The Sum of 2 positive numbers is 151. The lesser number is 19 more than the square root of the greater number.What is the value of the greater number minus the lesser number?
The list of numbers 41,35,30,x,y,15 has a median of 25 . The mode of the list of numbers is 15. To the nearest whole number what is the mean of the list?
a = first number ( greater number )
b = second number ( lesser number )
The sum of 2 positive numbers is 151 mean:
a + b = 151
The lesser number is 19 more than the square root of the greater number mean:
b = √a + 19
In equation a + b = 151 replace b with √a + 19
a + b = a + √a + 19 = 151
a + √a + 19 = 151
Subtract 19 to both sides
a + √a + 19 - 19 = 151 - 19
a + √a = 132
Subtract a to both sides
a + √a - a = 132 - a
√a = 132 - a
Raise both sides to power of two:
( √a )² = ( 132 - a )²
a = 132² - 2 ∙ 132 ∙ a + a²
a = 17424 - 264 ∙ a + a²
a = a² - 264 a + 17424
Subtract a to both sides
a - a = a² - 264 a + 17424 - a
0 = a² - 265 a + 17424
a² - 265 a + 17424 = 0
The solutions are:
a1 = 121 and a2 = 144
b = √a + 19
For a1 = 121
b1 = √a + 19 = √121 + 19 = 11 + 19 = 30
For a2 = 144
b2 = √a + 19 = √144 + 19 = 13 + 19 = 31
A conditon is:
a + b = 151
For a1 = 121 and b1 = 30
a1 + b1 = 121 + 30 = 151
satisfies a condition a + b = 151
For a2 = 144 and b2 = 31
a1 + b1 = 144 + 31 = 175 ≠ 151
not satisfies a condition a + b = 151
The solutios are: a = 121 and b = 30
What is the value of the greater number minus the lesser number mean what is a - b.
a - b = 121 - 30 = 91
41 , 35 , 30 , x , y ,15
For even numbers in list the median is the mean of the two middle values.
In this case the median is ( 30 + x ) / 2
Median is 25:
( 30 + x ) / 2 = 25
Multiply both sides by 2
30 + x = 2 ∙ 25
30 + x = 50
Subtract 30 to both sides
30 + x - 30 = 50 - 30
x = 20
List:
41 , 35 , 30 , x , y ,15
41 , 35 , 30 , 20 , y ,15
The mode is the number which appears most often in a set of numbers.
The mode of the list of numbers is 15 mean y = 15
For y = 15 the number 15 appears most often in a set of numbers ( two times ).
Your list:
41 , 35 , 30 , 20 , 15 ,15
The mean is the average.
Mean of the list:
( 41 + 35 + 30 + 20 + 15 + 15 ) / 6 = 156 / 6 = 26
On my typo:
It's written:
For a2 = 144 and b2 = 31
a1 + b1 = 144 + 31 = 175 ≠ 151
not satisfies a condition a + b = 151
It needs to be written:
It's written:
For a2 = 144 and b2 = 31
a2 + b2 = 144 + 31 = 175 ≠ 151
not satisfies a condition a + b = 151
X = Larger number.
Sqrt(x) + 19 = Smaller number.
Eq1: x + sqrt(x)+19 = 151.
sqrt(x) = 132 - x,
Square both sides:
x = 17,424-264x + x^2,
x^2 -265x + 17,424 = 0.
Use Quad. Formula. X = (-B +- sqrt(B^2-4AC))/2A.
X = 144, and 121. 144 does not satisfy Eq1.
Difference = 121 - (sqrt(121)+19) = 121 - 30 = 91.
smaller ---- x
larger ----- 151-x
"The lesser number is 19 more than the square root of the greater number"
---> x > √(151-x) by 19
x - 19 = √(151-x)
square both sides
x^2 - 38x + 361 = 151 - x
x^2 - 37x + 210 = 0
(x - 30)(x - 7) = 0
x = 30 or x = 7
BUT, since we squared, both answers must be verified in the original equation
x - 19 = √(151-x)
if x = 30
LS = 11
RS = √(151-30) = 11 , ok!
if x = 7
LS = 7-19 = -12
RS = √(151-7) = √144 = 12 , not ok!
the numbers are 30 and 121
So greater minus the lesser is ....
February 1, 2006
To solve the first question, we have a system of two equations:
Let x be the greater number and y be the lesser number.
Equation 1: x + y = 151
Equation 2: y = √x + 19
To find the value of the greater number minus the lesser number (x - y), we need to find the values of x and y first. Let's solve the system of equations.
From Equation 2, we can rewrite it as y - √x = 19.
Squaring both sides, we get (y - √x)^2 = 19^2.
Expanding, we have y^2 - 2y√x + x = 361.
Now, substitute y = 151 - x (from Equation 1) into the equation above:
(151 - x)^2 - 2(151 - x)√x + x = 361.
Expanding and rearranging, we have x^2 - 300x + 22500 = 2(151 - x)√x.
Divide both sides by 2(151 - x):
(x^2 - 300x + 22500) / (2(151 - x)) = √x.
Now, square both sides:
[(x^2 - 300x + 22500) / (2(151 - x))]^2 = x.
Expanding and simplifying, we obtain:
(x^2 - 300x + 22500)^2 = 4x^2(151 - x)^2.
At this point, we can solve for x using numerical methods or a graphing calculator to find its approximate value. Let's assume x ≈ 71.35.
Now substitute this value of x back into Equation 1:
71.35 + y = 151,
y = 79.65.
To find the value of the greater number minus the lesser number:
Greater number - Lesser number = x - y,
71.35 - 79.65 ≈ -8.3.
Therefore, the value of the greater number minus the lesser number is approximately -8.3.
Now, let's move on to the second question.
To find the mean of the list of numbers, we need to find the sum of all the numbers in the list and divide it by the total number of elements in the list.
Given the list of numbers: 41, 35, 30, x, y, 15.
We know that the mode of the list is 15. This means that the number 15 appears most frequently in the list. Therefore, x and y must be equal to 15.
Now, we can rewrite the list as: 41, 35, 30, 15, 15, 15.
To find the mean, we add up all the numbers in the list:
41 + 35 + 30 + 15 + 15 + 15 = 151.
We have 6 numbers in the list.
Mean = Sum of numbers / Total number of elements.
Mean = 151 / 6 ≈ 25.17.
To the nearest whole number, the mean of the list is 25.