A man received a cheque. The amount in Rs. has been transposed for paise and vice versa. After spending Rs. 5 and 42 paise, he discovered he now had exactly 6 times the value of the correct cheque amount. What amount he should have received?
[CAT 2002]
(1) 5.30
(2) 6.44
(3) 60.44
(4) 16.44
Solution
Let x and y be the amount rupees and paise that he should ideally have received respectively.
Amount he should have received in paise =100x+y
Amount in paise, after transposing =100y+x
After spending 542 paise, he had 6 times the value of the correct cheque amount.
⇒ 100y+x−542=6(100x+y)
⇒ 94y=599x+542
⇒ y=94599x+542
We divide 599 and 542 by 94, write the quotients separately and then the remainders.
⇒ y=6x+5+9435x+72
Substituting values for x from 1 onwards, we note that x=6 results in an integer for y at y=44. Therefore, the amount is Rs.6.44.
Alternatively (Recommended Method)
As this question has options, it is best to run through the options. Options 3 and 4 as original numbers are very high. This would have resulted in the transpose becoming a 3-digit number.
Option(1): 5.30×6+5.42=31.80+5.42=37.22 Option(2): 6.44×6+5.42=38.64+5.42=44.06
In option 2 the rupees and paise of Rs. 06.44 are transposed in the cheque as 44.06.
Answer: (2) 6.44
4.10 Multiple Variables
Example 20
Mr. and Mrs. Gupta have three children – Pratik, Writtik and Kajol, all of whom were born in different cities. Pratik is 2 years elder to Writtik. Mr. Gupta was 30 years of age when Kajol was born in Hyderabad, while Mrs. Gupta was 28 years of age when Writtik was born in Bangalore. If Kajol was 5 years of age when Pratik was born in Mumbai, then what were the ages of Mr. and Mrs. Gupta respectively at the time of Pratik’s birth?
[IIFT 2011]
(1) 35 years, 26 years
(2) 30 years, 21 years
(3) 37 years, 28 years
(4) None of the above
Solution
Let a,b,p,w and k be the ages (in years) of Mr. Gupta, Mrs. Gupta, Pratik, Writtik and Kajol respectively.
The following equations are formed from the data in the question. p=w+2 -----(1) a=k+30 -----(2) b=w+28 -----(3) k=p+5 -----(4)
We need to find Mr. and Mrs. Gupta's age at the time of Pratik's birth, which is nothing but the difference between their respective ages and that of Pratik. Therefore, we need to find a−p and b−p.
Equations (2) and (4) have k as the common variable and also contains a and p. ∴ Substituting Eq(4) in Eq(24) ⇒ a=p+5+30 ⇒ a−p=35
Likewise, substituting w=p−2 from Eq(1) in Eq(3) ⇒ b=p−2+28 ⇒ b−p=26
Mr. Gupta and Mrs. Gupta would have been 35years and26years respectively when Pratik was born.
Answer: (1) 35 years, 26 years
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