Quadratic - Distance / Partial Solution

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A and B traveled to meet each other. A left C at the same time B left D. They traveled the direct route from C to D. Upon arriving A traveled 18 more miles than B. A could have made B's time in 15 3/4 days, but B could have made A's time in 28 days. Calculate distance between C and D.

C M D
---------------------------------------------------------------------------
A ---------------------------------> <-----------------------------------B

Let X = CM = distance A traveled.

Let X - 18 = MD = distance B traveled.

X - 18 / 15 3/4 = distance A travels in 1 day.

x / 28 = distance B travels in 1 day.

Note : travel times are equal.

How do I solve ?

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@Randy Dandy,

Randy Dandy wrote:

X - 18 / 15 3/4 = distance A travels in 1 day. <-- this is A's velocity

x / 28 = distance B travels in 1 day. <-- this is B's velocity

If you divide A's distance travelled by A's velocity you get an expression for "days travelled". If you divide B's distance travelled by B's velocity you get a second expression for "days travelled". Equate the two expressions for days travelled and solve for X.

1 Reply

@Kolyo,

I don't understand.

1 Reply

@Randy Dandy,

Start by recognizing this:

(Days A spends travelling) * (Distance A travels per day) = (Distance A travels)

Then divide both sides by (Distance A travels per day) and you get:

(Days A spends travelling) = (Distance A travels) / (Distance A travels per day).

Similarly,

(Days B spends travelling) = (Distance B travels) / (Distance B travels per day).

Thus, you can get an expression involving x for (Days A spends travelling) and an expression involving x for (Days B spends travelling).

These two expressions equal each other because A and B spend the same amount of time travelling.

1 Reply

@Kolyo,

I still don't understand it.

1 Reply

@Randy Dandy,

What specifically don't you understand?

1 Reply

@engineer,

I don't understand Koylo's explanation.

1 Reply

@Randy Dandy,

We want to help and Koylo is IMO the most accomplished math person who floats around here, but you are going to have to let us know what you don't get. Do you understand that time x speed = distance?

1 Reply

@engineer,

Yes

1 Reply

@Randy Dandy,

Distance = Speed x Time

Distance A travels / Speed A goes = Time A takes
Distance B travels / Speed B goes = Time B takes

We know that ( The Time A takes = The Time B takes) by definition because they meet somewhere in the middle between points C and D.

Because Ta = Tb we therefore know that

Da / Sa = Db / Sb which is what you equate and solve the quadratic equation.

X / ( speed of A) = (X-18) ( speed of B)

Quote:

SPEED OF A = X - 18 / 15 3/4 = distance A travels in 1 day.

SPEED OF B = x / 28 = distance B travels in 1 day.

x = 72

Do you like calculus too?

1 Reply

@knaivete,

That didn't help. I still can't solve it.

I did not take calculus.

1 Reply

@Randy Dandy,

Quote:

That didn't help. I still can't solve it.

Quote:

X / ( speed of A) = (X-18) ( speed of B)

Quote:
SPEED OF A = X - 18 / 15 3/4 = distance A travels in 1 day.

SPEED OF B = x / 28 = distance B travels in 1 day.

x = 72

X times(speed of B) = (X -18) times (speed of A)

X times X/28 = (X -18) times (X -18)/ 15.75 AND SIMPLIFYING BY CROSS MULTIPLING GIVES

(63/4)X^2 = 28(X -18)(X -18) AND SIMPLIFYING BY MULTIPLING BY 4 GIVES

63X SQUARED = 112 (X -18) SQUARED and simplifying by dividing by 7 gives

9X SQUARED = 16 (X-18) SQUARED and simplifying by taking the square root of both sides gives

plus or minus 3X = 4(X- 18) so one root is

3X = 4X - 72 therefore

X = 72

Can you solve for total distance?

You could have multiplied everything out and then used the formula for finding the roots of a quadratic equation . (ax^2 + bx + c = 0 )

1 Reply

@knaivete,

"Can you solve for total distance?" No

" You could have multiplied everything out and then used the formula for finding the roots of a quadratic equation . (ax^2 + bx + c = 0 )"

I don't know how. I can't solve it.

1 Reply

@Randy Dandy,

In the "partial solution" to the question it was noted that

Quote:

A and B traveled to meet each other. A left C at the same time B left D. ... Upon arriving A traveled 18 more miles than B... Calculate distance between C and D.

Let X ... = distance A traveled.

Let X - 18 ... = distance B traveled.

One of the roots of the quadratic equation was shown to be :

X = 72

Therefore A travelled 72 miles.

B travelled (X -18) miles = 54 miles

The total distance travelled was 126 miles.

I thought it was amusing how multiples of 18 figured in the answer, what say you? 7 times 6 times 3 is 126 unless I'm all at 6's and 7's

1 Reply

@knaivete,

"I thought it was amusing how multiples of 18 figured in the answer, what say you?" Yes.

I still don't understand the solution, but thanks.

1 Reply

@Randy Dandy,

My pleasure. And thank you for such an excellent question.

If you read your question again and the my posts leading up to the solution, what is the first point in my posts that you do not follow?

1 Reply

@knaivete,

I don't follow these steps:

"X times(speed of B) = (X -18) times (speed of A)

X times X/28 = (X -18) times (X -18)/ 15.75 AND SIMPLIFYING BY CROSS MULTIPLING GIVES

(63/4)X^2 = 28(X -18)(X -18) AND SIMPLIFYING BY MULTIPLING BY 4 GIVES

63X SQUARED = 112 (X -18) SQUARED and simplifying by dividing by 7 gives

9X SQUARED = 16 (X-18) SQUARED and simplifying by taking the square root of both sides gives

plus or minus 3X = 4(X- 18) so one root is

3X = 4X - 72 therefore

X = 72"

"You could have multiplied everything out and then used the formula for finding the roots of a quadratic equation "

1 Reply

@Randy Dandy,

Simplifying, by doing the same thing to both sides of an equation leaves the equation unchanged.

http://www.purplemath.com/modules/solvquad.htm

1 Reply

@knaivete,

Helpful link. Thanks.

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Quadratic - Distance / Partial Solution

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