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CAT 2025 Lesson : Linear Equations - Transposed & Multiple Variables

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4.9 Interchanged/Transposed

Example 19

A man received a cheque. The amount in Rs. has been transposed for paise and vice versa. After spending Rs. 55 and 4242 paise, he discovered he now had exactly 66 times the value of the correct cheque amount. What amount he should have received? [CAT 2002] (11) 5.305.30            (22) 6.446.44            (33) 60.4460.44            (44) 16.4416.44           

Solution

Let xx and yy be the amount rupees and paise that he should ideally have received respectively. Amount he should have received in paise =100x+y= 100x + y

Amount in paise, after transposing =100y+x= 100y + x

After spending 542542 paise, he had 66 times the value of the correct cheque amount.
⇒ 100y+x−542=6(100x+y) 100y + x - 542 = 6(100x + y)
⇒ 94y=599x+542 94y = 599x + 542

⇒ y=599x+54294 y = \dfrac{599x + 542}{94}

We divide 599599 and 542542 by 9494, write the quotients separately and then the remainders.

⇒ y=6x+5+35x+7294 y = 6x + 5 + \dfrac{35x + 72}{94}

Substituting values for xx from 11 onwards, we note that x=6x = 6 results in an integer for yy at y=44y = 44. Therefore, the amount is Rs. 6.44\bm{6.44}.

Alternatively (Recommended Method)

As this question has options, it is best to run through the options. Options 33 and 44 as original numbers are very high. This would have resulted in the transpose becoming a 33-digit number.

Option (1)\bm{(1)}: 5.30×6+5.42=31.80+5.42=37.225.30 \times 6 + 5.42 = 31.80 + 5.42 = 37.22
Option (2)\bm{(2)}: 6.44×6+5.42=38.64+5.42=44.066.44 \times 6 + 5.42 = 38.64 + 5.42 = 44.06
In option 22 the rupees and paise of Rs. 06.4406.44 are transposed in the cheque as 44.0644.06.

Answer: (22) 6.446.44


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 22 years elder to Writtik. Mr. Gupta was 3030 years of age when Kajol was born in Hyderabad, while Mrs. Gupta was 2828 years of age when Writtik was born in Bangalore. If Kajol was 55 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] (11) 3535 years, 2626 years       
(22) 3030 years, 2121 years       
(33) 3737 years, 2828 years       
(44) None of the above           

Solution

Let a,b,p,wa, b, p, w and kk 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+2p = w + 2 -----(11)
a=k+30a = k + 30 -----(22)
b=w+28b = w + 28 -----(33)
k=p+5k = p + 5 -----(44)

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−pa - p and b−pb - p.

Equations (22) and (44) have kk as the common variable and also contains aa and pp.
∴\therefore Substituting Eq(44) in Eq(2424) ⇒ a=p+5+30 a = p + 5 + 30 ⇒ a−p=35a - p = \bm{35}

Likewise, substituting w=p−2w = p - 2 from Eq(11) in Eq(33) ⇒ b=p−2+28 b = p - 2 + 28 ⇒ b−p=26b - p = \bm{26}

Mr. Gupta and Mrs. Gupta would have been 35\bm{35} years and 26\bm{26} years respectively when Pratik was born.

Answer: (11) 3535 years, 2626 years


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