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Parent Category: Math League Website

Published: 16 September 2008

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Parent Category: Math League Website

Published: 16 September 2008

Algebra Course 1 Contest

Math League's Algebra Course 1 Contests are a great way to motivate students learning algebra for the first time. The questions range from basic algebra skills, to more difficult problems, requiring creative solutions by applying algebra course 1 techniques. A wide variety of word problems are included on each contest, in addition to computational problems, to stress the value of applied algebra techniques. These contests are provided for intraschool competition, and come with certificates of merit for your school's high scoring students, and one of our High School Contest problem books for your school's top scorer. Here's a quote from one of last year's Algebra Course 1 students who participated in our contest: "I hope you keep making those tests. It really gets your brain to work" -Amber L. Contest Format: Each contest consists of 30 multiple-choice questions that you can do in 30 minutes. On each 3-page contest, the questions on the 1st page are generally straightforward, those on the 2nd page are moderate in difficulty, and those on the 3rd page are more difficult. The questions require no more knowledge than that of a first year high school algebra course.

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

 

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Parent Category: Math League Website

Published: 16 September 2008

6th, 7th, and 8th Grade Contests

Math League's 6th, 7th, and 8th grade contests challenge students and schools in interschool league competitions. Students in each league compete for the highest scores, while schools compete for the highest team score: the total of the top 5 scores in each school. Each contest's questions cover material appropriate to each grade level. Questions may cover: basic topics, plus exponents, fractions, reciprocals, decimals, rates, ratios, percents, angle measurement, perimeter, area, circumference, basic roots, patterns, sequences, integers, triangles and right triangles, and other topics, depending on the grade level. Detailed solution sheets demonstrate the methods used to solve each problem. These contests encourage a variety of problem-solving skills and methods, to improve students' abilities and understanding of mathematical connections, while having fun! Contest Format: Each contest consists of 35 multiple-choice questions that you can do in 30 minutes. On each 3-page contest, the questions on the 1st page are generally straightforward, those on the 2nd page are moderate in difficulty, and those on the 3rd page are more difficult. There is a 6th Grade Score Report, and a 7th and 8th Grade Score Report sent to schools in each league after the contest.

 

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

Details

Parent Category: Math League Website

Published: 16 September 2008

Details

Parent Category: Math League Website

Published: 16 September 2008

Positive and negative numbers

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

About Positive and Negative Numbers

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

The following fractions are all equal:

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

The following mixed numbers are all equal:

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

The Number Line

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

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

Examples:

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

Absolute Value of Positive and Negative Numbers

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

Examples:

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

Adding Positive and Negative Numbers

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

Examples:

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

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

Example:

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

Example:

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

Example:

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

Example:

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

Subtracting Positive and Negative Numbers

Subtracting a number is the same as adding its opposite.

Examples:

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

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

Multiplying Positive and Negative Numbers

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

Examples:

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

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

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

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

To multiply any number of numbers:

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

Example:

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

Dividing Positive and Negative Numbers

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

Examples:

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

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

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

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

Coordinates

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

Example:

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

Comparing Positive and Negative Numbers

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

Examples:

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

Reciprocals of Negative Numbers

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

Examples:

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

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

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Parent Category: Math League Website

Published: 16 September 2008

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Parent Category: Math League Website

Published: 16 September 2008

Metric units and Measurement

Length
Volume
Mass
Time
Temperature
Decimals in measurement

Length

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

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

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

Volume

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

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

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

Mass

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

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

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

Time

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

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

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

Temperature

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

Decimals in measurement

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

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

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Parent Category: Math League Website

Published: 16 September 2008

Integers

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

Positive and Negative Integers

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

Example:

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

The Number Line

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

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

Examples:

9 > 4, 6 > -9, -2 > -8, and 0 > -5

Absolute Value of an Integer

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

Examples:

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

Adding Integers

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

Examples:

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

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

Example:

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

Example:

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

Example:

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

Example:

53 + (-53) = ?

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

Subtracting Integers

Subtracting an integer is the same as adding its opposite.

Examples:

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

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

Multiplying Integers

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

Examples:

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

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

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

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

To multiply any number of integers:

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

Example:

4 × (-2) × 3 × (-11) × (-5) = ?

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

Dividing Integers

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

Examples:

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

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

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

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

Integer coordinates

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

Example:

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

Comparing Integers

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

Examples:

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

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Parent Category: Math League Website

Published: 16 September 2008

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 one-fourth of that number.)

10% = 1/10 = 0.1
(10% of any number equals one-tenth of that number.)

1% = 1/100 = 0.01
(1% of any number equals one-hundredth 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 six-sided 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.

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: 1-1, 1-2, 1-3, 1-4, 1-5, 1-6, 2-2, 2-3, 2-4, 2-5, 2-6, 3-3, 3-4, 3-5, 3-6, 4-4, 4-5, 4-6, 5-5, 5-6, and 6-6. (We think of a 1 and a 2, a 1-2, 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.

Details

Parent Category: Math League Website

Published: 16 September 2008

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.