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# The current in a river is 4 mph. A boat can travel 20 mph in still wat

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Re: The current in a river is 4 mph. A boat can travel 20 mph in still wat [#permalink]
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Hi All,

This question is a multi-step rate question, so you'll need to use the Distance Formula, but you can actually go about solving it in a couple of different ways.

From the prompt, we know that the rate "with" the current is 24 miles/hour and the rate "against" the current is 16 miles/hour. Since 24/16 = 1.5, there's an interesting way to convert this information into a 'unit' that we can use to our advantage...

IF...we traveled upstream for 24 miles, then downstream for 24 miles, we would spend....

"with" the current
D = (R)(T)
24 = (24 mph)(T)
1 hour = T

"against" the current
D = (R)(T)
24 = (16 mph)(T)
1.5 hours = T

TOTAL Time = 1 + 1.5 = 2.5 hours to travel 24 miles in each direction. This data is useful, since the question asks how far the boat can travel "with" the current if the TOTAL trip takes 10 hours.

10 hours/2.5 hours = 4 "blocks" of 24 miles

(4)(24) = 96 miles in each direction.

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Re: The current in a river is 4 mph. A boat can travel 20 mph in still wat [#permalink]
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Another approach is using the term 'round trip' and equating distances

$$R1*T1 = R2*T2$$
Given $$T1 + T2 = 10$$
Speed upstream is 16 (deduct opposing current)
Speed downstream is 24 (add current)

$$24*T1=16(10-T1)$$
$$T1=4$$

Hence $$D1 = R1*T1 = 24*4 = 96$$
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Re: The current in a river is 4 mph. A boat can travel 20 mph in still wat [#permalink]
Bunuel wrote:
The current in a river is 4 mph. A boat can travel 20 mph in still water. How far up the river can the boat travel if the round trip is to take 10 hours?

A. 69 miles
B. 88 miles
C. 96 miles
D. 100 miles
E. 112 miles

let t=upstream hours
24 mph*(10-t) hours=16t miles
t=6 hours
6 hours*16 mph=96 upstream miles
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Re: The current in a river is 4 mph. A boat can travel 20 mph in still wat [#permalink]
Let D be the distance upstream.
D/20-4 + D/20+4 = 10
D(1/16 + 1/24) = 10
D(3+2/48) = 10
D/48 = 2
D = 96.
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Re: The current in a river is 4 mph. A boat can travel 20 mph in still wat [#permalink]
GMATPrepNow wrote:
Bunuel wrote:
The current in a river is 4 mph. A boat can travel 20 mph in still water. How far up the river can the boat travel if the round trip is to take 10 hours?

A. 69 miles
B. 88 miles
C. 96 miles
D. 100 miles
E. 112 miles

Boat's speed UPSTREAM = 20mph - 4mph = 16mph
Boat's speed DOWNSTREAM = 20mph + 4mph = 24mph

(travel time UPSTREAM) + (travel time DOWNSTREAM) = 10 hours

time = distance/speed
We know the speeds, but not the distance each way
So, let d = distance EACH EACH

Our word equation becomes....
d/16 + d/24 = 10
Eliminate the fractions by multiplying each side by 28 (the least common multiple of 16 and 24)
When we do this, we get: 3d + 2d = 480
Simplify: 5d = 480
Solve: d = 96

So, the distance EACH WAY is 96 miles

Cheers,
Brent

I really like your algebraic approach "word equation".
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Re: The current in a river is 4 mph. A boat can travel 20 mph in still wat [#permalink]
Bunuel wrote:
The current in a river is 4 mph. A boat can travel 20 mph in still water. How far up the river can the boat travel if the round trip is to take 10 hours?

A. 69 miles
B. 88 miles
C. 96 miles
D. 100 miles
E. 112 miles

another way to solve the problem would be to think a little logically here.

We know that the distance upstream and the distance downstream is the same. therefore, upstream=downstream (eq1)

Also, if total time is 10, then assuming Upstream as T, Downstream will be 10-T. (Not T-10 since 10 is the total)

Using R*T=D formula we then get,
Upstream= 16*T=D (eq2)
Downstream= 24*(10-T)=D (eq3)

Now Using eq1 we get eq2=eq3. Solving that,

16t=24(10-t)
16t=240-24t
40t=240
t=6

substituting t=6 in the Upstream equation eq2 we get 96.

To verify, we can also substitute in eq3 which should also give 96.

Hope it helps.
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Re: The current in a river is 4 mph. A boat can travel 20 mph in still wat [#permalink]
Hi,
shouldn't it be 192 as the stem asked about the distance of the round trip? A - B & B - A so in this way total distance should be 2D. I have done it with a tabular form. I saw the speed in the ration 6:4 so time would be 4:6. and then found the D as 96 and the total distance of the round trip to be 192.
Thanks Bunuel sriharimurthy
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Re: The current in a river is 4 mph. A boat can travel 20 mph in still wat [#permalink]
ashish0027 wrote:
Hi,
shouldn't it be 192 as the stem asked about the distance of the round trip? A - B & B - A so in this way total distance should be 2D. I have done it with a tabular form. I saw the speed in the ration 6:4 so time would be 4:6. and then found the D as 96 and the total distance of the round trip to be 192.
Thanks Bunuel sriharimurthy

Hi ashish0027,

The prompt does NOT ask for the distance of the round-trip; it asks how far the boat can go 'up river' (meaning in just one direction). It takes 10 hours total to 'up river' and THEN back 'down river', but again - the question is not asking about the round-trip.

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Re: The current in a river is 4 mph. A boat can travel 20 mph in still wat [#permalink]

Whenever you get a problem where the Distance or Time is the SAME, the easiest way will most likely be to equate both the sides and get the variable.

Up Speed=16 KMPH
Down Speed=24 KMPH

Up Speed*T1=Down Speed*(10-T1)
$$16*T1=24(10-T1)$$
$$T1=6$$

$$Distance Up speed = Time*Speed\\ =16*6\\ =96$$
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Re: The current in a river is 4 mph. A boat can travel 20 mph in still wat [#permalink]
Why does plugging in numbers for distance not work?
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Re: The current in a river is 4 mph. A boat can travel 20 mph in still wat [#permalink]
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Re: The current in a river is 4 mph. A boat can travel 20 mph in still wat [#permalink]
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