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Vol. 1, No. 1, June 2012 | At Right Angles 17
Shape, Size, Number and Cost
Analyzing Business through Math
How much math can there be in two bowls of gulab jamun? Prithwijit De models,
estimates, calculates and presents a convincing argument on which of the two
products earns the sweetseller a greater profit.
Gulab jamuns are a popular sweet in India, often sold singly
or two to a bowl. However, I was recently intrigued when the
sweet shop across the road from my institute’s canteen intro- duced a bowl of three gulab jamuns at a competitive price. The
sweet lovers in my office immediately changed loyalties but the
price conscious stayed with the canteen.
What motivated this competitive strategy from the shop across
the road? Being a mathematician, I naturally had to solve the
problem and in typical fashion, I called my canteen walla
‘Mr. X’ and the sweet shop owner ‘Mr. Y’. Here is my mathema- tised version of the situation.
Mr. X sells two pieces of gulab jamun at `p1 per piece, in a cylin- drical cup of cross-sectional radius R. The pieces are spherical
in shape and they touch each other externally and the cup in- ternally. Mr. Y, a competitor of Mr. X, sells three pieces of gulab
jamun in a cup of the same shape and size used by Mr. X, and he
charges a price of `p2 per piece. Snapshots of their offerings
are displayed in Figure 1, and top-down views are shown in
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18 At Right Angles | Vol. 1, No. 1, June 2012
Figure 2. Certainly p2 has to be less than p1 since
price is directly proportional to size if the mate- rial is kept the same. A person buying from Mr. X
would be paying 2p1 and a person buying from Mr.
Y would be paying 3p2.
Listen to some of the conversations I overheard
while working on a particularly difficult problem
in differential geometry: “Mr. X is charging `6
for a gulab jamun! So one bowl with two gulab
jamuns from his shop costs only `12!” “A bowl at
Mr. Y’s shop costs `15 but I get 3 pieces. So it’s
only `5 per gulab jamun and the bowls are of the
same size.” “Yes ...but the 3 gulab jamuns in the
same bowl are smaller! I get 2 bigger gulab ja- muns at a cheaper price! ” As you can see we have
serious and weighty discussions in my office.
Thinking deeply about this, I finally reduced the
problem to two main questions.
1. Which of the two cups contains a greater
amount of sweet?
2. Which sweetseller makes a greater profit
if the cost of producing unit volume of the
sweet is the same?
The sweets are sold in a cylindrical cup of
cross-sectional radius R. The pieces are spherical
in shape and they touch each other externally
and the cup internally.
The radius r1 of each piece in Figure 2 (I) is 2
The total volume V1 of sweet in the cup is there- fore:
2 . R R
3 3 r r c ` j m =
To find the volume of a piece in Figure 2 (II) we must
find the radius of each sphere.
In the two dimensional representation of the
configuration (Figure 2 (II)) the spheres become
circles and the cylindrical cross-section turns into
a circle circumscribing the three circles. The task
of finding the radius of the sphere thus reduces to
finding the common radii of the inner circles. The
triangle formed by joining the centres of the inner
circles (Figure 3) is an equilateral triangle whose
centroid is the centre of the large circle. If r2 is the
radius of an inner circle then the length of the side
of the triangle is 2r2 and
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Vol. 1, No. 1, June 2012 | At Right Angles 19
R r r r . r
2 3 2 2 2
2 = + = + = + ` jc m^ h e o
Thus the volume of a piece is
R 3 3
r e o
and the total volume V2 of sweet offered is
Now one asks, who is giving more sweet and by
how much? Note that
(1) 36 3
(2 3 ) 0.83, 1.2. V
2 = + . .
So Mr. Y is giving about 20% more sweet than Mr.
X. From a buyer’s perspective this is a good deal.
S/he may be paying more per cup but the per
piece price is still less as p2<p1.
If Mr. X charges `6, then a bowl costs `12. If Mr. Y
charges `5, then a bowl costs `15 but it will have
20.48% more sweet than a bowl from Mr. X and
the price per piece is still less at Mr. Y’s.
But from a seller’s perspective is it really worth it?
Assuming that the cost of producing unit volume
of the sweet is the same, c (say), in both cases, is it
possible for Mr. Y to price a piece in such a way so
as to ensure greater profit than Mr. X?
The selling price per unit volume is the cost of one
bowl divided by the volume of sweet given, so the
profit at each shop is
cost of one bowl
volume of sweet given
– cost per unit volume.
To find out which shop makes greater profit, we
therefore study the following inequality:
(2) – – c c > . 3 2
This is equivalent to
2 2 c m
By virtue of (1) this is equivalent to p2 >0.8p1.
Thus, if Mr. Y chooses p2 such that 0.8p1<p2<p1,
then he makes greater profit than Mr. X despite
reducing the size and price of the sweet.
For instance, if Mr. X charges `6 per piece then
Mr. Y can set the price anywhere between `4.80
and `6 per piece in order to beat his rival in the
money-making game (see Table 1). The buyer, in
all probability, will be happy to pay less per piece
and get three instead of two, as the more the mer- rier is likely to be his/her motto.
I am grateful to Sneha Titus and Shailesh Shirali for
significant improvement in the quality of presentation
of the content of this article.
Prithwijit De is a member of the Mathematical Olympiad Cell at Homi Bhabha Centre for Science
Education (HBCSE), TIFR. He loves to read and write popular articles in mathematics as much as he
enjoys mathematical problem solving. His other interests include puzzles, cricket, reading and music.
He may be contacted at firstname.lastname@example.org.
Remember that the volume of the bowl is the
same in both cases; so the buyer will probably
go to Mr. Y in order to get 3 gulab jamuns at a
slightly higher price. Mr. Y gets more customers
and a greater pro�it on each bowl!
p Cost/bowl Cost/bowl 2 p1