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Write a question that can be solved using the equation 5x=45.
:
How many x's can you buy for $45, if they cost $5 each
5x = 45
x = 45/5
x = 9 x's

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A person is biking 8m/s when the bike passes a car.
The car starts from rest and accelerates at 3m/s.
How long will it take for the car to catch up with the bike.
:
Let t = time for the car to overtake the bike
(this also is the time that the car accelerates at 3 m/sec
:
Write a distance equation, both will travel the same distance, dist = speed * time
:
* t = 8t
Multiply both sides by 2, results:
3t * t = 16t
:
3t^2 = 16
Divide both sides by t
3t = 16
t =
t = 5 seconds for the car to catch the bike
;
:
Check this;
Car will accelerate to: 3*5.33 = 16 m/sec
From a standing stop, it's average speed will be half, 8 m/sec
Both will travel 8m/sec for 5.33 min or about 42.7 meters

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Is it possible to have a shape with an area of 24 and a perimeter of 20 or less?
The shape cannot be a rectangle. What does the shape look like?
:
What about a circle?
find r
= 24
r =
r = 2.764
:
Find the circumference

:
C = 17.34, well below 20

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the measure of a supplementary angle is 5 times that of the angle's complement.
what is the measure of the complementary angle?
:
Let a = the angle
then
(180-a) = it's supplement
and
(90-a) = it's complement
:
The equation for the statement:
"the measure of a supplementary angle is 5 times that of the angle's complement."
180 - a = 5(90 - a)
180 - a = 450 - 5a
-a + 5a = 450 - 180
4a = 270
a =
a = 67.5 degrees
but it says," what is the measure of the complementary angle? "
90 - 67.5 = 22.5 degrees is the complementary angle

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solve the following using the substitution method and the elimination method.
:
4x+3y=48
3x+2y=34
:
Use the 2nd equation for substitution
3x + 2y = 34
2y = -3x + 34
Divide both sides by 2
y = x +
y = -1.5x + 17
:
Substitute (-1.5x+17) for y in the 1st equation
4x + 3(-1.5x + 17) = 48
4x - 4.5x + 51 = 48
4x - 4.5y = 48 - 51
-.5x = -3
x has to be positive, multiply both sides by -1
.5x = 3
x =
x = 6
find y using y = -1.5x+17
y = -1.5(6) + 17
y = -9 + 17
y = +8
Check solutions in the 2nd equation; 3x+2y=34
3(6) + 2(8) = 34
18 + 16 = 34
34 = 34; confirms our solutions
:
:
By elimination
4x+3y=48
3x+2y=34
Multiply the 1st equation by 2, and the 2nd equation by 3
8x + 6y = 96
9x + 6y = 102
------------------subtraction eliminates y, find x
-1x = -6
x = 6; mult by -1
:
You can find y using either equation, replacing x with 6

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Clue 1 each digit increases by one. Example 45,678.
Clue 2 if you add all six numbers, the answer is 20 .
:
let x = the lowest number
then: (x+1), (x+2), (x+3), (x+4) are next 4 number
:
They add up to 20
x + (x+1), (x+2), (x+3), (x+4) = 20
5x + 10 = 20
5x = 20 - 10
x =
x = 2 is the lowest number
;
The 5 digit number, 23456 (you couldn't do this with a 6 digit number)

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The sum of three consecutive integers is 279 Find the integers.
Set up an equation to describe each situation.
:
The three consecutive integers: x, (x+1), (x+2)
:
Their sum is 279, therefore:
x + (x+1) + (x+2) = 279
Combine like terms
x + x + x + 1 + 2 = 279
3x + 3 = 279
3x = 279 - 3
3x = 276
x =
x = 92
:
The integers, 92,93,94, add them up and see if this is right

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If Andy bikes at a steady rate of 15 miles per hour,
how many miles will he bike in four minutes?
:
Find how many miles in 1 min:
= .25 miles (which is what I think you meant)
:
Find how many miles in 4 minutes: 4 * .25 = 1 mi in 4 minutes

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a rectangular flower garden measuring 25 feet by 15 feet,
orders 3 cubic yards of pre-mixed cement, all of which is to be used to
create a border of uniform width around the garden.
If the border is to have a depth of 3 inches, how wide will the border be?
:
Find the area of the garden
25 * 15 = 375 sq/ft
:
Find the area of the border by dividing 3 cu/yds by 3 inches
Change 3 cu/yds to cu/ft: 3 * 27 = 81 cu/ft
Change 3 inches to .25 ft
= 324 sq/ft is the area of the border
:
The total area of the garden and the border
375 + 324 = 699 sq/ft
:
Let x = the width of the border
:
The equation for the total area
(2x+25)*(2x+15) = 699
FOIL
4x^2 + 30x + 50x + 375 = 699
4x^2 + 80x + 375 - 699 = 0
4x^2 + 80x - 324 = 0
Use the quadratic formula to find x

In this equation; a=4; b=80; c=-324

:

:

:

The positive solution only wanted here

x =
x = 3.45 ft is the width of the border
:
:
:
Check solution, 2x = 6.9
(6.9+25)*(6.9+15) = 699 sq/ft, total area
Subtract the area of the garden: 699 - 375 = 324, the area of the border
Find the volume of the border (mult by 3" which is .25 ft: 324*.25 = 81 cu/ft
Find 81 cu/ft in cu/yds: = 3 cu/yds

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If ellen and sue leave school for the library at 3pm.
Ellen is going 12 miles per hour n sue is going 9 miles an hour.
Ellen gets there 15 mins before sue.
What time does ellen get there.
How far is the library
:
Change 15 min to .25 hr
:
Let d = distance to the library
:
Write a time equation; time = dist/speed
:
Sue's time - Ellen's time = 15 min
- = .25
Multiply both sides by 36, results:
4d - 3d = .25(36)
d = 9 mi is the distance to the library
:
Find Ellen's time to get to the library
= hr which is 45 min
3:00 + :45 = 3:45 time when Ellen get's there

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if the sum of a number and five is tripled, the result is one less than twice the number. find the number?
:
x = "a number"
:
Just write an expression for each phrase:
:
"sum of a number & 5 is tripled",
3(x+5)
:
"the result"
=
:
"1 less than twice the number"
2x - 1
:
Put is all together to form an equation and solve it:
3(x+5) = 2x - 1
:
3x + 15 = 2x - 1
:
3x - 2x = -1 - 15
:
x = -16
:
:
Now in the original statement see if that is true
"the sum of a number and five is tripled, the result is one less than twice the number"
3(-16+5) = 2(-16) - 1
3(-11) = -32 - 1
-33 = -33; equality confirms our solution of x = -16
:
:
did this shed some light on this for you?

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A long distance runner starts at the beginning of a trail and runs at a rate of 6 miles per hour.
An hour and a half later, a cyclist starts at the beginning of the trail and
travels at a rate of 12 miles per hour.
What is the amount of time that the cyclist travels before overtaking the runner?
:
Let t = travel time of the cyclist
then
(t+1.5) = travel time of the runner (he start 1.5 hr before the cyclist)
:
When the cyclist overtakes the runner, they will have traveled the same distance
Write a distance equation: dist = speed * time
:
Cyclist dist = runner's dist
12t = 6(t+1.5)
12t = 6t + 9
12t - 6t = 9
3t = 9
t =
t = 3 hrs for the cyclist to catch the runner
:
:
We can confirm this solution by finding the actual distance each traveled;
(they should be equal)
12*3 = 27 mi
6(3+1.5) = 27 mi also

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Set up an equation and solve the problem.
Suppose that the width of a certain rectangle is 6 inches less than its length.
The area is numerically 16 less than twice the perimeter.
Find the length and width of the rectangle.
:
Let x = the width
then
(x+6) = the length
And the area
A = x(x+6)
A = x^2 + 6x
and the perimeter
P = 2(x+6) + 2x
P = 2x + 12 + 2x
P = 4x + 12
:
"The area is numerically 16 less than twice the perimeter."
x^2 + 6x = 2(4x + 12) - 16
x^2 + 6x = 8x + 24 - 16
x^2 + 6x = 8x + 8
Arrange as a quadratic equation on the left:
x^2 + 6x - 8x - 8 = 0
x^2 - 2x - 8 = 0
Factors to
(x-4)(x+2) = 0
Positive solution is what we want here
x = 4 inches is the width
then
4 + 6 = 10 inches is the length
:
:
Check solution
Find the perimeter
2(10) + 2(4) = 28 in
Find the area
10*4 = 40 sq/in
:
Check in the statement:
"The area is numerically 16 less than twice the perimeter."
40 = 2(28) - 16
40 = 56 - 16; confirms our solution of x = 4

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A man left A at 6:00 a.m. expecting to reach B at 9:00 a.m.
But after walking one hour, he was delayed for half an hour and so he had to
increase his rate by 1 mile per hour to reach B at 9:00 a.m.
Find his speed before the delay and the distance between A and B.
:
Let d = distance from A to B
Given that he left at 6, and expected to be there at 9, therefore:
= original speed
:
He walked for 1 hr, then stopped for .5 hrs, therefore
he continued at a speed of ( + 1) for 1.5 hrs
:
dist = speed * time
total dist = slow speed dist + faster speed dist
d = +
d = +
d =
multiply both sides by 3
3d = 2.5d + 3(1.5)
3d - 2.5d = 4.5
.5d = 4.5
d =
d = 9 miles from A to B
:
Find the original speed:
= 3 mph
:
:
Confirm this by finding the distance at each speed
1 * 3 = 3 miles
1.5 * 4 = 6 mile
----------------
total dist: 9 mi

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how would i translate into mathematical expression ...
8 more than 14 and a given number z is 37 ...thank you
:
just write what it says:
8 + 14 + z = 37

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One positive number is one-fifth of another number.
The difference between the two numbers is 316, find the numbers.
:
Let n = one number
then
5n = another number
:
5n - n = 316
4n = 316
n =
n = 79, then the other number: 5(79) = 395
:
:
Check 395 - 79 = 316

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1. Basketball team lost 40% of its first 40 games.
how many of its remaining 20 games must it win to raise its winning percentage to 65%?
:
Let x = no. of remaining games that need to be won for a winning percent of 65
:
Find how many game already lost: .4(40) 16 games lost
then 40 - 16 = 24 games won so far
:
Total games played 40 + 20 = 60
;
Games won/total games:
= .65
multiply both sides for 60
x + 24 = 60 * .65
x + 24 = 39
x = 39 - 24
x = 15 games of the remaining games need to be won
:
Check the percentage, 24 + 15 = 39 total games won
= .65 *100 = 65%
:
:
2. Write an equation for each statement;
The sum of 3 numbers is 94,
x + y + z = 94
:
the first number is 3/5 of the second number
x = y
:
the second number is 4/3 of the third.
y = z
We want z to be in terms of y also
Mult both sides by 3
4z = 3y
divide both sides by 4
z = y
:
Go back to the 1st equation: x + y + z = 94
Replace x and z
y + y + y = 94
Get rid of those annoying denominators, multiply each term by 20, results
4(3)y + 20y + 5(3)y = 20(94)
12y + 20y + 15y = 1880
47y = 1880
y =
y = 40
Find x
x = *40
x = 24
Find z
z = *40
z = 30
:
See if this is true: 24 + 40 + 30 = 94, so it's ok
:
What are the three numbers? 24, 40, 30
:
;
3. In one bag of chicken feeds, there are 24 chickens eating it in 45 days.
In how many days do 15 chickens consume the feeds?
:
Let's coin a term called "chicken-days"
It takes: 24 * 45 = 1080 chicken-days to consume 1 bag of feed
:
Let d = no. of days required by 15 chickens, a simple equation
15d = 1080
d =
d = 72 days for 15 chickens to consume a bag
:
:
I hope this has unconfused you. Could you follow the reasoning on these.
Any questions? let me know in the comment.

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a cylindrical chemical storage tank must have a height 4 meters greater than
the radius of the top of the tank.
Determine the radius of the top and the height of the tank if the tank must
have a volume of 15.71 cubic meters.
:
Let r = the radius of the tank
It says,"must have a height 4 meters greater than the radius." therefore
(r+4) = the height of the tank
:
Volume of a cylinder: V =
Therefore
= 15.71
divide both sides by pi, results
r^2(r+4) =
r^3 + 4r^2 = 5
r^3 + 4r^2 - 5 = 0
just from looking at this you know that r=1, then the height = 5
:
:
Check this
V =
v = 15.707 ~15.71

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Let x = the 1st angle, let y = the 2nd angle, let z = the 3rd angle
:
Write an equation for each phrase
;
" draw a triangle so that the first angle is 5/4 of the second"
x = y
Multiply both sides by 4
4x = 5y
divide both sides by 5
x = y
or we can just write it:
y = .8x
:
"but is greater than the third by 40."
z = x - 40
;
x + y + z = 180
Replace y with .8x; replace z with (x-40)
x + .8x + x - 40 = 180
2.8x = 180 + 40
x =
x = 78.57 degrees is the 1st angle
and
.8(78.57) = 62.65 degrees is the 2nd angle
and
78.57-40 = 38.57 degrees is the 3rd angle
:
:
Check solution
78.57 + 62.65 + 38.57 = 179.79 ~ 180

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A marksman fires at a target 420 meters away and hears the bullet strike 2
seconds after he pulled the trigger.
An observer 525 meters away from the target and 455 meters from the marksman
hears the bullet strike the target 1 second after he hears the report of the rifle.
Find the velocity of the bullet and the velocity of the sound.
:
Let s = speed of sound
Let v = velocity of the bullet
:
The Marksman equation
+ = 2 sec
:
The observer equation (bullet is traveling while rifle report is traveling)
+ - = 1
:
+ = 1
:
Use elimination here
+ = 2
+ = 1
--------------------------------subtraction eliminates v, find s
= 1
s = 350 m/sec is the speed of sound
:
Find v
+ = 1
+ = 1
Multiply by 5v
v + 5(420) = 5v
2100 = 5v - v
2100 = 4v
v =
v = 525 m/sec is the bullet velocity
:
:
Confirm these values in the Marksman equation
+ =
.8 + 1.2 = 2 sec

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It takes 12 seconds for two trains to pass one another when moving in opposite directions.
If they are in the same direction, it will take the faster train 3 minutes to pass the slower one.
If the trains are 130 meters and 120 meters long, respectively, what are the rates of the two trains?
:
Let a = speed of the faster train
let b = speed of the slower
:
Trains meeting each other
The sum of the speed of the two trains (in meters/sec)
speed = dist/time
a + b =
a + b = 20.83 m/sec is the sum of their speeds
:
Trains going in the same direction (change 3 min to 180 sec
a - b =
a - b = 1.39 m/sec is the difference of their speed
:
Use elimination
a + b = 20.83
a - b = 1.39
----------------adding eliminates b, find a
2a = 22.22
a =
a = 11.11 m/sec is the speed of train a
In km/hr: ~ 40 km/hr
:
then using the sum equation find b
11.11 + b = 20.83
b = 20.83 - 11.11
b = 9.72 m/sec is the speed of train b
In km/hr ~ 35 km/hr

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A long rectangular sheet of metal 10 in. wide is to be made into a gutter by turning up sides of equal length perpendicular to the sheet. Find the length that must be turned up to produce a gutter with a maximum cross-sectional area.
:
Let x = the depth of the gutter
Let y = the width of the gutter
:
This may illustrate an end view of the gutter: L|_w_|L
then
2L + w = 10 in; the width of the sheets of aluminum
w = (10-2L); use this form for substitution
and
L * w = A; the cross sectional area of the end of the gutter
:
Using the area equation replace w with (10-2L)
:
A = L*(10-2L)
A = -2L^2 + 10L
a quadratic equation, max area occurs at the axis of symmetry, x = -b/(2a)
in this problem x = L; b = 10; a =-2
L =
L =
L = +2.5 inches turned up to create max area
:
:
w = 10 -(2.5)
w = 5 inches is the width
:
Max area would be; 2.5 * 5 = 12.5 sq/inches

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Solve the equations: first: x+3y=5
second: 3x-y=5
:
We are going to use elimination here, multiply 2nd equation by 3, add to the first
x + 3y = 5
9x -3y = 15
----------------adding eliminates y, find x
10x = 20
x =
x = 2
:
Find y using the 1st equation; replace x with 2
2 + 3y = 5
3y = 5 - 2
3y = 3
y = 1
;
:
Check solution in the 2nd equation replace x & y with 2 and 1
3(2) - 1 = 5; confirms our solution
:
Did you understand what we did here?

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Jordan can run 200 meters in the time it takes Bill to run 190 m.
If they run at the same pace, and Jordan starts 10 m behind the start line,
Who will win?
Just for convenience assume;
it takes 20 seconds for J to run 200 m
it takes 20 seconds for B to run 190 m
then; speed = dist/time
= 10 m/sec is J's speed
and
= 9.5 m/sec is B's speed
:
J will cover 210 m in = 21 sec
B will cover 200 m in = 21.05 sec, J wins by an eyelash
:
How far back does Jordan need to start so Bill will win?
Let x = how far back for B to win
= 21.10 sec (B wins by .05 sec)
200 + x = 211; multiplied both sides by 10
x = 211 - 200
x = 11 m behind and B will win by an eyelash

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In a company printing work is done by two machine 5% of the work done by machine A are wasted ,
and 40% of work done by machine B are wasted.
if the 80% of the total work is done by machine A. then what is the total percentage of wastage.
:
Let 1 = total amt done by both machines
Machine A does 80%
Machine B does 20%
:
Using decimals
Machine A wastes .05(.8) = .04 of total wasted
Machine B wastes .40(.2) = .08 of total wasted
---------------------------------
total from both machines: .12 * 100 = 12% wasted

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A change machine contains nickels, dimes, and quarters.
There are 75 coins in the machine and the value of the coins is $7.25.
If there are 5 times as many nickels as dimes,
find the number of coins of each type in the machine.
;
Write an equation for each statement:
:
"There are 75 coins in the machine"
n + d + q = 75
:
"the value of the coins is $7.25."
.05 + .10d + .25q = 7.25
:
"there are 5 times as many nickels as dimes"
n = 5d
Arrange in the same form
n - 5d = 0
:
Using the coefficients of these 3 equations, write a matrix
(in the 3rd equation q=0
:
1, 1, 1, 75
.05, .10, .25, 7.25
1 - 5, 0, 0
:
Solve this 3 by 4 matrix using your favorite method
:
My favorite is the matrix feature on the Ti83, I got
n = 50
d = 10
q = 15

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A woodcarver made a model of a chair. The equation 0.08h=3 can be used to
find the actual height h of the chair in inches.
What was the actual height of the chair?
:
You can check your selection easily, just substitute 37.5 for h and see if equality prevails
:
.08(37.5) = 3
3 = 3, so you're right

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A and B run at constant velocities along a circular track.
The circumference of the track is 1320 meters.
If they run in opposite directions, they meet every 2 minutes;
if they run in the same direction, they are together every 22 minutes.
Find their velocities.
:
Find the sum of their velocities,
when they meet, they will have traveled a total of 1320 meters in two minutes
Speed = dist/time
(V1 + V2) =
V1 + V2 = 660 m/min
:
When they travel the same directions
V1 - V2 =
V1 - V2 = 60 m/min
:
Use elimination
V1 + V2 = 660
V1 - V2 = 60
-----------------addition eliminates V2
2V1 = 720
V1 =
V1 = 360 m/sec
and
V2 = 660 - 360
V2 = 300 m/sec
:
:
We can confirm this using the same direction scenario
Faster runner travels 22 * 360 = 7920 meters
Slower runner travels 22 * 300 = 6600 meters
----------------------------------------------
difference is exactly 1 lap or 1320 meters

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Let A = A's present age
Let B = B's present age
;
In a years, A will be n times as old as B then;
A + a = n(B + a)
A + a = nB + na
A = nB + na - a; is A's present age

and b years ago, A was m times as old as B.
A - b = m(B - b)
Find B
m(B - b) = A - b
B - b =
B = + b; is B's present age

:

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Chuck travel 124 miles in the time it takes Dana to travel 108 miles.
If Chucks rate is 4mph more than Danas, and they travel the same length in time, what is chucks speed.
:
Let s = Chuck's speed
then
(s-4) = Dana's speed
;
The times are equal, write a time equation: time = dist/speed
=
Cross multiply
124(s-4) = 108s
124s - 496 = 108s
124s - 108s = +496
16s = 496
s =
s = 31 mph is Chuck's speed
:
:
Check solution by finding the times, they should be equal
(27 mph is D.s speed)
124/31 = 4 hrs
108/27 = 4 hrs also, confirms our solution of x = 31
:
:
2)Airplane flies from Metro to Gotham with a tailwind that increases is
90mph and on the way back decreases the speed 90 mph.
with wind on their back it takes 2.05 hours compared to with wind in face 3.46 hours.
What's the distance between the two cities?
:
Find the airspeed with out the wind
Let s = speed in still air
then
(s+90) = effective speed with the wind
and
(s-90) = effective speed against the wind
:
The distances are the same each way, write a distance equation; d = speed * time
3.46(s-90) = 2.05(s+90)
3.46s - 311.4 = 2.05s + 184.5
3.46s - 2.05s = 184.5 + 311.4
1.41s = 495.9
s =
s = 351.7 mph air speed
:
Find the dist
3.46(351.7-90) = 905.5 miles between the two cities
:
Confirm this on the return trip
2.05(351.7+90) = 905.5 miles also

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You need to mix 6% peroxide solution with 9% peroxide solution to obtain 36
ounces of an 8% peroxide solution.
How many ounces of each solution is needed?
:
Let x = amt of 9% solution
then
(36-x) = amt of 6% solution
:
The equation
.09x + .06(36-x) = .08(36)
:
Find x,