product - The answer when two or more values are multiplied together

1 integer is 7 times another. If the product of the 2 integers is 448, then find the integers.

1 integer is 7 times another. If the product of the 2 integers is 448, then find the integers.
Let the first integer be x and the second integer be y. We have the following two equations:
[LIST=1]
[*]x = 7y
[*]xy = 448
[/LIST]
Substitute (1) into (2), we have:
(7y)y = 448
7y^2 = 448
Divide each side by 7
y^2 = 64
y = -8, 8
We use 8, since 8*7 = 56, and 56*8 =448. So the answer is [B](x, y) = (8, 56)[/B]

1/2 of the product x and y

1/2 of the product x and y
The product x and y:
xy
1/2 of the product:
[B]xy/2[/B]

110 subtracted from the product of 244 and w is the product of r and 177 increased by 266

110 subtracted from the product of 244 and w is the product of r and 177 increased by 266
The product of 244 and w:
244w
110 subtracted from the product of 244 and w
244w - 110
the product of r and 177
177r
the product of r and 177 increased by 266
177r + 266
The word [I]is[/I] means equal to, so we set 244w - 110 equal to 177r + 266
[B]244w - 110 = 177r + 266[/B]

12 is multiplied by some number, that product is reduced by 9, and the total is equal to 37

12 is multiplied by some number, that product is reduced by 9, and the total is equal to 37
The phrase [I]some number[/I] means an arbitrary variable, let's call it x.
12 multiplied by this number:
12x
The product of 12x is reduced by 9
12x - 9
The phrase [I]the total is equal to[/I] means an equation, so we set 12x - 9 equal to 37:
[B]12x - 9 = 37[/B]

12 plus the product of 4 and a number is greater than 72

A number means an arbitrary variable, let's call it x.
The product of 4 and a number is 4x.
12 plus that product is 4x + 12
Is greater than means an inequality, so we set the entire expression greater than 72
4x + 12 > 72

13 is the product of 5p and 5

13 is the product of 5p and 5
the product of 5p and 5 means we multiply 5p by 5:
5p * 5
25p
The word [I]is[/I] means equal to, so we set 25p equal to 13
[B]13 = 25p
25p = 13[/B]

2 consecutive odd integers such that their product is 15 more than 3 times their sum

2 consecutive odd integers such that their product is 15 more than 3 times their sum.
Let the first integer be n. The next odd, consecutive integer is n + 2.
We are given the product is 15 more than 3 times their sum:
n(n + 2) = 3(n + n + 2) + 15
Simplify each side:
n^2 + 2n = 6n + 6 + 15
n^2 + 2n = 6n + 21
Subtract 6n from each side:
n^2 - 4n - 21 = 0
[URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2-4n-21%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']Type this problem into our search engine[/URL], and we get:
n = (-3, 7)
If we use -3, then the next consecutive odd integer is -3 + 2 = -1. So we have [B](-3, -1)[/B]
If we use 7, then the next consecutive odd integer is 7 + 2 = 9. So we have [B](7, 9)[/B]

2 number Word Problems

This calculator handles word problems in the format below:

* Two numbers have a sum of 70 and a product of 1189 What are the numbers?

* Two numbers have a sum of 70. Their difference 32

* Two numbers have a sum of 70 and a product of 1189 What are the numbers?

* Two numbers have a sum of 70. Their difference 32

2-thirds of the sum of 5 and a plus the product of 3 and z

2-thirds of the sum of 5 and a plus the product of 3 and z
The sum of 5 and a
5 + a
2-thirds of this sum:
2(5 + a)/3
The product of 3 and z:
3z
The word [I]plus[/I] means we add the two terms together:
[B]2(5 + a)/3 + 3z[/B]

3 more than the product of 7 and a number x is less than 26

The product of 7 and a number x is written as 7x.
3 more than that product is written as 7x + 3.
Finally, that entire expression is less than 26, so we have:
7x + 3 < 26 as our algebraic expression.

5 times the product of 2 numbers a and b

5 times the product of 2 numbers a and b
The product of 2 numbers a and be means we multiply the variables together:
ab
5 times the product means we multiply ab by 5:
[B]5ab[/B]

50 is more than the product of 4 and w

50 is more than the product of 4 and w
Take this algebraic expression in pieces:
The product of 4 and w mean we multiply the variable w by 4:
4w
The phrase [I]is more than[/I] means an inequality using the (>) sign, where 50 is greater than 4w:
[B]50 > 4w[/B]

6 subtracted from the product of 5 and a number is 68

6 subtracted from the product of 5 and a number is 68
Take this algebraic expression in parts.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
The product of 5 and this number is:
5x
We subtract 6 from 5x:
5x - 6
The phrase [I]is[/I] means an equation, so we set 5x - 6 equal to 68
[B]5x - 6 = 68[/B]

8 increased by the product of a number and 7 is greater than or equal to -18

Take this in parts:
First, the phrase, "a number" means we pick an arbitrary variable, let's call it x.
The product of a number and 7 is 7x.
8 increased by the product of 7x means we add them together.
7x + 8
Finally that entire expression is greater than [U]or equal to[/U] -18
[B]7x + 8 >=-18[/B]

8 more than the product of x and 2 equals 4

8 more than the product of x and 2 equals 4
The product of x and 2:
2x
8 more than this, means we add 8:
2x + 8
Set this equal to 4:
[URL='https://www.mathcelebrity.com/1unk.php?num=2x%2B8%3D4&pl=Solve']2x + 8 = 4[/URL] <-- Algebraic expression
to solve for x, type this into the search engine and we get [B]x = -2[/B].

9 less than the product of the profit, p, and 6

9 less than the product of the profit, p, and 6
[U]The product of the profit p and 6:[/U]
6p
[U]9 less than the product:[/U]
[B]6p - 9[/B]

9 subtracted from the product of 3 and a number is greater than or equal to 16

9 subtracted from the product of 3 and a number is greater than or equal to 16
[LIST=1]
[*]The phrase [I]a number[/I] means an arbitrary variable, let's call it x
[*]The product of 3 and a number means we multiply 3 times x: 3x
[*]9 subtracted from the product: 3x - 9
[*]The phrase is greater than or equal to means an inequality. So we set up an inequality with >= for the greater than or equal to sign in relation to 3x - 9 and 16
[/LIST]
Our algebraic expression (inequality) becomes:
[B]3x - 19 >= 16[/B]

A bag contains 3 red marbles and 4 blue marbles. a marble is taken at random and replaced. another m

A bag contains 3 red marbles and 4 blue marbles. a marble is taken at random and replaced. Another marble is taken from the bag. Work out the probability that the two marbles taken from the bag are the same color.
[LIST]
[*]Total number of marbles in the bag is 3 + 4 = 7.
[*]The problem asks for the probability of (RR) [I]or[/I] (BB).
[*]It's worthy to note we are replacing the balls after each draw, which means we always have 7 to draw from
[/LIST]
Since each draw is independent, we take the product of each event for the total event probability.
P(RR) = 3/7 * 3/7 = 9/49
P(BB) = 4/7 * 4/7 = 16/49
We want to know P(RR) + P(BB)
P(RR) + P(BB) = 9/49 + 16/49 = 25/49
[MEDIA=youtube]26F9vjsgNGs[/MEDIA]

A chicken farm produces ideally 700,000 eggs per day. But this total can vary by as many as 60,000 e

A chicken farm produces ideally 700,000 eggs per day. But this total can vary by as many as 60,000 eggs. What is the maximum and minimum expected production at the farm?
[U]Calculate the maximum expected production:[/U]
Maximum expected production = Average + variance
Maximum expected production = 700,000 + 60,000
Maximum expected production = [B]760,000[/B]
[U]Calculate the minimum expected production:[/U]
Minimum expected production = Average - variance
Minimum expected production = 700,000 - 60,000
Minimum expected production = [B]640,000[/B]

A company has a fixed cost of $34,000 and a production cost of $6 for each unit it manufactures. A u

A company has a fixed cost of $34,000 and a production cost of $6 for each unit it manufactures. A unit sells for $15
Set up the cost function C(u) where u is the number of units is:
C(u) = Cost per unit * u + Fixed Cost
C(u) = [B]6u + 34000[/B]
Set up the revenue function R(u) where u is the number of units is:
R(u) = Sale price per unit * u
R(u) = [B]15u[/B]

A company is planning to manufacture a certain product. The fixed costs will be $474778 and it will

A company is planning to manufacture a certain product. The fixed costs will be $474778 and it will cost $293 to produce each product. Each will be sold for $820. Find a linear function for the profit, P , in terms of units sold, x .
[U]Set up the cost function C(x):[/U]
C(x) = Cost per product * x + Fixed Costs
C(x) = 293x + 474778
[U]Set up the Revenue function R(x):[/U]
R(x) = Sale Price * x
R(x) = 820x
[U]Set up the Profit Function P(x):[/U]
P(x) = Revenue - Cost
P(x) = R(x) - C(x)
P(x) = 820x - (293x + 474778)
P(x) = 820x - 293x - 474778
[B]P(x) = 527x - 474778[/B]

A Farmer Sell products at the market in 38- pound crates. If he sells 100 crates . How many pounds o

A Farmer Sell products at the market in 38- pound crates. If he sells 100 crates . How many pounds of produce has he sold
[U]Calculate the pounds of produce:[/U]
Pounds of Produce = Number of Crates * pounds per crate
Pounds of Produce = 100 crates * 38 pounds per crates
Pounds of Produce = [B]3,800 pounds of produce[/B]3

A manufacturer has a monthly fixed cost of $100,000 and a production cost of $10 for each unit produ

A manufacturer has a monthly fixed cost of $100,000 and a production cost of $10 for each unit produced. The product sells for $22/unit.
The cost function for each unit u is:
C(u) = Variable Cost * Units + Fixed Cost
C(u) = 10u + 100000
The revenue function R(u) is:
R(u) = 22u
We want the break-even point, which is where:
C(u) = R(u)
10u + 100000 = 22u
[URL='https://www.mathcelebrity.com/1unk.php?num=10u%2B100000%3D22u&pl=Solve']Typing this equation into our search engine[/URL], we get:
u =[B]8333.33[/B]

A manufacturer has a monthly fixed cost of $100,000 and a production cost of $12 for each unit produ

A manufacturer has a monthly fixed cost of $100,000 and a production cost of $12 for each unit produced. The product sells for $20/unit
[U]Cost Function C(u) where u is the number of units:[/U]
C(u) = cost per unit * u + fixed cost
C(u) = 12u + 100000
[U]Revenue Function R(u) where u is the number of units:[/U]
R(u) = Sale price * u
R(u) = 20u
Break even point is where C(u) = R(u):
C(u) = R(u)
12u + 100000 = 20u
To solve for u, we [URL='https://www.mathcelebrity.com/1unk.php?num=12u%2B100000%3D20u&pl=Solve']type this equation into our search engine[/URL] and we get:
u = [B]12,500[/B]

A manufacturer has a monthly fixed cost of $100,000 and a production cost of $14 for each unit produ

A manufacturer has a monthly fixed cost of $100,000 and a production cost of $14 for each unit produced. The product sells for $20/unit.
Let u be the number of units. We have a cost function C(u) as:
C(u) = Variable cost * u + Fixed Cost
C(u) = 14u + 100000
[U]We have a revenue function R(u) with u units as:[/U]
R(u) = Sale Price * u
R(u) = 20u
[U]We have a profit function P(u) with u units as:[/U]
Profit = Revenue - Cost
P(u) = R(u) - C(u)
P(u) = 20u - (14u + 100000)
P(u) = 20u - 14u - 100000
P(u) = 6u - 1000000

A manufacturer has a monthly fixed cost of $25,500 and a production cost of $7 for each unit produce

A manufacturer has a monthly fixed cost of $25,500 and a production cost of $7 for each unit produced. The product sells for $10/unit.
Set up cost function where u equals each unit produced:
C(u) = 7u + 25,500
Set up revenue function
R(u) = 10u
Break Even is where Cost equals Revenue
7u + 25,500 = 10u
Plug this into our [URL='http://www.mathcelebrity.com/1unk.php?num=7u%2B25500%3D10u&pl=Solve']equation calculator[/URL] to get [B]u = 8,500[/B]

A manufacturer has a monthly fixed cost of $52,500 and a production cost of $8 for each unit produce

A manufacturer has a monthly fixed cost of $52,500 and a production cost of $8 for each unit produced. The product sells for $13/unit.
Using our [URL='http://www.mathcelebrity.com/cost-revenue-profit-calculator.php?fc=52500&vc=8&r=13&u=20000%2C50000&pl=Calculate']cost-revenue-profit calculator[/URL], we get the following:
[LIST]
[*]P(x) = 55x - 2,500
[*]P(20,000) = 47,500
[*]P(50,000) = 197,500
[/LIST]

a number added to the product of y and x

a number added to the product of y and x
Since we're already using the variables x and y, we choose another arbitrary variable for the phrase [I]a number.[/I]
a
The product of y and x isL
xy
Then add a:
[B]a + xy[/B]

A random sample of n = 10 flash light batteries with a mean operating life X=5 hr. And a sample stan

A random sample of n = 10 flash light batteries with a mean operating life X=5 hr. And a sample standard deviation S = 1 hr. is picked from a production line known to produce batteries with normally distributed operating lives. What's the 98% confidence interval for the unknown mean of the working life of the entire population of batteries?
[URL='http://www.mathcelebrity.com/normconf.php?n=10&xbar=5&stdev=1&conf=98&rdig=4&pl=Small+Sample']Small Sample Confidence Interval for the Mean test[/URL]
[B]4.1078 < u < 5.8922[/B]

A rational number is such that when you multiply it by 7/3 and subtract 3/2 from the product, you ge

A rational number is such that when you multiply it by 7/3 and subtract 3/2 from the product, you get 92. What is the number?
Let the rational number be x. We're given:
7x/3 - 3/2 = 92
Using a common denominator of 3*2 = 6, we rewrite this as:
14x/6 - 9/6 = 92
(14x - 9)/6 = 92
Cross multiply:
14x - 9 = 92 * 6
14x - 9 = 552
To solve for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=14x-9%3D552&pl=Solve']type this equation into our search engine [/URL]and we get:
x = [B]40.07[/B]

A school dance committee is to consist of 2 freshmen, 3 sophomores, 4 juniors, and 5 seniors. If 6 f

A school dance committee is to consist of 2 freshmen, 3 sophomores, 4 juniors, and 5 seniors. If 6 freshmen, 9 sophomores, 7 juniors, and 7 seniors are eligible to be on the committee, in how many ways can the committee be chosen?
We want combinations for freshmen, sophomores, juniors, and seniors.
[LIST]
[*]Freshmen choices: [URL='https://www.mathcelebrity.com/permutation.php?num=6&den=2&pl=Combinations']6 C 2[/URL] = 15
[*]Sophomore choices: [URL='https://www.mathcelebrity.com/permutation.php?num=9&den=3&pl=Combinations']9 C 3[/URL] = 84
[*]Junior choices: [URL='https://www.mathcelebrity.com/permutation.php?num=7&den=4&pl=Combinations']7 C 4[/URL] = 35
[*]Senior choices: [URL='https://www.mathcelebrity.com/permutation.php?num=7&den=5&pl=Combinations']7 C 5 [/URL]= 21
[/LIST]
The number of committees we can choose is the product of combinations of freshmen, sophomores, juniors, and seniors.
Total Committees = Freshmen choices * Sophomore choices * Junior choices * Senior choices
Total Committees = 15 * 84 * 35 * 21
Total Committees = [B]926,100[/B]

A={2,8,1} and B={4,3,1}.find the Cartesian product A×B.

A={2,8,1} and B={4,3,1}.find the Cartesian product A×B.
Click [URL='http://www.mathcelebrity.com/cartprod.php?num1=2%2C8%2C1&num2=4%2C3%2C1&pl=Cartesian+Product']here[/URL] to find the answer

Algebraic Expressions

This calculator builds algebraic expressions based on word representations of numbers using the four operators and the words that represent them(increased,product,decreased,divided,times)
Also known as Mathematical phrases

An organic juice bottler ideally produces 215,000 bottle per day. But this total can vary by as much

An organic juice bottler ideally produces 215,000 bottle per day. But this total can vary by as much as 7,500 bottles. What is the maximum and minimum expected production at the bottling company?
Our production amount p is found by adding and subtracting our variance amount:
215,000 - 7,500 <= p <= 215,000 + 7,500
[B](min) 207,500 <= p <=222,500 (max)[/B]

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We are engaged in the production of registered TOEFL, IELTS, ESOL, CELTA / DELTA and other English certificates.
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Binomial Multiplication (FOIL)

Multiplies out the product of 2 binomials in the form (a + b)(c + d) with 1 unknown variable.

This utilizes the First-Outside-Inside-Last (F.O.I.L.) method.

This utilizes the First-Outside-Inside-Last (F.O.I.L.) method.

Boolean Algebra Multiplication

Determines the product of two expressions using boolean algebra.

C times the product b and a

C times the product b and a
[U]The product b and a:[/U]
ab
[U]c times the product:[/U]
[B]abc[/B]

Cartesian Product

Given a Set A and Set B, this calculates the Cartesian Product A × B

Cross Product

Given two vectors A and B in R^{3}, this calculates the cross product A × B as well as determine if the two vectors are parallel

Digit Product

Calculates a digit product for a number.

Divide 73 into two parts whose product is 402

Divide 73 into two parts whose product is 40
Our first part is x
Our second part is 73 - x
The product of the two parts is:
x(73 - x) = 40
Multiplying through, we get:
-x^2 + 73x = 402
Subtract 40 from each side, we get:
-x^2 + 73x - 402 = 0
This is a quadratic equation. To solve this, we type it in our search engine, choose "solve Quadratic", and we get:
[LIST=1]
[*]x = [B]6[/B]
[*]x = [B]67[/B]
[/LIST]

Estimating Reasonableness of Products

Given a product of 2 numbers and an estimated product, this will check to see if it is reasonable

From a standard 52 card deck, how many 6-card hands will have 2 spades and 4 hearts?

From a standard 52 card deck, how many 6-card hands will have 2 spades and 4 hearts?
We want the product of 13C2 * 13C4 since we have 13 possible spades choose 2 and 13 possible hearts choose 4
[LIST]
[*]Spades: 13C2 from our [URL='http://www.mathcelebrity.com/permutation.php?num=13&den=2&pl=Combinations']combinations calculator[/URL] = 78
[*]Hearts: 13C4 from our [URL='http://www.mathcelebrity.com/permutation.php?num=13&den=4&pl=Combinations']combinations calculator[/URL] = 715
[/LIST]
(78)(715) = [B]55,770[/B]

Fundamental Rule of Counting

Given a set of items, this calculates the total number of groups/choices that can be formed using the rule of product.

Gross Domestic Product (GDP)

Solves for all items of the Gross Domestic Product (GDP) equation:

GDP

Consumption (C)

Investment (I)

Government Spending (G)

Exports (X)

Imports (I).

GDP

Consumption (C)

Investment (I)

Government Spending (G)

Exports (X)

Imports (I).

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]

Imaginary Numbers

Calculates the imaginary number i where i = √-1 raised to any integer power as well as the product of imaginary numbers of quotient of imaginary numbers

Joanie multiplied 0.78 times 0.34 and got the product 26.52. What error did she make

Joanie multiplied 0.78 times 0.34 and got the product 26.52. What error did she make
[B]She didn't move the decimal point over 2 spots[/B]:
0.78 * 0.34 = 0.2652

Karen bought a bucket of popcorn at the movies for $5. She also bought some candy for $2 each. Karen

Karen bought a bucket of popcorn at the movies for $5. She also bought some candy for $2 each. Karen has to spend less than $15 on the popcorn and candy. Which inequality can be used to find c, the number of candies that Karen could have bought?
Since the candy cost is the product of price and quantity, we have:
2c + 5 < 15
To solve this inequality for c, we [URL='https://www.mathcelebrity.com/1unk.php?num=2c%2B5%3C15&pl=Solve']type it in our math engine [/URL]and we get:
[B]c < 5[/B]

Length (l) is the same as width (w) and their product is 64.

Length (l) is the same as width (w) and their product is 64.
We're given 2 equations:
[LIST=1]
[*]lw = 64
[*]l = w
[/LIST]
Substitute equation (2) into equation (1):
w * w = 64
w^2 = 64
[B]w = 8[/B]
Since l = w, then [B]l = 8[/B]

Let P(n) and S(n) denote the product and the sum, respectively, of the digits of the integer n. For

Let P(n) and S(n) denote the product and the sum, respectively, of the digits of the integer n. For example, P(23) = 6 and S(23) = 5. Suppose N is a two-digit number such that N = P(N) + S(N). What could N be? Is there more than one answer?
For example, for 23 P(23) = 6 and S(23) = 5, but 23 could not be the N that we want since 23 <> 5 + 6
Let t = tens digit and o = ones digit
P(n) = to
S(n) = t + o
P(n) + S(n) = to + t + o
N = 10t + o
Set them equal to each other N = P(N) + S(N)
10t + o = to + t + o
o's cancel, so we have
10t = to + t
Subtract t from each side, we have
9t = to
Divide each side by t
o = 9
So any two-digit number with 9 as the ones digit will work:
[B]{19,29,39,49,59,69,79,89,99}[/B]

Melinda is paid 17000 per year. She is also paid a sales commission of 5% of the value of her sales.

Melinda is paid 17000 per year. She is also paid a sales commission of 5% of the value of her sales. Last year she sold 344000 worth of products. What percent of her total income was her commission?
Calculate Melinda's commission:
344,000 * 0.05 = 17,200
Calculate her total income for the year
Total Income = Base Pay + Commission
Total Income = 17,000 + 17,200
Total Income = 34,200
Calculate the percent of her income which is commission:
Commission Income Percent = 100 * 17,200/34,200
Commission Income Percent = 100 * 0.5029
[B]Commission Income Percent = 50.29%[/B]

multiply k by 5.8, and then subtract 3.09 from the product

multiply k by 5.8, and then subtract 3.09 from the product
Take this algebraic expression in pieces:
[U]Multiply k by 5.8:[/U]
5.8k
[U]Then subtract 3.09 from the product[/U]
[B]5.8k - 3.09[/B]

n is equal to the product of 7 and the sum of m and 6

n is equal to the product of 7 and the sum of m and 6
The sum of m and 6:
m + 6
The product of 7 and this sum:
7(m + 6)
We set this expression equal to n:
[B]7(m + 6) = n[/B]

Nine less than the product of 2 and y is not less than 15

The product of 2 and y means we multiply
2y
Nine less than that product means we subtract 9
2y - 9
Finally, the entire expression is not less than 15, which means 15 or more. We use greater than or equal to
[B]2y - 9 >= 15
[/B]
If you want to solve this inequality and write the interval notation, use [URL='http://www.mathcelebrity.com/1unk.php?num=2y-9%3E%3D15&pl=Solve']our calculator[/URL].

n^2+n = odd

n^2+n = odd
Factor n^2+n:
n(n + 1)
We have one of two scenarios:
[LIST=1]
[*]If n is odd, then n + 1 is even. The product of an odd and even number is an even number
[*]If n is even, then n + 1 is odd. The product of an even and odd number is an even number
[/LIST]

n^2-n = even

n^2-n = even
Factor n^2-n:
n(n - 1)
We have one of two scenarios:
[LIST=1]
[*]If n is odd, then n - 1 is even. The product of an odd and even number is an even number
[*]If n is even, then n - 1 is odd. The product of an even and odd number is an even number
[/LIST]

product of 8 and the sum of 6 and 3y

product of 8 and the sum of 6 and 3y
the sum of 6 and 3y
6 + 3y
product of 8 and the sum of 6 and 3y
[B]8(6 + 3y)[/B]

product of a number and its reciprocal is increased by 7

product of a number and its reciprocal is increased by 7
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
Its reciprocal means we take the reciprocal of x:
1/x
product of a number and its reciprocal:
x * 1/x
x/x
The x's cancel giving us:
1
is increased by 7 means we add 7:
1 + 7
[B]8[/B]

Product of Consecutive Numbers

Finds the product of (n) consecutive integers, even or odd as well. Examples include:

product of 2 consecutive integers

product of 2 consecutive numbers

product of 2 consecutive even integers

product of 2 consecutive odd integers

product of 2 consecutive even numbers

product of 2 consecutive odd numbers

product of two consecutive integers

product of two consecutive odd integers

product of two consecutive even integers

product of two consecutive numbers

product of two consecutive odd numbers

product of two consecutive even numbers

product of 3 consecutive integers

product of 3 consecutive numbers

product of 3 consecutive even integers

product of 3 consecutive odd integers

product of 3 consecutive even numbers

product of 3 consecutive odd numbers

product of three consecutive integers

product of three consecutive odd integers

product of three consecutive even integers

product of three consecutive numbers

product of three consecutive odd numbers

product of three consecutive even numbers

product of 4 consecutive integers

product of 4 consecutive numbers

product of 4 consecutive even integers

product of 4 consecutive odd integers

product of 4 consecutive even numbers

product of 4 consecutive odd numbers

product of four consecutive integers

product of four consecutive odd integers

product of four consecutive even integers

product of four consecutive numbers

product of four consecutive odd numbers

product of four consecutive even numbers

product of 5 consecutive integers

product of 5 consecutive numbers

product of 5 consecutive even integers

product of 5 consecutive odd integers

product of 5 consecutive even numbers

product of 5 consecutive odd numbers

product of five consecutive integers

product of five consecutive odd integers

product of five consecutive even integers

product of five consecutive numbers

product of five consecutive odd numbers

product of five consecutive even numbers

product of 2 consecutive integers

product of 2 consecutive numbers

product of 2 consecutive even integers

product of 2 consecutive odd integers

product of 2 consecutive even numbers

product of 2 consecutive odd numbers

product of two consecutive integers

product of two consecutive odd integers

product of two consecutive even integers

product of two consecutive numbers

product of two consecutive odd numbers

product of two consecutive even numbers

product of 3 consecutive integers

product of 3 consecutive numbers

product of 3 consecutive even integers

product of 3 consecutive odd integers

product of 3 consecutive even numbers

product of 3 consecutive odd numbers

product of three consecutive integers

product of three consecutive odd integers

product of three consecutive even integers

product of three consecutive numbers

product of three consecutive odd numbers

product of three consecutive even numbers

product of 4 consecutive integers

product of 4 consecutive numbers

product of 4 consecutive even integers

product of 4 consecutive odd integers

product of 4 consecutive even numbers

product of 4 consecutive odd numbers

product of four consecutive integers

product of four consecutive odd integers

product of four consecutive even integers

product of four consecutive numbers

product of four consecutive odd numbers

product of four consecutive even numbers

product of 5 consecutive integers

product of 5 consecutive numbers

product of 5 consecutive even integers

product of 5 consecutive odd integers

product of 5 consecutive even numbers

product of 5 consecutive odd numbers

product of five consecutive integers

product of five consecutive odd integers

product of five consecutive even integers

product of five consecutive numbers

product of five consecutive odd numbers

product of five consecutive even numbers

product of r plus 7 and 4

product of r plus 7 and 4
r plus 7 means we add 7 to r:
r + 7
The product means we multiply the expression r + a 7 by 4:
[B]4(r + 7)[/B]

product of x and y decreased by their sum

product of x and y decreased by their sum
Product of x and y:
xy
Their sum:
x + y
Product of x and y decreased by their sum:
[B]xy - (x + y)[/B]

Prove 0! = 1

Prove 0! = 1
Let n be a whole number, where n! represents the product of n and all integers below it through 1.
The factorial formula for n is:
n! = n · (n - 1) * (n - 2) * ... * 3 * 2 * 1
Written in partially expanded form, n! is:
n! = n * (n - 1)!
[U]Substitute n = 1 into this expression:[/U]
n! = n * (n - 1)!
1! = 1 * (1 - 1)!
1! = 1 * (0)!
For the expression to be true, 0! [U]must[/U] equal 1. Otherwise, 1! <> 1 which contradicts the equation above

r less than 164 is 248 more than the product of 216 and r

r less than 164 is 248 more than the product of 216 and r
[U]r less than 164:[/U]
164 - r
[U]The product of 216 and r:[/U]
216r
[U]248 more than the product of 216 and r[/U]
216r + 248
[I]The word is means an equation, so we set 164 - r equal to 216r + 248[/I]
[B]164 - r = 216r + 248[/B]

r squared plus the product of 3 and s plus 5

r squared plus the product of 3 and s plus 5
r squared means we raise r to the power of 2
r^2
The product of 3 and s means we multiply s by 3:
3s
plus 5 means we add
3s + 5
R squared plus means we add r^2:
[B]r^2 + 3s + 5[/B]

raise r to the 8th power then find the product of the result and 3

raise r to the 8th power then find the product of the result and 3
Raise r to the 8th power means we raise r with an exponent of 8:
r^8
The product of the result and 3 means we muliply r^8 by 3
[B]3r^8[/B]

Rational Exponents - Fractional Indices

This calculator evaluates and simplifies a rational exponent expression in the form a^{b/c} where a is any integer *or* any variable [a-z] while b and c are integers. Also evaluates the product of rational exponents

Sales Price Variance

Calculates the Sales Price Variance and Total Variance for a group of products

Sam and Jeremy have ages that are consecutive odd integers. The product of their ages is 783. Which

Sam and Jeremy have ages that are consecutive odd integers. The product of their ages is 783. Which equation could be used to find Jeremy's age, j, if he is the younger man.
Let Sam's age be s. Let' Jeremy's age be j. We're given:
[LIST=1]
[*]s = j + 2 <-- consecutive odd integers
[*]sj = 783
[/LIST]
Substitute (1) into (2):
(j + 2)j = 783
j^2 + 2j = 783
Subtract 783 from each side:
j^2 + 2j - 783 = 0 <-- This is the equation to find Jeremy's age.
To solve this, [URL='https://www.mathcelebrity.com/quadratic.php?num=j%5E2%2B2j-783%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']we type this quadratic equation into the search engine[/URL] and get:
j = 27, j = -29.
Since ages cannot be negative, we have:
[B]j = 27[/B]

Serial numbers for a product are to be using 3 letters followed by 4 digits. The letters are to be t

Serial numbers for a product are to be using 3 letters followed by 4 digits. The letters are to be taken from the first 8 letters of the alphabet with no repeats. The digits are taken from numbers 0-9 with no repeats. How many serial numbers can be generated
The serial number is organized with letters (L) and digits (D) like this:
LLLDDDD
Here's how we get the serial number:
[LIST=1]
[*]The first letter can be any of 8 letters A-H
[*]The second letter can be any 7 of 8 letters A-H
[*]The third letter can be any 6 of 8 letters A-H
[*]The fourth digit can be any of 10 digits 0-9
[*]The fifth digit can be any 9 of 10 digits 0-9
[*]The sixth digit can be any 8 of 10 digits 0-9
[*]The seventh digit can be any 7 of 10 digits 0-9
[/LIST]
We multiply all possibilities:
8 * 7 * 6 * 10 * 9 * 8 * 7
[B]1,693,440[/B]

Serial numbers for a product are to made using 4 letters followed by 4 numbers. If the letters are t

Serial numbers for a product are to made using 4 letters followed by 4 numbers. If the letters are to be taken from the first 5 letters of the alphabet with repeats possible and the numbers are taken from the digits 0 through 9 with no repeats, how many serial numbers can be generated?
First 5 letters of the alphabet are {A, B, C, D, E}
The 4 letters can be chosen as possible:
5 * 5 * 5 * 5
The number are not repeatable, so the 4 numbers can be chosen as:
10 * 9 * 8 * 7 since we have one less choice with each pick
Grouping letters and numbers together, we have the following serial number combinations:
5 * 5 * 5 * 5 * 10 * 9 * 8 * 7 = [B]3,150,000[/B]

Set Notation

Given two number sets A and B, this determines the following:

* Union of A and B, denoted A U B

* Intersection of A and B, denoted A ∩ B

* Elements in A not in B, denoted A - B

* Elements in B not in A, denoted B - A

* Symmetric Difference A Δ B

* The Concatenation A · B

* The Cartesian Product A x B

* Cardinality of A = |A|

* Cardinality of B = |B|

* Jaccard Index J(A,B)

* Jaccard Distance J_{σ}(A,B)

* Dice's Coefficient

* If A is a subset of B

* If B is a subset of A

* Union of A and B, denoted A U B

* Intersection of A and B, denoted A ∩ B

* Elements in A not in B, denoted A - B

* Elements in B not in A, denoted B - A

* Symmetric Difference A Δ B

* The Concatenation A · B

* The Cartesian Product A x B

* Cardinality of A = |A|

* Cardinality of B = |B|

* Jaccard Index J(A,B)

* Jaccard Distance J

* Dice's Coefficient

* If A is a subset of B

* If B is a subset of A

Seven subtracted from the product of 3 and a number is greater than or equal to -26

Seven subtracted from the product of 3 and a number is greater than or equal to -26
[LIST=1]
[*]A number means an arbitrary variable, let's call it x.
[*]The product of 3 and a number is written as 3x
[*]Seven subtracted from 3x is written as 3x - 7
[*]Finally, that entire expression is greater than or equal to -26: [B]3x - 7 >= - 26[/B]
[/LIST]

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Square Roots and Exponents

Given a number (n), or a fraction (n/m), and/or an exponent (x), or product of up to 5 radicals, this determines the following:

* The square root of n denoted as √n

* The square root of the fraction n/m denoted as √n/m

* n raised to the x^{th} power denoted as n^{x} (Write without exponents)

* n raised to the x^{th} power raised to the yth power denoted as (n^{x})^{y} (Write without exponents)

* Product of up to 5 square roots: √a√b√c√d√e

* Write a numeric expression such as 8x8x8x8x8 in exponential form

* The square root of n denoted as √n

* The square root of the fraction n/m denoted as √n/m

* n raised to the x

* n raised to the x

* Product of up to 5 square roots: √a√b√c√d√e

* Write a numeric expression such as 8x8x8x8x8 in exponential form

subtract the product of 5 and x from 7

subtract the product of 5 and x from 7
The product of 5 and x means we multiply 5 by x:
5x
We subtract this product, 5x, from 7
[B]7 - 5x[/B]

Sum of two consecutive numbers is always odd

Sum of two consecutive numbers is always odd
Definition:
[LIST]
[*]A number which can be written in the form of 2 m where m is an integer, is called an even integer.
[*]A number which can be written in the form of 2 m + 1 where m is an integer, is called an odd integer.
[/LIST]
Take two consecutive integers, one even, and one odd:
2n and 2n + 1
Now add them
2n + (2n+ 1) = 4n + 1 = 2(2 n) + 1
The sum is of the form 2n + 1 (2n is an integer because the product of two integers is an integer)
Therefore, the sum of two consecutive integers is an odd number.

Suppose a computer chip manufacturer knows from experience that in an average production run of 5000

Suppose a computer chip manufacturer knows from experience that in an average production run of 5000 circuit boards, 100 will be defective. How many defective circuit boards can be expected in a run of 24,000 circuit boards?
100 defective / 5000 circuit boards * 24,000 circuit boards = [B]480 defective circuit boards[/B]

take away the product of 12 and p from 25

take away the product of 12 and p from 25
The product of 12 and p means we multiply 12 by p:
12p
Take away this product means we subtract 12p from 25:
[B]25 - 12p[/B]

Ten subtracted from the product of 9 and a number is less than ?24

Ten subtracted from the product of 9 and a number is less than ?24.
A number means an arbitrary variable, let's call it x
x
The product of 9 and a number:
9x
Ten subtracted from that
9x - 10
Finally, is less than means we set our entire expression less than -24
[B]9x - 10 < -24[/B]

the cube of the product of 3 and x

the cube of the product of 3 and x
The product of 3 and x:
3x
Cube this product means raise it to the power of 3:
(3x)^3 = [B]27x^3[/B]

The difference between the product of 4 and a number and the square of a number

The difference between the product of 4 and a number and the square of a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
The product of 4 and a number:
4x
The square of a number means we raise x to the power of 2:
x^2
The difference between the product of 4 and a number and the square of a number:
[B]4x - x^2[/B]

The fixed costs to produce a certain product are 15,000 and the variable costs are $12.00 per item.

The fixed costs to produce a certain product are 15,000 and the variable costs are $12.00 per item. The revenue for a certain product is $27.00 each. If the company sells x products, then what is the revenue equation?
R(x) = Revenue per item x number of products sold
[B]R(x) = 27x[/B]

The function f(x) = e^x(x - 3) has a critical point at x =

The function f(x) = e^x(x - 3) has a critical point at x =
The critical point is where the derivative equals 0.
We multiply through for f(x) to get:
f(x) = xe^x - 3e^x
Using the product rule on the first term f'g + fg', we get:
f'(x) = xe^x + e^x - 3e^x
f'(x) = xe^x -2e^x
f'(x) = e^x(x - 2)
We want f'(x) = 0
e^x(x - 2) = 0
When [B]x = 2[/B], then f'(x) = 0

The product 18 And q

The product 18 And q
[B]18q[/B]

the product of 2 less than a number and 7 is 13

the product of 2 less than a number and 7 is 13
Take this algebraic expression in [U]4 parts[/U]:
Part 1 - The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
Part 2 - 2 less than a number means we subtract 2 from x
x - 2
Part 3 - The phrase [I]product[/I] means we multiply x - 2 by 7
7(x - 2)
Part 4 - The phrase [I]is[/I] means an equation, so we set 7(x - 2) equal to 13
[B]7(x - 2) = 13[/B]

the product of 3 and the sum of m and 2n

the product of 3 and the sum of m and 2n
The sum of m and 2n means we add 2n to m:
m + 2n
The product of 3 means we multiply the sum m + 2n by 3:
[B]3(m + 2n)[/B]

the product of 8 and 15 more than a number

the product of 8 and 15 more than a number.
Take this algebraic expression in pieces.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
15 more than x means we add 15 to x:
x + 15
The product of 8 and 15 more than a number means we multiply 8 by x + 15
[B]8(x + 15)[/B]

The product of 8 and a number k is greater than 4 and no more than 16

Let's take this by pieces.
The product of 8 and a number k is written as: 8k.
Since it's greater than 4, but not more than 16, we include this in the middle of an inequality statement.
4 < 8k <= 16
Notice no more than has an equal sign, it means less than or equal to.
Greater does not include an equal sign.

The product of a number b and 3 is no less than 12.

The product of a number b and 3 is no less than 12.
A number b is just written as b. So we have:
The product of b and 3 is no less than 12.
take this in parts:
[LIST]
[*]The product of b and 3: 3b
[*]The phrase [I]is no less than[/I] means an inequality, so we have greater than or equal to. We set 3b greater than or equal to 12
[/LIST]
[B]3b >= 12[/B]

the product of k and 70, minus 15

the product of k and 70, minus 15
Take this algebraic expression in pieces:
The product of k and 70 means we multiply 70 times k
70k
The word [I]minus[/I] means we subtract 15 from 70k
[B]70k - 15[/B]

The product of the 2 numbers x and y

The product of the 2 numbers x and y
The phrase [I]product [/I]means we multiply the two variables, x and y.
[B]xy[/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 product of x and 7 is not greater than 21

The product of x and 7 is not greater than 21
The product of x and 7:
7x
Is not greater than means less than or equal to, so we have our algebraic expression:
7x <= 21
If you want to solve this inequality and interval notation, use our [URL='http://www.mathcelebrity.com/interval-notation-calculator.php?num=7x%3C%3D21&pl=Show+Interval+Notation']calculator[/URL].

The product of x and u is not greater than 21

The product of x and u is not greater than 21
The product of x and u
xu
Not greater than means less than or equal to:
xu <= 21

the reciprocal of the product a and b

the reciprocal of the product a and b
Take this algebraic expression in pieces:
The product a and b means we multiply a times b
ab
The [I]reciprocal[/I] means we take 1 over ab
[B]1/ab[/B]

the sum of 3 numbers divided by its product

the sum of 3 numbers divided by its product
The phrase [I]3 numbers[/I] means we choose [I]3[/I] arbitrary variables. Let's call them x, y, z.
The sum of of these 3 numbers is:
x + y + z
The phrase [I]its product[/I] means we multiply all 3 arbitrary variables together:
xyz
Now, the phrase [I]divided by[/I] means we divide x + y + z by xyz:
[B](x + y + z)/xyz[/B]

The sum of a and b divided by their product

The sum of a and b divided by their product
The sum of a and b means we add b to a:
a + b
The product of a and b means we multiply a by b:
ab
To get our final algebraic expression, we divide the sum (a + b) by the product ab:
[B](a + b)/ab[/B]

the sum of a and b, divided by the product of c and d

the sum of a and b, divided by the product of c and d
The sum of a and b, means we add b to a
a + b
The product of c and d means we multiply c by d
cd
Divided by means we divide a + b by cd
[B](a + b)/cd[/B]

The sum of the product and quotient of the numbers x and y

The sum of the product and quotient of the numbers x and y
the product of the numbers x and y
xy
The quotient of the numbers x and y
x/y
The sum of the product and quotient of the numbers x and y
[B]xy + x/y[/B]

To create an entry code, you must first choose 2 letters and then, 4 single-digit numbers. How ma

To create an entry code, you must first choose 2 letters and then, 4 single-digit numbers. How many different entry codes can you create?
List total combinations using the product of all possibilities:
26 letters * 26 letters * 10 digits (0-9) * 10 digits (0-9) * 10 digits (0-9) * 10 digits (0-9)
[B]6,760,000[/B]

translate the product of -1 and a number in mathematics expression

translate the product of -1 and a number in mathematics expression
The phrase [I]a number[/I] means an arbitrary variable. Let's call it x.
The product of -1 and the number;
[B]-x[/B]

twice the product of p q and r

twice the product of p q and r
The product of p q and r means we multiply all 3 variables together:
pqr
The word [I]twice[/I] means we multiply pqr by 2:
[B]2pqr[/B]

twice the product of p q and r

twice the product of p q and r
The product of p q and r:
pqr
Twice means we multiply pqr by 2:
[B]2pqr[/B]

twice the square of the product of x and y

twice the square of the product of x and y
Take this algebraic expression in pieces:
[LIST]
[*]The product of x and y means we multiply x and y: xy
[*]The square of the product means we raise xy to the power of 2: (xy)^2 = x^2y^2
[*]Twice the square means we multiply the square by 2: [B]2x^2y^2[/B]
[/LIST]

twice the square of the product of x and y

twice the square of the product of x and y
[LIST]
[*]The product of x and y: xy
[*]The square of the product means we raise xy to the power of 2: (xy)^2
[*]Twice the square means we multiply by 2
[/LIST]
[B]2(xy)^2
or
2x^2y^2[/B]

Units of Output (Service Output) Depreciation

Given an asset value, salvage value, production units, and units per period, this calculates the depreciation per period using the units of output depreciation (service output depreciation)

v is equal to the product of 7 and the sum of u and 6

v is equal to the product of 7 and the sum of u and 6
[LIST]
[*]Sum of u and 6: u + 6
[*]the product of 7 and the sum of u and 6: 7(u + 6)
[*]We set this expression equal to v:
[/LIST]
[B]v = 7(u + 6)[/B]

Vectors

Given 2 vectors A and B, this calculates:

* Length (magnitude) of A = ||A||

* Length (magnitude) of B = ||B||

* Sum of A and B = A + B (addition)

* Difference of A and B = A - B (subtraction)

* Dot Product of vectors A and B = A x B

A ÷ B (division)

* Distance between A and B = AB

* Angle between A and B = θ

* Unit Vector U of A.

* Determines the relationship between A and B to see if they are orthogonal (perpendicular), same direction, or parallel (includes parallel planes).

* Cauchy-Schwarz Inequality

* The orthogonal projection of A on to B, proj_{B}A and and the vector component of A orthogonal to B → A - proj_{B}A

Also calculates the horizontal component and vertical component of a 2-D vector.

* Length (magnitude) of A = ||A||

* Length (magnitude) of B = ||B||

* Sum of A and B = A + B (addition)

* Difference of A and B = A - B (subtraction)

* Dot Product of vectors A and B = A x B

A ÷ B (division)

* Distance between A and B = AB

* Angle between A and B = θ

* Unit Vector U of A.

* Determines the relationship between A and B to see if they are orthogonal (perpendicular), same direction, or parallel (includes parallel planes).

* Cauchy-Schwarz Inequality

* The orthogonal projection of A on to B, proj

Also calculates the horizontal component and vertical component of a 2-D vector.

What is an Exponent

This lesson walks you through what an exponent is, the product rule for exponents, the quotient rule for exponents, the 0 power rule, the power of a power rule for exponents

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]

Which of the following is equivalent to 3(2x + 1)(4x + 1)?

Which of the following is equivalent to 3(2x + 1)(4x + 1)?
[LIST]
[*]A) 45x
[*]B) 24x^2 + 3
[*]C) 24x^2 + 18x + 3
[*]D) 18x^2 + 6
[/LIST]
First, [URL='https://www.mathcelebrity.com/binomult.php?term1=2x%2B1&term2=4x%2B1&pl=Expand+Product+of+2+Binomials+using+FOIL']multiply the binomials[/URL]:
We get 8x^2 + 6x + 1
Now multiply this polynomial by 3:
3(8x^2 + 6x + 1) = [B]24x^2 + 18x + 3, answer C[/B]

y minus 10 is equal to the product of y and 8

y minus 10 is equal to the product of y and 8.
Take this algebraic expression in 3 parts:
Part 1: y minus 10
Subtract 10 from the variable y
y - 10
Part 2: The product of y and 8
We multiply 8 by the variable y
8y
Part 3: The phrase [I]is equal to[/I] means an equation, so we set y - 10 equal to 8y
[B]y - 10 = 8y[/B]

You are selling fertilizer to female farmers in Ghana. There are 22,600,000 people in Ghana, and 60%

You are selling fertilizer to female farmers in Ghana. There are 22,600,000 people in Ghana, and 60% are of working age. Within that working-age group, women account for 53%. Of the working-age females, 42% of them are employed in farming. What is the total number of potential customers for your fertilizer?
[U]Our sample population is found by this product:[/U]
Female farmers of working age in Ghana = Total people in Ghana *[I] Working Age[/I] * Women of working Age * Farmers
Since 60% = 0.6, 53% = 0.53, and 42% = 0.42, we have
Female farmers of working age in Ghana = 22,600,000 * 0.6 * 0.53 * 0.42
Female farmers of working age in Ghana = [B]3,018,456[/B]

Zero Multiplication Property

Demonstrates the Zero Multiplication property using a number. Also called the Zero Product Property.
Numerical Properties