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you have the formula wrong for conversion from Fahrenheit to Celsius.

the correct formula is:

C = (5/9) * (F-32)

when F is 32 (freezing), C = 0 as it should be.

when F is 212 (boiling), C = 100 as it should be.

now that we have the correct formula for Centigrade, we can derive the correct formula for fahrenheit.

start with:

C = (5/9) * (F-32)

multiply both sides of this formula by (9/5) to get:

(9/5) * C = F - 32

add 32 to both sides of this formula to get:

(9/5) * C + 32 = F

formula for F is:

F = (9/5) * C + 32

to test this formula we use freezing and boiling points again.

F = (9/5) * 0 + 32 = 32 (freezing point).

F = (9/5) * 100 + 32 = 180 + 32 = 212 (boiling point).

formula is good and we can use it to solve the problem.

when C = 0, F = 32 which we already solved.

when C = 30, F = (9/5) * 30 + 32 = 54 + 32 = 86.

your answer is

temperature range in celsius is 0 to 30.

equivalent temperature range in fahrenheit is 32 to 86.

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An airplane flew for 8 hours at an air speed of x miles per hour (mph), and for 7 more hours at 325 mph. If the average airspeed for the entire flight was 350 mph, which of the following equations could be used to find x ?

general formula is rate * time = distance.

let r = rate
let h = time in hours
let d = distance

total time it took was 8 + 7 = 15 hours.

average speed for the whole flight was 350 miles per hour.

formula for the entire flight becomes:

350 * 15 = d

formula for the first leg of the trip is equal to:

x * 8 = d1 where d1 is the distance flown for the first leg of the trip.

formula for the second leg of the trip is equal to :

325 * 7 = d2 where d2 is the distance flown for the second leg of the trip.

since we know that d1 + d2 = d, then we have a general formula that states

x * 8 + 325 * 7 = 350 * 15

that looks very much like selection E which is:

8x + 7(325) = 15(350)

your answer is selection E.

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total possible number of way is:

4 * 2 * 6 = 8 * 6 = 48

that would be selection d.

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p(admission) = .25

interviews from 6 institutions

probability that the candidate will get admission in at 3 institutions?

this would be equal to:

probability of getting admission in exactly 3 plus:
probability of getting admission in exactly 4 plus:
probability of getting admission in exactly 5 plus:
probability of getting admission in exactly 6.

this is also equal to:

1 minus:
probability of getting admission in exactly 0 plus:
probability of getting admission in eacttly 1 plus:
probability of getting admission in exactly 2.

we'll do both to show you how it works, but in practice, you would normally choose the option that takes less work to get the result from.

probability of getting exactly 0 admissions = 1 * .75^6

probability of getting exactly 1 admission = 6 * .25 * .75^5

probability of getting exactly 2 admissions = 15 * .25^2 * .75^4

probability of getting exactly 3 admissions = 20 * .25^3 * .75^3

probability of getting exactly 4 admissions = 15 * .25^4 * .75^2

probability of getting exactly 5 admissions = 6 * .25^5 * .75

probability of getting exactly 6 admissions = 1 * .25^6

the multipliers are derived as follows:

you have a total of 6 to draw from.

you want to know how many ways you can draw different sets from this number.

the number of ways you can draw 0 is 1 based on the combination formula of c(6,0) = 6! / (6!*0!)

the number of ways you can draw sets of 1 is 6 based on the combination formula of c(6,1) = 6! / (5!*1!)

the number of ways you can draw sets of 2 is 15 based on the combination formula of c(6,2) = 6! / (4!*2!)

the number of ways you can draw sets of 3 is 20 based on the combination formula of c(6,3) = 6! / (3!*3!)

the number of ways you can draw sets of 4 is 15 based on the combination formula of c(6,4) = 6! / (2!*4!)

the number of ways you can draw sets of 5 is 6 based on the combination formula of c(6,5) = 6! / (1!*5!)

the number of ways you can draw sets of 6 is 1 based on the combination formula of c(6,6) = 6! / (0!*6!)

note that 0! is always equal to 1.

the total probability should always be equal to 1.

the numbers from all the probabilities come out as follows:

p(0) = 1 * .75^6 = .177978516
p(1) = 6 * .25 * .75^5 = .355957031
p(2) = 15 * .25^2 * .75^4 = .296630859
p(3) = 20 * .25^3 * .75^3 = .131835938
p(4) = 15 * .25^4 * .75^2 = .032958984
p(5) = 6 * .25^5 * .75 = .004394531
p(6) = 1 * .25^6 = .000244141

probability of getting all possible combinations is equal to 1 as it should be.

probability of getting 0 or 1 or 2 = p(0) + p(1) + p(2) = .830566406

probability of getting 3 or 4 or 5 or 6 = p(3) + p(4) + p(5) + p(6) = .169433594

your answer is .169433594.

this was achieved by taking p(0) + p(1) + p(2) and subtracting the result from 1.

1 minus (p(0) + p(1) + p(2) = 1 - (.17... + .35... + .29...) = 1 - .830566406 = .169433594.

it was also achieved by taking p(3) + p(4) + p(5) + p(6) = .13... + .03... + .004... + .0002... = .169433594.

either way you get the same answer.

as an example of the ways that a particlar probability can happen, consider the following example:

what is the probability of getting exactly 2 admissions from 6 interviews if the probability of getting an admission is .25.

the probability of getting exactly 2 admissions is equal to .25^2 * .75^4 = .019775391.

this can happen in c(6,2) ways = 6! / (4!*2!) = 15 ways.

those possible ways are:

sn = school number (ex: s1 = school number 1)
1 = admitted
0 = not admitted

way        s1 s2 s3 s4 s5 s6
1          1  1  0  0  0  0
2          1  0  1  0  0  0
3          1  0  0  1  0  0
4          1  0  0  0  1  0
5          1  0  0  0  0  1
6          0  1  1  0  0  0
7          0  1  0  1  0  0
8          0  1  0  0  1  0
9          0  1  0  0  0  1
10         0  0  1  1  0  0
11         0  0  1  0  1  0
12         0  0  1  0  0  1
13         0  0  0  1  1  0
14         0  0  0  1  0  1
15         0  0  0  0  1  1

the probability for each one of these happening is .25^2 * .75^4.
they are all mutually exclusive (either one way or the other, never both at the same time).
the total probability is therefore equal to the sum of each individual one happening which is p(way 1) + p(way 2) ..... + p(way 25) which is eaual to:
15 * .24^2 * .75^4.

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p = 6500
f = 20600
g = 1200

formula would be:

f = p + (n * g)

f = future value
p = present value
g = growth per year
n = number of years

we are solving for n.

formula becomes:

20600 = 6500 + (1200 * n)

subtract 6500 from both sides of the equation to get

20600 - 6500 = 1200*n

simplify to get:

14100 = 1200*n

divide both sides by 1200 to get:

14100 / 1200 = n

simplify to get:

11.75 = n

it will take 11.75 years for the population to reach 20600.

.75 years * 12 months per year = 9 months

that's equivalent to 11 years and 9 months.

11.75 * 1200 = 14100 + 6500 = 20600 confirming the answer is good.

the answer is:

it will take 11.75 years for a population of 6500 to reach 20600 with a growth rate of 1200 people per year.

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you want to find the point inside the triangle that is equidistance from all the vertex of the triangle.

I believe you want the circumcenter of the triangle.

this is the center of a circle that circumscribes the triangle.

as such, all vertices of the triangle are on the circumference of the circle which means they are the same distance to the center of the circle.

that point is equidistant from each vertex of the triangle.

here's a link that shows you what I mean.

http://www.mathopenref.com/trianglecircumcenter.html

that point is the intersection of the perpendicular bisectors of each side of the triangle.

that point can be inside or outside the triangle.

just play with the figure and you can see how that happens.

here's another look at it.

http://faculty.evansville.edu/ck6/tcenters/class/ccenter.html

in your triangle SAB, you have:

SA is 8 cm.
AB is 8 cm.
SB is 11.4 cm.

this should be an isosceles triangle since 2 sides of it are equal.

the legs are SA and AB and the base is SB.

you state that angle S is 28 degrees and angle A is 90 degrees and angle B is 45 degrees.

that can't be.

sum of the interior angles of a triangle is 180 degrees.

sum of the angles you provided is 163 degrees.

also, if the sides of a triangle are equal, then their corresponding angles are also equal.

This does not ring true with your statement about the sides and the angles.

something is wrong with your problem setup.

check it again and correct it and resubmit the problem.

in any case, if you want the point that is equidistance from the vertices of the triangle, then you are looking for the intersection of the perpendicular bisectors of each of the sides because that give you the center of the circle that circumscribes the triangle.

that point is equidistant from the vertices of the triangle.

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rate * time = distance.

distance is 500 miles.
rate is 152.672 miles per hour.

substitute in this equation to get:

152.672 * time = 500

divide both sides of this equation by 152.672 to get:

time = 500/152.672 = 3.27499476 hours.

answer should be 3.275 to the nearest thousandth of an hour.

here's where you went wrong:

your statement:

152.672*t=500 and then to get t on one side subtracted 152.672 from both sides. The answer I got was 347.328. Where did I go wrong? Please help. This is for my college class Intermediate Algebra.

152.672*t = 500 is correct.

you should have divided both sides of the equation by 152.672 rather than subtracted 152.672 from both sides of the equation to get 152.672 from one side of the equation to the other.

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(A) In terms of s, what is the speed of the faster car?

faster car travels at s + 10 miles per hour.

(B) In terms of S, how long does the slower car take to complete the trip?

rate * time = distance.

this makes time = distance / rate.

slower car takes (120/s) hours to complete the trip.

(C) In terms of s, how long does the faster car take to complete the trip?

faster car takes (120 / (s+10)) hours to complete the trip.

(D) Express the conditions "the faster car completes the journey in one hour less than the slower car" as an equation involving s using the answers to part b and c.

the time it takes the slower car is expressed by the equation:

h[s] = 120/s where h represents the time in hours.

the time it takes the faster car is expressed by the equation:

h[f] = 120/(s+10) where h represents the time in hours.

the statement that it takes the faster car one less hour to complete the trip can be expressed by the equation h[f] = h[s] - 1.

replace h[f] and h[s] with their respective equivalents and you get:

120/(s+10) = 120/s - 1


(E) Solve the equation in part d to find the speed of the slower car.

the equation in part D is:

120/(s+10) = 120/s - 1

multiply both sides of this equation by s * (s+10) to get:

120 * s = 120 * (s+10) - (1 * (s+10) * s)

simplify to get:

120 * s = 120 * s + 1200 - (s^2 + 10*s))

simplify further to get:

120 * s = 120 * s + 1200 - s^2 - 10*s

subtract 120 * s from both sides of this equation to get:

0 = 120 * s - 120 * s + 1200 - s^2 - 10*s

combine like terms to get:

0 = 1200 - s^2 - 10*s

add s^2 to both sides of this equation and add 10*s to both sides of this equation and subtract 1200 from both sides of this equation to get:

s^2 + 10*s - 1200 = 0

factor this quadratic equation to get:

(s+40) * (s-30) = 0

solve for s to get s = -40 or s = 30

since s can't be negative, then s must equal 30.

speed of the slower car is 30 miles per hour.

speed of the faster car is 40 miles per hour.

since time = distance / rate, then:

the time it takes the slower car = 120/30 = 4 hours.

the time it takes the faster car = 120/40 = 3 hours.

the faster car takes 1 hour less than the slower car so the equation is satisfied.

answer to part E is that the speed of the slower car is 30 miles per hour.

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the length of the side of this cone will be 5 cm.

the length of the altitude of this cone will be 4 cm.

this should form a 3/4/5 right triangle.

this makes the radius of the base of the cone equal to 3 cm.

volume of the cone is equal to 1/3 * pi * r^2 * h.

this becomes 1/3 * pi * 9 * 4 = 1/3 * pi * 36 = 12 * pi cubic cm.

surface area of the cone is equal to pi * r * (r + s).

this becomes pi * 3 * (3 + 5) = pi * 3 * 8 = 24 * pi square cm.

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volume of sphere is 4/3 * pi * r^3.

r = 3 cm

volume of sphere is 4/3 * pi * (3)^3 = 4/3 * 27 * pi = 4*9 * pi = 36 * pi cubic cm.

each small spherical ball has a diameter of .6 cm.

this means that each small spherical ball has a radius of .3 cm because radius = diameter / 2.

volume of each small spherical ball is equal to 4/3 * pi * r^2.

since r = .3, then volume of each small spherical ball is equal to 4/3 * pi * (.3)^3 = 4/3 * pi * .027 = .036 * pi cubic cm.


divide 36 * pi by .036 * pi and you get:

36 * pi / .036 * pi = 36 / .036 = 1000

you can make 1000 small spherical balls out of the one big spherical ball.

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volume of a sphere is equal to 4/3 * pi * r^3

volume of a cone is equal to 1/3 * pi * r^2 * h

the cone is 6 feet high and the entire ice cream cone is 9 feet.

this means that the radius of the sphere is 3 feet.

this is also the radius of the base of the cone.

volume of the sphere is 4/3 * pi * 3^3 = 4/3 * pi * 27 = 4 * pi * 9 = 36 * pi.

take half of that to get the top half of the sphere only and you get the volume of the icream on top of the cone is equal to 18 * pi.

volume of the cone is 1/3 * pi * r^2 * h

r = 3 and h = 6

volume of the cone is 1/3 * pi * (3)^2 * 6 = 18 * pi.

volume of the ice cream cone is equal to 18 * pi + 18 * pi = 36 * pi.

36 * pi = 113.0973355 cubic feet.

click on the following link to see a picture of the cone.

http://theo.x10hosting.com/problems/258330.jpg

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you multiply the decimal by 100%

example

1 * 100% = 100%

.5 * 100% = 50%

.1 * 100% = 10%

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slope intercept form of the equation for a line is y = m*x + b where m is the slope and b is the y-intercept.

the given line is 9x - 5y = 6

that equation is in standard form.

to convert to slope intercept form, solve for y.

equation is:

9x - 5y = 6

subtract 9x from both sides of this equation to get:

-5y = -9x + 6

divide both sides of this equation by 5 to get:

-y = (-9/5)*x + (6/5)

multiply both sides of this equation by -1 to get:

y = (9/5)*x - (6/5)

you now have the equation in slope intercept form.

the slope is (9/5) and the y-intercept is -(6/5)

graph of this equation looks like this:



that just gets you the graph of the line you want to be parallel to.

your line will be parallel to this line so the slope will be the same.

the slope intercept form of the equation for your line will be:

y = (9/5)*x + b

you do not know what b is but you can solve for b by substituting the x value of your point for x and the y value of your point for y and then solve for b.

your point that is on the line parallel to the given line is (2,-4).

that's in (x,y) coordinate form.

x = 2
y = -4

substitute these in the equation for your line that is parallel to the given line.

that equation is:

y = (9/5)*x + b

substitute 2 for x and -4 for y to get:

-4 = (9/5)*2 + b

simplify to get:

-4 = (18/5) + b

subtract (18/5) from both sides of the equation to get:

-4 - (18/5) = b

-4 is equivalent to -20/5 because if you divide -20 by 5, you get -4.

solve for b to get

b = (-20/5) - (18/5) = (-38/5)

equation for you line becomes:

y = (9/5)*x - (38/5) after you substitute (-38/5) for b.

graph of this equation in addition to the given line is shown below:

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the graph of this equation will look like this:



the values along the x-axis are radians.

the values along the y-axis are your answer.

you can translae radians to degrees as follows:

degrees = radians * 57.29577951

example:

when x = 6.283185307, then degrees = 6.283185307 * 57.29577951 = 360.

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divide by 100%

examples:

100% / 100% = 1

50% / 100% = .5

10% / 100% = .1

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Movie tickets cost $25 for adults, and $5.00 for children. Last night a total of 305 tickets were sold totalling $6025, How many of each ticket was sold?

let a = number of adult tickets sold.
let c = number of child tickets sold.

25 * a + 5 * c = 6025

a + c = 305

you have 2 equations that have to be solved simultaneously (same solution for both equations).

solve for a or c in the second equation and then use that to solve for the remaining variable in the first equation.

a + c = 305

solve for a to get:

a = 305 - c

substitute for a in the first equation.

first equation is:

25 * a + 5 * c = 6025

substitute 305 - c for a to get:

25 * (305 - c) + 5 * c = 6025

simplify to get:

7625 - 20*c = 6025

subtract 7625 from both sides of the equation to get:

-20*c = 6025 - 7625 = -1600

divide both sides of the equation by -20 to get:

c = -1600/-20 = 80

since a + c = 305, then a must be equal to 305 - 80 = 225

225 + 80 = 305

25 * 225 + 80 * 5 = 5625 + 400 = 6025

numbers look good.

answer is:

number of adult tickets is 225
number of child tickets is 80

the number of tickets sold is a + c = 305

the amount of money made is 25*a + 5*c = 6025

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(a+b)/2 = 10 (first equation)

(b+10)/2 = (c/2) (second equation)

first equation states that the average of a and b is 10.

second equation states that the average of b and 10 is c/2

question is what is the average of a and c.

take the second equation and solve for b.

second equation is:

(b+10)/2 = c/2

multiply both sides of this equation by 2 to get:

b + 10 = c

subtract 10 from both sides of this equation to get:

b = c - 10

substitute for b in the first equation.

first equation is:

(a+b)/2 = 10

multiply both sides of this equation by 2 to get:

a + b = 20

subsstitute (c-10) for b to get:

a + c - 10 = 20

add 10 to both sides of this equation to get:

a + c = 30

divide both sides of this equation by 2 to get:

(a+c)/2 = 15

average of (a+c) is equal to 15.

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probability that a passenger will not show up 10% of the time means the passenger will show up 90% of the time.

any plane is overbooked when at least one passenger is denied.

for the smaller plane this happens when all 10 passengers show up for the 9 seats.

the probability of this happening is .9^10 = .34867844.

for the larger plan this happens when all 20 passengers show up or when 19 of the 20 passengers show up for the 18 seats.

probability of all 20 passengers showing up for the larger plane is .9^20 = .121576655.

probability that exactly 19 passengers show up for the larger plane is .1^1 * .9^19 * C(20,1).

C(20,1) is the number of possible combinations of selecting 1 out of 20.

C(20,1) = 20!/(19!*1!) = (20*19!)/19! = 20

proability that exactly 19 passengers will show up for the larger plane is therefore .1^1 * .9^19 * 20 = .270170344.

probability that all 18 passengers or exactly 19 passengers will show up for the larger plane is equal to .121576655 + .270170344 = .391746998

based on the probabilities, the larger plane should be overbooked more often than the smaller plane.

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y^2 = 3x is not a function.

solve for y to get:

y = +/- sqrt(3x)

for it to be a function there has to be one and only one value of y for each value of x.

if there is at least one instance where there are more than 1 value of y for each value of x, then it is a relation and not a function.

you give the graph of this equation the vertical line test.

if the vertical line cuts through more than one point on the graph of the equation, then it is not a function.

graph of this equation is shown below:



on this graph, you can see that a vertical line passing through the graph of the equation will cut through the equation twice for each value of x at all points except at the vertex.

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expression is:



you process operations within parentheses first starting from the inner parentheses and working to the outer parentheses.

your expressions within the inner parentheses can't be processed any further.

they are:

and and

your terms in the outer parentheses need to be processed next.

they are:



you would need to divide by first.

that gets you

the expression within the outer parenthese becomes:



you process this next by removing the inner parentheses from the outer parentheses to get:

which becomes:



you replace the original expression within the outer parentheses by this to make your original equation of:

becomes:



you multiply each term within the parentheses by 2*x to get:



if you did this right, then you should be able to get the same results in the original equation and the final equation for any value of x.

take x = 5

final equation of becomes:



original equation of becomes:

which becomes:

which becomes:

which becomes:

which becomes:

1000

since they both come up with the same answer, the final equation is equivalent to the original equation which indicates we simplified the equation properly.

the simplified equation is:

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30 students total.
7 received an a in history.
13 received an a in math.
4 received an a in both courses.
how many students did not receive an a in either course.

20 total received an a in history or math.

of these, 4 received an a in history and math.

those 4 are being counted twice.

once in history and once in math.

you need to subtract 4 from the number who got an a in history or an a in math to get a total of 16 students who got an a in history or in math.

this leaves 30 - 16 = 14 students who did not get an a in either course.

here's the breakdown.

14 students did not get an a in either course.
3 students got an a in history only.
9 students got an a in math only.
4 students got an a in both history and math.
total is 30 students.

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if you are traveling 45 miles per hour you will have traveled 45 miles in 1 hour.

since 1 hour = 60 minutes, this means you are traveling .75 miles in 1 minute.

that is because 60 minutes * .75 mies per minute = 45 miles

in 4 minutes you will have traveled 4 minutes * .75 miles per minute = 3 miles.

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cross section area of the full roll is equal to:

pi * r^2 = pi * 5^2 = pi * 25.

this is not all tissue however.

the core of the roll is empty so we have to subtract that.

cross section area of the core of the roll is equal to:

pi * r^2 = pi * 1^2 = pi.

if we subtract pi from 25 * pi we get 24 * pi which is the cross section area of the tissue on the roll when it is full.

the thickness of the paper layer is equal to 4 cm

half the thickness of the paper layer is therefore equal to 2 cm.

the radius of half the thickness of the paper layer on the roll is therefore 2 + 1 = 3 cm.

the cross section area of the half a roll of tissue paper is equal to:

pi*r^2 = pi*3^2 = pi*9

we have to subtract the core area from this which we know is pi, so we get:

9*pi = pi = 8*pi.

cross section area of a full roll is 24*pi.

cross section area of a half roll is 8*pi.

fraction of the original amount of paper on the roll is equal to 8*pi / 24*pi = 1/3

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sum of the terms of an arithmetic series is given by the formula:

s[n] = n * (a[1] + a[n])/2

s[n] = sum of the n terms of the sequence.
a[1] = first term in the sequence.
a[n] = nth term in the sequence.
n = number of terms in the sequence.

the formuula for the nth term in the sequence is:

a[n] = a[1] + (n-1) * d

d is the common difference.

you will use both these formulas to find the answer.

you are given:

sum of the first 9 terms in a sequence is 162
sum of the first 12 terms in the same sequence is 288.

the formnula for the sum of the first n terms in a sequence is:

s[n] = n * (a[1] + a[n])/2

substitute for the first 9 terms to get:

162 = 9 * (a[1] + a[9])/2 (first equation)

substitute for the first 12 terms to get:

288 = 12 * (a[1] + a[12])/2 (second equation)

multiply both sides of the first equation by 2 and divide both sides of the first equation by 9 to get:

a[1] + a[9] = 36 (third equation)

multiply both sides of the second equation by 2 and divide both sides of the second equation by 12 to get:

a[1] + a[12] = 48 (fourth equation)

subtract third equation from fourth equation to get:

a[12] - a[9] = 12

since 12 - 9 = 3, there are 3 terms between a[9] and a[12].

12/3 = 4 which means the common difference between each term is 4.

the nth term in each sequence is given by the formula:

a[n] = a[1] + (n-1) * d

we now know that d = 4

we solve for a[9] to get:

a[9] = a[1] + 8*4 = a[1] + 32

we take our first equation and plug in our new found values.

first equation is:

162 = 9 * (a[1] + a[9])/2 (first equation)

substitute a[1] + 32 for a[9] to get:

162 = 9 * (a[1] + a[1] + 32)/2

combine like terms to get:

162 = 9 * (2* a[1] + 32)/2

multiply both sides of this equation by 2 and divide both sides of this equation by 9 to get:

36 = 2 * a[1] + 32

subtract 32 from both sides of this equation to get:

36-32 = 2 * a[1] which becomes:

4 = 2 * a[1].

divide both sides of this equation by 2 to get:

a[1] = 2.

we now know a[1] and we know d so we can construct the sequence.

formula for nth term in the sequence is:

a[n] = a[1] + (n-1) * d

since d = 4,...

the 9th term in the sequence is:

a[9] = 2 + 8*4 = 2 + 32 = 34

the 12th term in the sequence is:

a[12] = 2 + 11*4 = 2 + 44 = 46

the sequence is:

         
term:         1  2  3   4   5   6   7   8   9   10  11  12

sequence:     2  6  10  14  18  22  26  30  34  38  42  46

2 is the 1st term.
34 is the 9th term.
46 is the 12th term.

common difference is 4.

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the positive digits are 1 through 9.

0 is a non-negative digits.

0 is not a positive digit.

when we sum up the digits it won't matter if we include 0 or not because the sum of all 0's is still 0.

when you count to 100, you have 1 set of tens digits and 1 set of units digits.

the total set of digits is shown in the following table:

00 10 20 30 40 50 60 70 80 90
01 11 21 31 41 51 61 71 81 91
02 12 22 32 42 52 62 72 82 92
03 13 23 33 43 53 63 73 83 93
04 14 24 34 44 54 64 74 84 94
05 15 25 35 45 55 65 75 85 95
06 16 26 36 46 56 66 76 86 96
07 17 27 37 47 57 67 77 87 97
08 18 28 38 48 58 68 78 88 98
09 19 29 39 49 59 69 79 89 99

the ones digits are on the right of each number.
the tens digits are on the left of each number.

example:

number 49:
9 is the ones digit
4 is the tens digit

to sum up all the digits, we sum up all the ones digits and all the tens digits separately and then add them together.

within each column, the ones digits add up to 45. that would be the sum of 1 through 9.

within each column, the tens digits add up to 10 * the digit shown. that would be 10 * 1, 10 * 2, etc.

you get the following number:

                     sum of tens digit     sum of units digit    total
from 0 to 9                  0                   45                45
from 10 to 19               10                   45                55
from 20 to 29               20                   45                65
from 30 to 39               30                   45                75
from 40 to 49               40                   45                85
from 50 to 59               50                   45                95
from 60 to 69               60                   45               105
from 70 to 79               70                   45               115
from 80 to 89               80                   45               125
from 90 to 99               90                   45               135
  
total                      450                  450               900

sume of all the positive digits less than 100 is equal to 900.

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that would be .85 * .8 * .75 * .7 = .357 = 35.7%

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round trip distance is 60 miles.

rate * time = distance

to travel 60 miles at 60 miles per hour, you have to take 1 hour for the round trip.

this would make the formula:

60 * 1 = 60 which would be true.

the total time it should take would be 1 hour.

you traveled the first leg at 30 miles per hour.

this means that the formula of rate * time = distance for the first leg becomes:

30 * x = 30.

this means that it took 1 hour to travel 30 miles.

this means that you will never be able to average 60 miles per hour for the round trip.

example:

you traveled 30 miles per hour for the first part of the trip.

that took you 1 hour.

you travel 3000 miles per hour for the return trip.

since rate * time = distance, then the formula becomes:

3000 * x = 30

x = 30/3000 = 3/300 = .01 hours

total time it takes is 1.01 hours.

that's 1 hour for the going leg and .01 hours for the return leg.

average rate for the round trip is based on the formula of rate * time = distance:

x * 1.01 = 60

rate you traveled is 60/1.01 = 59.40594059 miles per hour.

you can approach 60 miles per hour overall but you will never quite make it because your divisor will always be a very small amount greater than 1.

this means that your numerator will always be a very small amount less than 60.

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25000 deposited today for 40 years.
pays 4% compounded annually.
income tax on the interest is 30%.

which of the following earns more?

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(A) the income tax on the interest that is earned is paid in one lump sum at the end of 40 years.

if you pay the income tax at the end of the 40 years, then you will pay as follows:

the formula is future worth of a present amount.

that formula is:

f = p * (1+r)^n

f = future value
p = present amount
r = 4% per year divided by 100% = .04 interest rate per year.
n = number of years = 40

formula becomes:

f = 25000 * (1.04)^40

this becomes:

f = 120025.51578

uncle sam takes 30% of the profit and you are left with your original investment plus 70% of the profit.

120025.51578 - 25000 = 95025.5157 * .7 = 66517.86099 + 25000 = 91517.86099 = $91,517.86

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(B)The income tax on the interest that is earned each year is paid at the end of each year.

you earn 4% per year and uncle sam takes 30% of that out each year.

you are netting 4% minus .3 * 4% = 4% minus 1.2% = 2.8% per year.

that's what you are earning each year after uncle sam takes out his share at the end of each year.

same formula applies.

f = p * (1+r)^n

p = 25000
r = 2.8% / 100% = .028 interest rate per year.
n = 40 years.

formula becomes:

f = 25000 * (1.028)^40

this becomes:

f = 75450.92962 = $75,450.93

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you make more money paying uncle sam at the end of the 40 years rather than at the end of each year.

this is because the money that you would have been paying uncle sam each year is sitting in your account earning more money each year.

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daily compounding with annual interest rate of 4%.

formula is f = p * (1+r)^n

f = future value = 45000

p is what you're trying to find.

I believe daily compounding assumes 30 days each month totaling 360 days each year for financial purposes.

If you're not happy with this you can adjust to whatever days in the year you wish to have.

It won't make much of a difference either way.

r = 4% / 100% = .04 / 360 = .000111111

number of years is multiplied by 360 to get number of days.

you get 360 * 5 = 1800

n = number of time periods = 1800 days.

formula becomes:

45000 = p * (1.00011111)^1800

this becomes:

45000 = p * 1221389186

solve for p to get:

p = 45000 / 1.221389186 = 36843.29329

you would have to invest $36,843.29 today to have $45,000 in 5 years at 4% per year compounded daily.

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following information is interesting to know but not necessary to solve your problem.

that was solved above the dashed line.

additional interesting information:

this figure is very close to what you would get with continuous compounding.

with continuous compounding, the formula would be:

f = p * e^(r*t)

e is the scientific constant of 2.718281828...

r = .04
t = 5 years

formula becomes:

45000 = p * e^(.04*5)

this becomes:

45000 = p * 1.221402758

solve for p to get:

p = 45000 / 1.221402758 = 36842.88389.

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don went 2 + 3/4 miles north to store.

7/8 miles south to gary's house.

2 + 3/4 miles south to the park.

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he rode 2 + 3/4 miles north and then came back south 2 + 3/4 + 7/8 miles.

don's house is 7/8 miles from the park.

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they are on the road in the following sequence:

              B --- T - C -- A

each dash is a kilometer.

Bob is 3 km away from Ted and 6 km away from Alice.

Carol is 1 km away from Ted and 2 km away from Alice.

Ted is 4 km away from Carol.