# 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?

1. 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 ....

2. 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 ).

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

3. 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

4. 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.

5. February 1, 2006

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