General
 Details
 Parent Category: Math League Website
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 xcoordinate 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 ycoordinate 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 xaxis and the yaxis 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
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Percent and Probability
Percent
What is a percent?
Percent as a fraction
Percent as a decimal
Estimating percents
Interest
Simple interest
Compound interest
Percent increase and decrease
Percent discount
Chances and probability
What is an event?
Possible outcomes of an event
Probability
What is a Percent?
A percent is a ratio of a number to 100. A percent can be expressed using the percent symbol %.
Example: 10 percent or 10% are both the same, and stand for the ratio 10:100.
Percent as a fraction
A percent is equivalent to a fraction with denominator 100.
Example: 5% of something = 5/100 of that thing.
Example: 2 1/2% is equal to what fraction?
Answer:
2 1/2% = (2 1/2)/100 = 5/200 = 1/40
Example: 52% most nearly equals which one of 1/2, 1/4, 2, 8, or 1/5?
Answer: 52% = 52/100. This is very close to 50/100, or 1/2.
Example: 13/25 is what %?
We want to convert 13/25 to a fraction with 100 in the denominator: 13/25 = (13 × 4)/(25 × 4) = 52/100, so 13/25 = 52%.
Alternatively, we could say: Let 13/25 be n%, and let us find n. Then 13/25 = n/100, so cross multiplying, 13 × 100 = 25 × n, so 25n = 13 × 100 = 1300. Then 25n ÷ 25 = 1300 ÷ 25, so n = 1300 ÷ 25 = 52. So 13/25 = n% = 52%.
Example: 8/200 is what %?
Method 1: 8/200 = (4 × 2)/(100 × 2), so 8/200 = 4/100 = 4%.
Method 2: Let 8/200 be n%. Then 8/200 = n/100, so 200 × n = 800, and 200n ÷ 200 = 800 ÷ 200 = 4, so n% = 4%.
Example: Write 80% as a fraction in lowest terms.
80% = 80/100, which is equal to 4/5 in lowest terms.
Percent as a decimal
Percent and hundredths are basically equivalent. This makes conversion between percent and decimals very easy.
To convert from a decimal to a percent, just move the decimal 2 places to the right. For example, 0.15 = 15 hundredths = 15%.
Example:
0.0006 = 0.06%
Converting from percent to decimal form is similar, only you move the decimal point 2 places to the left. You must also be sure, before doing this, that the percentage itself is expressed in decimal form, without fractions.
Example:
Express 3% in decimal form. Moving the decimal 2 to the left (and adding in 0's to the left of the 3 as place holders,) we get 0.03.
Example:
Express 97 1/4% in decimal form. First we write 97 1/4 in decimal form: 97.25. Then we move the decimal 2 places to the left to get 0.9725, so 97 1/4% = 0.9725. This makes sense, since 97 1/4% is nearly 100%, and 0.9725 is nearly 1.
Estimating percents
When estimating percents, it is helpful to remember the fractional equivalent of some simple percents.
100% = 1
(100% of any number equals that number.)
50% = 1/2 = 0.5
(50% of any number equals half of that number.)
25% = 1/4 = 0.25
(25% of any number equals onefourth of that number.)
10% = 1/10 = 0.1
(10% of any number equals onetenth of that number.)
1% = 1/100 = 0.01
(1% of any number equals onehundredth of that number.)
Because it is very easy to switch between a decimal and a percent, estimating a percent is as easy as estimating a fraction as a decimal, and converting to a percent by multiplying by 100.
Example:
Estimate 19 as a percent of 80.
As a fraction, 19/80 20/80 = 1/4 =0.25 = 25%. The step used to estimate the percent occurred when we estimated 19/80 as 20/80.
The exact percent is actually 23.75%, so the estimate of 25% is only 1.25% off. (About 1 part in 100.)
Example:
Estimate 7 as a percent of 960.
As a fraction, 7/960 7/100 = 0.007 = 0.7%. The step used to estimate the percent occurred when we estimated 7/960 as 7/1000.
The exact percent, to the nearest thousandth of a percent, is actually 0.729%.
To estimate the percent of a number, we may convert the percent to a fraction, if useful, to estimate the percent.
Example:
Estimate 13% of 72.
Twice 13% is 26%, which is very close to 25%, and 25%=1/4. We may multiply both sides by 1/2 to get an estimate for 13%: 13% 12.5% = 1/2 × 25% = 1/2 × 1/4 = 1/8. Using our estimate of 1/8 for 13%, 1/8 × 72 = 9, so we get an estimate of 9 for 13% of 72.
If we had calculated this exactly, 13% of 72 equals 9.36. It may look like we did a lot more work to get the estimate of 9 that just multiplying 72 by 0.13, but with practice, keeping in mind some simple percents and the fractions they are equal to will enable you to estimate some number combinations very quickly.
Example:
Estimate 9.6% of 51.
Method 1: We could estimate 9.6% of 50. It would be easy to estimate 9.6% of 100, which is just 9.6. Since 50 is half of 100, we can just take half of 9.6, which is 4.8. The actual value of 9.6% of 51 is 4.896, so an estimate of 4.8 is pretty good.
Method 2: We could estimate 10% of 51, which is just 5.1. This is not as close an estimate as method 1, but is still a good estimate of the actual answer of 4.896.
Interest
Interest is a fee paid to borrow money. It is usually charged as a percent of the total amount borrowed. The percent charged is called the interest rate. The amount of money borrowed is called the principal. There are two types of interest, simple interest and compound interest.
Example: A bank charges 7% interest on a $1000 loan. It will cost the borrower 7% of $1000, which is $70, for each year the money is borrowed. Note that when the loan is up, the borrower must pay back the original $1000.
Simple Interest
Simple interest is interest figured on the principal only, for the duration of the loan. Figure the interest on the loan for one year, and multiply this amount by the number of years the money is borrowed for.
Example: A bank charges 8% simple interest on a $600 loan, which is to be paid back in two years. It will cost the borrower 8% of $600, which is $48, for each year the money is borrowed. Since it is borrowed for two years, the total charge for borrowing the money will be $96. After the two years the borrower will still have to pay back the original $600.
Compound Interest
Compound interest is interest figured on the principal and any interest owed from previous years. The interest charged the first year is just the interest rate times the amount of the loan. The interest charged the second year is the interest rate, times the sum of the loan and the interest from the first year. The interest charged the third year is the interest rate, times the sum of the loan and the first two years' interest amounts. Continue figuring the interest in this way for any additional years of the loan.
Example: A bank charges 8% compound interest on a $600 loan, which is to be paid back in two years. It will cost the borrower 8% of $600 the first year, which is $48. The second year, it will cost 8% of $600 + $48 = $648, which is $51.84. The total amount of interest owed after the two years is $48 + $51.84 = $99.84. Note that this is more than the $96 that would be owed if the bank was charging simple interest.
Example: A bank charges 4% compound interest on a $1000 loan, which is to be paid back in three years. It will cost the borrower 4% of $1000 the first year, which is $40. The second year, it will cost 4% of $1000 + $40 = $1040, which is $41.60. The third year, it will cost 4% of $1040 + $41.60 = $1081.60, which is $43.26 (with rounding). The total amount of interest owed after the three years is $40 + $41.60 + 43.26 = $124.86.
Percent increase and decrease
Percent increase and decrease of a value measure how that value changes, as a percentage of its original value.
Example: A collectors' comic book is worth $120 in 1994, and in 1995 its value is $132. The change is $132  $120 = $12, an increase in price of $12; since $12 is 10% of $120, we say its value increased by 10% from 1994 to 1995.
Example: A bakery makes a chocolate cake that has 8 grams of fat per slice. A new change in the recipe lowers the fat to 6 grams of fat per slice. The change is 8g  6g = 2g, a decrease of 2 grams; since 2 grams is 25% of 8, we say that the new cake recipe has 25% less fat, or a 25% decrease in fat.
Example: Amy is training for the 1500 meter run. When she started training she could run 1500 meters in 5 minutes and 50 seconds. After a year of practice her time decreased by 8%. How fast can she run the race now? Her old time was 5 × 60 + 50 = 350 seconds, and 8% of 350 is 28, so she can run the race in 350  28 = 322 seconds (5 minutes and 22 seconds).
Example: A fishing magazine sells 110000 copies each month. The company's president wants to increase the sales by 6%. How many extra magazines would they have to sell to reach this goal? This problem is easy, since it only asks for the change in sales: 6% of 110000 equals 6600 more magazines.
Percent Discount
A discount is a decrease in price, so percent discount is the percent decrease in price.
Example: Chocolate bars normally cost 80 cents each, but are on sale for 40 cents each, which is 50% of 80, so the chocolate is on sale at a 50% discount.
Example: A compact disc that sells for $12 is on sale at a 20% discount. How much does the disc cost on sale? The amount of the discount is 20% of $12, which is $2.40, so the sale price is $12.00  $2.40 = $9.60.
Example: Movie tickets sell for $8.00 each, but if you buy 4 or more you get $1.00 off each ticket. What percent discount is this? We figure $1 as a percentage of $8: $1.00/$8.00 × 100% = 12.5%, so this is a 12.5% discount.
What is an event?
An event is an experiment or collection of experiments.
Examples:
The following are examples of events.
1) A coin toss.
2) Rolling a die.
3) Rolling 5 dice.
4) Drawing a card from a deck of cards.
5) Drawing 3 cards from a deck.
6) Drawing a marble from a bag of different colored marbles.
7) Spinning a spinner in a board game.
8) Tossing a coin and rolling a die.
Possible Outcomes of an Event
Possible outcomes of an event are the results which may occur from any event. (Remember, they may not occur.)
Examples:
The following are possible outcomes of events.
1) A coin toss has two possible outcomes. The outcomes are "heads" and "tails".
2) Rolling a regular sixsided die has six possible outcomes. You may get a side with 1, 2, 3, 4, 5, or 6 dots.
3) Drawing a card from a regular deck of 52 playing cards has 52 possible outcomes. Each of the 52 playing cards is different, so there are 52 possible outcomes for drawing a card.
4) How many different outcomes are there for the color of marble that may be drawn from a bag containing 3 red, 4 green, and 5 blue marbles? This event has 3 possible outcomes. You may get a red marble, a green marble, or a blue marble. Even if the marbles are different sizes, the outcome we are considering is the color of the marble that is drawn.
5) How many different outcomes are there for the colors of two marbles that may be drawn from a bag containing 3 red, 4 green, and 5 blue marble? This event has 6 possible outcomes: you may get two reds, two greens, two blues, a red and blue, a red and green, or a blue and green.
6) Rolling two regular dice, one of them red and one of them blue, has 36 possible outcomes. The outcomes are listed in the table below.
Red Die  
Result:  Red1  Red2  Red3  Red4  Red5  Red6  
Blue1  Blue1, Red1  Blue1, Red2  Blue1, Red3  Blue1, Red4  Blue1, Red5  Blue1, Red6  
Blue2  Blue2, Red1  Blue2, Red2  Blue2, Red3  Blue2, Red4  Blue2, Red5  Blue2, Red6  
Blue Die 
Blue3  Blue3, Red1  Blue3, Red2  Blue3, Red3  Blue3, Red4  Blue3, Red5  Blue3, Red6 
Blue4  Blue4, Red1  Blue4, Red2  Blue4, Red3  Blue4, Red4  Blue4, Red5  Blue4, Red6  
Blue5  Blue5, Red1  Blue5, Red2  Blue5, Red3  Blue5, Red4  Blue5, Red5  Blue5, Red6  
Blue6  Blue6, Red1  Blue6, Red2  Blue6, Red3  Blue6, Red4  Blue6, Red5  Blue6, Red6 
Note that the event tells us how to think of the outcomes. Even though there are 12 different marbles in example 4, the event tells us to count only the color of the die, so there are three outcomes. In example 6, the two dice are different, and there are 36 possible outcomes. Suppose we don't care about the color of the dice in example 6. Then we would only see 21 different outcomes: 11, 12, 13, 14, 15, 16, 22, 23, 24, 25, 26, 33, 34, 35, 36, 44, 45, 46, 55, 56, and 66. (We think of a 1 and a 2, a 12, as being the same as a 2 and a 1.)
Probability of an Outcome
The probability of an outcome for a particular event is a number telling us how likely a particular outcome is to occur. This number is the ratio of the number of ways the outcome may occur to the number of total possible outcomes for the event. Probability is usually expressed as a fraction or decimal. Since the number of ways a certain outcome may occur is always smaller or equal to the total number of outcomes, the probability of an event is some number from 0 through 1.
Example:
Suppose there are 10 balls in a bucket numbered as follows: 1, 1, 2, 3, 4, 4, 4, 5, 6, and 6. A single ball is randomly chosen from the bucket. What is the probability of drawing a ball numbered 1? There are 2 ways to draw a 1, since there are two balls numbered 1. The total possible number of outcomes is 10, since there are 10 balls.
The probability of drawing a 1 is the ratio 2/10 = 1/5.
Example:
Suppose there are 10 balls in a bucket numbered as follows: 1, 1, 2, 3, 4, 4, 4, 5, 6, and 6. A single ball is randomly chosen from the bucket. What is the probability of drawing a ball with a number greater than 4? There are 3 ways this may happen, since 3 of the balls are numbered greater than 4. The total possible number of outcomes is 10, since there are 10 balls. The probability of drawing a number greater than 4 is the ratio 3/10. Since this ratio is larger than the one in the previous example, we say that this event has a greater chance of occurring than drawing a 1.
Example:
Suppose there are 10 balls in a bucket numbered as follows: 1, 1, 2, 3, 4, 4, 4, 5, 6, and 6. A single ball is randomly chosen from the bucket. What is the probability of drawing a ball with a number greater than 6? Since none of the balls are numbered greater than 6, this can occur in 0 ways. The total possible number of outcomes is 10, since there are 10 balls. The probability of drawing a number greater than 6 is the ratio 0/10 = 0.
Example:
Suppose there are 10 balls in a bucket numbered as follows: 1, 1, 2, 3, 4, 4, 4, 5, 6, and 6. A single ball is randomly chosen from the bucket. What is the probability of drawing a ball with a number less than 7? Since all of the balls are numbered greater than 7, this can occur in 10 ways. The total possible number of outcomes is 10, since there are 10 balls. The probability of drawing a number less than 7 is the ratio 10/10 = 1.
Note in the last two examples that a probability of 0 meant that the event would not occur, and a probability of 1 meant the event definitely would occur.
Example:
Suppose a card is drawn at random from a regular deck of 52 cards. What is the probability that the card is an ace? There are 4 different ways that the card can be an ace, since 4 of the 52 cards are aces. There are 52 different total outcomes, one for each card in the deck. The probability of drawing an ace is the ratio 4/52 = 1/13.
Example:
Suppose a regular die is rolled. What is the probability of getting a 3 or a 6? There are a total of 6 possible outcomes. Rolling a 3 or a 6 are two of them, so the probability is the ratio of 2/6 = 1/3.
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Ratio and Proportion
Ratio
Comparing ratios
Proportion
Rate
Converting rates
Average rate of speed
Ratio
A ratio is a comparison of two numbers. We generally separate the two numbers in the ratio with a colon (:). Suppose we want to write the ratio of 8 and 12.
We can write this as 8:12 or as a fraction 8/12, and we say the ratio is eight to twelve.
Examples:
Jeannine has a bag with 3 videocassettes, 4 marbles, 7 books, and 1 orange.
1) What is the ratio of books to marbles?
Expressed as a fraction, with the numerator equal to the first quantity and the denominator equal to the second, the answer would be 7/4.
Two other ways of writing the ratio are 7 to 4, and 7:4.
2) What is the ratio of videocassettes to the total number of items in the bag?
There are 3 videocassettes, and 3 + 4 + 7 + 1 = 15 items total.
The answer can be expressed as 3/15, 3 to 15, or 3:15.
Comparing Ratios
To compare ratios, write them as fractions. The ratios are equal if they are equal when written as fractions.
Example:
Are the ratios 3 to 4 and 6:8 equal?
The ratios are equal if 3/4 = 6/8.
These are equal if their cross products are equal; that is, if 3 × 8 = 4 × 6. Since both of these products equal 24, the answer is yes, the ratios are equal.
Remember to be careful! Order matters!
A ratio of 1:7 is not the same as a ratio of 7:1.
Examples:
Are the ratios 7:1 and 4:81 equal? No!
7/1 > 1, but 4/81 < 1, so the ratios can't be equal.
Are 7:14 and 36:72 equal?
Notice that 7/14 and 36/72 are both equal to 1/2, so the two ratios are equal.
Proportion
A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal.
3/4 = 6/8 is an example of a proportion.
When one of the four numbers in a proportion is unknown, cross products may be used to find the unknown number. This is called solving the proportion. Question marks or letters are frequently used in place of the unknown number.
Example:
Solve for n: 1/2 = n/4.
Using cross products we see that 2 × n = 1 × 4 =4, so 2 × n = 4. Dividing both sides by 2, n = 4 ÷ 2 so that n = 2.
Rate
A rate is a ratio that expresses how long it takes to do something, such as traveling a certain distance. To walk 3 kilometers in one hour is to walk at the rate of 3 km/h. The fraction expressing a rate has units of distance in the numerator and units of time in the denominator.
Problems involving rates typically involve setting two ratios equal to each other and solving for an unknown quantity, that is, solving a proportion.
Example:
Juan runs 4 km in 30 minutes. At that rate, how far could he run in 45 minutes?
Give the unknown quantity the name n. In this case, n is the number of km Juan could run in 45 minutes at the given rate. We know that running 4 km in 30 minutes is the same as running n km in 45 minutes; that is, the rates are the same. So we have the proportion
4km/30min = n km/45min, or 4/30 = n/45.
Finding the cross products and setting them equal, we get 30 × n = 4 × 45, or 30 × n = 180. Dividing both sides by 30, we find that n = 180 ÷ 30 = 6 and the answer is 6 km.
Converting rates
We compare rates just as we compare ratios, by cross multiplying. When comparing rates, always check to see which units of measurement are being used. For instance, 3 kilometers per hour is very different from 3 meters per hour!
3 kilometers/hour = 3 kilometers/hour × 1000 meters/1 kilometer = 3000 meters/hour
because 1 kilometer equals 1000 meters; we "cancel" the kilometers in converting to the units of meters.
Important:
One of the most useful tips in solving any math or science problem is to always write out the units when multiplying, dividing, or converting from one unit to another.
Example:
If Juan runs 4 km in 30 minutes, how many hours will it take him to run 1 km?
Be careful not to confuse the units of measurement. While Juan's rate of speed is given in terms of minutes, the question is posed in terms of hours. Only one of these units may be used in setting up a proportion. To convert to hours, multiply
4 km/30 minutes × 60 minutes/1 hour = 8 km/1 hour
Now, let n be the number of hours it takes Juan to run 1 km. Then running 8 km in 1 hour is the same as running 1 km in n hours. Solving the proportion,
8 km/1 hour = 1 km/n hours, we have 8 × n = 1, so n = 1/8.
Average Rate of Speed
The average rate of speed for a trip is the total distance traveled divided by the total time of the trip.
Example:
A dog walks 8 km at 4 km per hour, then chases a rabbit for 2 km at 20 km per hour. What is the dog's average rate of speed for the distance he traveled?
The total distance traveled is 8 + 2 = 10 km.
Now we must figure the total time he was traveling.
For the first part of the trip, he walked for 8 ÷ 4 = 2 hours. He chased the rabbit for 2 ÷ 20 = 0.1 hour. The total time for the trip is 2 + 0.1 = 2.1 hours.
The average rate of speed for his trip is 10/2.1 = 100/21 kilometers per hour.
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Space figures and basic solids
Space figures
Crosssection
Volume
Surface area
Cube
Cylinder
Sphere
Cone
Pyramid
Tetrahedron
Prism
Space Figure
A space figure or threedimensional figure is a figure that has depth in addition to width and height. Everyday objects such as a tennis ball, a box, a bicycle, and a redwood tree are all examples of space figures. Some common simple space figures include cubes, spheres, cylinders, prisms, cones, and pyramids. A space figure having all flat faces is called a polyhedron. A cube and a pyramid are both polyhedrons; a sphere, cylinder, and cone are not.
CrossSection
A crosssection of a space figure is the shape of a particular twodimensional "slice" of a space figure.
Example:
The circle on the right is a crosssection of the cylinder on the left.
The triangle on the right is a crosssection of the cube on the left.
Volume
Volume is a measure of how much space a space figure takes up. Volume is used to measure a space figure just as area is used to measure a plane figure. The volume of a cube is the cube of the length of one of its sides. The volume of a box is the product of its length, width, and height.
Example:
What is the volume of a cube with sidelength 6 cm?
The volume of a cube is the cube of its sidelength, which is 6^{3} = 216 cubic cm.
Example:
What is the volume of a box whose length is 4cm, width is 5 cm, and height is 6 cm?
The volume of a box is the product of its length, width, and height, which is 4 × 5 × 6 = 120 cubic cm.
Surface Area
The surface area of a space figure is the total area of all the faces of the figure.
Example:
What is the surface area of a box whose length is 8, width is 3, and height is 4? This box has 6 faces: two rectangular faces are 8 by 4, two rectangular faces are 4 by 3, and two rectangular faces are 8 by 3. Adding the areas of all these faces, we get the surface area of the box:
8 × 4 + 8 × 4 + 4 × 3 + 4 × 3 + 8 × 3 + 8 × 3 =
32 + 32 + 12 + 12 +24 + 24=
136.
Cube
A cube is a threedimensional figure having six matching square sides. If L is the length of one of its sides, the volume of the cube is L^{3} = L × L × L. A cube has six squareshaped sides. The surface area of a cube is six times the area of one of these sides.
Example:
The space figure pictured below is a cube. The grayed lines are edges hidden from view.
Example:
What is the volume and surface are of a cube having a sidelength of 2.1 cm?
Its volume would be 2.1 × 2.1 × 2.1 = 9.261 cubic centimeters.
Its surface area would be 6 × 2.1 × 2.1 = 26.46 square centimeters.
Cylinder
A cylinder is a space figure having two congruent circular bases that are parallel. If L is the length of a cylinder, and r is the radius of one of the bases of a cylinder, then the volume of the cylinder is L × pi × r^{2}, and the surface area is 2 × r × pi × L + 2 × pi × r^{2}.
Example:
The figure pictured below is a cylinder. The grayed lines are edges hidden from view.
Sphere
A sphere is a space figure having all of its points the same distance from its center. The distance from the center to the surface of the sphere is called its radius. Any crosssection of a sphere is a circle.
If r is the radius of a sphere, the volume V of the sphere is given by the formula V = 4/3 × pi ×r^{3}.
The surface area S of the sphere is given by the formula S = 4 × pi ×r^{2}.
Example:
The space figure pictured below is a sphere.
Example:
To the nearest tenth, what is the volume and surface area of a sphere having a radius of 4cm?
Using an estimate of 3.14 for pi,
the volume would be 4/3 × 3.14 × 4^{3} = 4/3 × 3.14 × 4 × 4 × 4 = 268 cubic centimeters.
Using an estimate of 3.14 for pi, the surface area would be 4 × 3.14 × 4^{2} = 4 × 3.14 × 4 × 4 = 201 square centimeters.
Cone
A cone is a space figure having a circular base and a single vertex.
If r is the radius of the circular base, and h is the height of the cone, then the volume of the cone is 1/3 × pi × r^{2} × h.
Example:
What is the volume in cubic cm of a cone whose base has a radius of 3 cm, and whose height is 6 cm, to the nearest tenth?
We will use an estimate of 3.14 for pi.
The volume is 1/3 × pi × 3^{2} × 6 = pi ×18 = 56.52, which equals 56.5 cubic cm when rounded to the nearest tenth.
Example:
The pictures below are two different views of a cone.
Pyramid
A pyramid is a space figure with a square base and 4 triangleshaped sides.
Example:
The picture below is a pyramid. The grayed lines are edges hidden from view.
Tetrahedron
A tetrahedron is a 4sided space figure. Each face of a tetrahedron is a triangle.
Example:
The picture below is a tetrahedron. The grayed lines are edges hidden from view.
Prism
A prism is a space figure with two congruent, parallel bases that are polygons.
Examples:
The figure below is a pentagonal prism (the bases are pentagons). The grayed lines are edges hidden from view.
The figure below is a triangular prism (the bases are triangles). The grayed lines are edges hidden from view.
The figure below is a hexagonal prism (the bases are hexagons). The grayed lines are edges hidden from view..
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Coordinates and similar figures
What is a coordinate?
Similar figures
Congruent figures
Rotation
Reflection
Folding
Symmetric figure
What is a Coordinate?
Coordinates are pairs of numbers that are used to determine points in a plane, 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 plane or the starting point for finding all other points. Any other point in the plane has a pair of coordinates (x,y). The x value or xcoordinate tells how far 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 ycoordinate tells how far 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 we like by simply using a pair of numbers.
Example:
The origin below is where the xaxis and the yaxis 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 (3,0), since it is 3 units to the right, and lies on the xaxis. Point C has coordinates (6.3,9), since it is 6.3 units to the right, and 9 units up from the origin. Point D has coordinates (9,2.5); it is 9 units to the right, and 2.5 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 (7,5.5); it is 7 units to the left, and 6 units up from the origin. Point G has coordinates (0,7) since it lies on the yaxis 7 units below the origin.
Similar Figures
Figures that have the same shape are called similar figures. They may be different sizes or turned somewhat.
Example:
The following pairs of figures are similar.
These pairs of figures are not similar.
Congruent Figures
Two figures are congruent if they have the same shape and size.
Example:
The following pairs of figures below are congruent. Note that if two figures are congruent, they must be similar.
The pairs below are similar but not congruent.
The pairs below are not similar or congruent.
Rotation
When a figure is turned, we call it a rotation of the figure. We can measure this rotation in terms of degrees; a 360 degree turn rotates a figure around once back to its original position.
Example:
For the following pairs of figures, the figure on the right is a rotation of the figure on the left.
Reflection
If we flip (or mirror) along some line, we say the figure is a reflection along that line.
Examples:
Reflections along a vertical line:
Reflections along a horizontal line:
Reflections along a diagonal line:
Folding
When we talk about folding a plane figure, we mean folding it as if it were a piece of paper in that shape. We might fold this into a solid figure such as a box, or fold the figure flat along itself.
Example:
Folding the figure on the left into a box:
Folding the figure on the left flat along the dotted line:
Symmetric Figure
A figure that can be folded flat along a line so that the two halves match perfectly is a symmetric figure; such a line is called a line of symmetry.
Examples:
The triangle below is a symmetric figure. The dotted line is the line of symmetry.
The square below is a symmetric figure. It has four different lines of symmetry shown below.
The rectangle below is a symmetric figure. It has two different lines of symmetry shown below.
The regular pentagon below is a symmetric figure. It has five different lines of symmetry shown below.
The circle below is a symmetric figure. Any line that passes through its center is a line of symmetry!
The figures shown below are not symmetric.
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Area and perimeter
AreaArea of a square
Area of a rectangle
Area of a parallelogram
Area of a trapezoid
Area of a triangle
Area of a circle
Perimeter
Circumference of a circle
Area
The area of a figure measures the size of the region enclosed by the figure. This is usually expressed in terms of some square unit. A few examples of the units used are square meters, square centimeters, square inches, or square kilometers.
Area of a Square
If l is the sidelength of a square, the area of the square is l^{2} or l × l.
Example:
What is the area of a square having sidelength 3.4?
The area is the square of the sidelength, which is 3.4 × 3.4 = 11.56.
Area of a Rectangle
The area of a rectangle is the product of its width and length.
Example:
What is the area of a rectangle having a length of 6 and a width of 2.2?
The area is the product of these two sidelengths, which is 6 × 2.2 = 13.2.
Area of a Parallelogram
The area of a parallelogram is b × h, where b is the length of the base of the parallelogram, and h is the corresponding height. To picture this, consider the parallelogram below:
We can picture "cutting off" a triangle from one side and "pasting" it onto the other side to form a rectangle with sidelengths b and h. This rectangle has area b × h.
Example:
What is the area of a parallelogram having a base of 20 and a corresponding height of 7?
The area is the product of a base and its corresponding height, which is 20 × 7 = 140.
Area of a Trapezoid
If a and b are the lengths of the two parallel bases of a trapezoid, and h is its height, the area of the trapezoid is
1/2 × h × (a + b) .
To picture this, consider two identical trapezoids, and "turn" one around and "paste" it to the other along one side as pictured below:
The figure formed is a parallelogram having an area of h × (a + b), which is twice the area of one of the trapezoids.
Example:
What is the area of a trapezoid having bases 12 and 8 and a height of 5?
Using the formula for the area of a trapezoid, we see that the area is
1/2 × 5 × (12 + 8) = 1/2 × 5 × 20 = 1/2 × 100 = 50.
Area of a Triangle
or
Consider a triangle with base length b and height h.
The area of the triangle is 1/2 × b × h.
To picture this, we could take a second triangle identical to the first, then rotate it and "paste" it to the first triangle as pictured below:
or
The figure formed is a parallelogram with base length b and height h, and has area b × ×h.
This area is twice that of the triangle, so the triangle has area 1/2 × b × h.
Example:
What is the area of the triangle below having a base of length 5.2 and a height of 4.2?
The area of a triangle is half the product of its base and height, which is 1/2 ×5.2 × 4.2 = 2.6 × 4.2 = 10.92..
Area of a Circle
The area of a circle is Pi × r^{2} or Pi × r × r, where r is the length of its radius. Pi is a number that is approximately 3.14159.
Example:
What is the area of a circle having a radius of 4.2 cm, to the nearest tenth of a square cm? Using an approximation of 3.14159 for Pi, and the fact that the area of a circle is Pi × r^{2}, the area of this circle is Pi × 4.2^{2} 3.14159 × 4.2^{2} =55.41…square cm, which is 55.4 square cm when rounded to the nearest tenth.
Perimeter
The perimeter of a polygon is the sum of the lengths of all its sides.
Example:
What is the perimeter of a rectangle having sidelengths of 3.4 cm and 8.2 cm? Since a rectangle has 4 sides, and the opposite sides of a rectangle have the same length, a rectangle has 2 sides of length 3.4 cm, and 2 sides of length 8.2 cm. The sum of the lengths of all the sides of the rectangle is 3.4 + 3.4 + 8.2 + 8.2 = 23.2 cm.
Example:
What is the perimeter of a square having sidelength 74 cm? Since a square has 4 sides of equal length, the perimeter of the square is 74 + 74 + 74 + 74 = 4 × 74 = 296.
Example:
What is the perimeter of a regular hexagon having sidelength 2.5 m? A hexagon is a figure having 6 sides, and since this is a regular hexagon, each side has the same length, so the perimeter of the hexagon is 2.5 + 2.5 + 2.5 + 2.5 + 2.5 + 2.5 = 6 × 2.5 = 15m.
Example:
What is the perimeter of a trapezoid having sidelengths 10 cm, 7 cm, 6 cm, and 7 cm? The perimeter is the sum 10 + 7 + 6 + 7 = 30cm.
Circumference of a Circle
The distance around a circle. It is equal to Pi () times the diameter of the circle. Pi or is a number that is approximately 3.14159.
Example:
What is the circumference of a circle having a diameter of 7.9 cm, to the nearest tenth of a cm? Using an approximation of 3.14159 for , and the fact that the circumference of a circle is times the diameter of the circle, the circumference of the circle is Pi × 7.9 3.14159 × 7.9 = 24.81…cm, which equals 24.8 cm when rounded to the nearest tenth of a cm.
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Figures and polygons
Polygon
Regular polygon
Vertex
Triangle
Equilateral triangle
Isosceles triangle
Scalene triangle
Acute triangle
Obtuse triangle
Right triangle
Quadrilateral
Rectangle
Square
Parallelogram
Rhombus
Trapezoid
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Circle
Convex
Polygon
A polygon is a closed figure made by joining line segments, where each line segment intersects exactly two others.
Examples:
The following are examples of polygons:
The figure below is not a polygon, since it is not a closed figure:
The figure below is not a polygon, since it is not made of line segments:
The figure below is not a polygon, since its sides do not intersect in exactly two places each:
Regular Polygon
A regular polygon is a polygon whose sides are all the same length, and whose angles are all the same. The sum of the angles of a polygon with n sides, where n is 3 or more, is 180° × (n  2) degrees.
Examples:
The following are examples of regular polygons:
Examples:
The following are not examples of regular polygons:
Vertex
1) The vertex of an angle is the point where the two rays that form the angle intersect.
2) The vertices of a polygon are the points where its sides intersect.
Triangle
A threesided polygon. The sum of the angles of a triangle is 180 degrees.
Examples:
Equilateral Triangle or Equiangular Triangle
A triangle having all three sides of equal length. The angles of an equilateral triangle all measure 60 degrees.
Examples:
Isosceles Triangle
A triangle having two sides of equal length.
Examples:
Scalene Triangle
A triangle having three sides of different lengths.
Examples:
Acute Triangle
A triangle having three acute angles.
Examples:
Obtuse Triangle
A triangle having an obtuse angle. One of the angles of the triangle measures more than 90 degrees.
Examples:
Right Triangle
A triangle having a right angle. One of the angles of the triangle measures 90 degrees. The side opposite the right angle is called the hypotenuse. The two sides that form the right angle are called the legs. A right triangle has the special property that the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. This is known as the Pythagorean Theorem.
Examples:
Example:
For the right triangle above, the lengths of the legs are A and B, and the hypotenuse has length C. Using the Pythagorean Theorem, we know that A^{2} + B^{2} = C^{2}.
Example:
In the right triangle above, the hypotenuse has length 5, and we see that 3^{2} + 4^{2} = 5^{2} according to the Pythagorean Theorem.
Quadrilateral
A foursided polygon. The sum of the angles of a quadrilateral is 360 degrees.
Examples:
Rectangle
A foursided polygon having all right angles. The sum of the angles of a rectangle is 360 degrees.
Examples:
Square
A foursided polygon having equallength sides meeting at right angles. The sum of the angles of a square is 360 degrees.
Examples:
Parallelogram
Parallelogram
A foursided polygon with two pairs of parallel sides. The sum of the angles of a parallelogram is 360 degrees.
Examples:
Rhombus
A foursided polygon having all four sides of equal length. The sum of the angles of a rhombus is 360 degrees.
Examples:
Trapezoid
A foursided polygon having exactly one pair of parallel sides. The two sides that are parallel are called the bases of the trapezoid. The sum of the angles of a trapezoid is 360 degrees.
Examples:
Pentagon
A fivesided polygon. The sum of the angles of a pentagon is 540 degrees.
Example:
A regular pentagon: 

Hexagon
A sixsided polygon. The sum of the angles of a hexagon is 720 degrees.
Examples:
A regular hexagon: 
An irregular hexagon: 
Heptagon
A sevensided polygon. The sum of the angles of a heptagon is 900 degrees.
Examples:
A regular heptagon:

An irregular heptagon: 
Octagon
An eightsided polygon. The sum of the angles of an octagon is 1080 degrees.
Examples:
A regular octagon:  An irregular octagon: 
Nonagon
A ninesided polygon. The sum of the angles of a nonagon is 1260 degrees.
Examples:
A regular nonagon:  An irregular nonagon: 
Decagon
A tensided polygon. The sum of the angles of a decagon is 1440 degrees.
Examples:
A regular decagon: 
An irregular decagon: 
Circle
A circle is the collection of points in a plane that are all the same distance from a fixed point. The fixed point is called the center. A line segment joining the center to any point on the circle is called a radius.
Example:
The blue line is the radius r, and the collection of red points is the circle.
Convex
A figure is convex if every line segment drawn between any two points inside the figure lies entirely inside the figure. A figure that is not convex is called a concave figure.
Example:
The following figures are convex.
The following figures are concave. Note the red line segment drawn between two points inside the figure that also passes outside of the figure.
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Angles and angle terms
What is an angle?
Degrees: measuring angles
Acute angles
Obtuse angles
Right angles
Complementary angles
Supplementary angles
Vertical angles
Alternate interior angles
Alternate exterior angles
Corresponding angles
Angle bisector
Perpendicular lines
What is an Angle?
Two rays that share the same endpoint form an angle. The point where the rays intersect is called the vertex of the angle. The two rays are called the sides of the angle.
Example: Here are some examples of angles.
We can specify an angle by using a point on each ray and the vertex. The angle below may be specified as angle ABC or as angle CBA; you may also see this written as ABC or as CBA. Note how the vertex point is always given in the middle.
Example: Many different names exist for the same angle. For the angle below, PBC, PBW, CBP, and WBA are all names for the same angle.
Degrees: Measuring Angles
We measure the size of an angle using degrees.
Example: Here are some examples of angles and their degree measurements.
Acute Angles
An acute angle is an angle measuring between 0 and 90 degrees.
Example:
The following angles are all acute angles.
Obtuse Angles
An obtuse angle is an angle measuring between 90 and 180 degrees.
Example:
The following angles are all obtuse.
Right Angles
A right angle is an angle measuring 90 degrees. Two lines or line segments that meet at a right angle are said to be perpendicular. Note that any two right angles are supplementary angles (a right angle is its own angle supplement).
Example:
The following angles are both right angles.
Complementary Angles
Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees. One of the complementary angles is said to be the complement of the other.
Example:
These two angles are complementary.
Note that these two angles can be "pasted" together to form a right angle!
Supplementary Angles
Two angles are called supplementary angles if the sum of their degree measurements equals 180 degrees. One of the supplementary angles is said to be the supplement of the other.
Example:
These two angles are supplementary.
Note that these two angles can be "pasted" together to form a straight line!
Vertical Angles
For any two lines that meet, such as in the diagram below, angle AEB and angle DEC are called vertical angles. Vertical angles have the same degree measurement. Angle BEC and angle AED are also vertical angles.
Alternate Interior Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle D are called alternate interior angles. Alternate interior angles have the same degree measurement. Angle B and angle C are also alternate interior angles.
Alternate Exterior Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle D are called alternate exterior angles. Alternate exterior angles have the same degree measurement. Angle B and angle C are also alternate exterior angles.
Corresponding Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle C are called corresponding angles. Corresponding angles have the same degree measurement. Angle B and angle D are also corresponding angles.
Angle Bisector
An angle bisector is a ray that divides an angle into two equal angles.
Example:
The blue ray on the right is the angle bisector of the angle on the left.
The red ray on the right is the angle bisector of the angle on the left.
Perpendicular Lines
Two lines that meet at a right angle are perpendicular.
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Basic terms
Lines
Points
Intersection
Line segments
Rays
Endpoints
Parallel lines
Lines
A line is one of the basic terms in geometry. We may think of a line as a "straight" line that we might draw with a ruler on a piece of paper, except that in geometry, a line extends forever in both directions. We write the name of a line passing through two different points A and B as "line AB" or as , the twoheaded arrow over AB signifying a line passing through points A and B.
Example: The following is a diagram of two lines: line AB and line HG.
The arrows signify that the lines drawn extend indefinitely in each direction.
Points
A point is one of the basic terms in geometry. We may think of a point as a "dot" on a piece of paper. We identify this point with a number or letter. A point has no length or width, it just specifies an exact location.
Example: The following is a diagram of points A, B, C, and Q:
Intersection
The term intersect is used when lines, rays, line segments or figures meet, that is, they share a common point. The point they share is called the point of intersection. We say that these figures intersect.
Example: In the diagram below, line AB and line GH intersect at point D:
Example: In the diagram below, line 1 intersects the square in points M and N:
Example: In the diagram below, line 2 intersects the circle at point P:
Line Segments
A line segment is one of the basic terms in geometry. We may think of a line segment as a "straight" line that we might draw with a ruler on a piece of paper. A line segment does not extend forever, but has two distinct endpoints. We write the name of a line segment with endpoints A and B as "line segment AB" or as . Note how there are no arrow heads on the line over AB such as when we denote a line or a ray.
Example: The following is a diagram of two line segments: line segment CD and line segment PN, or simply segment CD and segment PN.
Rays
A ray is one of the basic terms in geometry. We may think of a ray as a "straight" line that begins at a certain point and extends forever in one direction. The point where the ray begins is known as its endpoint. We write the name of a ray with endpoint A and passing through a point B as "ray AB" or as . Note how the arrow heads denotes the direction the ray extends in: there is no arrow head over the endpoint.
Example: The following is a diagram of two rays: ray HG and ray AB.
Endpoints
An endpoint is a point used to define a line segment or ray. A line segment has two endpoints; a ray has one.
Example: The endpoints of line segment DC below are points D and C, and the endpoint of ray MN is point M below:
Parallel Lines
Two lines in the same plane which never intersect are called parallel lines. We say that two line segments are parallel if the lines that they lie on are parallel. If line 1 is parallel to line 2, we write this as
line 1  line 2
When two line segments DC and AB lie on parallel lines, we write this as
segment DC  segment AB.
Example: Lines 1 and 2 below are parallel.
Example: The opposite sides of the rectangle below are parallel. The lines passing through them never meet.
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Fractions
Prime numbers
Greatest common factor
Least common multiple
What is a fraction?
Equivalent fractions
Comparing fractions
Converting and reducing fractions
Lowest terms
Improper fractions
Mixed numbers
Converting mixed numbers to improper fractions
Converting improper fractions to mixed numbers
Writing a fraction as a decimal
Rounding a fraction to the nearest hundredth
Adding and subtracting fractions
Adding and subtracting mixed numbers
Multiplying fractions and whole numbers
Multiplying fractions and fractions
Multiplying mixed numbers
Reciprocal
Dividing fractions
Dividing mixed numbers
Simplifying complex fractions
Repeating decimals
Prime Numbers
A whole number greater than one that is divisible by only 1 and itself. The numbers 2, 3, 5, 37, and 101 are some examples of prime numbers.
Greatest Common Factor
The greatest common factor of two or more whole numbers is the largest whole number that divides each of the numbers.
There are two methods of finding the greatest common factor of two numbers.
Method 1: List all the factors of each number, then list the common factors and choose the largest one.
Example:
36: 1, 2, 3, 4, 6, 9, 12, 18, 36
54: 1, 2, 3, 6, 9, 18, 27, 54
The common factors are: 1, 2, 3, 6, 9, and 18.
The greatest common factor is: 18.
Method 2: List the prime factors, then multiply the common prime factors.
Example:
36 = 2 × 2 × 3 × 3
54 = 2 × 3 × 3 × 3
The common prime factors are 2, 3, and 3.
The greatest common factor is 2 × 3 × 3 = 18..
Least Common Multiple
The least common multiple of two or more nonzero whole numbers is the smallest whole number that is divisible by each of the numbers. There are two common methods for finding the least common multiple of 2 numbers.
Method 1:
List the multiples of each number, and look for the smallest number that appears in each list.
Example:
Find the least common multiple of 12 and 42. We list the multiples of each number:
12: 12, 24, 36, 48, 60, 72, 84, ...
42: 42, 84, 126, 168, 190, ...
We see that the number 84 is the smallest number that appears in each list.
Method 2:
Factor each of the numbers into primes. For each different prime number in either of the factorizations, follow these steps:
1. Count the number of times it appears in each of the factorizations.
2. Take the largest of these two counts.
3. Write down that prime number as many times as the count in step 2.
To find the least common multiple take the product of all of the prime numbers written down in steps 1, 2, and 3.
Example:
Find the least common multiple of 24 and 90. First, we find the prime factorization of each number.
24 = 2 × 2 × 2 × 3
90 = 2 × 3 × 3 × 5
The prime numbers 2, 3, and 5 appear in the factorizations. We follow steps 1 through 3 for each of these primes.
The number 2 occurs 3 times in the first factorization and 1 time in the second, so we will use three 2's.
The number 3 occurs 1 time in the first factorization and 2 times in the second, so we will use two 3's.
The number 5 occurs 0 times in the first factorization and 1 time in the second factorization, so we will use one 5.
The least common multiple is the product of three 2's, two 3's, and one 5.
2 × 2 × 2 × 3 × 3 × 5 = 360
Example:
Find the least common multiple of 14 and 49. First, we find the prime factorization of each number.
14 = 2 × 7
49 = 7 × 7
The prime numbers 2 and 7 appear in the factorizations. We follow steps 1 through 3 for each of these primes.
The number 2 occurs 1 times in the first factorization and 0 times in the second, so we will use one 2.
The number 7 occurs 1 time in the first factorization and 2 times in the second, so we will use two 7's.
The least common multiple is the product of one 2 and two 7's.
2 × 7 × 7 = 98
Examples:
Some other least common multiples are listed below.
The least common multiple of 12 and 9 is 36.
The least common multiple of 6 and 18 is 18.
The least common multiple of 2, 3, 4, and 5 is 60.
What is a Fraction?
A fraction is a number that expresses part of a group.
Fractions are written in the form or a/b, where a and b are whole numbers, and the number b is not 0. For the purposes of these web pages, we will denote fractions using the notation a/b, though the preferred notation is generally .
The number a is called the numerator, and the number b is called the denominator.
Examples:
The following numbers are all fractions
1/2, 3/7, 6/10, 4/99
Example:
The fraction 4/6 represents the shaded portion of the circle below. There are 6 pieces in the group, and 4 of them are shaded.
Example:
The fraction 3/8 represents the shaded portion of the circle below. There are 8 pieces in the group, and 3 of them are shaded.
Example:
The fraction 2/3 represents the shaded portion of the circle below. There are 3 pieces in the group, and 2 of them are shaded.
Equivalent Fractions
Equivalent fractions are different fractions which name the same amount.
Examples:
The fractions 1/2, 2/4, 3/6, 100/200, and 521/1042 are all equivalent fractions.
The fractions 3/7, 6/14, and 24/56 are all equivalent fractions.
We can test if two fractions are equivalent by crossmultiplying their numerators and denominators. This is also called taking the crossproduct.
Example:
Test if 3/7 and 18/42 are equivalent fractions.
The first crossproduct is the product of the first numerator and the second denominator: 3 × 42 = 126.
The second crossproduct is the product of the second numerator and the first denominator: 18 × 7 = 126.
Since the crossproducts are the same, the fractions are equivalent.
Example:
Test if 2/4 and 13/20 are equivalent fractions.
The first crossproduct is the product of the first numerator and the second denominator: 2 × 20 = 40.
The second crossproduct is the product of the second numerator and the first denominator: 4 × 13 = 52.
Since the crossproducts are different, the fractions are not equivalent. Since the second crossproduct is larger than the first, the second fraction is larger than the first.
Comparing Fractions
1. To compare fractions with the same denominator, look at their numerators. The larger fraction is the one with the larger numerator.
2. To compare fractions with different denominators, take the cross product. The first crossproduct is the product of the first numerator and the second denominator. The second crossproduct is the product of the second numerator and the first denominator. Compare the cross products using the following rules:
a. If the crossproducts are equal, the fractions are equivalent.
b. If the first cross product is larger, the first fraction is larger.
c. If the second cross product is larger, the second fraction is larger.
Example:
Compare the fractions 3/7 and 1/2.
The first crossproduct is the product of the first numerator and the second denominator: 3 × 2 = 6.
The second crossproduct is the product of the second numerator and the first denominator: 7 × 1 = 7.
Since the second crossproduct is larger, the second fraction is larger.
Example:
Compare the fractions 13/20 and 3/5.
The first crossproduct is the product of the first numerator and the second denominator: 5 × 13 = 65.
The second crossproduct is the product of the second numerator and the first denominator: 20 × 3 = 60.
Since the first crossproduct is larger, the first fraction is larger.
Converting and Reducing Fractions
For any fraction, multiplying the numerator and denominator by the same nonzero number gives an equivalent fraction. We can convert one fraction to an equivalent fraction by using this method.
Examples:
1/2 = (1 × 3)/(2 × 3) = 3/6
2/3 = (2 × 2)/(3 × 2) = 4/6
3/5 = (3 × 4)/(5 × 4) = 12/20
Another method of converting one fraction to an equivalent fraction is by dividing the numerator and denominator by a common factor of the numerator and denominator.
Examples:
20/42 = (20 ÷ 2)/(42 ÷ 2) = 10/21
36/72 = (36 ÷ 3)/(72 ÷ 3) = 12/24
9/27 = (9 ÷ 3)/(27 ÷ 3) = 3/9
When we divide the numerator and denominator of a fraction by their greatest common factor, the resulting fraction is an equivalent fraction in lowest terms.
Lowest Terms
A fraction is in lowest terms when the greatest common factor of its numerator and denominator is 1. There are two methods of reducing a fraction to lowest terms.
Method 1:
Divide the numerator and denominator by their greatest common factor.
12/30 = (12 ÷ 6)/(30 ÷ 6) = 2/5
Method 2:
Divide the numerator and denominator by any common factor. Keep dividing until there are no more common factors.
12/30 = (12 ÷ 2)/(30 ÷ 2) = 6/15 = (6 ÷ 3)/(15 ÷ 3) = 2/5
Improper Fractions
Improper fractions have numerators that are larger than or equal to their denominators.
Examples:
11/4, 5/5, and 13/2 are improper fractions.
Mixed Numbers
Mixed numbers have a whole number part and a fraction part.
Examples:
are mixed numbers also written as 2 3/4 and 6 1/2. In these web pages, we denote mixed numbers in the form a b/c.
Converting Mixed Numbers to Improper Fractions
To change a mixed number into an improper fraction, multiply the whole number by the denominator and add it to the numerator of the fractional part.
Examples:
2 3/4 = ((2 × 4) + 3)/4 =11/4
6 1/2 = ((6 × 2) + 1)/2 = 13/2
Converting Improper Fractions to Mixed Numbers
To change an improper fraction into a mixed number, divide the numerator by the denominator. The remainder is the numerator of the fractional part.
Examples:
11/4 = 11 ÷ 4 = 2 r3 = 2 3/4
13/2 = 13 ÷ 2 = 6 r1 = 6 1/2
Writing a Fraction as a Decimal
Method 1  Convert to an equivalent fraction whose denominator is a power of 10, such as 10, 100, 1000, 10000, and so on, then write in decimal form.
Examples:
1/4 = (1 × 25)/(4 × 25) = 25/100 = 0.25
3/20 = (3 × 5)/(20 × 5) = 15/100 = 0.15
9/8 = (9 × 125)/(8 × 125) = 1125/1000 = 1.125
Method 2  Divide the numerator by the denominator. Round to the decimal place asked for, if necessary.
Example:
13/4 = 13 ÷ 4 = 3.25
Example:
Convert 3/7 to a decimal.
Round to the nearest thousandth.
We divide one decimal place past the place we need to round to, then round the result.
3/7 = 3 ÷ 7 = 0.4285…
which equals 0.429 when rounded to the nearest thousandth.
Example:
Convert 4/9 to a decimal.
Round to the nearest hundredth.
We divide one decimal place past the place we need to round to, then round the result.
4/9 = 4 ÷ 9 = 0.4444…
which equals 0.44 when rounded to the nearest hundredth.
Rounding a Fraction to the Nearest Hundredth
Divide to the thousandths place. If the last digit is less than 5, drop it. This is particularly useful for converting a fraction to a percent, if we want to convert to the nearest percent.
1/3 = 1 ÷ 3 = 0.333… which rounds to 0.33
If the last digit is 5 or greater, drop it and round up.
2/7 = 2 ÷ 7 = 0.285 which rounds to 0.29
Adding and Subtracting Fractions
If the fractions have the same denominator, their sum is the sum of the numerators over the denominator. If the fractions have the same denominator, their difference is the difference of the numerators over the denominator. We do not add or subtract the denominators! Reduce if necessary.
Examples:
3/8 + 2/8 = 5/8
9/2  5/2 = 4/2 = 2
If the fractions have different denominators:
1) First, find the least common denominator.
2) Then write equivalent fractions using this denominator.
3) Add or subtract the fractions. Reduce if necessary.
Example:
3/4 + 1/6 = ?
The least common denominator is 12.
3/4 + 1/6 = 9/12 + 2/12 = 11/12.
Example:
9/10  1/2 = ?
The least common denominator is 10.
9/10  1/2 = 9/10  5/10 = 4/10 = 2/5.
Example:
2/3 + 2/7 = ?
The least common denominator is 21
2/3 + 2/7 = 14/21 + 6/21 = 20/21.
Adding and Subtracting Mixed Numbers
To add or subtract mixed numbers, simply convert the mixed numbers into improper fractions, then add or subtract them as fractions.
Example:
9 1/2 + 5 3/4 = ?
Converting each number to an improper fraction, we have 9 1/2 = 19/2 and 5 3/4 = 23/4.
We want to calculate 19/2 + 23/4. The LCM of 2 and 4 is 4, so
19/2 + 23/4 = 38/4 + 23/4 = (38 + 23)/4 = 61/4.
Converting back to a mixed number, we have 61/4 = 15 1/4.
The strategy of converting numbers into fractions when adding or subtracting is often useful, even in situations where one of the numbers is whole or a fraction.
Example:
13  1 1/3 = ?
In this situation, we may regard 13 as a mixed number without a fractional part. To convert it into a fraction, we look at the denominator of the fraction 4/3, which is 1 1/3 expressed as an improper fraction. The denominator is 3, and 13 = 39/3. So 13  1 1/3 = 39/3  4/3 = (394)/3 = 35/3, and 35/3 = 11 2/3.
Example:
5 1/8  2/3 = ?
This time, we may regard 2/3 as a mixed number with 0 as its whole part. Converting the first mixed number to an improper fraction, we have 5 1/8 = 41/8. The problem becomes
5 1/8  2/3 = 41/8  2/3 = 123/24  16/24 = (123  16)/24 = 107/24.
Converting back to a mixed number, we have 107/24 = 4 11/24.
Example:
92 + 4/5 = ?
This is easy. To express this as a mixed number, just put the whole number and the fraction side by side. The answer is 92 4/5.
Multiplying Fractions and Whole Numbers
To multiply a fraction by a whole number, write the whole number as an improper fraction with a denominator of 1, then multiply as fractions.
Example:
8 × 5/21 = ?
We can write the number 8 as 8/1. Now we multiply the fractions.
8 × 5/21 = 8/1 × 5/21 = (8 × 5)/(1 × 21) = 40/21
Example:
2/15 × 10 = ?
We can write the number 10 as 10/1. Now we multiply the fractions.
2/15 × 10 = 2/15 × 10/1 = (2 × 10)/(15 × 1) = 20/15 = 4/3
Multiplying Fractions and Fractions
When two fractions are multiplied, the result is a fraction with a numerator that is the product of the fractions' numerators and a denominator that is the product of the fractions' denominators.
Example:
4/7 × 5/11 = ?
The numerator will be the product of the numerators: 4 × 5, and the denominator will be the product of the denominators: 7 × 11.
The answer is (4 × 5)/(7 × 11) = 20/77.
Remember that like numbers in the numerator and denominator cancel out.
Example:
14/15 × 15/17 = ?
Since the 15's in the numerator and denominator cancel, the answer is
14/15 × 15/17 = 14/1 × 1/17 = (14 × 1)/(1 × 17) = 14/17
Example:
4/11 × 22/36 = ?
In the solution below, first we cancel the common factor of 11 in the top and bottom of the product, then we cancel the common factor of 4 in the top and bottom of the product.
4/11 × 22/36 = 4/1 × 2/36 = 1/1 × 2/9 = 2/9
Multiplying Mixed Numbers
To multiply mixed numbers, convert them to improper fractions and multiply.
Example:
4 1/5 × 2 2/3 = ?.
Converting to improper fractions, we get 4 1/5 = 21/5 and 2 2/3 = 8/3. So the answer is
4 1/5 × 2 2/3 = 21/5 × 8/3 = (21 × 8)/(5 × 3) = 168/15 = 11 3/15.
Examples:
3/4 × 1 1/8 = 3/4 × 9/8 = 27/32.
3 × 7 3/4 = 3 × 31/4 = (3 × 31)/4 = 93/4 = 23 1/4.
Reciprocal
The reciprocal of a fraction is obtained by switching its numerator and denominator. 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 a fraction and its reciprocal, the product is always 1.
Example:
Find the reciprocal of 31/75. We switch the numerator and denominator to find the reciprocal: 75/31.
Example:
Find the reciprocal of 12 1/2. First, convert the mixed number to an improper fraction: 12 1/2 = 25/2. Next, we switch the numerator and denominator to find the reciprocal: 2/25.
Dividing Fractions
To divide a number by a fraction, multiply the number by the reciprocal of the fraction.
Examples:
7 ÷ 1/5 = 7 × 5/1 = 7 × 5 = 35
1/5 ÷ 16 = 1/5 ÷ 16/1 = 1/5 × 1/16 = (1 × 1)/(5 × 16) = 1/80
3/5 ÷ 7/12 = 3/5 × 12/7 = (3 × 12)/(5 × 7) = 36/35 or 1 1/35
Dividing Mixed Numbers
To divide mixed numbers, you should always convert to improper fractions, then multiply the first number by the reciprocal of the second.
Examples:
1 1/2 ÷ 3 1/8 = 3/2 ÷ 25/8 = 3/2 × 8/25 = (3 × 8)/(2 × 25) = 24/50
1 ÷ 3 3/5 = 1/1 ÷ 18/5 = 1/1 × 5/18 = (1 × 5)/(1 × 18) = 5/18
3 1/8 ÷ 2 = 25/8 ÷ 2/1 = 25/8 × 1/2 = (25 × 1)/(8 × 2) = 25/16 or 1 9/16.
Simplifying Complex Fractions
A complex fraction is a fraction whose numerator or denominator is also a fraction or mixed number.
Example of complex fractions:
otherwise written as (1/4)/(2/3), (3/7)/100, 11/(2/3), and (23 1/5)/(2/3).
To simplify complex fractions, change the complex fraction into a division problem: divide the numerator by the denominator.
The first of these examples becomes
(1/4)/(2/3) = 1/4 ÷ 2/3 = 1/4 × 3/2 = 3/8.
The second of these becomes
(3/7)/100 = 3/7 ÷ 100 = 3/7 × 1/100 = 3/700.
The third of these becomes
11/(2/3) = 11 ÷ 2/3 = 11 × 3/2 = 33/2 = 16 1/2.
The fourth of these becomes
(23 1/5)/(2/3) = 23 1/5 ÷ 2/3 = 116/5 ÷ 2/3 = 116/5 × 3/2 = 174/5 = 34 4/5.
Repeating Decimals
Every fraction can be written as a decimal.
For example, 1/3 is 1 divided by 3.
If you use a calculator to find 1 ÷ 3, the calculator returns 0.333333... This is called a repeating decimal. To represent the idea that the 3's repeat forever, one uses a horizontal bar (overstrike) as shown below:
Example:
What is the repeating decimal for 1/7 ? Dividing 7 into 1, we get 0.142857142..., and we see the pattern begin to repeat with the second 1, so .