positive - numbers or values greater than 0

-3x to the negative one power

-3x to the negative one power
Raising to a negative power means taking 1 over the same expression to the positive power"
(-3x)^-1 = 1/-3x = [B]-1/3x[/B]

7 times a positive number n is decreased by 3, it is less than 25

7 times a positive number n is decreased by 3, it is less than 25
7 times a positive number n:
7n
Decreased by 3:
7n - 3
The phrase [I]it is less than [/I]means an inequality. So we relate 7n - 3 less than 25 using the < sign to get our algebraic expression of:
[B]7n - 3 < 25[/B]

7.5 degrees C to -5.21 degrees C. Kayla and Jason were discussing the changes in temperature. Kayla

7.5 degrees C to -5.21 degrees C. Kayla and Jason were discussing the changes in temperature. Kayla said, "Wow, I can't believe the temperature fell 12.71 degrees C." Jason said, "What are you talking about? The temperature only fell 2.29 degrees C." Who is correct
[B]Kayla[/B] is correct. The temperature change TC is:
TC = High Temp - Low Temp
TC = 7.5 - (-5.21)
TC = 7.5 + 5.21 ([I]Since a minus negative is positive)[/I]
TC = 12.71

A 3 hour river cruise goes 15 km upstream and then back again. The river has a current of 2 km an ho

A 3 hour river cruise goes 15 km upstream and then back again. The river has a current of 2 km an hour. What is the boat's speed and how long was the upstream journey?
[U]Set up the relationship of still water speed and downstream speed[/U]
Speed down stream = Speed in still water + speed of the current
Speed down stream = x+2
Therefore:
Speed upstream =x - 2
Since distance = rate * time, we rearrange to get time = Distance/rate:
15/(x+ 2) + 15 /(x- 2) = 3
Multiply each side by 1/3 and we get:
5/(x + 2) + 5/(x - 2) = 1
Using a common denominator of (x + 2)(x - 2), we get:
5(x - 2)/(x + 2)(x - 2) + 5(x + 2)/(x + 2)(x - 2)
(5x - 10 + 5x + 10)/5(x - 2)/(x + 2)(x - 2)
10x = (x+2)(x-2)
We multiply through on the right side to get:
10x = x^2 - 4
Subtract 10x from each side:
x^2 - 10x - 4 = 0
This is a quadratic equation. To solve it, [URL='https://www.mathcelebrity.com/quadratic.php?num=x%5E2-10x-4%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']we type it in our search engine[/URL] and we get:
Speed of the boat in still water =X=5 +- sq. Root of 29 kmph
We only want the positive solution:
x = 5 + sqrt(29)
x = 10.38
[U]Calculate time for upstream journey:[/U]
Time for upstream journey = 15/(10.38 - 2)
Time for upstream journey = 15/(8.38)
Time for upstream journey = [B]1.79[/B]
[U]Calculate time for downstream journey:[/U]
Time for downstream journey = 15/(10.38 + 2)
Time for downstream journey = 15/(12.38)
Time for downstream journey = [B]1.21[/B]

A chest of treasure was hidden in the year 64 BC and found in 284 AD. For how long was the chest hid

A chest of treasure was hidden in the year 64 BC and found in 284 AD. For how long was the chest hidden
BC stands for Before Christ. Year 0 is when Christ was born. AD stands for After Death
On a number line, the point of Christ's birth is 0.
So BC is really negative
AD is positive
So we have:
284 - -64
284 + 64
[B]348 years[/B]

A customer withdrew $100 from a bank account. The customer then deposited $33 the next day. Write an

A customer withdrew $100 from a bank account. The customer then deposited $33 the next day. Write and then evaluate an expression to show the net effect of these transactions.
Withdrawals are negative since we take money away
Deposits are positive since we add money
So we have:
[LIST]
[*]100 withdrawal = -100
[*]33 deposit = +33
[/LIST]
Our balance is:
-100 + 33 = [B]-67 net[/B]

A football team gained 4 yards on a play,lost 8 on the next play ,then gained 2 yards on the third p

A football team gained 4 yards on a play,lost 8 on the next play ,then gained 2 yards on the third play write and addition expression
Gains are expressed with positives (+) and losses are expressed with negatives (-):
[LIST]
[*]Gained 4 years: +4
[*]Lost 8 on the next play: -8
[*]Gained 2 yards on the third play: +2
[/LIST]
Expression:
[B]+4 - 8 + 2 = -2[/B]

A professor assumed there was a correlation between the amount of hours people were expose to sunlig

A professor assumed there was a correlation between the amount of hours people were expose to sunlight and their blood vitamin D level. The null hypothesis was that the population correlation was__
a. Positive 1.0
b. Negative 1.0
c. Zero
d. Positive 0.50
[B]c. Zero[/B]
Reason: Since the professor wanted to assume a correlation (either positive = 1.0 or negative = -1.0), then we take the other side of that assumption for our null hypothesis and say that there is no correlation (Zero)

A rectangular garden is 5 ft longer than it is wide. Its area is 546 ft2. What are its dimensions?

A rectangular garden is 5 ft longer than it is wide. Its area is 546 ft2. What are its dimensions?
[LIST=1]
[*]Area of a rectangle is lw. lw = 546ft^2
[*]We know that l = w + 5.
[/LIST]
Substitute (2) into (1)
(w + 5)w = 546
w^2 + 5w = 546
Subtract 546 from each side
w^2 + 5w - 546 = 0
Using the positive root in our [URL='http://www.mathcelebrity.com/quadratic.php?num=w%5E2%2B5w-546%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we get [B]w = 21[/B].
This means l = 21 + 5.
[B]l = 26[/B]

A submarine at an elevation of -185 meters descends to 3 times that elevation. Then, it elevates 90

A submarine at an elevation of -185 meters descends to 3 times that elevation. Then, it elevates 90 meters. What is the submarines new elevation?
3 times the current elevation is:
3 * -185 = -555
Elevating 90 meters means we have a positive change:
-555 + 90 = [B]-465[/B]

Absolute Value

Free Absolute Value Calculator - Add, subtract, multiply or divide any two numbers with absolute value signs. Positive Difference.

Addition and Multiplication Multiples

Free Addition and Multiplication Multiples Calculator - Shows all addition and multiplication multiples up to 20 for a positive integer

Approximate Square Root Using Exponential Identity

Free Approximate Square Root Using Exponential Identity Calculator - Calculates the square root of a positive integer using the Exponential Identity Method

are all integers whole numbers true or false

are all integers whole numbers true or false
[B]False
[/B]
[LIST]
[*]All whole numbers are integers but not all integers are whole numbers.
[*]Whole numbers are positive integers. Which means negative integers are not whole numbers
[*]-1 for instance is an integer, but not a whole number
[/LIST]

Bakshali Method

Free Bakshali Method Calculator - Calculates the square root of a positive integer using the Bakshali Method

Base Change Conversions

Free Base Change Conversions Calculator - Converts a positive integer to Binary-Octal-Hexadecimal Notation or Binary-Octal-Hexadecimal Notation to a positive integer. Also converts any positive integer in base 10 to another positive integer base (Change Base Rule or Base Change Rule or Base Conversion)

Bill raises the flag 15 feet above the ground. Then he lowers it 8 feet and raises it another 2 feet

Bill raises the flag 15 feet above the ground. Then he lowers it 8 feet and raises it another 2 feet. how far above the ground is the flag now?
We tally up the positives and negatives:
Raise = +15
Lower = -8
Raise = +2
Total = +15 - 8 + 2
Total = [B]9 feet above the ground[/B]

Construct a data set of seven temperature readings where the mean is positive and the median is nega

Construct a data set of seven temperature readings where the mean is positive and the median is negative.
[B]{-20,-10.-5,-2,-1,20,40}[/B]
[URL='https://www.mathcelebrity.com/statbasic.php?num1=-20%2C-10%2C-5%2C-2%2C-1%2C20%2C40&num2=+0.2%2C0.4%2C0.6%2C0.8%2C0.9&pl=Number+Set+Basics']Using our mean and median calculator[/URL], we see that:
[B]Mean = 3.142857 (positive)
Median = -2[/B]

difference between 2 positive numbers is 3 and the sum of their squares is 117

difference between 2 positive numbers is 3 and the sum of their squares is 117
Declare variables for each of the two numbers:
[LIST]
[*]Let the first variable be x
[*]Let the second variable be y
[/LIST]
We're given 2 equations:
[LIST=1]
[*]x - y = 3
[*]x^2 + y^2 = 117
[/LIST]
Rewrite equation (1) in terms of x by adding y to each side:
[LIST=1]
[*]x = y + 3
[*]x^2 + y^2 = 117
[/LIST]
Substitute equation (1) into equation (2) for x:
(y + 3)^2 + y^2 = 117
Evaluate and simplify:
y^2 + 3y + 3y + 9 + y^2 = 117
Combine like terms:
2y^2 + 6y + 9 = 117
Subtract 117 from each side:
2y^2 + 6y + 9 - 117 = 117 - 117
2y^2 + 6y - 108 = 0
This is a quadratic equation:
Solve the quadratic equation 2y2+6y-108 = 0
With the standard form of ax2 + bx + c, we have our a, b, and c values:
a = 2, b = 6, c = -108
Solve the quadratic equation 2y^2 + 6y - 108 = 0
The quadratic formula is denoted below:
y = -b ± sqrt(b^2 - 4ac)/2a
[U]Step 1 - calculate negative b:[/U]
-b = -(6)
-b = -6
[U]Step 2 - calculate the discriminant ?:[/U]
? = b2 - 4ac:
? = 62 - 4 x 2 x -108
? = 36 - -864
? = 900 <--- Discriminant
Since ? is greater than zero, we can expect two real and unequal roots.
[U]Step 3 - take the square root of the discriminant ?:[/U]
?? = ?(900)
?? = 30
[U]Step 4 - find numerator 1 which is -b + the square root of the Discriminant:[/U]
Numerator 1 = -b + ??
Numerator 1 = -6 + 30
Numerator 1 = 24
[U]Step 5 - find numerator 2 which is -b - the square root of the Discriminant:[/U]
Numerator 2 = -b - ??
Numerator 2 = -6 - 30
Numerator 2 = -36
[U]Step 6 - calculate your denominator which is 2a:[/U]
Denominator = 2 * a
Denominator = 2 * 2
Denominator = 4
[U]Step 7 - you have everything you need to solve. Find solutions:[/U]
Solution 1 = Numerator 1/Denominator
Solution 1 = 24/4
Solution 1 = 6
Solution 2 = Numerator 2/Denominator
Solution 2 = -36/4
Solution 2 = -9
[U]As a solution set, our answers would be:[/U]
(Solution 1, Solution 2) = (6, -9)
Since one of the solutions is not positive and the problem asks for 2 positive number, this problem has no solution

During the first lesson on coordinate grids, Audrey is asked by her teacher to find the ordered pair

During the first lesson on coordinate grids, Audrey is asked by her teacher to find the ordered pair (2,5) on the coordinate grid. Starting from (0,0) which movement should Audrey complete first?
We move [B]2 spaces right (positive x), and then 5 spaces up (positive 5)[/B]

Eulers Totient (φ)

Free Eulers Totient (φ) Calculator - Given a positive integer (n), this calculates Euler's totient, also known as φ

Factoring and Root Finding

Free Factoring and Root Finding Calculator - This calculator factors a binomial including all 26 variables (a-z) using the following factoring principles:

* Difference of Squares

* Sum of Cubes

* Difference of Cubes

* Binomial Expansions

* Quadratics

* Factor by Grouping

* Common Term

This calculator also uses the Rational Root Theorem (Rational Zero Theorem) to determine potential roots

* Factors and simplifies Rational Expressions of one fraction

* Determines the number of potential*positive* and *negative* roots using Descarte’s Rule of Signs

* Difference of Squares

* Sum of Cubes

* Difference of Cubes

* Binomial Expansions

* Quadratics

* Factor by Grouping

* Common Term

This calculator also uses the Rational Root Theorem (Rational Zero Theorem) to determine potential roots

* Factors and simplifies Rational Expressions of one fraction

* Determines the number of potential

Factorization

Free Factorization Calculator - Given a positive integer, this calculates the following for that number:

1) Factor pairs and prime factorization and prime power decomposition

2) Factors and Proper Factors 3) Aliquot Sum

1) Factor pairs and prime factorization and prime power decomposition

2) Factors and Proper Factors 3) Aliquot Sum

Find all numbers whose absolute value is -3

Find all numbers whose absolute value is -3
[B][U][I]None[/I][/U][/B]. Absolute values are always positive, so no number has a negative absolute value.

Find an angle (theta) with 0<(theta)<360° or 0<(theta)<(2*pi) that has the same sine value as 80°

Find an angle (theta) with 0<(theta)<360° or 0<(theta)<(2*pi) that has the same sine value as 80°.
The sine is positive in Quadrant I and Quadrant II. So we find the reference angle for 80°.
It's 180 - 80 = [B]100°[/B].
This is our answer. Sin(80°) = Sin(100°)

find the difference between a mountain with an altitude of 1,684 feet above sea level and a valley

find the difference between a mountain with an altitude of 1,684 feet above sea level and a valley 216 feet below sea level.
Below sea level is the same as being on the opposite side of zero on the number line. To get the difference, we do the following:
1,684 - (-216)
Since subtracting a negative is a positive, we have:
1,684 + 216
[B]1,900 feet[/B]

Find two consecutive integers if the sum of their squares is 1513

Find two consecutive integers if the sum of their squares is 1513
Let the first integer be n. The next consecutive integer is (n + 1).
The sum of their squares is:
n^2 + (n + 1)^2 = 1513
n^2 + n^2 + 2n + 1 = 1513
2n^2 + 2n + 1 = 1513
Subtract 1513 from each side:
2n^2 + 2n - 1512 = 0
We have a quadratic equation. We [URL='https://www.mathcelebrity.com/quadratic.php?num=2n%5E2%2B2n-1512%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']type this into our search engine[/URL] and get:
n = (-27, 28)
Let's take the positive solution.
The second integer is: n + 1
28 + 1 = 29

Find two consecutive positive integers such that the difference of their square is 25

Find two consecutive positive integers such that the difference of their square is 25.
Let the first integer be n. This means the next integer is (n + 1).
Square n: n^2
Square the next consecutive integer: (n + 1)^2 = n^2 + 2n + 1
Now, we take the difference of their squares and set it equal to 25:
(n^2 + 2n + 1) - n^2 = 25
Cancelling the n^2, we get:
2n + 1 = 25
[URL='https://www.mathcelebrity.com/1unk.php?num=2n%2B1%3D25&pl=Solve']Typing this equation into our search engine[/URL], we get:
n = [B]12[/B]

Find two consecutive positive integers such that the sum of their squares is 25

Find two consecutive positive integers such that the sum of their squares is 25.
Let the first integer be x. The next consecutive positive integer is x + 1.
The sum of their squares equals 25. We write this as::
x^2 + (x + 1)^2
Expanding, we get:
x^2 + x^2 + 2x + 1 = 25
Group like terms:
2x^2 + 2x + 1 = 25
Subtract 25 from each side:
2x^2 + 2x - 24 = 0
Simplify by dividing each side by 2:
x^2 + x - 12 = 0
Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2%2Bx-12%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we get x = 3 or x = -4. The problem asks for positive integers, so we discard -4, and use 3.
This means, our next positive integer is 3 + 1 = 4. So we have [B](3, 4) [/B]as our answers.
Let's check our work:
3^2 + 4^2 = 9 + 16 = 25

Fractions and Mixed Numbers

Free Fractions and Mixed Numbers Calculator - Given (improper fractions, proper fraction, mixed numbers, or whole numbers), this performs the following operations:

* Addition (Adding)

* Subtraction (Subtracting)

* Positive Difference (Absolute Value of the Difference)

* Multiplication (Multiplying)

* Division (Dividing: complex fraction division is included)

* Compare Fractions

* Simplifying of proper and improper fractions as well as mixed numbers. Fractions will be reduced down as far as possible (Reducing Fractions).

* Reciprocal of a Fraction

* Find all fractions between two fractions

* reduce a fraction

* Addition (Adding)

* Subtraction (Subtracting)

* Positive Difference (Absolute Value of the Difference)

* Multiplication (Multiplying)

* Division (Dividing: complex fraction division is included)

* Compare Fractions

* Simplifying of proper and improper fractions as well as mixed numbers. Fractions will be reduced down as far as possible (Reducing Fractions).

* Reciprocal of a Fraction

* Find all fractions between two fractions

* reduce a fraction

if a and b are odd then a + b is even

if a and b are odd then a + b is even
Let a and b be positive odd integers of the form:
[LIST]
[*]a = 2n + 1
[*]b = 2m + 1
[/LIST]
a + b = 2n + 1 + 2m + 1
a + b = 2n + 2m + 1 + 1
Combing like terms, we get:
a + b = 2n + 2m + 2
a + b = 2(n + m) + 2
Let k = n + m
a + b = 2k + 2
[B]Therefore a + b is even[/B]

If a, b, and c are positive integers such that a^b = x and c^b = y, then xy = ?

If a, b, and c are positive integers such that a^b = x and c^b = y, then xy = ?
A) ac^b
B) ac^2b
C) (ac)^b
D) (ac)^2b
E) (ac)^b^2
xy = a^b * c^b
We can use the Power of a Product Rule
a^b * c^b = (ac)^b
Therefore:
xy = [B](ac)^b - Answer C[/B]

If n is odd, then 3n + 2 is odd

Look at the Contrapositive: If n is even, then 3n + 2 is even...
Suppose that the conclusion is false, i.e., that n is even.
Then n = 2k for some integer k.
Then we have:
3n + 2 = 3(2k) + 2
3n + 2 = 6k + 2
3n + 2 = 2(3k + 1).
Thus 3n + 2 is even, because it equals 2j for an integer j = 3k + 1.
So 3n + 2 is not odd.
We have shown that ¬(n is odd) ? ¬(3n + 2 is odd),
therefore, the contrapositive (3n + 2 is odd) ? (n is odd) is also true.

if x2 is added to x, the sum is 42

If x2 is added to x, the sum is 42.
x^2 + x = 42
Subtract 42 from both sides:
x^2 + x - 42 = 0
We have a quadratic equation. Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2%2Bx-42%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic equation solver[/URL], we get:
[B]x = 6 and x = -7
[/B]
Since the problem does not state positive number solutions, they are both answers.

in a city, the record monthly high temperature for March is 56°F. The record monthly low for March i

in a city, the record monthly high temperature for March is 56°F. The record monthly low for March is -4°F. What is the range of temperatures for the month of March
Range = High - Low
Range = 56 - -4
Range = 56 + 4 [I]since double negative is positive[/I]
Range = [B]60[/B]

In a golf tournament Ned was 4 under par on Saturday and 5 over or on Sunday. What is the difference

In a golf tournament Ned was 4 under par on Saturday and 5 over or on Sunday. What is the difference in his score on Sunday compared to his score on Saturday
Givens and opening thoughts:
[LIST]
[*]Think of par as 0 or average.
[*]Under par is negative
[*]Over par is positive
[*]We have 4 under par as -4
[*]We have 5 over par as +5
[/LIST]
The difference is found by subtracting:
+5 - -4
+5 + 4
[B]9 strokes[/B]

Lanette walked forward 9 steps, backward 15 steps, forward 7 steps, and backward 8 steps. How many s

Lanette walked forward 9 steps, backward 15 steps, forward 7 steps, and backward 8 steps. How many steps is lanette from where she started?
We use (+) for forward steps and (-) for backward steps
[LIST]
[*]Forward 9, backward 15 is: +9 - 15 = -6
[*]Forward 7 steps is: -6 + 7 = +1
[*]Backward 8 steps = +1 - 8 = -7
[/LIST]
Since negative is backward and positive is forward, we see Lanette is [B]7 steps backwards [/B]from her starting point.

Modified Internal Rate of Return (MIRR)

Free Modified Internal Rate of Return (MIRR) Calculator - Given a set of positive/negative cash flows, a finance rate, and a reinvestment rate, this calculates the modified internal rate of return

Mount McKinley in Alaska, the highest mountain in North America, is 20,320 feet above sea level. Dea

Mount McKinley in Alaska, the highest mountain in North America, is 20,320 feet above sea level. Death Valley, the lowest point, is 280 feet below sea level. What is the difference in height between Mount McKinley and Death Valley?
Regarding height with respect to sea level...
[LIST]
[*]Above sea level is written as positive height
[*]Below sea level is written as negative height
[/LIST]
So we have:
+20,320 - -280
+20,320 + 280
[B]20,600[/B]

Mt. Everest, the highest elevation in Asia, is 29,028 feet above sea level. The Dead Sea, the lowest

Mt. Everest, the highest elevation in Asia, is 29,028 feet above sea level. The Dead Sea, the lowest elevation, is 1,312 feet below sea level. What is the difference between these two elevations?
Above sea level is listed as positive (+)
Below sea level is listed as negative (-)
We have:
Difference = +29,028 - (-1312)
Difference = 29028 + 1312
[URL='https://www.mathcelebrity.com/longdiv.php?num1=29028&num2=1312&pl=Add']Difference[/URL] = [B]30,340[/B]

natural numbers that are factors of 16

natural numbers that are factors of 16
Natural numbers are positive integers starting at 1.
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}
Of these, [URL='https://www.mathcelebrity.com/factoriz.php?num=16&pl=Show+Factorization']the only factors of 16[/URL] are:
{[B]1, 2, 4, 8, 16}[/B]

Newton Method

Free Newton Method Calculator - Calculates the square root of a positive integer using the Newton Method

Number Information

Free Number Info Calculator - Calculates number info for a positive integer

One number is 8 times another number. The numbers are both positive and have a difference of 70.

One number is 8 times another number. The numbers are both positive and have a difference of 70.
Let the first number be x, the second number be y. We're given:
[LIST=1]
[*]x = 8y
[*]x - y = 70
[/LIST]
Substitute(1) into (2)
8y - y = 70
[URL='https://www.mathcelebrity.com/1unk.php?num=8y-y%3D70&pl=Solve']Plugging this equation into our search engine[/URL], we get:
[B]y = 10[/B] <-- This is the smaller number
Plug this into Equation (1), we get:
x = 8(10)
[B]x = 80 [/B] <-- This is the larger number

One number is equal to the square of another. Find the numbers if both are positive and their sum is

One number is equal to the square of another. Find the numbers if both are positive and their sum is 650
Let the number be n. Then the square is n^2. We're given:
n^2 + n = 650
Subtract 650 from each side:
n^2 + n - 650 = 0
We have a quadratic equation. [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2%2Bn-650%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']We type this into our search engine[/URL] and we get:
n = 25 and n = -26
Since the equation asks for a positive solution, we use [B]n = 25[/B] as our first solution.
the second solution is 25^2 = [B]625[/B]

One positive number is one-fifth of another number. The difference between the two numbers is 192, f

One positive number is one-fifth of another number. The difference between the two numbers is 192, find the numbers.
Let the first number be x and the second number be y. We're given two equations:
[LIST=1]
[*]x = y/5
[*]x + y = 192
[/LIST]
Substitute equation 1 into equation 2:
y/5 + y = 192
Since 1 equals 5/5, we rewrite our equation like this:
y/5 = 5y/5 = 192
We have fractions with like denominators, so we add the numerators:
(1 + 5)y/5 = 192
6y/5 = 192
[URL='https://www.mathcelebrity.com/prop.php?num1=6y&num2=192&den1=5&den2=1&propsign=%3D&pl=Calculate+missing+proportion+value']Type this equation into our search engine[/URL], and we get:
[B]y = 160[/B]
Substitute this value into equation 1:
x = 160/5
x = [B]32[/B]

Opposite Numbers

Free Opposite Numbers Calculator - Given a positive or negative integer (n), this calculates the opposite number of n

positive even numbers less than 10

positive even numbers less than 10
First, list out all positive even numbers less than 10.
Less than 10 means we do [U]not[/U] include 10.
[B]{2, 4, 6, 8}
[MEDIA=youtube]5YsPQo_2dpI[/MEDIA][/B]

Positive numbers less than 4

Less than means we do not include 4.
But positive means greater than 0, so we have:
{1, 2, 3}

Positive numbers less than 4

Update, this has been added to our shortcuts.
You can type any expression in the form, positive numbers less than x where x is any integer.
You can also type positive numbers greater than x where x is any integer.
Same with less than or equal to and greater than or equal to.

Positivity Rate

Free Positivity Rate Calculator - This calculator determines the positivity rate using positive tests and total tests

Quotient-Remainder Theorem

Free Quotient-Remainder Theorem Calculator - Given 2 positive integers n and d, this displays the quotient remainder theorem.

Reference Angle

Free Reference Angle Calculator - Calculates the reference angle for a given angle. Also known as the positive acute angle.

Roman Numeral Conversions

Free Roman Numeral Conversions Calculator - Converts a Positive integer less than 4000 to a Roman Numeral.

Converts a Roman Numeral with a positive value less than 4000 to a number.

Converts a Roman Numeral with a positive value less than 4000 to a number.

s = tu^2 for u

s = tu^2 for u
Divide each side by t
u^2 = s/t
Take the square root of each side
[LIST]
[*]u = sqrt(s/t)
[*]u = -sqrt(s/t)
[/LIST]
We have two answers due to negative number squared is positive

Sarah has $250 in her account. She withdraws $25 per week. How many weeks can she withdraw money fro

Sarah has $250 in her account. She withdraws $25 per week. How many weeks can she withdraw money from her account and still have money left?
Let w be the number of weeks. We have the following equation for the Balance after w weeks:
B(w) = 250 - 25w [I]we subtract for withdrawals[/I]
The ability to withdrawal money means have a positive or zero balance after withdrawal. So we set up the inequality below:
250 - 25w >= 0
To solve this inequality for w, we [URL='https://www.mathcelebrity.com/1unk.php?num=250-25w%3E%3D0&pl=Solve']type it in our search engine[/URL] and we get:
w <= [B]10
So Sarah can withdrawal for up to 10 weeks[/B]

Six is the principal square root of 36

Six is the principal square root of 36
The two square roots of 36 are:
[LIST]
[*]+6
[*]-6
[/LIST]
The positive square root is known as the principal square root, therefore, this is [B]true[/B].

Tally Marks

Free Tally Marks Calculator - Shows the tally mark representation (tallies) for a positive integer.

The difference between two positive numbers is 5 and the square of their sum is 169

The difference between two positive numbers is 5 and the square of their sum is 169.
Let the two positive numbers be a and b. We have the following equations:
[LIST=1]
[*]a - b = 5
[*](a + b)^2 = 169
[*]Rearrange (1) by adding b to each side. We have a = b + 5
[/LIST]
Now substitute (3) into (2):
(b + 5 + b)^2 = 169
(2b + 5)^2 = 169
[URL='https://www.mathcelebrity.com/community/forums/calculator-requests.7/create-thread']Run (2b + 5)^2 through our search engine[/URL], and you get:
4b^2 + 20b + 25
Set this equal to 169 above:
4b^2 + 20b + 25 = 169
[URL='https://www.mathcelebrity.com/quadratic.php?num=4b%5E2%2B20b%2B25%3D169&pl=Solve+Quadratic+Equation&hintnum=+0']Run that quadratic equation in our search engine[/URL], and you get:
b = (-9, 4)
But the problem asks for [I]positive[/I] numbers. So [B]b = 4[/B] is one of our solutions.
Substitute b = 4 into equation (1) above, and we get:
a - [I]b[/I] = 5
[URL='https://www.mathcelebrity.com/1unk.php?num=a-4%3D5&pl=Solve']a - 4 = 5[/URL]
[B]a = 9
[/B]
Therefore, we have [B](a, b) = (9, 4)[/B]

The difference of 2 positive numbers is 54. The quotient obtained on dividing the 1 by the other is

The difference of 2 positive numbers is 54. The quotient obtained on dividing the 1 by the other is 4. Find the numbers.
Let the numbers be x and y. We have:
[LIST]
[*]x - y = 54
[*]x/y = 4
[*]Cross multiply x/y = 4 to get x = 4y
[*]Now substitute x = 4y into the first equation
[*](4y) - y = 54
[*]3y = 54
[*]Divide each side by 3
[*][B]y = 18[/B]
[*]If x = 4y, then x = 4(18)
[*][B]x = 72[/B]
[/LIST]

The distance between X and 8 is less than 14

Distance implies the positive difference between 2 points. Therefore, we use absolute value:
|x - 8| < 14
Note, we use less than since 14 is not included.

The January record high temperature in Austin, Texas is 92 degrees. The January record low temperatu

The January record high temperature in Austin, Texas is 92 degrees. The January record low temperature is -1 degree. What is the difference between the record high and low temperatures?
Difference = High - Low
Difference = 92 - -1
Difference = 92 + 1. <-- Since minus negative equals positive
Difference = [B]93 degrees[/B]

The length of a wooden frame is 1 foot longer than its width and its area is equal to 12ft²

The length of a wooden frame is 1 foot longer than its width and its area is equal to 12ft²
The frame is a rectangle. The area of a rectangle is A = lw. So were given:
[LIST=1]
[*]l = w + 1
[*]lw = 12
[/LIST]
Substitute equation (1) into equation (2) for l:
(w + 1) * w = 12
Multiply through and simplify:
w^2 + w = 12
We have a quadratic equation. To solve for w, we type this equation into our search engine and we get two solutions:
w = 3
w = -4
Since width cannot be negative, we choose the positive result and have:
w = [B]3[/B]
To solve for length, we plug w = 3 into equation (1) above and get:
l = 3 + 1
l = [B]4[/B]

The marianas trench is 10415m below the sea level. Directly above it a helicopter is hovering 3200m

The marianas trench is 10415m below the sea level. Directly above it a helicopter is hovering 3200m above sea level. How far is the helicopter from the trench?
Below sea level is negative:
-10415
Above sea level is positive:
+3200
The distance is found by:
+3200 - -10415
+3200 + 10415
[B]13,615[/B]

The product of two positive numbers is 96. Determine the two numbers if one is 4 more than the other

The product of two positive numbers is 96. Determine the two numbers if one is 4 more than the other.
Let the 2 numbers be x and y.
We have:
[LIST=1]
[*]xy = 96
[*]x = y - 4
[/LIST]
[U]Substitute (2) into (1)[/U]
(y - 4)y = 96
y^2 - 4y = 96
[U]Subtract 96 from both sides:[/U]
y^2 - 4y - 96 = 0
[U]Factoring using our quadratic calculator, we get:[/U]
(y - 12)(y + 8)
So y = 12 and y = -8. Since the problem states positive numbers, we use [B]y = 12[/B].
Substituting y = 12 into (2), we get:
x = 12 - 4
[B]x = 8[/B]
[B]We have (x, y) = (8, 12)[/B]

The square of a number is always nonnegative.

The square of a number is always nonnegative.
This is true, and here is why:
Suppose you have a positive number n.
n^2 = n * n
A positive times a positive is a positive
Suppose you have a negative number -n
(-n)^2 = -n * -n = n^2
A negative times a negative is a positive.

The square of a number is positive

The square of a number is positive
N ca be positive or negative, so test both scenarios:
Take a positive number n.
n^2 = n^2 * n^2 or Positive * Positive which is positive
Take a negative number n
(-n)^2 = -n * -n or Negative * Negative which is positive
(-n)^2 = n^2

The Square of a positive integer is equal to the sum of the integer and 12. Find the integer

The Square of a positive integer is equal to the sum of the integer and 12. Find the integer
Let the integer be x.
[LIST]
[*]The sum of the integer and 12 is written as x + 12.
[*]The square of a positive integer is written as x^2.
[/LIST]
We set these equal to each other:
x^2 = x + 12
Subtract x + 12 from each side:
x^2 - x - 12 = 0
We have a quadratic function. [URL='https://www.mathcelebrity.com/quadratic.php?num=x%5E2-x-12%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']Run it through our search engine[/URL] and we get x = 3 and x = -4.
The problem asks for a positive integer, so we have [B]x = 3[/B]

The square of a positive integer minus twice its consecutive integer is equal to 22. find the intege

The square of a positive integer minus twice its consecutive integer is equal to 22. Find the integers.
Let x = the original positive integer. We have:
[LIST]
[*]Consecutive integer is x + 1
[*]x^2 - 2(x + 1) = 22
[/LIST]
Multiply through:
x^2 - 2x - 2 = 22
Subtract 22 from each side:
x^2 - 2x - 24 = 0
Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2-2x-24%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we get:
x = 6 and x = -4
Since the problem states [U]positive integers[/U], we use:
x = 6 and x + 1 = 7
[B](6, 7)[/B]

The sum of a number and its square is 72. find the numbers?

The sum of a number and its square is 72. find the numbers?
Let the number be n. We have:
n^2 + n = 72
Subtract 72 from each side:
n^2 + n - 72 = 0
Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=n%5E2%2Bn-72%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we have:
[B]n = 8 or n = -9
[/B]
Since the numbers do not state positive or negative, these are the two solutions.

The sum of the squares of two consecutive positive integers is 61. Find these two numbers.

The sum of the squares of two consecutive positive integers is 61. Find these two numbers.
Let the 2 consecutive integers be x and x + 1. We have:
x^2 + (x + 1)^2 = 61
Simplify:
x^2 + x^2 + 2x + 1 = 61
2x^2 + 2x + 1 = 61
Subtract 61 from each side:
2x^2 + 2x - 60 = 0
Divide each side by 2
x^2 + x - 30
Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2%2Bx-30&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic equation calculator[/URL], we get:
x = 5 and x = -6
The question asks for [I]positive integers[/I], so we use [B]x = 5. [/B]This means the other number is [B]6[/B].

Two numbers have a sum of 20. Determine the lowest possible sum of their squares.

Two numbers have a sum of 20. Determine the lowest possible sum of their squares.
If sum of two numbers is 20, let one number be x. Then the other number would be 20 - x.
The sum of their squares is:
x^2+(20 - x)^2
Expand this and we get:
x^2 + 400 - 40x + x^2
Combine like terms:
2x^2 - 40x + 400
Rewrite this:
2(x^2 - 20x + 100 - 100) + 400
2(x - 10)^2 - 200 + 400
2(x?10)^2 + 200
The sum of squares of two numbers is sum of two positive numbers, one of which is a constant of 200.
The other number, 2(x - 10)^2, can change according to the value of x. The least value could be 0, when x=10
Therefore, the minimum value of sum of squares of two numbers is 0 + 200 = 200 when x = 10.
If x = 10, then the other number is 20 - 10 = 10.

What can we conclude if the coefficient of determination is 0.94?

What can we conclude if the coefficient of determination is 0.94?
[LIST]
[*]Strength of relationship is 0.94
[*]Direction of relationship is positive
[*]94% of total variation of one variable(y) is explained by variation in the other variable(x).
[*]All of the above are correct
[/LIST]
[B]94% of total variation of one variable(y) is explained by variation in the other variable(x)[/B]. The coefficient of determination explains ratio of explained variation to the total variation.

When 3 consecutive positive integers are multiplied, the product is 16 times the sum of the 3 intege

When 3 consecutive positive integers are multiplied, the product is 16 times the sum of the 3 integers. What is the difference of the product minus the sum?
Let the 3 consecutive positive integers be:
[LIST=1]
[*]x
[*]x + 1
[*]x + 2
[/LIST]
The product is:
x(x + 1)(x + 2)
The sum is:
x + x + 1 + x + 2 = 3x + 3
We're told the product is equivalent to:
x(x + 1)(x + 2) = 16(3x + 3)
x(x + 1)(x + 2) = 16 * 3(x + 1)
Divide each side by (x + 1)
x(x + 2) = 48
x^2 + 2x = 48
x^2 + 2x - 48 = 0
Now subtract the sum from the product:
x^2 + 2x - 48 - (3x + 3)
[B]x^2 - x - 51[/B]

When 4 is subtracted from the square of a number, the result is 3 times the number. Find the positiv

When 4 is subtracted from the square of a number, the result is 3 times the number. Find the positive solution.
Let the number be n. We have:
n^2 - 4 = 3n
Subtract 3n from each side:
n^2 - 3n - 4 = 0
[URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2-3n-4%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']Typing this quadratic equation into the search engine[/URL], we get:
n = (-1, 4)
The problem asks for the positive solution, so we get [B]n = 4[/B].

Which of the following is the probability that subjects do not have the disease, but the test result

Which of the following is the probability that subjects do not have the disease, but the test result is positive?
a. Miss rate
b. False positive rate
c. Base rate
d. Disease rate
[B]b. [URL='http://sites.stat.psu.edu/~lsimon/stat250/sp00/solutions/misc/diagtests.htm']False positive rate[/URL][/B]

X is a natural number greater than 6

I saw this ticket come through today.
The answer is x > 6.
Natural numbers are positive numbers not 0. So 1, 2, 3, ...
Let me build this shortcut into the calculator.
Also, here is the[URL='http://www.mathcelebrity.com/interval-notation-calculator.php?num=x%3E6&pl=Show+Interval+Notation'] interval notation[/URL] for that expression.

You and your friend are playing a number-guessing game. You ask your friend to think of a positive n

You and your friend are playing a number-guessing game. You ask your friend to think of a positive number, square the number, multiply the result by 2, and then add three. If your friend's final answer is 53, what was the original number chosen?
Let n be our original number.
Square the number means we raise n to the power of 2:
n^2
Multiply the result by 2:
2n^2
And then add three:
2n^2 + 3
If the friend's final answer is 53, this means we set 2n^2 + 3 equal to 53:
2n^2 + 3 = 53
To solve for n, we subtract 3 from each side, to isolate the n term:
2n^2 + 3 - 3 = 53 - 3
Cancel the 3's on the left side, and we get:
2n^2 = 50
Divide each side of the equation by 2:
2n^2/2 = 50/2
Cancel the 2's, we get:
n^2 = 25
Take the square root of 25
n = +-sqrt(25)
n = +-5
We are told the number is positive, so we discard the negative square root and get:
n = [B]5[/B]

You conduct 50,000 tests, 1500 people test positive, what's the positivity rate?

You conduct 50,000 tests, 1500 people test positive, what's the positivity rate?
[U]Our Positivity Rate formula is below:[/U]
Positivity Rate = 100% * positive tests / Total tests
[U]Plugging in our numbers from the problem, we get:[/U]
Positivity Rate = 100% * 1500/50000
Positivity Rate = 100% * 0.03
Positivity Rate = [B]3%[/B]