Q9: Word Problems B Flashcards
If A is a subset of B, what does that mean?
All the elements of the set A are in set B.
Example 1: if A is the set {1,8} and B is the set {1, 3, 5, 8}, A is a subset of B
Example 2: if A represents all male employees, and B represents all employees, A is a subset of B.
Data sufficiency:
Of 20 houses built last year in city Y, how many were occupied at the end of the year?
(1) Of all the houses in city Y, 50 percent were occupied at the end of last year
(2) A total of 100 houses in city Y were occupied at the end of last year
E- Not sufficient
Read very carefully! The key words are “last year” in the stem. We need to find number in the orange circle, which is a subset of the 20 built last year. The clues talk about ALL houses, not just the ones built last year, so they don’t help us.
What is the definition of “Median”?
The “middle” value, when numbers are listed from least to greatest:
If there are an odd number of terms, it will be the center term:
1, 3, 6, 8, 10 –> Median is 6
If there are an even number of terms, it will be the average of the two middle terms:
1, 3, 5, 8 –> the Median is between 3 and 5 – > it is the midpoint between them, or average, which is 4.
If there are 15 terms listed from least to greatest, which value will be the median?
Example: With 5 terms, the 3rd is the median
The 8th term.
Formula for odd number of terms: (n+1) / 2
What is the median of the following terms?
2
1
5
8
3.5
First, list from least to greatest:
1, 2, 5, 8
Even number of terms, so median is average of 2 and 5
(2+5) / 2 = 3.5
What is the median of the following:
x+11
x+3
x
x+15
x+4
x+4
List from least to greatest:
x, x+3, x+4, x+11, x+15
The middle is x+4
What is the formula for “average” (also called “mean”)?
What is the formula for “Sum”?
Average = Sum / number of terms
A = S/n
or, S = An
example: the Average of {2, 6, 7} = (2+6+7)/3
= 15/3 = 5
If the average of 8 scores is 21, what is the total of all the scores?
168
S = An = 8*21 = 168
If k = 4, what is the average of:
k
k+2
k-3
3k+1
Total = 6k = 6*4 = 24
A = S/n = 24/4 = 6
If the average of the set {1, 4, 3, 7, 8, x} is 5, what is x?
7
Sum = Average * number of terms
S = An = 5*6 = 30
1+4+3+7+8+x = 30
23+x = 30
x = 7
For the given set, how much higher is the mean (average) than the median?
{3, 4, 4, 5, 11, 15)
2.5
Mean: A = S/n = 42 / 6 = 7
Median for even # of terms = average of middle 2 values
= average (midpoint) of 4 and 5 = 4.5
7 - 4.5 = 2.5
John’s 10th sale, $1200, raises his average sale to $300. What was his average sale before the 10th sale?
$200
After 10th sale: S = An = 10*300 = 3000
Before 10th sale, there were 9 sales. n = 9
The sum was 3000 - 1200 = 1800
A = S / n = 1800 / 9 = $200
What does “Standard Deviation” mean?
(you don’t need the the formula, just the concept)
Standard Deviation measures the spread or variation of the data in a set.
It indicates how far the data points are from the average (mean)
What does a standard deviation of 0 signify?
A standard deviation of 0 means all numbers in the set are equal.
Which set has a larger standard deviation?
A: {2, 4, 6}
B: {3, 4, 5}
A
Both sets have the same average, 4, but the numbers in A are more spread out from the average.
If each term in a set is increased by 5, does the set’s standard deviation increase, decrease, or stay the same?
Stay the same
If the set were plotted on a number line, it would still be spread out in the same pattern, with the same amount of spread between the terms, but it would just slide 5 units to the right.
If each term in a set is multiplied by 1/2, does the set’s standard deviation increase, decrease, or stay the same? (assume that the set includes different numbers)
Decrease
As seen below, the data will get closer together. Each point’s distance from the mean will be cut in half.
If the set’s numbers were all the same, the standard deviation wouldn’t change, because it would be 0 for both.
How many integers are there between 5 and 11, inclusive?
7
A common mistake is to subtract and get 6
Inclusive means it includes the endpoints, 5 and 11.
5, 6, 7, 8, 9, 10, 11
Our formula is Last - First + 1
How many multiples of 5 are there between 74 and 128?
What is the formula?
11
(Last - First) / Increment + 1
First is 75. Last is 125. Increment is 5.
(125 - 75) / 5 + 1 = 11
For an evenly spaced set, such as {3, 5, 7, 9} what is true about the mean and median?
They are equal.
An evenly spaced set means the difference is the same between each term.
What is the formula for the mean and median of an evenly spaced set?
What is the mean and median of {6, 12, 18, 24, 30, 36} ?
(First + Last) / 2
(6+36) / 2 = 21
For an evenly spaced set, the mean and median are the same
What is the formula for the sum of consecutive integers?
For an evenly spaced set:
Formula: Sum = Average * Number of Terms
Average = (First + Last) / 2
Number of Terms = Last - First + 1
What is the sum of all the integers between 10 and 50, inclusive?
1230
For an evenly spaced set:
Formula: Sum = Average * Number of Terms
Average = (First + Last) / 2 = (10 + 50) / 2 = 30
Number of Terms = Last - First + 1 = 50 - 10 + 1 = 41
41*30 = 1230
What is the formula for the sum of consecutive integers between 1 and n, inclusive?
For the sum of integers from 1 to n,
Sum = Average * Number of Terms is:
Using the formula, what is the sum of the integers between 1 and 10, inclusive?
