1. ## Fractional Equations????

1) An intake pipe can fill a certain tank in 6 hours when the outlet pipe is closed, but with the outlet pipe open it takes 9 hours. How long would it take the outlet pipe to fill an empty tank?

2) Members of the ski club contributed equally to obtain $1800 for a holiday trip. When 6 members found that they could not go, their contributions were refunded and each remaining member then had to pay$10 more to raise the $1800. How many went on the trip? 3) Because of traffic Maria could average only 40 km/h for the first 20% of her trip, but she averaged 75 km/h for the whole trip. What was her average speed for the last 80% of her trip? 4) A number x is the harmonic mean of a and b if 1/x is the average of 1/a and 1/b. Find two positive numbers that differ by 12 and have harmonic mean 5 I don't know how to do these please help?? I am really clueless on how to do these..... I don't know where to start I know I need a variable but I don't know how to make it all work 2. 3) Because of traffic Maria could average only 40 km/h for the first 20% of her trip, but she averaged 75 km/h for the whole trip. What was her average speed for the last 80% of her trip? as this is my first time making a reply on this forum, someone correct me if i'm wrong please. We know Maria was averaging 40 km/h for the first 20% of her trip. [40 x 20%] We know that she averaged x km/h for the remaining 80% of her trip. [x x 80%] We know those two numbers together give us 75 km/h. so we add those figures: (40 x 0.2)+(x x 0.8) = 75 km/h 8 + 0.8x = 75 km/h -8 + 8 + 0.8x = 75 - 8 0.8x = 67 3. ## Re: Fractional Equations???? Originally Posted by dunit0001 1) An intake pipe can fill a certain tank in 6 hours when the outlet pipe is closed, but with the outlet pipe open it takes 9 hours. How long would it take the outlet pipe to fill an empty tank? 2) Members of the ski club contributed equally to obtain$1800 for a holiday trip. When 6 members found that they could not go, their contributions were refunded and each remaining member then had to pay $10 more to raise the$1800. How many went on the trip?

3) Because of traffic Maria could average only 40 km/h for the first 20% of her trip, but she averaged 75 km/h for the whole trip. What was her average speed for the last 80% of her trip?

4) A number x is the harmonic mean of a and b if 1/x is the average of 1/a and 1/b. Find two positive numbers that differ by 12 and have harmonic mean 5

I don't know how to do these please help?? I am really clueless on how to do these.....

I don't know where to start I know I need a variable but I don't know how to make it all work
3) Because of traffic Maria could average only 40 km/h for the first 20% of her trip, but she averaged 75 km/h for the whole trip. What was her average speed for the last 80% of her trip?
Assume

The distance = d

speed for the last 80% = v

then

time for first 20% = t_1 = 0.2*d/40 hour = d/200 hr

time for last 80% = t_2 = 0.8*d/v hour = 4d/(5v) hr

total time = t_1 + t_2 = d(0.005 + 0.8/v)

total average speed = d/[d(0.005 + 0.8/v)]

Now continue...

From the complexity (and variety) of these problems - it looks like a test or a prep for a test. If you have no clue how to do these problems - you should talk to your teacher and re-evaluate your priorities.

4. ## Re: Fractional Equations????

Originally Posted by dunit0001
1) An intake pipe can fill a certain tank in 6 hours when the outlet pipe is closed, but with the outlet pipe open it takes 9 hours. How long would it take the outlet pipe to fill an empty tank?

I don't know where to start I know I need a variable but I don't know how to make it all work

An "outlet" pipe would take water OUT of the tank, right? So, I don't see how an outlet pipe could POSSIBLY fill the tank. Please check to see if you've typed the question correctly.

5. ## Re: Fractional Equations????

Hello, dunit0001!

3) Because of traffic Maria could average only 40 km/h for the first 20% of her trip.
But she averaged 75 km/h for the whole trip.
What was her average speed for the last 80% of her trip?

Let $d$ = distance (length of her trip in kilometers).

She drove $0.2d$ km at 40 kph.
. . This took her: $\:\frac{0.2d}{40$ hours.

She drove $0.8d$ km at $x$ kph.
.This took her: $\:\frac{0.8d}{x}$ hours.

Hence, her total driving time was: $\:\frac{0.2d}{40}\,+\,\frac{0.8d}{x}$ hours.

Her average speed is: $\:\frac{\text{Total distance}}{\text{Total time}} \;=\;\L\frac{d}{\frac{0.2d}{40}\.+\,\frac{0.8d}{x} } \;=\;\frac{40x}{0.2x\,+\,32}$$\text{ km/h}$

There is our equation! . . . $\L\;\;\frac{40x}{0.2x\,+\,32} \:=\:75$

Then we have: $\:40x \:=\:75(0.2x\,+\,32)$

. . $40x \:=\:15x\,+\,2400$

. . $25x\:=\:2400$

. . $x \:=\:\fbox{96\text{ km/h}}$

6. ## Re: Fractional Equations????

Hello, dunit0001!

Here's #2 . . .

2) Members of the ski club contributed equally to obtain $1800 for a holiday trip. When 6 members found that they could not go, their contributions were refunded and each remaining member then had to pay$10 more to raise the \$1800.
How many went on the trip?

$N$ people paid $x$ dollars each: $\:Nx \:=\:1800\;\;\Rightarrow\;\;x\:=\:\frac{1800}{N}\;$[1]

When 6 members dropped out, only $N-6$ went on the trip.
. . and they each paid $x\,+\,10$ dollars.
. . . . $(N\,-\,6)(x\,+\,10) \:=\:1800\;$[2]

Substitute [1] into [2]: $\:(N \,-\,6)\left(\frac{1800}{N}\,+\,10\right) \:=\:1800$

. . and we have: $\:1800\,+\,10N\,-\,\frac{10800}{N}\,-\,60\:=\:1800\;\;\Rightarrow\;\;10N\,-\,60\,-\,\frac{10800}{N} \:=\:0$

Multiply by $\frac{N}{10}:\;\;N^2\,-\,6N\,-\,1080\:=\:0$

This factors: $\:(N\,-\,36)(N\,+\,30)\:=\:0$

. . and has roots: $\:N \:=\:36,\:-30$

Therefore, 36 members originally signed up for the trip.

. . but only $N\,-\,6\:=\:\fbox{30}$ members went on the trip.

7. ## Re: Fractional Equations????

Hello again, dunit0001!

4) A number $x$ is the harmonic mean of $a$ and $b$ if $\frac{1}{x}$ is the average of $\frac{1}{a}$ and $\frac{1}{b}.$

Find two positive numbers that differ by 12 and have harmonic mean 5.

Let's translate that ugly definition . . .

$\L\frac{1}{x} \;=\;\frac{\frac{1}{a}\,+\,\frac{1}{b}}{2} \;=\;\frac{a\,+\,b}{2ab} \;\;\Rightarrow\;\;x \;=\;\frac{2ab}{a\,+\,b}$

We have two numbers that differ by 12.
. . Let $a$ = the smaller number,
. . $a\,+\,12$ = the larger number.

Their harmonic mean is: $\L\:\frac{2a(a\,+\,12)}{a\,+\,(a\,+\,12)} \;=\;\frac{2a(a\,+\,12)}{2a\,+\,12} \;=\;\frac{2a(a\,+\,12)}{2(a\,+\,6)} \;=\;\frac{a(a\,+\,12)}{a\,+\,6}$

We're told that the harmonic mean is 5: $\L\:\frac{a(a\,+\,12)}{a\,+\,6} \:=\:5$

We have: $\:a(a\,+\,12)\;=\;5(a\,+\,6)\;\;\Rightarrow\;\;a^2 \,+\,12a\;=\;5a\,+\,30$

This is a quadratic: $\:a^2\,+\,7a\,-\,30\;=\;0$

. . which factors: $\:(a\,-\,3)(a\,+\,10)\;=\;0$

. . and has roots: $\:a\;=\;3,\,-10$

Therefore, the two positive numbers are: $\:\fbox{3 \text{ and }15}$

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