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The equation of a line can be written as y=mx+c where m=gradient(slope) and c is the y value where the line cuts the vertical y axis.
Your slope is 2 so we have y=2x+c.(i)
Now we need to find the value of c.
In ANY question you must use all the info given(although with some vindictive posers of questions you may get 'red herrings').
Here we are told that the line passes through (6,4) and so replacing x and y by 6 and 4 respectively in (i)we get:
4=2*6+c=12+c.
Hence c=-8.
Your equation then is y=2x-8ANS

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First of all the question itself is incorrect!
It should read:'Calculate -2.3-(-8.8)'.
If you follow the question as asked then the question is '-[-2.3-(-8.8)]'
This leads to -[-2.3+8.8]=2.3-8.8= -6.5.
This answer is not in the list given!
Whoever set you this problem should be disciplined for not making the question clear!
If we ignore the command 'subtract' then the correct answer is 6.5

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You must use vectors.

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m(the gradient)=vertical distance between the two points/horizontal distance between the two points.
So m=(Y-y)/(X-x) where the two points have coordinates of (X,Y) and (x,y).
So here (X,Y)=(-1,-7) and (x,y)=(1,3).
Hence the gradient m=(-7-3)/(-1-1)=-10/-2=5.
Now the general equation of a straight line is y=mx+c where m is the gradient and c is the y intercept(the y value where the line crosses the vertical Y axis).
So far we have: y=5x+c.(i)
We need to find the value of c.
Since the line passes through (1,3) then we can replace x and y in our equation by 1 and 3 respectively.
This gives (i) as: 3=5*1+c
So 3=5+c
Subtract 5 from each side of the equation and we get: 3-5=c
So c=-2
Hence the required equation is y=5x-2 ANS
(N.B. You should verify your answer by replacing the x by -1 and the y by -7)

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Since the hotel is 9 times the height of the hoop then the hotel's shadow must be 9 times the length of the hoop's shadow.
So the length of the hotel's shadow is 9*7=63 feet.ANS

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A/z+2 + B/2z-3 = [A(2z-3) + B(z+2)]/(z+2)(2z-3)
= (2Az-3A + Bz+2B)/(2z^2-3z+4z-6)
= (2Az+Bz-3A+2B)/(2z^2+z-6)
But we are given that this is equal to (5z-11)/(2z^2+z-6) and since the denominators(the bottom part of each expression) are equal then it follows that the numerators(the top part of each expression) are also equal.
Hence
2Az+Bz-3A+2B = 5z-11.
The left hand side can be written as (2A+B)z-(3A-2B) [remember - sign in front of a bracket means that the sign inside changes so that -3A+2B= -(3A-2B)].
And so we now have:
(2A+B)z-(3A-2B) = 5z - 11.
This must mean that 2A+B = 5 (i) and 3A-2B = 11 (ii) [This process is called equating the coefficients.]
Now multiply (i) by 2 [in order to get 2B] and (i) becomes:
4A+2B = 10 (i)
Now by adding this new form of equation (i) to equation (ii) we get:
7A +0 = 21 [4A+3A,2B add -2B and 10+11].
So 7A = 21
Divide both sides by 7:
A = 3 ANS
To find B replace A by 3 in the original form of equation (i) and we get:
2*3 + B = 5
Hence 6 + B = 5
Subtract 6 from both sides and we get:
B = -1 ANS

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From the information given angles WXV and ZXY are opposite angles at the point of intersection X.
Hence these two angles are equal.
So 4s - 9 = 2s + 17
Add 9 to both sides:
4s = 2s + 26
Subtract 2s from both sides:
2s = 26
Divide both sides by 2:
s = 13 QED

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7e^(5x) - 1 = 4e^(5x) + 8
Add 1 to each side
7e^(5x) = 4e^(5x) + 9
Subtract 4e^(5x) from both sides
3e^(5x) = 9
Divide both sides by 3
e^(5x) = 3
Take logs of both sides
In[e^(5x)] = In(3)
So 5xIn(e) = In(3)
So 5x = In(3) (since In(e) = 1)
Divide both sides by 5
Hence x = [In(3)]/5 = [1.098612289]/5 = 0.219722457 ( to 9 dec. places) ANS

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You need to know that in a right angled triangle the sum of the squares of the "legs" is equal to the square of the longest side(the hypotenuse).
This is known as Pythagoras's Theorem and apparently it was used in the building of the Pyramids!
In order to find the perimeter(the distance all round) of the triangle we need to know the length of the third side and since it is a right angled triangle we can use this theorem.
Using algebra the theorem states: a^2+b^2=c^2 where a and b are the "legs" and c is the hypotenuse.
In your question a=3ins and b=4ins and so 3^2+4^2=c^2.
Hence c^2= 9+16=25.
So c==5 (strictly speaking = plus or minus 5 but in the context of the question c must =5,since it is a length(a scalar quantity)). So the sides are 3,4 and 5 inches.
Hence the perimeter is 3+4+5=12 inches ANS

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x^2+8=57
Subtract 8 from each side
x^2=49
Square root both sides
x=+7 or -7.
So the answer is c.

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The total length is 20 inches.
Let the two lengths be x and 20-x inches
Using calculus:
The area of the square is (x/4)^2 and the area of the circle is pi*radius^2.
Now the circumference = 2*pi*r and C=20-x.
Hence r=(20-x)/(2*pi)= 10/pi-x/2pi
So total area is A=(x/4)^2+pi*(10/pi-x/2pi)^2=(x^2)/16+pi*(100/(pi)^2+x^2/4pi^2-10x/pi^2)
A= x^2/16 + 100/pi + x^2/4pi - 10x/pi
Maximum or minimum values are given when dA/dx=0 and so:
2x/16+2x/4pi - 10/pi = 0
x/8+x/2pi - 10/pi = 0
Multiply by 8pi
x*pi + 4x - 80 = 0 = 0
x*(pi + 4) - 80 = 0
x = 80/(pi + 4)
x = 80/(22/7 + 28/7) = 80/(50/7) = (8*7/5 = 56/5 = 11.2
So the lengths are 11.2 inches and 8.8 inches ANS

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Fine up to .
So we'll go from there.
x=
Square both sides:
.
Now multiply both sides by
We get:

Now subtract from both sides:
We get (i)
Let and (i) becomes:

Now using the formula for the solution of i.e.
where , and
we get:

But and so:

Hence .
Now there are TWO possible answers for the solution to the equation but only ONE is correct for the inverse function and it is:
ANS.
(NB. I'll leave it up to you to work out what represents.
You should also use your calculator to check the inverse function answer - I suggest, for example, that you replace x by 4 or a similar small value).

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Looking at the polynomial we can see that 4w is a COMMON FACTOR and so
-4w^3 - 16w^2 + 20w = 4w[-w^2 - 4w + 5]
= -4w[w^2 + 4w - 5]
Now to factorise w^2 + 4w - 5 we need to look for two numbers that when added together give you 4 and when multiplied together give you -5.
These numbers are 5 and -1.
So -4w[w^2 + 4w - 5] = -4w[(w + 5)(w-1)]ANS

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To do this you need to remember that
Area of triangle=(base*height)/2 and you should be able to factorise a quadratic trinomial expression such as ax^2+bx+c where a,b and c are given numbers(constants).You will need to be able to factorise x^2+5x+6.
Let the base be b metres(this is how we spell this word in "English" English).
Then Area of Triangle = [b*height]/2
Since the height is given as (x+3) metres then
Area=[b*(x+3)]/2.
But you are told that the area is x^2+5x+6
So [b*(x+3)]/2 is the same as x^2+5x+6
And so we can write
[b*(x+3)]/2=x^2+5x+6
Now the Left Hand Side of the equal sign is in a factorised form ,that is it is written as one thing times another thing (b times (x+3))
So to be able to find out what b is we need to factorise the Right Hand Side which is x^2+5x+6.
Suppose the factors are x+a and x+b then (x+a)*(x+b) =x^2 +(a+b)x+ab [Multiplying the brackets together]
Now compare x^2+(a+b)x+ab (i) to x^2+5x+6 (ii).
You can see that a+b in expression (i) is in the same place as the number 5 is in expression (ii)
and where the ab is in (i) we have the number 6 in (ii)
And so we can write
a+b=5 and
a*b=6 [this is called comparing coefficients]
So to be able to write x^2+5x+6 as (x+a)*(x+b) we need to find two numbers (a and b) so that when you add them together you get 5 and when you multiply them together you get 6.
The numbers are then a=3 and b=2 (3+2=5 and 3*2=6)
This means then that we can write the equation [b*(x+3)]/2=x^2+5x+6 as
[b*(x+3)]/2=(x+3)*(x+2)
Now share both sides by(x+3) and we get
b/2=(x+2)
Now multiply both sides by 2 and we get
b=2*(x+2)
Now multiply the bracket out(multiply everything in the bracket by 2) and we get
b=2x+4
And so the base is 2x+4 metres ANS

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The general series is S=a+(a+d)+(a+2d)+(a+3d) +....+(l-d))+l where a is the first term,d is the common difference and l is the last term.
If we write this in reverse we get S=l+(l-d)+(l-2d)+(l-3d)+....+(a+d)+a.
So we have,
S=a+(a+d)+(a+2d)+(a+3d)+....+(l-d)+l (i)
and
S=l+(l-d)+(l-2d)+(l-3d)+....+(a+d)+a (ii)
Now add (i) and (ii) together, making sure that you add corresponding terms together and you get
2S=(a+l)+(a+l)+(a+l)....+(a+l)+(a+l).
And so
2S= n(a+l) (because there are n lots of (a+l))
So
S= [n(a+l)]/2 (iii)
But your last term or nth term can be written as a+(n-1)d so
l=a+(n-1)d
Now replace l in (iii) and we get
S=[n(a+a+(n-1)d)]/2
This simplifies to
S=[n(2a+(n-1)d)]/2 Q.E.D.
(N.B. l is the letter L and 1 is the number ONE)

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Let m be Milcah's age now and j be Joshua's age now.
From what we are told about the ages 3 years ago we get the equation
m-3=3*(j-3) (i)
and from what we are told about the ages 4 years from now we get the equation
m+4=j+4+8, that is m+4=j+12. If we subtract 4 from each side of this equation we get that m=j+8. (ii)
Now we can replace m in equation (i) by j+8 so that equation (i) becomes
j+8-3=3*(j-3)
So j+5=3*(j-3)
Now multiply the bracket out and we get
j+5=3j-9
Now subtract j from both sides and we get
5=2j-9
Now add 9 to both sides and we get
14=2j
This means that j=7.
To find m replace j by 7 in equation(ii) and we get m=7+8 =15
So Milcah is 15 years old and Joshua is 7 years old.ANS
(NB I will leave it up to you to check the answers)

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You need to know about the difference of two squares.
a^2 - b^2 = (a+b)*(a-b) or more simply,(a+b)(a-b).
In your question think of 16x^2 as (4^2)*x^2 or more simply,(4^2)(x^2).
Now (4^2)(x^2)=(4*x)^2 or more simply (4x)^2.
So 16x^2=(4x)^2.
Also think of 25 as (5)^2.
So now we can write: 16x^2-25=(4x)^2-(5)^2.
Now the difference of two squares tells us that
a^2-b^2=(a+b)(a-b).
Compare your (4x)^2-(5)^2 to a^2-b^2.
If you look carefully you can see that a is in the same place as 4x and b is in the same place as 5.
So now we can replace a and b by 4x and 5 respectively.
This means that
a^2-b^2=(a+b)(a-b) can be written as (4x)^2-(5)^2=(4x+5)(4x-5)ANS

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You need to work out 1/4+1/8+1/2.
The bottom numbers of the fractions (the DENOMINATORS) are 2,4 and 8 and they have an LCM (Lowest Common Multiple or the smallest number that is in EACH of the times tables of 2,4 and 8) of 8.
Since the LCM is 8 change your fractions to eighths and then you can add them together.
1/4 = 2/8 ( 1/4= 1/4*(2/2)= (1*2)/4*2)=2/8 ).
1/8=1/8 (no change needed )
and 1/2 =4/8 ( 1/2=1/2*(4/4)= (1*4)/(2*4)=4/8 ).
So 1/4+1/8+1/2 = (2/8)+(1/8)+(4/8) = 7/8.
And so 7/8ths of a teaspoon are needed.ANS

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Since the hemisphere has height(radius) 3 inches then the height of the cylinder must be 9 inches(since you are told that the total height is 12 inches).
Now the formula for the volume of a cylinder is pi*(radius)^2*(height) and so the cylinder has volume 3.142*3^2*9=254.469 cubic inches.
Now the formula for the volume of a sphere is (4*pi*(radius)^3)/3.
This means that the formula for a hemisphere is a half of this.
And so the volume of a hemisphere is (2*pi*(radius)^3)/3.
So our hemisphere must have a volume of (2*3.142*(3)^3)/3 = (6.284*27)/3 = 56.556 cubic inches.
The total volume then is 254.469+56.556 cubic inches =311.025 cu.ins.ANS

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Let the two numbers be x and y.
Now their sum(total)is 36 so x+y=36 (i)
and their difference(what you get when you take one number away from the other) is 12 so x-y=12 (ii) (note: we could have said y-x=12 instead,it doesn't matter which way round you do it you will still be able to find the right answers).
So we have: x+y=36 (i)
and
x-y=12 (ii)
Now add the two equations together and we get 2x=48 (x+x=2x,+y+-y=y-y=0 and 36+12=48)
and so x=24 (a half of 48), which means that y=12 (since x+y=36).
So the two numbers are 24 and 12ANS

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Let the 5th score be x.
Now the average(mean)= (total of all values)/(the number of values).
Now the total of all values in your question is 80+96+78+87+x and the number of values in your question is 5. You are also told that the average value has to be 84.
So the average 84=(80+96+78+87+x)/5.
This means that 84=(341+x)/5.
Now multiply both sides of this equation by 5 and you get:
84*5=341+x
Which leads to 420=341+x
Now take away 341 from both sides of the equation and you get:
420-341=x
And so x=420-341=79.
The 5th test score then is 79ANS

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slope(gradient)of straight line between two points on the line is vertical distance between the two points divided by the horizontal distance between the two points.
So if the two points on the line have coordinates of (X,Y) and (x,y) then
the slope=(Y-y)/(X-x).
In your question the two points on the line are (-9,6) and (-4,5) and so here
X=-9,x=-4 and Y=6,y=5.
Replacing the letters X,x,Y and y by their number values the formula becomes
the slope=(6-5)/(-9--4).
Now the top part=1 and the bottom part =-9--4 becomes -9+4 (because --=+).
The value of -9+4 is -5 (by using the number line,start at -9 and move 4 spaces to the right and you end up at -5).
So the slope is 1/-5=-0.2ANS

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The number of students=56+42-15=98-15=83ANS
Perhaps the best way to see where this answer comes from is by using a VENN DIAGRAM.
Alternatively,
Since 15 went to both camps then 42-15=27 went to the youth camp only.
Also since 15 went to both then 56-15=41 went to the family camp only.
So the total number of students=those who went to youth camp only+those who went to the family camp only+those who went to both=27+41+15=83ANS

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Let w be the walking speed, then w+2 must be the running speed (since we're told that his running speed is 2m/s more than his walking speed).
Let the walking time be t seconds and the running time be T seconds

Now using these facts: distance= average speed*time,
the distance is 15 metres in each case( walking and running),
the factors of 15 are 5 and 3 and
t+T= 8 (given)

we can find w BY INSPECTION as follows:
Since 15=3*5=w*t
and
15=5*3=(w+2)*T then it follows that w=3,t=5 and T=3.
So the walking speed,w,is 3m/sec.ANS

You can of course solve the question using algebra but why do that with such a trivial question.
In an exam situation time is of the essence and a solution BY INSPECTION is totally valid provided you show what it is based upon.
Using algebra and the definitions given above a possible solution might be as follows:
15=w*t so t=15/w
Similarly,15=(w+2)*T so T=15/(w+2).
But t+T=8 and so (15/w)+(15/(w+2))=8
This leads to (1/w)+(1/(w+2))=8/15
Adding the two algebraic fractions together we get:
(w+2+w)/w(w+2) = 8/15
Hence 2w+2 = 8 (i) and w(w+2) = 15 (ii)
Considering 2w+2 = 8 it follows that w+1 = 4 (dividing both sides by 2)
And so w = 3
That is, the walking speed is 3m/sec.ANS

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To do this question you must be aware of the following.
The general equation of a straight line is written as y=mx+c where m is the slope(or gradient)of the line and c is the y coordinate where the line cuts the y axis( c is also called the y intercept).
In the more difficult questions you must also know how to re-arrange equations.
Here is a simple example.
Suppose the equation of a straight line is y=3x+2.
If we compare this to the general equation y=mx+c then we can see that the number next to x(called the coefficient of x) is 3 in our equation and this is in the place taken up by m in the general equation.
So we can say that m=3. But m is the slope of the line so the slope of the line that has equation y=3x+2 is 3.
Also comparing both equations we can see that c=2. This means that the line with equation y=3x+2 cuts(or crosses) the vertical y axis at 2 units up from the origin.
Examples: The line with equation y=5x+7 has a slope of 5 and and cuts the y axis at 7 up.
The line with equation y=-4x-3 has a gradient of -4 and cuts the y axis at -3(3 below the origin).
So to identify the slope and the y coordinate where a line crosses the y axis your equation must be in the form y=mx+c.
If the equation you are given is not in this form then you have to re-arrange it so that it is in the form y=mx+c.
Example: Find the slope of the straight line that has the equation 4y-9x=12
So 4y-9x=12 has to be re-arranged into the form y=mx+c before we can say what its slope is.
Solution: Given 4y-9x=12
Add 9x to both sides
This gives 4y=12+9x or 4y=9x+12.
But we need to have just y(or 1y) by itself on the left-hand side of the equal sign.
To get this we must now share(divide)both sides of the equation 4y=9x+12 by 4.
When we do this we get that 4y/4=9x/4+12/4.
This simplifies to: y=2.25x+3 (since 9/4=2.25)
So now we have the equation in the form y=mx+c and we can see that the number next to the x is 2.25 so the slope of the line is 2.25ANS.

Now for your equation 2x-5y=3.
If we add 5y to both sides of the equation we get: 2x-5y+5y=3+5y
This simplifies to 2x=3+5y or 2x=5y+3
Now swap sides to get: 5y+3=2x
We now need to get rid of the +3 on the left-hand side so that we are just left with 5y on the left-hand side.
To do this we must subtract 3 from both sides so that we get:5y+3-3=2x-3.
This simplifies to 5y=2x-3.
(We're neally there, just one more step!)
We need to have just y or 1y(same thing) on the left-hand side and so that 5 next to the y has to be changed to a 1.
We do this by sharing(dividing)both sides by 5 so that we get: 5y/5=2x/5-3/5.
This simplifies to y=0.4x-0.6 (since 5/5=1, 2/5=0.4 and 3/5=0.6).
So we (finally) have y=0.4x-0.6.
This means that the slope of your line is 0.4 and it cuts the y axis at 0.6 below the origin (below because its -0.6 in the equation and not +0.6).
So looking at the list of answers that you have to choose from for the slope the correct answer is d)2/5 (since 0.4=2/5).

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Before you can answer your question you need to know about the DISTRIBUTIVE LAW.
We will start by using NO algebra.
What is 5*6?
Yes, it's 30
Now what is 20+10?
Again it's 30.
So doesn't that mean that 5*6=20+10? (i)
Now think about the 20.
Isn't this 5*4? And the 10,isn't this 5*2?
So doesn't this mean that we can write (i) as: 5*6=5*4+5*2 instead of 5*6=20+10?
So we've now got 5*6=5*4+5*2 (ii)
Now think about the 6.
Doesn't 6=(4+2)?
So now we can write (ii) as: 5*(4+2)=5*4+5*2 instead of 5*6=5*4+5*2.
Now look carefully at what we've just written:
5*(4+2)=5*4+5*2
To DISTRIBUTE something is to spread it and that is exactly what we have done here.We have spread the multiplication by 5 over the 4 and the 2.
Put more simply:
YOU MULTIPLY EVERYTHING IN THE BRACKET BY WHATEVER IS IN FRONT OF THE BRACKET
(BUT REMEMBER whatever is in front has to be linked to the bracket by a times sign).
In algebra if we think of a,b and c as being numbers we can write the Distributive Law as:
a*(b+c)=a*b+a*c or a(b+c)= ab+ac
So we have removed the bracket by multiplying everything in the bracket by the number a.
So in your question we use the distributive property by multiplying everything in the bracket by 2.
So we get 2(1+w)=2*(1+w)=2*1+2*w.
We then simplify this so that 2*1+2*w= 2+2w.
So 2+2w is the answer (we cannot simplify any further because we cannot add unlike terms together).

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To solve this question you should be aware of the following result.
"Given two quantities y and x where y varies directly with x then the relationship between y and x can be written as: y=kx where k is a constant".
Since the bounce varies DIRECTLY with the height from which it is dropped then the relationship between the height and bounce can be expressed as
b=k*h (i)
where h= the height from which the ball is dropped,b=the height of the bounce and k= a constant value.
In order to find the equation that links b to h we must find the value of the constant k.
To do this we use the rest of the given information which is that when the ball is dropped from 5 feet the bounce is 2.8 feet.
This means that when h=5,b is equal to 2.8.
Knowing this means that we can now replace h and b in equation (i) by these values so that the equation becomes:
2.8=k*5 or 5k=2.8
Now share both sides of the equation by 5 and we get:
k=2.8/5=5.6/10=0.56
Hence the equation b=k*h can now be written as b=0.56h ANS

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Let the walking rate be x(km/h).
This means that after 1 hour she has walked x km.
So after 2 hours she will have walked 2x km.
We are told that she then cycles at three times her walking rate.
This means that her cycling rate =3*x=3x(km/h).
But we are told that she only cycles for 1/3 of an hour and so her cycling distance is 1/3 of 3x km. which is 1x or x km.
So she walks 2x km and cycles x km,giving a total travelling distance of 3x km.
But we are told that the total distance travelled is 12km.
Hence 3x=12.
Now share each side of this equation by 3 and we get:
x=4.
So her walking rate is 4km/h.ANS
(NB. In English English we usually say "cycled" rather than "bicycled" and we spell the US "traveled" as "travelled"!).

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You will need to be able to use the formula: distance = speed*time.
Always give the unknowns appropriate labels. Here, since the journey is in two stages, two different times are involved and we will need to find their values before we can find the total distance.
Call them T(time taken for 1st stage) and t(time taken for 2nd stage).
Units: Since the speeds are given in miles per hour you will have to change the time given,2hours and 15 minutes, into hours before you can use the formula mentioned above so that the total time is now written as 2 and 1 quarter hours or 2.25hrs.
We are now ready to begin.

1st Stage:
Using the formula, distance=speed*time we can replace the speed by the number 5 and the time by the letter T.
This gives us our first equation:

1st stage distance = 5*T=5T.(i)

2nd Stage:
In the same way we can get our second equation:
2nd stage distance = 8*t=8t.(ii)
We now need to find the total distance,that is,
5T+8t(iii).
Since the the distance of the 2nd stage is 2*distance of 1st stage then
8t=2*5T=10T(from (i) and (ii))
or 10T=8t.(iv)
Sharing both sides by 10 equation(iv)becomes:
T=0.8t
Now since the total time is 2.25hrs then
T+t=2.25.(v).
But T=0.8t so (v) becomes:
0.8t+t=2.25.
This simplifies to
1.8t=2.25(since t=1t).
Sharing both sides by 1.8 we get:
t=2.25/1.8=1.25hrs.
So T+1.25=2.25(from (v))
Hence T=2.25-1.25=1hr.
Finally then replacing T and t by their values in equation(iii) which represents the total distance we get:
5T+8t=5*1+8*1.25=5+10=15miles.
So the Walkathon was a distance of 15miles.