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  • Directions for questions The famous Indian nuclear physicist, Padma Vibhusan Dr Raja Ramanna, while researching during India’s first nuclear bomb test -- Project l

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The famous Indian nuclear physicist, Padma Vibhusan Dr Raja Ramanna, while researching during India’s first nuclear bomb test -- Project ‘Smiling Buddha’, hit upon the secret of preparing a new radioactive material (Indianium X) using two secret compounds – compound Abra and compound Cadabra.

For preparing two kilograms of Indianium X, 880 gm of Abra and 540 gm of Cadabra were required. The preparation of the material happened only in batches of two kilograms and preparation of any batch of Indianium X could begin only if the quantity of secret compounds available were sufficient to prepare two kilograms of the material. Only when there was a shortfall in any secret compound because of which it was not possible to prepare the batch of Indianium X, that secret compound was purchased from the international market with the help of the Indian intelligence agency (RAW).

Dr Ramanna always noted down the details regarding the number of kilograms of Indianium X he prepared on each month and the type and quantity of secret compound he had to purchase, if any. Given below is an excerpt from his notes for the year 1970 :

Month of 1970 Jan Feb Mar Apr May Jun July Aug Sep Oct
Indianium X prepared (kg) 14 16 10 10 14 14 12 12 14 12

Abra purchased (kg)

 nil nil 10 nil 12 nil 16 nil nil 6
Cadabra purchased (kg) nil nil nil 8 nil 6 nil 8 nil nil

Question:- Which of the following could be the amount of Abra Dr Ramanna already had as stock before he started to prepare the first two kilogram batch of Indianium X in Jan 1970 ? 

 

Option: 1

17.82 kg


Option: 2

18.24 kg


Option: 3

18.36 kg

 


Option: 4

18.52 kg


Answers (1)

best_answer

Given below is the table which complies with the cumulative quantity of Indianium X prepared by Dr Ramanna as per his excerpt :

Month of 1970 Jan Feb Mar Apr May Jun July Aug Sep Oct
Indianium X prepared (kg) 14 16 10 10 14 14 12 12 14 12
Cumulative quantity of Indianium X prepared (kg)

`14

30

40

50

64

78

90

102

116

128

Abra purchased (kg)

 nil nil 10 nil 12 nil 16 nil nil 6
Cadabra purchased (kg) nil nil nil 8 nil 6 nil 8 nil nil

 Now, for preparing two kilograms of Indianium X, 880 gm of Abra and 540 gm of Cadabra were required.
Also Indianium X could be prepared only in batches of two kilograms.

a) Abra

Month of 1970 Jan Feb Mar Apr May Jun July Aug Sep Oct
Indianium X prepared (kg) 14 16 10 10 14 14 12 12 14 12
Cumulative quantity of Indianium X prepared (kg)

`14

30

40

50

64

78

90

102

116

128

Abra purchased (kg)

 nil nil 10 nil 12 nil 16 nil nil 6

The first consignment of Abra was purchased in Mar 1970.
To prepare 40 kg of Indianium X, there must have been at least (40/2)*0.88 = 17.60 kg of Abra and less than 17.60+0.88 = 18.48 kg of Abra. (If there was any more Abra, more batches of Indianium X could have been prepared before purchasing Abra). Hence, the range of Abra initially available would be >/= 17.60 to < 18.48 kg. After the production of 40 kg of soap, stock of Abra left would be >/= 0 to < 0.88 kg.
Then 10 kg of Abra was purchased at the end of Mar 1970. So stock of Abra after purchase >/= 10.00 to < 10.88 kg. 10+14 = 24 kg of Indianium X was prepared after the purchase and before the second purchase of Abra in May 1970.

To prepare 24 kg of Indianium X, there must have been at least (24/2)*0.88 = 10.56 kg of Abra and less than 10.56+0.88 = 11.44 kg of Abra. Hence, the actual range of Abra available before this production would be >/= 10.56 to < 10.88 kg. After the production of 24 kg of Indianium X, stock of Abra left would be >/= 0 to < 0.32 kg.

Then 12 kg of Abra was purchased at the end of May 1970. So stock of Abra after purchase >/= 12.00 to < 12.32 kg. 14+12 = 26 kg of Indianium X was prepared after the purchase and before the third purchase of Abra in July 1970.

To prepare 26 kg of Indianium X, there must have been at least (26/2)*0.88 = 11.44 kg of Abra and less than 11.44+0.88 = 12.32 kg of Abra. Hence, the actual range of Abra available before this production would be >/= 12.00 to < 12.32 kg. After the production of 26 kg of Indianium X, stock of Abra left would be >/= 0.56 to < 0.88 kg.

Then 16 kg of Abra was purchased at the end of July 1970. So stock of Abra after purchase >/= 16.56 to < 16.88 kg. 12+14+12 = 38 kg of Indianium X was prepared after the purchase and before the fourth purchase of Abra in Oct 1970.

To prepare 38 kg of Indianium X, there must have been at least (38/2)*0.88 = 16.72 kg of Abra and less than 16.72+0.88 = 17.60 kg of Abra. Hence, the actual range of Abra available before this production would be >/= 16.72 to < 16.88 kg. After the production of 38 kg of Indianium X, stock of Abra left would be >/= 0 to < 0.16 kg.

Then 6 kg of Abra was purchased at the end of Oct 1970. So stock of Abra after purchase >/= 6.00 to < 6.16 kg.

Considering the maximum stock and vis-à-vis requirement at the various stages of production (as discussed above), the maximum possible stock of Abra at the beginning of Jan 1970 is 

< 18.48 + (10.88 – 10.88) + (12.32 – 12.32) + (16.88 – 16.88),

that is < 18.48 kg + 0 + 0 + 0, 

that is < 18.48 kg

Also considering the minimum stock and vis-à-vis requirement at the various stages of production (as discussed above), the minimum possible stock of Abra at the beginning of Jan 1970 is 

>/= 17.60 + (10. 56 – 10.00) + (12.00 – 12.00) + (16.72 – 16.56),

that is >/= 17.60 + 0.56 – 0 + 0.16,

that is >/= 18.32 kg

b) Cadabra:

Month of 1970 Jan Feb Mar Apr May Jun July Aug Sep Oct
Indianium X prepared (kg) 14 16 10 10 14 14 12 12 14 12
Cumulative quantity of Indianium X prepared (kg)

`14

30

40

50

64

78

90

102

116

128
Cadabra purchased (kg) nil nil nil 8 nil 6 nil 8 nil nil

 The first consignment of Cadabra was purchased in Apr 1970.

To prepare 50 kg of Indianium X, there must have been at least (50/2)*0.54 = 13.50 kg of Cadabra and less than 13.50+0.54 = 14.04 kg of Cadabra. (If there was any more Cadabra, more batches of Indianium X could have been prepared before purchasing Cadabra). Hence, the range of Cadabra initially available would be >/= 13.50 to < 14.04 kg. After the production of 50 kg of Indianium X, stock of Cadabra left would be >/= 0 to < 0.54 kg.

Then 8 kg of Cadabra was purchased at the end of Apr 1970. So stock of Cadabra after purchase >/= 8.00 to < 8.54 kg. 14+14 = 28 kg of Indianium X was prepared after the purchase and before the second purchase of Cadabra in June 1970.

To prepare 28 kg of Indianium X, there must have been at least (28/2)*0.54 = 7.56 kg of Cadabra and less than 7.56+0.54 = 8.10 kg of Cadabra. Hence, the range of Cadabra initially available would be >/= 8.00 to < 8.10 kg. After the production of 28 kg of Indianium X, stock of Cadabra left would be >/= 0.44 to < 0.54 kg.

Then 6 kg of Cadabra was purchased at the end of June 1970. So stock of Cadabra after purchase >/= 6.44 to < 6.54 kg. 12+12 = 24 kg of Indianium X was prepared after the purchase and before the third purchase of Cadabra in Aug 1970.
To prepare 24 kg of Indianium X, there must have been at least (24/2)*0.54 = 6.48 kg of Cadabra and less than 6.48+0.54 = 7.02 kg of Cadabra. Hence, the range of Cadabra initially available would be >/= 6.48 to < 6.54 kg. After the production of 24 kg of Indianium X, stock of Cadabra left would be >/= 0 to < 0.06 kg.

Then 8 kg of Cadabra was purchased at the end of Aug 1970. So stock of Cadabra after purchase >/= 8.00 to < 8.06 kg. 14+12 = 26 kg of Indianium X was prepared after this with the existing stock.

To prepare 26 kg of Indianium X, there must have been at least (26/2)*0.54 = 7.02 kg of Cadabra and less than 7.02+0.54 = 7.56 kg of Cadabra. Hence, the range of Cadabra initially available would be >/= 8.00 to < 8.06 kg. After the production of 26 kg of Indianium X, stock of Cadabra left would be >/= 0.98 to < 1.04 kg.

Considering the maximum stock and vis-à-vis requirement at the various stages of production (as discussed above),  the maximum possible stock of Cadabra at the beginning of Jan 1970 is 

< 14.04 + (8.10 – 8.54) + (6.54 – 6.54) + (8.06 – 8.06),

that is < 14.04 – 0.44 + 0 + 0,

that is < 13.60 kg

Also considering the minimum stock and vis-à-vis requirement at the various stages of production (as discussed above), the minimum possible stock of Cadabra at the beginning of Jan 1970 is 

>/= 13.50 + (8.00 – 8.00) + (6.48 – 6.44) + (8.00 – 8.00),

that is >/= 13.50 + 0 + 0.04 + 0,

that is >/= 13.54 kg

We found out that at the start of Jan 1970, the range of the possible stock of Abra was >/= 18.32 kg and < 18.48 kg.

Out of the options only 18.36 kg lies in the range.

Hence 18.36 kg could be the amount of Abra Dr Raja Ramanna already had as stock before he started to prepare the first 2 kg batch of Indianium X in Jan 1970.

Posted by

Shailly goel

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