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Vol. 1, No. 1, June 2012 | At Right Angles 17

Shape, Size, Number and Cost

A Sweet

Seller’s Trick

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.

Prithwijit De

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

feature

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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

1 R.

The total volume V1 of sweet in the cup is there- fore:

2 . R R

3

4

2 3

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

Fig. 1

Fig. 2

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Vol. 1, No. 1, June 2012 | At Right Angles 19

R r r r . r

r 3

2

2

3 2

3

2

3

2 3 2 2 2

2

2 = + = + = + ` jc m^ h e o

Thus the volume of a piece is

3

4

2 3

R 3 3

+

r e o

and the total volume V2 of sweet offered is

4

2 3

R 3

3

+

re o

Now one asks, who is giving more sweet and by

how much? Note that

(1) 36 3

(2 3 ) 0.83, 1.2. V

V

V

V

2

1

3

1

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

V

p

V

p

2

2

1

1

This is equivalent to

> 3

2

p

p

V

V

1 1

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.

Acknowledgement

I am grateful to Sneha Titus and Shailesh Shirali for

significant improvement in the quality of presentation

of the content of this article.

Fig. 3

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 de.prithwijit@gmail.com.

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!

14.40

15.00

16.50

p Cost/bowl Cost/bowl 2 p1

12 4.80

5.00

5.50

6