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distance(d) equals rate(r) times time(t) or d=rt; r=d/t and t=d/r
Let r=speed (rate) of the river
rate of the boat downstream=25+r
rate of the boat upstream=25-r
Time to travel downstream=(distance downstream)/(rate downstream)=240/(25+r)
Time to travel upstream=(distance upstream)/(rate upstream)=160/(25-r)
Now we are told that these two times are equal. So:
240/(25+r)=160/(25-r) multiply both sides by (25+r)(25-r)
240(25-r)=160(25+r) get rid of parens
6000-240r=4000+160r subtract 6000 and also 160r from both sides
6000-6000-240r-160r=4000-6000+160r-160r collect like terms
-400r=-2000 divide both sides by -400
r=5 mph-------------------rate (speed) of the river
CK
240/30=160/20
8=8
Hope this helps---ptaylor

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distance(d)=rate(r) times time(t) or d=rt;t=d/r and r=d/t
Let t=time that elapses after John leaves until he catches Martina
We know that when John leaves, Martina has already travelled 24*2 or 48 miles (d=rt)
Distance Martina has travelled when John catches her=48+24t
Distance John has travelled when he catches Martina=48t
Now we know that John will have caught up with Martina when they have both travelled the same distance.
So our equation to solve is:
48+24t=48t subtract 24t from both sides
48+24t-24t=48t-24t collect like terms
48=24t divide both sides by 24
t=2 hrs---------------------time that elapses after John leaves until he catches Martina
We are also told that John leaves at 11:00AM.
So 11:00AM plus 2 hours=1:00PM ---------time that John catches Martina
CK
48+24*2=48*2
48+48=96
96=96
Hope this helps---ptaylor

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NOW:
x=John's age
3x=father's age
3x-3=mother's age
TEN YEARS FROM NOW:
x+10=John's age
3x+10=father's age
(3x-3)+10=mother's age
SO THE SUM OF THEIR AGES IN TEN YEARS TIME IS:
x+10+3x+10+(3x-3)+10 get rid of parens
x+10+3x+10+3x-3+10 collect like terms
7x+27---------------ans

Hope this helps---ptaylor

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Let x=height of Washington Monument: (x-58)----58 ft shorter
Then x+497=height of Eiffel Tower: ((x+497)-58)---------58 ft shorter
Now we are told that:
((x+497)-58)=2(x-58) get rid of parens
x+497-58=2x-116 simplify
x+439=2x-116 add 116 to and subtract x from both sides
x-x+439+116=2x-x-116-116 collect like terms
555=x or
x=555 ft--------------------------height of Washington Monument
x+497=555+497=1052 ft---------- height of Eiffel Tower
CK
1052-58=2*(555-58)
994=2*497
994=994

Hope this helps-----ptaylor

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Let x=amount that Brian earns as a tutor---weekly wages
Then (x/2)=amount that Brian earns pumping gas
So x+(x/2)= Brian's total weekly wages
And (x/4)=amount that Brian's sister earns
Now we are told that:
x+(x/2)=(x/4)+150 multiply both sides by 4
4x+2x=x+600 subtract x from both sides
4x+2x-x=x-x+600 collect like terms
5x=600 divide both sides by 5
x=$120------Brian's weekly wages as a tutor
(x/2)=(120/2)=$60-------------------Brian's weekly wages pumping gas
(x/4)=(200/4)=$50 ----------- Brian's sister's weekly wages
CK
$120+$60=$30+$150
$180=$180
Hope this helps---ptaylor

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Let x=amount that needs to be drained and replaced with pure antifreeze
Now we know that the amount of pure antifreeze left after we drain x amount out (0.40(5-x))plus the amount of pure antifreeze added (x) has to equal the amount of pure antifreeze in the final mixture (0.50*5). So our equation to solve is:
0.40(5-x)+x=0.50*5 get rid of parens and simplify
2-0.40x+x=2.5 subtract 2 from both sides
2-2-0.40x+x=2.5-2 collect like terms
0.60x=0.5 divide both sides by 0.60
x=0.833333-- gal----------amount that needs to be drained and replaced with pure antifreeze
CK
0.40(5-0.83333333333)+0.8333333333333=2.5
1.666666666666+0.8333333333333333333=2.5
~2.5=2.5

Hope this helps----ptaylor

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Let x=the number
Then 5x= five times the number
And 5x+1=five times the number increased by 1 and we are told that this equals 36
So:
5x+1=36 subtract 1 from both sides
5x+1-1=36-1 collect like terms
5x=35 divide both sides by 5
x=7
Ck
5*7+1=36
36=36
Hope this helps--ptaylor

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Lets deal in inches to make things simple to start with:
Let x=Dana's adult height
4ft 8in=48+8=56in--------------Dana's height at the age of 10
Now we are told that:
56 in is 84.4% of x------------So:
0.844x=56 divide both sides by 0.844
x=66.3507 in-----------------------Dana's adult height
66.3507/12=5.529 ft or 5ft 6.3in
NOW LETS LOOK AT IT YOUR WAY:
Her adult height (x) minus her height at 10 years old (56) equals = the amount that she grew and that equals 15.6% of her adult height
x-56=0.156x and this turns out to be the same equation as before
Hope this helps------ptaylor

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Let x=miles driven by the 15 mpg car
Then 15x=amount of gas used by the 15 mpg car
And 65-x=miles driven by the 40 mpg car
Then 40(65-x)=amount of gas used by the 40 mpg car
Now we are told that together the two cars use 1600 gallons. So our equation to solve is:
15x+40(65-x)=1600 get rid of parens
15x+2600-40x=1600 subtract 2600 from both sides
15x+2600-2600-40x=1600-2600 collect like terms
-25x=-1000 divide both sides by -25
x=40 miles--------------------miles driven by the 15 mpg car
65-x=65-40=25 miles -----------miles driven by the 40 mpg car
CK
40*15+25*40=1600
600+1000=1600
1600=1600
Hope this helps----ptaylor

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The answer is a).
a)
3w+6-w=4w-2(w-3) get rid of parens
3w+6-w=4w-2w+6 collect like terms on each side of the equation
2w+6=2w+6---------------this is an identity---true for all values of w
b)
8-(5v+4)=-5v-4 get rid of parens
8-5v-4=-5v-4 collect like terms on each side of the equation
-5v+4=-5v-4
This problem does not have a solution. There are no values for v that will satisfy this equation.
c)
5y+7=5y-7
This problem does not have a solution. There are no values for y that will satisfy this equation.
d)
4m-6=5m+9-m collect like terms on each side of the equation
4m-6=4m+9
This problem does not have a solution. There are no values for m that will satisfy this equation.
Hope this helps---ptaylor

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Well, lets begin by letting S=one of the sides of the square. Now we know that the area of a square is S*S or S^2 and we are told that this equals 82 ft^2. So:
S^2=82 ft^2 take the square root of both sides
S=+or-SQRT(82) forget the negative value because lengths are positive
S=9.055 now to the nearest hundreth, S would be either 9.06 or 9.05 depending on our convention for rounding.

Hope this helps----ptaylor

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Let x=required amount of yogurt that is 3% fat
Then 60-x=required amount of no-fat yogurt
Now we know that the amount of pure fat in the 3% yogurt (0.03x) plus the amount of pure fat in the no-fat yogurt (0) has to equal the amount of pure fat in the final 1% yogurt mixture(0.01*60) so or equation to solve is:
0.03x=0.01*60 simplify
0.03x=0.6 divide both sides by 0.03
x=20 lbs-----------------------------amount of yogurt that is 3% fat
60-x=60-20=40 lb-------------------------amount of no-fat yogurt needed
CK
0.03*20=0.01*60
0.6=0.6

Hope this helps---ptaylor

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Let x=amount of 25% copper sulfate solution needed
Then 140-x=amount of 60% copper sulfate solution needed
Now we know that the amount of pure copper sulfate in the 25% solution (0.25x) plus the amount of pure copper sulfate in the 60% solution (0.60(140-x)) has to equal the amount of pure copper sulfate in the final mixture(0.30*140). So our equation to solve is:
0.25x+0.60(140-x)=).30*140 get rid of parens
0.25x+84-0.60x=42 subtract 84 from both sides
0.25x+84-84-0.60x=42-84 collect like terms
-0.35x=--42 divide both sides by -0.35
x=120 ml ---------------------------------- amount of 25% solution
140-x=140-120=20 ml------------------------------amount of 60% solution
ck
0.25*120+0.60(140-120)=0.30*140 simplify
30+12=42
42=42
Hope this helps---ptaylor

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Let x=cost of a case of juice
And y=cost of a case of bottled water
Now we are told that:
6x+y=$135---------------eq1----cases and cost for the math club
4x+2y=$110---------------eq2-----cases and cost for the science club
Multiply eq1 by 2 and we get:
12x+2y=$270-------------------eq1a
4x+2y=$110-----------------------eq2
subtract eq2 from eq1a and we get:
8x=$160 divide both sides by 8
x=$20----------------cost of a case of juice
substitute x=$20 into eq2 and we have:
4*$20+2y=$110 or
$80+2y=$110 subtract $80 from both sides
$80-$80+2y=$110-$80 collect like terms
2y=$30 divide both sides by 2
y=$15----------------------------cost of a case of bottled water
CK
6*20+1*15=135
120+15=135
135=135
also
4*20+2*15=110
80+30=110
110=110
Hope this helps----ptaylor

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FIRST OF ALL, YOU NEED TO USE GROUPING SYMBOLS AS YOUR PROFESSOR SUGGESTS. I HAVE RE-WRITTEN THE FORMULA BELOW USING GROUPING SYMBOLS TO CLARIFY THE OPERATIONS THAT ARE REQUIRED:
x = {-b±√[b2 – 4ac]} / 2a
SECONDLY, YOU NEED TO GET THE QUADRATIC INTO STANDARD FORM:
x^2 - 3x = 7x -2 SUBTRACT 7X FROM AND ADD 2 TO BOTH SIDES
x^2-3x-7x+2=7x-7x-2+2 COLLECT LIKE TERMS
x^2-10x+2=0---QUADRATIC IN STANDARD FORM (AX^2+BX+C=0)
FROM HERE ON OUT IT'S JUST A MATTER OF PLUGGING IN NUMBERS AND FOLLOWING YOUR PROFESSOR'S INSTRUCTIONS. YOU SHOULD BE ABLE TO HANDLE IT.
Hope this helps----ptaylor

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How do u convert fractions into decimals?

You divide the denominator of the fraction into the numerator


Ex1-------1/4= 1 divided by 4 =0.25
Ex2--------1/3=1 divided by 3=0.333333333333333333---
Ex3-------2/3=2 divided by 3=0.6666666666666666666-----
Ex4----------2/4=2 divided by 4=0.50
Generally, when we are dealing with decimalss, we round them off depending on the level of accuracy that we need.
In example 2, if we are content with accuracy to the nearest tenth, then the answer is 0.3; If we want an accuracy to the nearest hundreth, then the answer is 0.33; If we want an accuracy to the nearest millionth, then the answer is 0.333333 and so forth.
In example 3, if we want an accuracy to the nearest thousandth, then the answer would be 0.667.
If someone tells you they walked 4 miles, then they could have walked anywhere between 3.5 miles and 4.5 miles (to the nearest mile) but if they tell you they walked 4.0 miles, they must have walked somewhere between 3.95 miles and 4.05 miles (to the nearest 1/10 of a mile)

Hope this helps---ptaylor

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x + y = 12---------------eq1
y = 2x ----------------------eq2
Lets substitute y=2x from eq2 into eq1 and we get:
x+2x=12 collect like terms
3x=12 divide both sides by 3
x=4
Now we can substitute x=4 into eq2 and we get:
y=2*4=8
So our answer is:
x=4 and
y=8
CK
From eq1:
x+y=12
4+8=12
12=12
From eq2:
y=2x
8=2*4
8=8

Hope this helps---ptaylor

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distance(d)=rate(r) times time(t) or d=rt; r=d/t and t=d/r
Let r=rate (speed) in still water.
distance upstream = distance downstream=6 mi
time to travel upstream + time to travel downstream=4 hrs
(Note: When travelling against the current, you must subtract the speed of the current from your speed in still water. Likewise, when travelling with the current, you must add the speed of the current to your speed in still water)
time to travel upstream=6/(r-2)
time to travel downstream=6/(r+2)
So, our equation to solve is:
6/(r-2)+6/(r+2)=4 multiply both sides by (r-2)(r+2) or (r^2-4)
6(r+2)+6(r-2)=4(r^2-4) get rid of parens
6r+12+6r-12=4r^2-16 subtract 4r^2 from and add 16 to both sides
6r+6r+12-12-4r^2+16=0 collect like terms
-4^r2+12r+16=0 multiply each term by -1
4r^2-12r-16=0 quadratic in standard form and it can be factored
(2r+2)(2r-8)=0
2r+2=0
r=-1---------------------disregard negative speed in this problem
and
2r-8=0
r=4 mph----------------------speed in still water
CK
time upstream=6/(4-2)=3 hrs
time downstream=6(4+2)=1 hr
3 hrs+1 hr=4 hr
4 hr=4 hr

Hope this helps----ptaylor

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Let x=number of liters of the 22% sugar solution needed
Then 680-x=number of liters of the 31% solution needed
Now we know that the pure sugar in the 22% solution (0.22x) plus the pure sugar in the 31% solution (0.31(680-x)) has to equal the pure sugar in the final mixture(0.27*680). So our equation to solve is:
0.22x+0.31(680-x)=0.27*680 get rid of parens
0.22x+210.8-0.31x=183.6 subtract 210.8 from both sides
0.22x+210.8-210.8-0.31x=183.6-210.8 collect like terms
-0.09x=-27.2 divide both sides by -0.09
x=302.222222222---- liters of 22% solution
680-x=680-302.222222222222222222---=377.77777777777777----liters of 31% solution
CK
302.2222*0.0.22+377.7777*0.31=680*0.27
66.4888888888888+117.11111111111111=183.6
183.599999999----~183.6

Hope this helps----ptaylor

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Let x=length of time it takes the small and medium pump working together
The small pump empties the tank at the rate of 1/14 tank per hour
The medium pump empties the tank at the rate of 1/8 tank per hour
Together they empty the tank at the rate of (1/14+1/8 or 4/56+7/56=11/56) tank per hour.
So, the question is, if both tanks working together can empty 11/56 tank per hour, how long will it take them to empty 56/56 or all of the tank?
(11/56)x=56/56 multiply each side by 56

11x=56 divide both sides by 11
x=5.090909--- hours
Hope this helps----ptaylor

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Let x=amount of pure acid needed
Now we know that the amount of pure acid in the acid that's added (x) plus the amount of pure acid in the 40% solution ((7)*(0.40)) equals the amount of pure acid in the final mixture ((0.70)(7+x)). So our equation to solve is:
x+7*0.40=0.70(7+x) get rid of parens
x+2.8=4.9+0.70x subtract 0.70x and also 2.8 from both sides
x+2.8-2.8-0.70x=4.9-2.8+0.70x-0.70x collect like terms
0.30x= 2.1 divide both sides by 0.30
x=7 gallons---------------------------amount of pure acid needed
CK
7+7*0.40=0.70(14)
7+2.8=9.8
9.8=9.8
Hope this helps---ptaylor

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Let x= one integer
Then x+2=the other
Now we are told that x+x+2=-298 subtract 2 from both sides
x+x+2-2=-298-2 collect like terms
2x=-300 divide both sides by 2
x=-150--------------one integer
x+2=-150+2=-148-----the other

CK
(-150)+(-148)=-298
-298=-298
Hope this helps---ptaylor

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Let x= one integer
Then x+2= the other
Now we are told that x+x+2=118 subtract 2 from both sides
x+x+2-2=118-2 collect like terms
2x=116 divide both sides by 2
x=58------------------------one integer
x+2=58+2=60------------------------the other
CK
58+60=118
118=118
Hope this helps---ptaylor

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Interest(I)=Principal(P) times Rate(R) times Time(T) or I=PRT
Let P-amount invested at 9%
(1) Interest accrued at 9%=P*0.09*1 or 0.09P

Then 15000-P=amount invested at 10%
(2) Interest accrued at 10%=(15000-P)*0.10*1 or 0.10(15000-P)
Now we are told that (1)+(2)=1432 or
0.09P+0.10(15000-P)=1432 get rid of parens
0.09P+1500-0.10P=1432 subtract 1500 from both sides
0.09P+1500-1500-0.10P=1432-1500 collect like terms
-0.01P=-68 divide both sides by -0.01
P=$6800---------------------amount invested at 9%
15000-P=15000-6800=$8200----------------amount invested at 10%
CK
6800*0.09+8200*0.10=1432
$612+$820=$1432
$1432=$1432
Hope this helps----ptaylor

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Interest(I)=Principal(P) times Rate(R) times Time(T) or I=PRT
Let P=amount invested at 12%
(1) Interest earned at 12%=P*0.12*1=0.12P
Then (17,500-P)=amount invested at 10%
(2) Interest earned at 10%=($17,500-P)*0.10*1=0.10($17,500-P)
Now we are told that (1)+(2)=$1,941, so our equation to solve is:
0.12P+0.10($17,500-P)=$1,941 get rid of parens
0.12P+$1,750-0.10P=$1,941 subtract $1,750 from both sides
0.12P+$1,750-$1,750-0.10P=$1,941-$1,750 collect like terms
0.02P=$191 divide both sides by 0.02
P=$9,550-------------------------------amount invested at 12%
($17,500-P)=$17,500-$9,550)=$7,950----------amount invested at 10%
CK
$9550*0.12+$7950*0.10=$1941
$1146+$795=$1941
$1941=$1941
Hope this helps----ptaylor

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YOU ARE ON THE RIGHT TRACK!!!! SOLVE YOUR EQUATIONS FOR t1 AND t2; ADD THEM TO GET TOTAL TIME AND OUR CALCULATIONS BEGIN TO AGREE.
distance(d) equals rate(r) times time(t) or d=rt;r=d/t and t=d/r
Let S=average speed on this trip
S=(Total distance travelled)/(Total time required)
(Note: Here's another problem that sort of underscores what average speed is all about: "You are going on a 60-mile trip. The first 30 miles, you travel at the rate of 30 mph. How fast must you travel the remaining 30 miles in order to average 60 mph???" ANSWER: Impossible!!! Why??? In order to average 60 mph, you must travel the entire 60 miles in an hour. Unfortunately, you have already used up that hour in the first 30 miles.)
Now back to your problem:
Let d=total distance travelled
Let d1=distance travelled for the first half of the trip
And let d2=distance travelled for the second half of the trip
*****Now we know that d1=d2. So total distance (d)=d1+d2=2d1 (or 2d2)
Let t=total time required
Let t1=time to travel the first half and this equals d1/30
Let t2=time to travel the second half and this equals d1/60
*****So Total time (t)=t1+t2=(d1/30+d1/60)=(2d1/60+d1/60)=(3d1/60)=d1/20
So S=2d1/(d1/20)---Now we multiply numerator and denominator by (20/d1) to get rid of the complex fraction and we get:
S=(2d1(20/d1))/(d1/20*20/d1)=(40d1/d1)/1 or 40 mph (the d1's cancel)
S=40 mph-------------------------------ans

Hope this helps---ptaylor

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Let x=original number of apples in the basket
1st person gets (1/3)x
2nd person gets(1/4)x
3rd person gets(1/8)x
4th person gets (1/5)x
5th person gets ---10 apples
6th person gets-------1 apple
After the first four persons get their apples, we'll find out what fraction of the apples are left for the 5th and 6th persons and we are told that whatever that fraction is, will equal 11:
(1/3)x+(1/4)x+(1/8)x+(1/5)x= Multiply each term by 120/120, the LCM:
(40/120)x+(30/120)x+(15/120)x+(24/120)x=(109/120)x
So, the apples that are left for the 5th & 6th persons equals:
(120/120)x-(109/120)x=(11/120)x ---and this equals 11.
(11/120)x=11 multiply both sides by 120
11x=1320 divide both sides by 11

x=120--------------------------number of apples
CK
(40/120)*120+(30/120)*120+(15/120)*120+(24/120)*120+11=120 or all the apples
40+30+15+24+11=120
109+11=120
120=120
Another way to work this problem is as follows:
(1/3)x+(1/4)x+(1/8)x+(1/5)x+11=x
(109/120)x-x=-11
-(11/120)x=-11
-11x=-1320
x=120

Hope this helps---ptaylor

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The possible factors of this quadratic are as follows:
(2x+1)(2x+3)
(4x+1)(x+3)
(4x+3)(x+1)
We will use the FOIL crutch (First, Inner, Outer, Last) to expand the above:
(2x+1)(2x+3)----4x^2+2x+6x+3=4x^2+8x+3;k=8
(4x+1)(x+3)-----4x^2+x+12x+3=4x^2+13x+3; k=13
(4x+3)(x+1)-----4x^2+3x+4x+3=4x^2+7x+3; k=7
So the answer is:
k=7, 8 or 13
There may be a more scientific approach to solving this problem, but if there is I'm not aware of it. Generally, when the A coefficient of a quadratic is >1, some level of trial and error is needed to get at the factors. There are some tricks, however, that simplifies the trial and error process. In this quadratic, for example, since all signs are positive then we know there cannot be any negative signs in the factors. Also, if the C term is positive and the B term is negative,then we know, right off, that both signs in each of the two factors must be negative. If both the B term and the C term are negative, then we know that one factor is positive and one is negative. Information such as this can frequently help to reduce the number of possibilities.
Hope this helps----ptaylor

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Let x=number of pounds of cashews needed
Value of the cashews=$7x
Value of the walnuts=$4*10
Value of the final mixture $5.50*(10+x)
Now we know that the value of the nuts before the mixture is equal to the value of the nuts after the mixture. So we have:
($ are understood)
7x+4*10=5.50*(10+x) get rid of parens
7x+40=55+5.5x subtract 40 and also 5.5x from both sides
7x+40-40-5.5x=55-40+5.5x-5.5x collect like terms
1.5x=15
x=10---------------------------------number of pounds of cashews needed
CK
7*10+4*10=5.5*20
70+40=110
110=110

Hope this helps----ptaylor

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distance(d)=rate(r) times time(t) or d=rt; r=d/t and t=d/r
Let t=number of hours that it takes the faster car to be 35 mi ahead of the slower car
Distance faster car travels=(rate of faster car)*t=55t
Distance slower car travels=(rate of slower car)*t=50t
Now the difference between these distances needs to equal 35 mi, so:
55t-50t=35
5t=35 divide both sides by 5
t=7 hours-------------ans
CK
in 7 hrs faster car travels 7*55 mi=385 mi
in 7 hrs slower car travels 7*50=350 mi
385-350=35
35=35
Hope this helps--ptaylor

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4. One more than the difference between 18 and 7 times a number is -9
This is what we have tried:
7n + 18 +1 = -9
7n + 19 = -9
7n = -28
n= -4
Is this correct?

YOU ARE VERY CLOSE!!!!!! WORD PROBLEMS ARE GENERALLY FAR EASIER IF YOU LAY OUT THE PROBLEM, STEP BY STEP, AS I HAVE DONE BELOW.
Let n=the number
7 times the number is 7n
Difference between 18 and 7 times a number: 18-7n
One more than the difference between 18 and 7 times a number: (18-7n)+1
--------and we are told that this equals -9. So:
18-7n+1=-9 simplify
19-7n=-9 subtract 19 from both sides
19-19-7n=-9-19 collect like terms
-7x=-28 divide both sides by -7
n=4------------the number

CK
18-(7*4)+1=-9
18-28+1=-9
-10+1=-9
-9=-9

Hope this helps---ptaylor