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each person eats 1/6 of a pizza.
6 people per pizza
90 pizzas
number of persons is equal to 6 * 90 = 540.

formula is:

number of people * amount of pizza eaten per person = number of pizzas eaten.

n = number of people
r = amount of pizza eaten per person
p = number of pizzas eaten

n * r = p

r = 1/6 because each person eats 1/6 of a pizza.

p = 90 (given)

n * r = p becomes:

n * (1/6) = 90

multiply both sides of this equation by 6 to get:

n = 90 * 6 which becomes:

n = 540

540 people eating 1/6 of a pizza each consume 90 pizzas (540 * 1/6 = 90)

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this is a geometric series with a common ratio of 1/4.

formula for sum of a geometric series is:
Sn = (a*(1-r^n)) / (1-r)
n = number of terms in the series.
S = sum of the terms in the series.
a = the first term in the series.
r = the common ratio in the series.

In this problem:

n = 8
a = 256
r = (.25)

Substitute in the formula to get:

Sn = (a(1-r^n)) / (1-r) becomes:

S8 = (256 * (1 - .25^8)) / (1 - .25)

Simplify to get:

S8 = (256 * (1 - .000015259) / (.75)

Simplify further to get:

S8 = (256 * .999984741) / (.75)

Solve to get:

S8 = 341.328125

You can verify this is correct by detailing each individual term from 1 to 8 and then summing them up.

You get:
S1 = 256
S2 = 64
S3 = 16
S4 = 4
S5 = 1
S6 = .25
S7 = .0625
S8 = .015625

Sum is equal to 341.328125

Answer is the same.

You're good.

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bracelet will be 100 grams with a 50% silver alloy.
artist has 60% silver alloy and 35% silver alloy.
he needs to combine these together to make the bracelet with 50% silver alloy.

x = number of grams of 60% silver alloy.
y = number of grams of 35% silver alloy.

x + y = 100 grams of bracelet

.60 * x = amount of silver in the first alloy.
.35 * y = amount of silver in the second alloy.

.60 * x + .35 * y = .50 * 100 grams of silver in the bracelet

your 2 equations that you need to solve simultaneously are:

x + y = 100

.60*x + .35*y = .50*100

we'll use substitution.

from the first equation, solve for y to get y = 100 - x

substitute 100-x for y in the second equation to get:

.60*x + .35*(100-x) = .5*100
simplify to get:
.60*x + .35*100 - .35*x = .5*100
simplify to get:
.60*x + 35 - .35*x = 50
combine like terms to get:
.25*x + 35 = 50
subtract 35 from both sides to get:
.25*x = 50-35
simplify to get:
.25*x = 15
divide both sides by .30 to get:
x = 15/.25 = 60
if x = 60, then y = 100 - 60 = 40

you need 60 grams of 60% alloy and 40 grams of 35% alloy.

60 + 40 = 100 grams of alloy in the bracelet (from x + y = 100).

.6*60 + .35*40 = 36 + 14 = 50 grams of silver in the bracelet (from .6*x + .25*y = .5*100).

Your answer is:

He needs 60 grams of 60% alloy and 40 grams of 35% alloy to make the bracelet.

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The probability that a group of 5 chosen out of a group of 10 will contain exactly 2 girls and 3 boys is equal to:
(C(6,3) * C(4,2)) / C(10,5)

That becomes 120 / 252 = .476190476

That is the number of ways you can get a group of 5 consisting of 2 girls and 3 boys divided by the total number of ways you can get a group of 5.

It can also be calculated another way.

The probability of getting bbbgg (3 boys picked first and then 2 girls) would be:

6/10 * 5/9 * 4/8 * 4/7 * 3/6

This becomes .0476190476

You then need to multiply that by the number of different ways the group of 5 can be chosen.

That would be C(5,2) = 10

The total probability is therefore .0476190476 * 10 = .476190476

The number of ways the group of 5 can be choen is shown below:

bbbgg
bbgbg
bbggb
bgbbg
bgbgb
bggbb
gbbbg
gbbgb
gbgbb
ggbbb

bbbgg means boy drawn first and second and third girl drawn fourth and fifth.
bbgbg means boy drawn first and second girl drawn third boy drawn fourth girl drawn fifth.
etc. until you get to:
ggbbb means girls drawn first and second boy drawn third fourth and fifth.

both ways will get you the same answer.

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y = (1/3)*x^2 - 2*x + 7
x = -b/2a
------------------
a = (1/3)
b = -2
c = 7
------------------
-b/2a = -(-2)/(2/3) = 2/(2/3) = 2*3/2 = 6/2 = 3
-------------------
you found x
now you need to find f(x) to get the y value
when x = 3, (1/3)*x^2 - 2*x + 7 = (1/3)*3^2 - 2*3 + 7
this equals 3 - 6 + 7 = 4
---------------------
your vertex should be at the point pair of (x,y) = 3,4)
-----------------------
the graph of this equation looks like this:

you can see from the graph that the vertex is at the point (x,y) = 3,4)

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200 pounds of chemicals cost $95.
mixing vats = $400 each.
Sales tax on all products is 6% of the total cost of the order.
Shipping costs for the order will be a flat $30.

question a:

she orders 4000 pounds of chemicals.
cost per pound of chemical = $95.00 / 200 = .475
.475 * 4000 = $1900.00 for the chemicals.
Total cost for the order for 4000 pounds of chemicals only is as follows:

Chemicals = $1900.00
Shipping = $30.00
Total without Tax = $1930.00
Tax on $1930.00 at 6% = $115.80
Total Cost for the Order = $1930.00 + $115.80 = $2045.80

In equation form, this would be:

((.475 * 4000) + 30) * 1.06 = total cost of the order.

Tax is assumed to apply to the total order (material plus shipping).

question b:

Cost per vat = $400.00

y = number of vats to order.

Total cost of the order for vats only would be:

(y * 400 + 30) * 1.06

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I presume the statement is:
log(4,x) >= log(4,x^2)
This means that the log of x to the base 4 is greater than or equal to log of x^2 to the base 4.
find x.
to find where they cross, you need to graph both equations.
we'll graph log(4,x) and log(4,x^2) to see if they cross anywhere.

from the graph, it looks like this occurs when goes from 0 to 1.
when x is negative doesn't apply because the graph of log(4,x) is only valid when x is greater than 0.
log(4,0) is not valid.
log(4,-x) is not valid.
you probably just need to graph them from x = 0 to x = 2.
The crossing point is at x = 1.
log(4,x) >= log(4,x^2) in the interval 0 < x <= 1
the graph of the log(4,x^2) is still not valid at x = 0.
the graph of the log(4,x^2) is valid when x is negative because x^2 is positive.

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x = sean's age.
x - 5 = peter's age.
x + x-5 <= 37
combine like terms to get:
2x - 5 <= 37
add 5 to both sides to get:
2x <= 42
divide both sides by 2 to get:
x <= 21
peter's age is x-5
the oldest peter can be is 21-5 = 16
when peter = 16, sean = 16 + 5 = 21
21 + 16 = 37

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You have 315 students.
There are 21 students per class.
How many classes are there?
Answer should be 15 classes because 15 * 21 = 315.

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He bought 2 bags for $8.28
There are 36 biscuits in each bag.
2 * 36 = 72
He bought 72 biscuits.
Cost per biscuit = $8.28 / 72 = .115 = 11.5 cents per biscuit.
.115 * 72 = $8.28

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I believe the answer is 8! which is equal to 8*7*6*5*4*3*2*1 which equals 40320.
If it were 3 posters, the answer would be 3! = 6
3! can be shown
8! is a lot more difficult to show.
We'll show 3!
Let the posters be a, b, and c.
The ways they can be shown are:
abc
acb
bac
bca
cab
cba
With 4 posters, the number of ways they can be shown is 4! = 24 possible ways.
Let the posters be a, b, c, and d
The ways they can be shown are:
abcd
abdc
acbd
acdb
adbc
adcb
bacd
badc
bcad
bcda
bdac
bdca
cabd
cadb
cbad
cbda
cdab
cdba
dabc
dacb
dbac
dbca
dcab
dcba
As the number of posters goes up, the number of possible permutations gets significntly higher.
The number you are looking for is 8! = 40320

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The formula for total cost is:
T = x + y*h
x = the membership cost.
y = the cost per hour.
For the non-member, the total cost is equal to:
T = 0 + 10*h which becomes:
T = 10*h
For the member, the total cost is equal to:
T = 300 + 4*h
You want to know at what point the total cost for the member becomes less than the total cost for the non-member.
That point is given by the equation:
300 + 4*h < 10*h
Subtract 4*h from both sides of this equation to get:
300 < 10*h - 4*h
Simplify to get:
300 < 6*h
Divide both sides of this equation by 6 to get:
300/6 < h
Simplify to get:
50 < h
50 < h also means h > 50.
The total membership cost will becomes less than the total non-member cost when the number of hours is greater than 50.
If h = 50, then the equation of:
300 < 10*h - 4*h becomes:
300 < 10*50 - 4*50
Simplify this to get:
300 < 500 - 200
Simplify further to get:
300 < 300
The statement is not true because h is not greater than 50.
Take any h greater than 50 and the equation should be true.
Take h = 51.
You get:
300 < 10*51 - 4*51
simplify this to get:
300 < 510 - 204
Simplify further to get:
300 < 306
The statement is now true.

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1/4 of the seniors have averages above 90.
The remainder is 3/4 of the seniors.
1/2 of the remainders have averages between 80 and 90.
That means that 1/2 * 3/4 of the seniors have averages between 80 and 90.
That means that 3/8 of the seniors have averages between 80 and 90.
The total number of seniors that have averages above 80 is equal to 1/4 + 3/8.
Since 1/4 = 2/8, then the total number of seniors that have averages above 80 is equal to 5/8.
That leaves 3/8 of the seniors that have averages below 80 because 5/8 + 3/8 = 8/8 = 1.

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Rate * Time = Units Produced
R*T=U
The number of units produced is 1 clean house.
U = 1
Working together, the rate of each individual is additive.
R1 = his rate
R2 = your rate
(R1 + R2) * T = 1
He takes 2 hours and 5 minutes to clean the house by himself.
5 minutes = 5/60 hours = 1/12 hours = .0833333333 hours.
He take 2.083333333 hours to clean the house by himself.
Working by himself, the formula becomes R1*T = U which becomes:
R1*2.08333333 = 1
Divide both sides of this equation by 2.0833333, and you get:
R1 = 1.2.0833333 = .48
his rate is equal to .48 of the house in 1 hour.
The formula for both of you cleaning the house together becomes:
(.48 + R2) * T = 1
The time it takes both of you to clean the house is 1 hours and 20 minutes.
20 minutes = 20/60 hours = .3333333 hours.
The time it takes both of you to clean the house is 1.3333333 hours.
The formula becomes:
(.48 + R2) * 1.3333333 = 1
Remove parentheses to get:
.48 * 1.33333 + R2 * 1.3333333 = 1
Simplify to get:
.64 + R2 * 1.3333333 = 1
Subtract .64 from both sides of the equation and then divide both sides of the equation by 1.3333333 to get:
R2 = (1 - .64) / 1.3333333
Simplify to get:
R2 = .36 / 1.3333333 which becomes:
R2 = .27
Your rate of cleaning the house is equal to .27 of the house in 1 hour.
The formula for both of you completing the house at the same time becomes:
(.48 + .27) * 1.33333333 = 1
Simplify this to get:
.75 * 1.3333333 = 1
Simplify further to get:
1 = 1
The formula is good.
You can clean .27 of the house in 1 hour.
The formula for you working alone becomes:
R2 * T = 1
Since R2 = .27, this becomes:
.27 * T = 1
Divide both sides of this equation by .27 to get:
T = 3.703703704 house.
You would take 3.703703704 hours to clean the house by yourself.
.703703704 hours * 60 = 42.222222 minutes.
If would take approximately 3 hours and 42 minutes for you to clean the house by yourself at that rate.

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1^3 = 1
2^3 = 8
3^3 = 27
4^3 = 64
5^3 = 125
6^3 = 216

The next term is 216.
Every term is equal to the next number in the sequence cubed.

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slope intercept form of the equation of a straight line is y = mx + b
m is the slope
b is the y intercept

your equation is y = (-3/8)*x + 4

this equation is already in slope intercept form.

the slope is (-3/8)

the y intercept is 4

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he got 44 hits in 117 at bats in 49 games.

ratio of hits to at bats is equal to 44/117

44/117 = .376068376.

rounded to the nearest thousandth that equals .376

ratio of hits to games is equal to 44/49.

44/49 = .897959184

based on the ratio of hits to games, he will get .897959184 * 162 = 145.4693878 hits in a 162 game season.

based on the ratio of hits to at bats, he will get .376068376 * 400 = 150.4273504 hits in 400 at bats.

these round to 145 hits in a 162 game season and 150 hits in 400 at bats.

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m is the slope and the line is passing through the point (x,y) = (-2,1)
This means that x = -2 and y = 1 at that point on the graph.
the slope intercept form of the equation of a straight line is y = m*x + b
m is the slope
b is the y-intercept
replace m with -3 to get:
y = -3x + b
replace y with 1 and x with -2 and then solve for b.
you get:
1 = -3*(-2) + b
this becomes:
1 = 6 + b
this results in b = -5
your equation becomes:
y = -3x - 5
graph of your equation looks like this:


you can see that the graph passes through the point (-2,1)
when x = -2, y = -3*(-2) - 5 = 6 - 5 = 1
since the slope is -3, then for every + 1 unit change in the value of x, the value of y should change by - 3 units.
when x = 0, y = -5.
when x = 1, y = -8.
when x = 1, y = -11.
etc.

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I get an answer of 7b^2 - 2b + 8
first you divide 35b^3 by 5b to get 7b^2
then you multiply (5b+5) by 7b^2 to get (35b^3 + 35b^2)
then you subtract (35b^3 + 35b^2) from (35b^3 + 25b^2) to get -10b^2
then you bring down the 30b to get (-10b^2 + 30b)
then you divide -10b^2 by 5b to get -2b
then you multiply (5b+5) by -2b to get (-10b^2 - 10b)
then you subtract (-10b^2 - 10b) from (-10b^2 + 30b) to get 40b
then you bring down the 47 to get (40b + 47)
then you divide 40b by 5b to get 8
then you multiply (5b+5) by 8 to get (40b + 40)
then you subtract (40b + 40) from (40b + 47) to get 7
that's your remainder.

the answer is 7b^2 - 2b + 8 with a remainder of 7

on paper, it looks like what is shown below.

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Answer should be:
(x-2) with a remainder of -34
remainder of -34 is equivalent to -34/(x+8)
answer is therefore x-2 - 34/(x+8)
the process works like this.
divide x^2 by x to get x because x*x = x^2
multiply (x+8) by x to get x^2 + 8x
subtract (x^2 + 8x) from (x^2 + 6x) to get -2x
bring down the -50 to get (-2x - 50)
divide -2x by x to get -2
multiply (x+8) by -2 to get (-2x - 16)
subtract (-2x -16) from (-2x - 50) to get -34
that's your remainder.
your answer is (x-2) with a remainder of -34
when you multiply (x-2) by (x+8) you will then need to add the remainder to get back to the original equation.
(x-2) * (x+8) = x^2 + 6x - 16
add -34 to that and you get x^2 + 6x - 50 which was your original equation.
on paper, the division will be as shown below:

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x = one of the integers
y = the other

x = 2*y
x + y <= 12

substitute for x in the second equation to get:

2*y + y <= 12

combine like terms to get:

3*y <= 12

divide both sides by 3 to get:

y <= 4

the greatest that y can be is 4.

if y is equal to 4, then x = 2*y means that x = 8

the greatest that x can be is 8.

when x = 8 and y = 4, then:

8 = 2*4 = ok
4 + 8 = 12 = ok

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6 foot long sides.
square ends.

find height if volume is 48 cubic feet

Volume of rectangular prism is h*l*w (height * length * width)

length is 6 feet.

height and width (form the ends) are the same measure, call it s.

volume is therefore h*s^2 (height * area of the base)

formula for volume is v = h*s^2

height is 6 feet, so formula becomes v = 6*s^2

volume is 48 cubic feet, so formula becomes 48 = 6*s^2

divide both sides of equation by 6 to get 8 = s^2

take square root of both sides of equation to get s = sqrt(8).

length is 6
width is sqrt(8)
height is sqrt(8)

volume is l*w*h = 6*sqrt(8)*sqrt(8) = 6*8 = 48 cubic feet.

sqrt(8) = sqrt(2*2*2) = 2*sqrt(2)

--------------------------------------

(b) Find the height if 44 square feet of glass is to be used.

aquarium is open topped.

length is still 6 feet.

ends are still square.

aquarium is open on top.

it therefore has 2 sides (one on each end) and 3 sides along the length.

since the ends are still square, then the width equal the height and we can still call them s rather than w and h.

the area of each end is still s^2.

2 times the area of each end is therefore equal to 2*s^2.

the area of each side along the length will be s*l.

we have 3 of them.

the total area of the sides along the length is therefore 3*s*l

the side along the length on the top of the aquarium is open and so its area is not counted for total area of glass.

total area of glass is the area of the 2 ends plus the area along the length for 3 of the 4 sides along the length..

that equals 2*s^2 + 3*s*l which equals 44 square feet.

formula becomes 44 = 2*s^2 + 3*s*l

we know that l = 6, so we substitute 6 for l to get:

44 = 2*s^2 + 3*s*6 which becomes 44 = 2*s^2 + 18*s

subtract 44 from both sides of this equation to get the standard form of a quadratic equation of:

2*s^2 + 18*s -44 = 0

divide both sides of this equation by 2 to get:

s^2 + 9*s - 22 = 0

factor this equation to get:

(s+11) * (s-2) = 0

solve for s to get s = -11 or s = 2

s can't be negative, so s has to be equal to 2.

that's your answer.

total surface area = 2*4 + 3*2*6 = 8 + 36 = 44 square inches.

a picture of your aquarium that might help is shown here:

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a b log means the log of a to the base b as far as I know.

this is written as log(b,a) which means the log of a to the base b.

y a b log means y = log of a to the base of b.

This is written as y = log(b,a) which means that y is equal to the log of a to the base of b.

The conversion of a log expression to an exponential expression is given by the equation of:

b^y = a if and only if log(b,a) = y

This means that the base b raised to the power of y is equal to a if and only if the log of a to the base b is equal to y.

Likewise, the conversion of an exponential expression to a log expression is given by the equation of:

log(b,a) = y if and only if b^y = a

An example is as follows:

We know that 10^3 is equal to 1000.

By the equation of b^y = a if and only if log(b,a) = y, we should be able to see that:

10^3 = 1000 if and only if log(10,1000) = 3

Since the LOG function of our calculator can give us the log of 1000 to the base of 10, we can easily see that log(1000) = 3.

You just enter 1000 into the calculator and then hit the log key. It should return a value of 3.

This assumes you have a calculator like the Texas Instruments or one with equivalent functionality.

The most frequently used bases for logs are 10 and LN.

LN stands for natural log.

the LOG function of the calculator does logs to the base of 10.
The LN function of the calculator does logs to the base of e.
the base of e is equivalent to the base of 2.718281828... because the letter e represents that value.

The value of the base e is used a lot in scientific studies.

3 commonly used log properties are:

log(a*b) = log(a) + log(b)
log(a/b) = log(a) - log(b)
log(a^b) = b*log(a)

log(a) implies the base of 10, and can also be written as log(10,a).

There is also a conversion formula that you can use to translate the log of any base to the base of 10 or e, so that the log can be solved using your calculator.

That formula is:

log(b,a) = log(c,a) / log(c,b)

This means that the log of a number to the base of b is equivalent to the log of that same number to the base of c divided by the log of the base b to the base of c.

An example will clarify:

You start with log(2,8)

We know that log(2,8) = 3 because 2^3 = 8

Suppose we didn't know that.

We could find it by converting to the base of 10.

log(2,8) = log(10,8) / log(10,2)

Use your calculator to find log(8).
Use your calculator to find log(2).

divide log(8) / log(2) and you should get 3.

Works every time and is a very useful tool if you have logs in unfamiliar bases and you want to use your calculator to solve them.

You can also do the same thing with the LN function of your calculator.

log(2,8) = ln(e,8) / ln(e,2)

Use the LN function of your calculator to find LN (8).
Use the LN function of your calculator to find LN (2).

Divide ln(8) by ln(2) and you should also get 3.

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In 5 hours, Ajay will have walked 5h * 6km/h = 30 km.
In 5 hours, Vijay will have walked 5h * 5km/h = 25 km.

Since they are walking in opposite directions, they will be 55 km apart.

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When b^2 - 4ac is equal to 0, you will have 1 solution.

The standard form of the quadratic equation is:

ax^2 + bx + c = 0

a is the coefficient of the x^2 term.
b is the coefficient of the x term.
c is the constant term.

To find the roots of this equation, you can use the quadratic formula.

the quadratic formula is:

x = -b +/- sqrt(b^2-4ac)
    --------------------
             2a

If b^2 - 4ac is equal to 0, then you have +/- 0 in the numerator and there is only one solution to the problem.

We can construct an example to show you what I mean.

To construct this example, let's pick a b term at random.

Let's say b is equal to 6.

That makes b^2 = 36

Divide 36 by 4 to get 9.

We need to get a * c to be equal to 9.

a * c will be equal to 9 when a = 1 and c = 9

We have:

a = 1
b = 6
c = 9

b^2 - 4ac becomes 36 - (4*1*9) which becomes 36 - 36 which becomes 0.

Our quadratic equation should have one solution only.

The standard form of the quadratic equation is ax^2 + bx + c = 0

Since a = 1, b = 6 and c = 9, this equation becomes:

x^2 + 6x + 9 = 0

We will now graph this equation as shown below:



You can see from the graph that the graph of the quadratic equation touches the x-axis at the point x = -3 and only at that point.

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Let > imply taller
Let < imply shorter

the first statement says that:

B > j + 3

This means that there is a positive value we can add to j + 3 to make it equal to B.

We'll call that value x.

We get:

B = j + 3 + x

The second statement says that:

P < B + 2

We can replace B with j + 3 + x to make that statement become:

P < j + 3 + x + 2

We can simplify this statement to be equal to:

P < j + x + 5

This means that there is a value we can add to P to make it equal to j + x + 5.

We'll call that value y.

The statement becomes:

P + y = j + x + 5

If y = x + 5, then the statement will become:

P + x + 5 = j + x + 5

If we subtract (x + 5) from both sides of the equation, we will then get:

P = j

So, if y = x + 5, then we get P = j.

So P = j is possible when y = x + 5.

Going back to the original equation, we have:

P + y = j + x + 5

If we let y be greater than x + 5, then we get another value (call it z), such that:

y = x + 5 + z

Our equation of P + y = j + x + 5 becomes:

P + x + 5 + z = j + x + 5

If we subtract (x + 5) from both sides of this equation, we get:

P + z = j

This implies that P < j

So we can have P = j and we can have P < j

Going back to the original equation again, we have:

P + y = j + x + 5

If we let y be smaller than x + 5, then we get another value (call it w), such that:

y + w = x + 5 which becomes y = x + 5 - w after we subtract w from both sides of the equation.

Our equation becomes:

P + x + 5 - w = j + x + 5

If we subtract x + 5 from both sides of this equation, we get:

P - w = j

If we add w to both sides of this equation, we get:

P = j + w

This implies that P > j

So, we can get all 3 conditions, depending on the relationship between x and y.

They are:

P = j when y = x + 5
P < j when y > x + 5
P > j when y < x + 5

We can put this into numbers in order to confirm that what we have determined algebraically is correct.

Our starting statements are:

B > j + 3
P < B + 2

We translated these into:

B = j + 3 + x (x > 0)
P + y = B + 2 (y > 0)

Let's let x = 5 and y = 10 (y = x + 5)

We get:

B = j + 3 + 5 which becomes B = j + 8
P + 10 = B + 2 which becomes P = B - 8

Substitute j + 8 for B and we get P = j + 8 - 8 which becomes P = j.

When y = x + 5, we get P = j.

Back to the original equations:

B > j + 3
P < B + 2

We translated these into:

B = j + 3 + x (x > 0)
P + y = B + 2 (y > 0)

Let's let x = 5 and y = 11 (y > x + 5)

We get:

B = j + 3 + 5 which becomes B = j + 8
P + 11 = B + 2

Substitute j + 8 for B in the second equation and we get:

P + 11 = j + 8 + 2 which becomes:
P + 11 = j + 10
Subtract 10 from both sides of the equation to get:
P + 1 = j
This implies that P is less than j (P < j).

Back to the original equations:

B > j + 3
P < B + 2

We translated these into:

B = j + 3 + x (x > 0)
P + y = B + 2 (y > 0)

Let's let x = 5 and y = 9 (y < x + 5)

We get:

B = j + 3 + 5 which becomes B = j + 8
P + 9 = B + 2

Substitute for B in the second equation to get:

P + 9 = j + 8 + 2 which becomes:
P + 9 = j + 10

Subtract 9 from both sides of the equation to get:

P = j + 1

This implies that P > j

Bottom line:

P can be taller than or equal to or shorter than j.

This all depends on the relationship between x and y.

In the translation of the equation B > j + 3 to B = j + 3 + x, x is the amount that B is greater than (taller than) j + 3.

In the translation of the equation P < B + 2, to P + y = B + 2, y is the amount that P is smaller than (shorter than) B + 2.

Making the inequality an equality allows the problem to be solved.

The creation of the difference variables of x, y, z, w allowed the equality to be modeled from the inequality.

x,y,z,w are all assumed to be greater than 0.

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Let x = yoshi's sister's age.
yoshi's age is x + 12.

6 years from now the sum of their age will be 32.

6 years from now yoshi's sister will be x + 6 years old.

6 years from now yoshi will be (x + 12) + 6 years old.

The sum of their ages will be x + 6 + (x + 12) + 6 = 2*x + 24 = 32

You have 2*x + 24 = 32

Solve for x to get x = 4

yoshi's sister's age today is 4.
in 6 years yoshi's sister will be 10 years old.

yoshi is 16 years old today.
in 6 years, yoshi will be 22 years old.

10 + 22 = 32

answer to the question is that yoshi's present age is 16 years old.

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There are multiple solutions to this problem because you have 1 equation in 2 unknowns to work with.

Let x = number of gallons of 15% sugar solution.
Let y = number of gallons of 30% sugar solution.

The formula is:

.15*x + .4*5 = .3*y

Simplify to get:

.15*x + 2 = .3*y

Divide both sides of this equation by .3 to get:

.5*x + 6.666666667 = y

This is the same as:

y = .5*x + 6.6666666667

For every value of x, you have a corresponding value of y that will satisfy the equation.

You can graph this equation.

It looks like this:



I'll pick 3 points at x = 5, 10, 15 to show you that the relationship holds for all values of x.

When x = 5, y = 9.166666667: .15*x + 2 = 2.75 and .3*y = 2.75.
When x = 10, y = 11.66666667: .15*x + 2 = 3.5 and .3*y = 3.5.
When x = 15, y = 14.166666667: .15*x + 2 = 4.25 and .3*y = 4.25.

The equation holds for all values of x.

The answer to your question is that:

x gallons of a 15% solution will produce .5*x + 6.66666667 gallons of a 30% solution.

This comes from the equation of y = .5*x + 6.66666667

The x is the number of gallons of 15% solution.
The y is the number of gallons of 30% solution.

If the question had said:

How many gallons of a 15% solution plus 5 gallons of a 40% solution are required to make 15 gallons of a 30% solution, then you could have solved the problem for an exact solution.

In that case, you get y = 15 and the equation of y = .5*x + 6.6666667 becomes:

15 = .5*x + 6.6666667

Now you have 1 equation in one unknown that can be solved.

You subtract 6.66666667 from both sides of the equation to get:

8.33333333 = .5*x

You divide both sides of the equation by .5 to get:

16.66666667 = x

Your answer, in that case, is that 16.66666667 gallons of a 15% solution plus 5 gallons of a 40% solution are required to make 15 gallons of a 30% solution.

.15*16.66666667 = 2.5
.4*5 = 2
2.5 + 2 = 4.5 gallons of solution from the 15% and 40% solution.
.3*15 = 4.5 gallons of solution from the 30% solution.

Answer is confirmed as correct.

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Equation for quarterly compounding would be:

y = 1000 * (1.013)^x where x is the number of quarterly compounding periods.

1000 is the initial investment
.013 is the annual interest rate of .052 divided by 4 to get a quarterly interest rate.
x is the number of quarterly compounding periods

The graph of the quarterly compounding equation is shown below:



Equation for continuous compounding would be:

y = 500 * e^(.048*x) where x is the number of years.

500 is the initial investment
.048 is the annual growth rate
x is the number of years
e is the scientific constant of 2.718281828...

The graph of the continuous compounding equation is shown below:



With quarterly compounding, the value of x represents the number of quarterly compounding periods (4 compounding periods per year). This means that the number of years is equal to the number of compounding periods divided by 4.

From the graph, it appears that the initial value of 1000 reaches 3000 in approximately 85 quarters which is equivalent to 21.25 years.

With continuous compounding, the value of x represents the number of years.

From the graph, it appears that the initial value of 500 reaches 3000 in approximately 37 or 38 years.

We can solve for this algebraically.

With quarterly compounding, the formula would become:

3000 = 500 * (1.013)^x

We use logs to solve this.

The answer is x = 85.05676124 which is pretty close to x = 85.

Don't forget that this is quarters, not years. To get years, you need to divide by 4.

With continuous compounding, the formula would become:

3000 = 500 * e^(.048*x)

We use logs to solve this as well.

The answer is x = 37.32832228

If you need to know how I solved this using logs, let me know and I'll send you the details of the solution.
Otherwise, I'm just showing you that the approximate values shown by the graph are close to the actual values that were calculated algebraically.

The key to graphing the quarterly compounding formula is that x represents the number of quarters, not the number of years.

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In a general expression, x represents any number.

To solve for a specific number, you replace x with that number in the equation.

example:

y = x^2 + 3*x - 5

If you want to know the value of y when x = 3, you replace x with 3 in the equation to get:

y = 3^2 + 3*3 - 5 = 9 + 9 - 5 = 18 - 5 = 13

When you get into functional notation (if not already), then you will see:

f(x) = x^2 + 3*x - 5

When x = 3, then you will see:

f(3) = 3^2 + 3*3 - 5

The 3 replaces the x to give you the value of the equation when x = 3.

If you want to know the value of y or f(x), when x = 5, then you replace x with 5, etc.

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If you invest $P, and you earn interest only on the amount you invested (the $P), then SIMPLE interest is earned.

If you invest $P and you earn interest not only on the $P, but also on the interest gained, then COMPOUND interest is earned.

Let P = principal,
r = annual nominal interest rate (in decimal form),
t = time of the investment (in years),
A = amount in the account after t years. THEN…

SIMPLE INTEREST

A = P + (P * r * t)

Example:

P = 10,000
r = .10
t = 5 years

A = 10,000 + (10,000 * .10 * 5) = 10,000 + 5,000 = 15,000

COMPOUND INTEREST

A = P * (1+r)^t

Example:

P = 10,000
r = .10
t = 5 years

A = 10,000 * (1.10)^5 = 10,000 * 1.61051 = 16,1051

You make more money with compound interest than you do with simple interest, assuming the same investment in the same time frame with the same rate of return.

LAST QUESTION

Ideally, what do you want to happen to the value of n in the compound interest formula if the formula represented the amount of money you have in the bank?

I don't see "n" anywhere in your question.

If "n" is the number of compounding periods per year, than, ideally, you would want n to be as large as possible.

If n = 1, then the number of compounding periods per year is 1.

If n = 12, then the number of compounding periods per year is 12.

When n is greater than 1, you multiply the number of years by n and you divide the nominal interest rate by n.

The nominal interest rate is the annual interest assuming only 1 compounding period per year.

In the compounding example above, n was equal to 1 (not shown in the formula.

with n in the formula, the formula becomes:

A = P * (1 + (r/n))^(t*n)

When n was equal to 1, you got:

A = P * (1 + r) ^ n

That's what we solved earlier.

When n is 12, the formula becomes:

A = P * (1 + (r/12))^(t*12)

Same compounding example but with number of compounding periods per year equal to 12.

P = 10,000
r = .1
t = 5

r/12 = .1/12 = .008333333333
t*12 = 5*12 = 60

formula becomes:

A = 10,000 * (1.0083333333) ^ 60 = 16453.08934

with monthly compounding, the future value is 16,453.08934
with yearly compounding, the future value is 16,16,105.1

Your savings are greater when the number of compounding periods per year are greater.

The most compounding periods per year you can get is with continuous compounding.

That formula is A = P * e^rt)

e is the well known scientific constant of 2.718281828.....

Here you use the annual interest rate and the number of years.

You get:

A = 10,000 * e^(.1*5) = 10,000 * e^.5 = 16,487.21271

That's the theoretical maximum number of compounding periods per year.

If you compounded daily, you would get close to that.

Assume daily compounding with 365 days per year.

P = 10,000
i = .1 / 365 = .000273973
t = 5 * 365 = 1825

A = P * (1.000273973)^1825 = 16486.08362

That's pretty close to the theoretical maximum, but not exactly.

Bottom Line is the more the compounding periods per year, the greater your savings will be.

Without knowing what you meant by "n", this is the best I can come up with based on what I know about compounding.

I can't think of what else "n" can be, given the formulas as presented.