River Boat Speed

Although Johnny is doing the CIE IGCSE Additional Mathematics programme, he is confused at times by questions in regular Mathematics which his teacher deems “not so hard,” such as the following:

“Captain Jack has a motorboat that travels at 20 km per hour in still water. Given that the current in the river is 5 km per hour, the boat’s downstream speed (ignoring friction) will be 20 + 5 = 25 km/h while its speed in the upstream direction will be 20 – 5 = 15 km/h. Suppose that Jack has just enough fuel for 3 hours. How far can he go downstream before he has to turn around to get back home?”

A first attempt

The first physics formula Johnny learnt was ‘Distance = Speed × Time’ or D = S × T. For brevity he decides to use subscripts to identity total, average, downstream or upstream variables, _{T, A, D,} and _U, respectively. Thus his formula for the boat’s downstream distance is D_D = S_D × T_D = 25 T_D. Similarly his upstream distance formula is D_U = S_U × T_U = 15 T_U. Johnny also knows average speed is total distance divided by the total time, giving him the equation S_A = D_T ÷ T_T. He reasons that the average speed, S_A, is (15 + 25)/2 = 20 and consequently the total distance must be 20 × 3 = 60 km. 

D_D + D_U = D_T

S_D × T_D + S_U × T_U = D_T

25 T_D + 15 T_U = 60

25 T_D + 15 (3 – T_D) = 60

25 T_D + 45 – 15 T_D = 60

10 T_D = 15

T_D = 1.5 hours 

 

Consequently Johnny tells his group the downstream distance will be 25 km/h × 1.5 hours = 37.5 km.

 

 Disagreement

 Mary is in the same discussion group. She says to Johnny: That can’t be right. I used ‘Trial and Improvement.’ The two distance variables must be equal, D_D = D_U. Given that the total time is 3 hours, I wrote T_U = 3 – T_D. Obviously the downstream time cannot be less than 0 hours and your own calculations demonstrate that T_D must be less than or equal to 1.5 hours. Using the vocabulary of functions, the domain of the downstream distance is 0 < T_D < 1.5 hours.

Mary’s trials are shown below.

 

Trial	TD	TU = 3 – TD	DD = 25TD	DU = 15TU	Error
1st	1.5 hours	1.5 hours	35.5 km	22.5 km	15 km
2nd	0 hours	3 hours	0 km	45 km	-45 km
3rd	1 hour	2 hours	25 km	30 km	-5 km
4th	1.25 hour	1.75 hour	31.25 km	26.25 km	+5 km
5th	1.125 hour	1.875 hour	28.125 km	28.125 km	0 km

Johnny replies:Your table certainly looks convincing. But the algebra I used is perfectly logical. How can there be a mistake?

At this point, as the teacher, I confirm that Mary is correct and ask the group to find the mistake in Johnny’s algebra.

 



Algebra revisited

 

Johnny’s experience tells him it is extremely difficult to proof-read for errors so he decides to take a slightly different approach. After discussion with Mary, he now realizes that he does not actually need the total distance. He only requires that:

D_D = D_U

25 T_D = 15 T_U

25 T_D = 15 (3 – T_D)

25 T_D = 45 – 15 T_D

40 T_D = 45

T_D = 45 ÷ 40 = 1.125 hours

D_D = 25 km/h × 1.125 h = 28.125 km

While he is able to confirm Mary’s answer obtained by ‘Trial and Improvement,’ nevertheless, he is somewhat dissatisfied. Algebra required rather more mental gymnastics and, at least initially, appeared to steer him in the wrong direction. He asks me “Is algebra worth the effort?”

My honest answer is, “We have no need of algebra if we are only seeking the answer to one particular question. Algebra’s real advantage is that it is very easily generalized. Keeping the total time at 3 hours but supposing the current is some other constant than ‘5’, you can readily modify the previous solution as follows.

D_D = D_U

S_D × T_D = S_U × T_U

S_D × T_D = S_U (3 – T_D)

S_D × T_D = 3 S_U – S_U × T_D

(S_D + S_U) T_D = 3 S_U

T_D = 3 S_U ÷ (S_D + S_U)

Would it be significantly harder to suppose we had fuel for H hours instead of 3 hours?

 

ICT

An approach very similar to Trial and Improvement uses the computer programme Geogebra to model the situation:

 

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Where did Johnny go wrong?

Like many students, Johnny is over-reliant on the arithmetic mean. For the purpose of determining the average speed, the Harmonic Mean is called for. How do you obtain the Harmonic Mean of two numbers a and b? 

 

Procedure	Algebra	Captain Jack’s speeds
Start with a and b	a, b	15, 25
Take the reciprocal of each number	1/a and 1/b	1/15, 1/25
Rewrite with a common denominator in lowest terms	b/ab, a/ab	1/15 = 5/75, 1/25 = 3/75
Find the arithmetic mean	(a + b)/2/ab	(5+3)/2/75 = 4/75
Take the reciprocal of the previous line	2ab/(a+b)	75/4

The total distance = (harmonic average of speed) × Total time = 75/4 × 3 = 225/4 = 56.25 km.

The downstream distance must be half the upstream distance or 28.125 km, as previously calculated.

(Please note the harmonic mean is not an explicit part of the IGCSE Maths Syllabus and consequently it is unlikely to be found on the examiner’s official mark scheme.)

Conclusion

As Grandma used to say, “There’s more than one way “to skin a cat.” (Cat befrienders are advised to replace the offending text with “poach an egg.”) Today’s message is, “There’s more than one way to solve a problem.” In the ‘real world,’ team skills and independent verification of results are important. Used correctly, algebra is indeed logical, but you should appreciate that an apparently small misunderstanding will lead to a false conclusion.  There is more to a mathematics education than the syllabus.