# Maria was travelling in her boat . . ., two equations – three unknowns

by VIJAY
(Chandigarh)

My Question is written below:

"MARIA WAS TRAVELLING IN HER BOAT WHEN THE WIND BLEW HER HAT OFF AND HAT STARTED FLOATING AT DOWNSTREAM. THE BOAT CONTINUED TO TRAVEL UPSTREAM FOR 12 MORE MINUTES BEFORE MARIA REALIZED THAT HER HAT HAD FALLEN OFF AND TURNED BACK DOWNSTREAM. SHE CAUGHT UP WITH THAT AS SOON AS IT REACHED THE STARTING POINT. FIND THE SPEED OF RIVER IF MARIA'S HAT FLEW OFF EXACTLY 3 KM FROM WHERE SHE STARTED"

### Comments for Maria was travelling in her boat . . ., two equations – three unknowns

 Oct 19, 2011 Linear Equation in Two Variables by: Staff --------------------------------------------------------------------------- Part II Note that there are three unknowns, but only two equations. Therefore, the problem may have multiple solutions (it doesn’t, but you don’t know that at this point). Solve for t in equation 2: (t + 0.2) * R = 3 Rt + 0.2R = 3 Rt + 0.2R - 0.2R = 3 - 0.2R Rt + 0 = 3 - 0.2R Rt = 3 - 0.2R t = (3 - 0.2R) / R substitute the equation for t into the first equation: t * (B + R) = 0.2 * (B - R) + 3 t * (B + R) = 0.2 * (B - R) + 3 [(3 - 0.2R) / R] * (B + R) = 0.2 * (B - R) + 3 Solve for R. [(3 - 0.2R) / R] * (B + R) = 0.2 * (B - R) + 3 [(3 - 0.2R) * (B + R) / R] = 0.2 * B - 0.2 * R + 3 [(3 - 0.2R) * (B + R) / R] * R = (0.2 * B - 0.2 * R + 3) * R (3 - 0.2R) * (B + R) * (R / R) = (0.2 * B - 0.2 * R + 3) * R (3 - 0.2R) * (B + R) * (1) = (0.2 * B - 0.2 * R + 3) * R (3 - 0.2R) * (B + R) = (0.2 * B - 0.2 * R + 3) * R 3B + 3R - 0.2R*B - 0.2R*R = (0.2 * B - 0.2 * R + 3) * R 3B + 3R - 0.2BR - 0.2R² = (0.2 * B - 0.2 * R + 3) * R 3B + 3R - 0.2BR - 0.2R² = 0.2 * B * R - 0.2 * R * R + 3 * R 3B + 3R - 0.2BR - 0.2R² + 0.2R² = 0.2BR - 0.2R² + 0.2R² + 3R 3B + 3R - 0.2BR + 0 = 0.2BR + 0 + 3R 3B + 3R - 0.2BR = 0.2BR + 3R 3B + 3R - 3R - 0.2BR = 0.2BR + 3R - 3R 3B + 0 - 0.2BR = 0.2BR + 0 3B - 0.2BR = 0.2BR 0.2BR = 3B - 0.2BR (0.2BR) / B = (3B - 0.2BR) / B 0.2R = 3 - 0.2R 0.2R + 0.2R = 3 - 0.2R + 0.2R 0.2R + 0.2R = 3 0.4R = 3 (0.4R) / 0.4 = 3 / 0.4 R = 3 / 0.4 R = 7.5 km per hour The final answer is: the speed of the river is 7.5 km per hour What about the speed of the boat? Solve for t (t + 0.2) * R = 3 Rt + 0.2R = 3 Rt + 0.2R - 0.2R = 3 - 0.2R Rt + 0 = 3 - 0.2R Rt = 3 - 0.2R (Rt) / R = (3 - 0.2R) / R t = (3 - 0.2R) / R R = 7.5 t = (3 - 0.2 * 7.5) / 7.5 t = 0.2 hours t * (B + R) = 0.2 * (B - R) + 3 0.2 * (B + 7.5) = 0.2 * (B - 7.5) + 3 0.2 * B + 0.2 * 7.5 = 0.2 * B - 0.2 * 7.5 + 3 0.2B + 1.5 = 0.2 * B - 1.5+ 3 0.2B + 1.5 = 0.2B + 1.5 The speed of the boat can be any real number: B ∈ ℝ In summary: R = 7.5 km per hour B ∈ ℝ Thanks for writing. Staff www.solving-math-problems.com

 Oct 19, 2011 Linear Equation in Two Variables by: Staff Part I Question: by VIJAY (Chandigarh) My Question is written below: "MARIA WAS TRAVELLING IN HER BOAT WHEN THE WIND BLEW HER HAT OFF AND HAT STARTED FLOATING AT DOWNSTREAM. THE BOAT CONTINUED TO TRAVEL UPSTREAM FOR 12 MORE MINUTES BEFORE MARIA REALIZED THAT HER HAT HAD FALLEN OFF AND TURNED BACK DOWNSTREAM. SHE CAUGHT UP WITH THAT AS SOON AS IT REACHED THE STARTING POINT. FIND THE SPEED OF RIVER IF MARIA'S HAT FLEW OFF EXACTLY 3 KM FROM WHERE SHE STARTED" Answer: B = speed of boat = unknown R = speed of river = unknown Time boat continued upstream after wind blew Maria’s hat into river = 12 minutes= 12/60 hours = 0.2 hours Distance from starting point when wind blew Maria’s hat into river = 3 km t = Time it took for the boat to reach the starting point after it turned around to recover Maria’s hat = unknown Boat’s distance from the starting point when it turned around to recover Maria’s hat B_d = 0.2 * (B - R) + 3 Distance the boat traveled to pick up Maria’s hat after it turns around: B_d = t * (B + R) Distance the hat floated downstream after it was blown into the water D_h = (t + 12) * R = 3 The final equations are: B_d = 0.2 * (B - R) + 3 B_d = t * (B + R) (t + 0.2) * R = 3 Since no further information is given, the first two equations should be combined. Since the distance is the same in both cases: t * (B + R) = 0.2 * (B - R) + 3 The final two equations are: t * (B + R) = 0.2 * (B - R) + 3 (t + 0.2) * R = 3 ---------------------------------------------------------------------------

 Oct 19, 2011 querry on equation no 3 by: Anonymous The equation no. 3 i.e (t+0.2)*R=3 seems to be confusing as time taken here (t+0.2) might be incorrect ,Because in question it is said that the time taken by HAT and by boat to reach the starting point are same. That means this equation should have been t*R=3 instead of (t+0.2)*R=3.

 Oct 19, 2011 Hat on the River by: Staff t is only PART of the TOTAL TIME the hat drifted with the current.t = Time it took for the boat to reach the starting point AFTER it turned around to recover Maria's hatThe total time the hat floated down the river is: t + 0.2 hours (t+0.2)*R=3 is correct.Thanks for writing.Staff www.solving-math-problems.com

 Apr 20, 2017 different time taken in eqn 2 and eqn 3 NEW by: Anonymous If you are taking t+0.2 in eqn 3 then why you are not taking t+2 in eqn 2.. Eqn 2 was the distance when the boat turn around and you said "t" is the only part of total time when boat turn around.

 Apr 20, 2017 different time taken in eqn 2 and eqn 3 NEW by: Anonymous If you are taking t+0.2 in eqn 3 then why you are not taking t+2 in eqn 2.. Eqn 2 was the distance when the boat turn around and you said "t" is the only part of total time when boat turn around.

 Aug 28, 2017 Simple way of solving this NEW by: Anonymous The answer seems to be much simpler than it is made. If the river was not flowing, Maria will take the same amount of time to get to the hat i.e. 12 min. The fact that it is flowing does not change this time. It only changes the distance Maria goes in that 12 min (up stream and down stream). So the total time is 24min. In that time, the hat has travelled 3Km back to the original point. Therefore, the speed of the river is 3Km/24min = 7.5km/h. You can do an analogy of a person on a cruise ship walking toward the end and dropping the hat somewhere and after realizing it 12 min later, returning to pick it up. In that time, the boat has traveled 3Km relative to the ground.

 Aug 28, 2019 query on simple way to solve it NEW by: Anonymous if the river was not flowing, how the hat will flow upto 3kms and the speed of the river will be 0.

 Sep 09, 2019 short trick NEW by: Indrakant let speed of the boat =b km/hr and speed of the river=w km/hr Hence , speed of the boat in upstream=b-w km/hr speed of the cap=w km/hr relative speed of the boat and cap=b-w+w=b km/hr distance traveled by cap in 12 min=b/5 km Relative speed of the boat when boat started travelling in downstream =b+w-w=b km/hr so the time to catch cap=1/5 hr+b/5*b hr =2/5 hr In 2/5 hr cap traveled 3 km distance hence speed of river=3*5/2=15/2 km/hr

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