Arithmetic Reasoning & Mathematics knowledge

**Integers, Odd and Even Numbers, Prime Numbers, Digits**

**Integers:**-4, -3, -2, -1, 0, 1, 2, 3, 4, …**Consecutive Integers**: Integers that follow in sequence; for example,22, 23, 24, 25. Consecutive Integers can be more generally represented by

n, n + 1 , n + 2, n +3, …

**Odd numbers**: -9, -7, -5, -3, -1, 1, 3, 5, 7, 9, …**Even numbers:**-8, -6, -4, -2, 0, 2, 4, 6, 8, … (Note: Zero is an even number)**Prime Numbers:**2, 3, 4, 7, 11, 13, 17, 19, … (Note: 1 is**NOT**a prime and 2 is the only even prime number)**Digits:**0, 1, 2, 3, 4, 5, 6, 7, 8, 9

| Multiplication |
---|---|

even + even = even | even x even = even |

odd + odd = even | even x odd = even |

even + odd = odd | odd x odd = odd |

Percent means hundredths or number out of 100. For example, 40

percent means 40/100 or .40 or 2/5.

Percent less than 100

**Problem 1:** If the sales tax on a $30 item is $1.80, what is the sales tax rate?

**Solution**: $1.80 = n/100 x $30

n = 6, so 6% is the sale tax rate

Percent Greater than 100

**Problem 2**: What number is 250% of 2?

**Solution**: n = 250/100 x 2

n = 5, so 5 is the number

Percent less than 1

**Problem 3**: 3 is 0.2 percent of what number?

**Solution**: 3 = 0.2/100 x n

n = 1,500 , so 1,500 is the number

Percent Increase/Decrease

**Problem 4**: If the price of a computer was decreased from $1,000 to $750, by what percent was the price decreased?

**Solution**: The price decrease is $250. The percent decrease is the value of n in the equation 250/1000 = n/100. The value of n is 25, so the price was decreased by 25%.

Notes: n% increase means increase/original = n/100;

n% decrease means decrease/original = n/100.

An average is a statistic that is used to summarize data. The most common type of average is the **arithmetic** mean. The average (arithmetic mean) of a list of n numbers is equal to the sum of the numbers divided by n. For example, the mean of 2, 3, 5, 7, and 13 is equal to

2 + 3 + 5 + 7 + 13 / 5 = 6

When the average of a list of n numbers is given, the sum of the numbers can be found. For example if the average of six numbers is 12, the sum of these six numbers is 12 x 6, or 72.

The **median** of a list of numbers is the number in the middle when the numbers are ordered from greatest to least or from least to greatest.

For example, the median of 3, 8, 2, 6, and 9 is 6 because when the numbers are ordered, 2, 3, 6, 8, 9, the number in the middle is 6. When there is an even number of values, the median is the same as the mean of the two middle numbers.

For example, the median of 6, 8, 9, 13, 14, and 16 is

9 + 13 / 2 = 11

The mode of a list of numbers is the number that occurs most often in the list. For example, 7 is the mode of 2, 7, 5, 8, 7, and 12. The numbers 10, 12, 14, 16, and 18 have no mode and the numbers 2, 4, 2, 8, 2, 4, 7, 4, 9, and 11 have two modes, 2 and 4.

Note: The mean, median, and mode can each be considered an average. On the test, the use of the word average refers the arithmetic mean and is induced by “average (arithmetic mean)”. The exception is when a question involves average speed (See problem 2 below). Questions involving the median and mode will have those terms stated as part of the questions text.

**Problem** **1**: In a group of 10 students, 7 are 13 years old and 3 are 17 years old. What is the average (arithmetic mean) age of these 10 students?

Solution: The solution is not the average of 13 and 17, which is 15. In this case the average is

7(13) + 3 (17) / 10 = 91 + 51 / 10 = 14.2 years

The expression “weighted average” comes from the fact that 13 gets a weight factor of 7, whereas 17 gets a

`weight`

factor of 3.

Problem 2: Jane traveled for 2 hours at a rate of 70 kilometers per hour and for 5 hours at a rate of 60 kilometers per hour. What was her average speed for the 7-hour time period?

Solution: In this situation, the average speed is:

Total Distance/Total Time

The total distance is 2(70) + 5 (60) = 440 km.

The total time is 7 hours. Thus the average speed was

440/7 = 62 6/7 kilometers per hour.

Note: In this example the average speed is not the average of the two separate speeds, which would be 65.

**Integers, Odd and Even Numbers, Prime Numbers, Digits**

**Integers:**-4, -3, -2, -1, 0, 1, 2, 3, 4, …**Consecutive Integers**: Integers that follow in sequence; for example,22, 23, 24, 25. Consecutive Integers can be more generally represented by

n, n + 1 , n + 2, n +3, …

**Odd numbers**: -9, -7, -5, -3, -1, 1, 3, 5, 7, 9, …**Even numbers:**-8, -6, -4, -2, 0, 2, 4, 6, 8, … (Note: Zero is an even number)**Prime Numbers:**2, 3, 4, 7, 11, 13, 17, 19, … (Note: 1 is**NOT**a prime and 2 is the only even prime number)**Digits:**0, 1, 2, 3, 4, 5, 6, 7, 8, 9

| Multiplication |
---|---|

even + even = even | even x even = even |

odd + odd = even | even x odd = even |

even + odd = odd | odd x odd = odd |

Percent means hundredths or number out of 100. For example, 40

percent means 40/100 or .40 or 2/5.

Percent less than 100

**Problem 1:** If the sales tax on a $30 item is $1.80, what is the sales tax rate?

**Solution**: $1.80 = n/100 x $30

n = 6, so 6% is the sale tax rate

Percent Greater than 100

**Problem 2**: What number is 250% of 2?

**Solution**: n = 250/100 x 2

n = 5, so 5 is the number

Percent less than 1

**Problem 3**: 3 is 0.2 percent of what number?

**Solution**: 3 = 0.2/100 x n

n = 1,500 , so 1,500 is the number

Percent Increase/Decrease

**Problem 4**: If the price of a computer was decreased from $1,000 to $750, by what percent was the price decreased?

**Solution**: The price decrease is $250. The percent decrease is the value of n in the equation 250/1000 = n/100. The value of n is 25, so the price was decreased by 25%.

Notes: n% increase means increase/original = n/100;

n% decrease means decrease/original = n/100.

An average is a statistic that is used to summarize data. The most common type of average is the **arithmetic** mean. The average (arithmetic mean) of a list of n numbers is equal to the sum of the numbers divided by n. For example, the mean of 2, 3, 5, 7, and 13 is equal to

2 + 3 + 5 + 7 + 13 / 5 = 6

When the average of a list of n numbers is given, the sum of the numbers can be found. For example if the average of six numbers is 12, the sum of these six numbers is 12 x 6, or 72.

The **median** of a list of numbers is the number in the middle when the numbers are ordered from greatest to least or from least to greatest.

For example, the median of 3, 8, 2, 6, and 9 is 6 because when the numbers are ordered, 2, 3, 6, 8, 9, the number in the middle is 6. When there is an even number of values, the median is the same as the mean of the two middle numbers.

For example, the median of 6, 8, 9, 13, 14, and 16 is

9 + 13 / 2 = 11

The mode of a list of numbers is the number that occurs most often in the list. For example, 7 is the mode of 2, 7, 5, 8, 7, and 12. The numbers 10, 12, 14, 16, and 18 have no mode and the numbers 2, 4, 2, 8, 2, 4, 7, 4, 9, and 11 have two modes, 2 and 4.

Note: The mean, median, and mode can each be considered an average. On the test, the use of the word average refers the arithmetic mean and is induced by “average (arithmetic mean)”. The exception is when a question involves average speed (See problem 2 below). Questions involving the median and mode will have those terms stated as part of the questions text.

**Problem** **1**: In a group of 10 students, 7 are 13 years old and 3 are 17 years old. What is the average (arithmetic mean) age of these 10 students?

Solution: The solution is not the average of 13 and 17, which is 15. In this case the average is

7(13) + 3 (17) / 10 = 91 + 51 / 10 = 14.2 years

The expression “weighted average” comes from the fact that 13 gets a weight factor of 7, whereas 17 gets a

`weight`

factor of 3.

Problem 2: Jane traveled for 2 hours at a rate of 70 kilometers per hour and for 5 hours at a rate of 60 kilometers per hour. What was her average speed for the 7-hour time period?

Solution: In this situation, the average speed is:

Total Distance/Total Time

The total distance is 2(70) + 5 (60) = 440 km.

The total time is 7 hours. Thus the average speed was

440/7 = 62 6/7 kilometers per hour.

Note: In this example the average speed is not the average of the two separate speeds, which would be 65.