ACT study guide Math: median

If the sum of n numbers is m and the average of the same n numbers is p, what is the value of n in terms of m and p?

The average is found by taking the total of the values and dividing by how many values there are. In this case, we would have an average of m/n=p, which simplifies to np = m. Next, to write n in terms m of m and p, divide both sides by p to get n=m/p.

If each element in a data set whose mean is x is multiplied by 3, and each resulting product is then reduced by 4, which of the following expressions gives the mean of the resulting data set in terms of x?

(For the sake of simplicity, let every element in the original set have the value x. If each element in the set is multiplied by 3, and then reduced by 4, each element then has the value 3x − 4. In a set where each value is 3x − 4, the mean is )3x − 4

The median of the following data set is 2. Which of the following is a possible value of x? x, 2, 1, 6, 7

(If the median is 2, this means that when the numbers are placed in order from smallest to largest, 2 will be the middle number. Of the given answer choices, this only occurs when) x = 2

The value of a number x is twice the value of y, and the average of the two numbers is 30. What is the value of x?

(If x = 2y, then 1/2x=y and x+y/2 = x+(1/2)x/2 =(3/2)x/2= 3/4x =30. Solving this equation yields the solution) x = 40

If the median of 1, x, y, 4, and z is only x, and y > z > 4, which of the following statements MUST be true?

(If x is the median, it must be in the middle of the list when the items are placed in order. This means two numbers must be smaller than x, and since z and y are larger than 4, the smaller numbers must be 1 and 4. Given this information, the correct order of the list would be 1, 4, x, z, y, and only the inequality in answer choice C holds.) x < z

A professor kept track of the attendance at his Monday-Wednesday-Friday class for one week. The average daily attendance was 32. How many students attended his class on Friday? Monday: 32 wednesday: 34 friday: ?

(If x is the number in attendance on Friday (which is unknown), we can use the formula Average =Sum Count of numbers to find x. Sum: The sum of the students who attended the three classes will be 32 + 34 + x Count of numbers: There are 3 total student counts (1 each for Monday, Wednesday, and Friday), so divide by 3. If the average student count over the 3 days is 32, then set the Average equation equal to 32. Our equation will therefore be: 32=32+34+x3 Multiply by 3: 96 = 32 + 34 + x Combine like terms: 96 = 66 + x Subtract 66: )30 = x

The average daily rainfall for the past six days is 3.22 inches. How many inches of rain must fall on the seventh day for the average daily rainfall over the past week to be 3.40 inches?

(If x is the total rain over the past six days and the average is 3.22, then x6= 3.22, so x = 19.32. For the average over seven days to be 3.40, it must be true that 19.32 + y/7 = 3.4, where y is the rainfall on the seventh day. This equation has a solution of) 4.48.

The average of four consecutive integers is 14.5. What is the sum of the largest and the smallest of the integers?

(Let x be the smallest of the integers. The remaining integers are therefore x + 1, x + 2, and x + 3. Given the average, we can say that x+(x+1)+(x+2)+(x+3)4=14.5, which is equivalent to 4x+64=14.5. Therefore, x+32=14.5, or x = 13. This means the largest integer is 13 + 3 = 16, and the sum of x and x + 3 is 13 + 16 =) 29.

The average of 5 numbers is 20. What is the sum of these same numbers?

(Let x represent the sum of the numbers. Then x5=20, and by cross multiplying, we see that) x = 100

Using the following set of data, determine the mean, mode, and median, respectively. 19, 20, 22, 22, 32

(Mean = average = (19 + 20 + 22 + 22 + 32)/5 = 23 Mode = most common value = 22 Median = middle value = 22) 22, 23, 22

The list of numbers 1, 1, x, y, 10, 14 is written in order of smallest to largest. If the median of the list is 5, which of the following numbers is a possible value of the product xy?

(Since the list has an even number of values, the median is the average of x and y: x+y/2=5, and the sum of x and y must be 10. Therefore the product of x and y must be the product of two numbers that add up to 10. )24

If x > 0, what is the median of the data set consisting of −2x, −5x, x, −8x, and 4x?

(Since we know that x is positive, the terms can be placed in order from smallest to largest: −8x, −5x, −2x, x, 4x. By definition, the median is the middle term in this list) -2x

If the sum of ten numbers is x, which of the following expressions represents the average of these ten numbers?

(The average of any set of numbers is the sum of those numbers divided by how many numbers are in the set.) x/10

The median of a list of seven distinct numbers is 3. If a number x > 3 is added to the list, which of the following will be true of the new list?

(The median is the middle number when the numbers in the list are placed in order. Since there are seven numbers in the original list, 3 must be the fourth number when the list is placed in order. When the new number is added, the new median will be the average of the fourth and fifth numbers. The numbers are distinct, so the fifth number must be larger than 3, and the median will therefore also) be larger than 3.

What is the median of the data list q, x, y, z, w if the inequality y < x < w < 8 < z < q is true?

(The median is the middle value if the list items are put in order from smallest to largest. Using the inequality, w would be the middle number of the data list.) w

To earn a commission for the workweek (Monday through Friday), a salesperson must have average daily sales of $250 or greater for the week. If a salesperson's sales for Monday through Thursday are $98, $255, $175, and $320, what is the LEAST whole number value of sales the salesperson needs to have on Friday to be eligible to earn a commission?

(The smallest average that would allow for the salesperson to get a commission is $250. Therefore, if x represents the person's sales on Friday, it must be that, and 848 + x = 250(5 days) = 1250. This equation has a solution of x = )402

A student has earned the following scores on four 100-point tests this marking period: 63, 72, 88, and 91. What score must the student earn on the fifth and final 100-point test of the marking period to earn an average test grade of 80 for the five tests?

(To find the score on the fifth 100-point test that will yield an average score of 80, first calculate the total of the four scores already obtained: 63 + 72 + 88 + 91 = 314. To obtain an average of 80 on 5 tests, the total score of all 5 tests must be 80 × 5, or 400. The score needed on the last test is equivalent to 400 − 314, or )86

The sum of ten numbers is 250. What is the average of the ten numbers?

250/10=25

The average of five numbers is 12.4. The average of four of these numbers is 11. What is the value of the fifth number?

Let the five numbers be represented by a, b, c, d, and e. Since the average of all five is 12.4, we know that a+b+c+d+e/5=12.4, or a + b + c + d + e = 12.4 × 5 = 62. We can say the four numbers that have an average of 11 are a, b, c, and d. This means that a+b+c+d/4=11, or a + b + c + d = 44. Combining these equations, (a + b + c + d) + e = 62 = 44 + e, so e = 62 − 44 = 18.

The average of a set of six numbers is 10. If 5 is added to each number in the set, what is the average of the new set of numbers?

Let x represent the sum of the six numbers. Then, x since the average is 10, x6=10, and x=60. When 5 is added to each number in the set, the new average is 60+(6×5)/6=90/6=15

What is the median of √2, √5, √3, √1, and √5?

The list written in order from smallest to largest is 1,√2,√3,√5,5. The median is the middle value of this list.

A set of numbers contains m numbers, one of which is even. If a number is randomly selected from the set, what is the probability it is NOT even?

The probability of an event not occurring is one minus the probability it will occur. 1−1/m = m−1/m.

What is the average of the numbers 5/4, 1/2, and x/2?

To find the average of any 3 numbers, add the numbers together and then divide the sum by 3. To add these 3 fractions together, you'll first need to find a common denominator between the fractions: 4. Now you'll need to multiply the 2nd and 3rd fraction by 2 on top and bottom to change the denominators to 4: Finally, add the numbers together and divide by 3 (or multiply by 3's reciprocal: ) 1/3(5/4+1/2+x/2)=2x+7/12