should this student participate?
- We only have 1 or 2 students in our school interested in these contests. May we participate?
- Our school wants to participate but we do not want our scores listed. May we participate on an unofficial basis?
- My child is being home-schooled. May my child participate in the contest?
Answers to Questions
Books and Copying
May I make copies of pages from your books of past contests to use with my students?
If you have registered for this year's contest, you may duplicate pages from our books for use with your class. If you are not registered for this year's contest, then you may not duplicate pages from the books.
Do the Volume 6 books contain the contests that appeared in previous volumes?
Later book volumes do not contain the contests published in previous contest problem books. You can view a complete listing of the contests published in each volume here.
Math Help
I have a math question. Can I e-mail you for help on my math homework from Math League?
We are happy to answer letters regarding questions that appear on our contests.
You are also welcome to browse our online Help Reference, which contains math reference information, with sample questions and solutions for grades 4-8.
We do not respond to math questions that do not appear on our contests. If you need help with homework or another question, visit "Ask Dr. Math" (K-12 grades only).
Can you mail or e-mail me the solutions for contest xxx?
Sorry, we do not have solutions for our contests in a suitable format for sending by e-mail. Our contest problem books have complete solutions, and can be ordered online. Just use any of the links on the left marked 'Shopping" to browse our list of contest books.
Fees and Billing
I received an invoice for my order but I haven't received the materials yet. Do I have to pay in advance?
As indicated on your invoice, payment is due upon receipt of all materials ordered. It is not necessary to pay any part of your order until you have received the complete order, although payment is certainly accepted (and welcomed) at the time you receive the invoice.
How much does the High School League cost for the school year?
The cost of entering the High School League for a school year is $100. There are no other fees associated with it.
I didn't order the high school contests but I received a set anyway. Must I pay for these?
Since the first high school contest occurs early in the school year, the League mails a set of contests to any advisor whose school was enrolled in the League last year. If you wish to participate in the League, please complete the registration form that was included with the contest package. The cost of participation is $100. If you do not wish to participate, please discard the materials in a secure manner.
Contests
How many contests are in a set?
High school contests come with 6 sets of 30 contest copies each: one set for each of the year's six contests.
All other contests consist of 30 contest copies per set. You may make additional copies of any contest on the day of the contest.
How do I check if my school is registered for contests?
A new contest registration index page has been added for teachers, administrators, and parents to check thier school's registration status. The web page is best viewed with a Javascript-enabled browser. The list is updated frequently. School registration listings will appear about 2 weeks after contest registration is mailed to us; about 1 week for schools using our online registration form.
I received an invoice for my order but I haven't received the materials yet. Do I have to pay in advance?
Payment is due upon receipt of all materials ordered, as indicated on your invoice. It is not necessary to pay any part of your order until you have received the complete order, although payment is certainly accepted (and welcomed) at the time you receive the invoice.
How do I obtain additional Certificates of Merit for my top students?
If your school needs more Certificates of Merit, send your name, school, and school mailing address to our mailer at:
Math League Certificates
P.O. Box 17
Tenafly, NJ 07670-0017
Include a self-addressed, stamped envelope (two stamps required) large enough to hold certificates.
My school is closed or on a special schedule the day of the contest. May we still participate?
You may reschedule the contest for either the alternate testing date listed or, with the permission of the League, for another date close to the official testing date.
My package for 4th Grade Contests (or 5th Grade or Algebra Course 1) did not include a score report form. How do I report the scores?
Our contests for 4th Grade, 5th Grade, and Algebra Course 1 are intramural contests that do not require the submission of scores to the League. Instead of giving plaques to the top two schools and top two students in the League as we do on all our other contests, we provide each school with a book prize to award to the top student in the school. We also include certificates of merit for other high-scoring students.
One of my best students was absent from school on the day of the contest. May this student take the contest later?
Although you may administer the contest to this student and award this student a certificate of merit if this student is the highest scoring student in your school, you may not submit the student's score to the League. Only students who take the test during the first testing are eligible for official recognition by the League.
I'm scheduled for the 6th, 7th and 8th grade contest. Must all grades take the test at the same time or on the same day? Must all my students take the contest at the same time?
For 4th grade, 5th grade, and Algebra Course 1, students may take the contest whenever it is convenient. You may give the 6th, 7th, and 8th grade contests on different dates. For each of these contests, students whose scores are reported to the League must all take the contest at the same time. If you give the 6th (or 7th or 8th) grade contest at several different times and on several different days, only the scores of the students who were present for the first testing session may be reported to the League.
I'm a new high school advisor this year, although my school was in the League last year. I have not yet received my high school package. What should I do?
Your school package was sent in September to last year's adviser. Please check and see if the package can be located. If not, e-mail us at www.mathleague.com (or call 1-201-568-6328) and we will ship another package immediately.
Some of my 8th grade students are studying algebra (or other high school math classes). Are they permitted to participate in the 8th grade contest?
Any student who has not yet completed the 8th grade may participate in the 8th grade contest, regardless of the math course in which they may be enrolled.
Should my 8th grade students enroll in the Algebra Course 1 contest or the 8th grade contest?
Students who take the algebra course 1 contest may also take any other contest the League sponsors.
I teach a 6th grade student who is doing 7th grade math. In which contest should this student participate?
A student may participate officially in only one of the contests for grades 6, 7, and 8. A 6th grade student taking 7th grade math may participate officially in either the 6th Grade Contest or the 7th Grade Contest, but not both. In this case, the choice of contest is not made by us: it's made by you and/or your student.
We only have 1 or 2 students in our school interested in these contests. May we participate?
There is no minimum number of students required to participate in any of these contests.
Our school wants to participate but we do not want our scores listed. May we participate on an unofficial basis?
Any school may choose to be an unofficial participant in the contests. Unofficial schools receive the same materials as official schools, but are not eligible for plaques. For grades 6, 7, and 8, only high scoring schools and students are listed on the score report summary published by the League. For high schools, the scores of all schools are listed in the score report summary unless a school requests not to be listed.
My child is being home-schooled. May my child participate in the contest?
Your child can participate on either an official or an unofficial basis. To participate unofficially, order our contest subscription package for homeschoolers, which will be sent to you in May for testing at home. You can compare your child's unofficial performance on the contest to scores of official participants by viewing the contest results at our Web site. If you want your child to be an official participant, you need to arrange with an accredited local school to order the official contest package, and proctor the contest for your child on the official contest date. All test materials would be mailed to your local school.
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