Logical Reasoning capsules by Vikas Saini - Set 10



  • Set 1

    The table below gives some information about the points scored by Arjun in AHC (All Hastin Championship), in each of five different mind games- Dice, Dance. Dupe, Digit and Dynasty.

    GameTotal problemsProblems attemptedSuccessful AttemptsFailuresNet Score
    Dice3529
    Dance30
    Dupe35176
    Digit4011
    Dynasty4530

    It is also known that,

    (i) For every successful attempt Arjun gets one point, for un-attempted he loses 1/6th point and for every failure attempt he loses 1/3rd point.
    (ii) He attempted 125 problems and scored 67 points.
    (iii) The number of failures of Arjun in digit is 1/6th of his total number of failures and twice his number of failures in dance.
    (iv) Arjun’s net score in dance is double of Dupe.

    Q1. What is the total number of failures of Arjun in the competition?
    Q2. What is the net score of Arjun in Dynasty ?
    Q3. What is the successful attempts in Dice ?

    Solution :-

    Total attempted = 125
    Not attempted = 185 – 125 = 60
    Points loss due to not attempted = 10
    Scores before = 67+10 = 77
    Suppose out of 125 problems, A were successful
    A x 1 + (125 – A)(-1/3) = 77
    A + A/3 = 77 + 125/3
    A = 89
    Failure attempts = 125 – 89 = 36
    Dupe, attempted 17 problems out of 35, where 6 were failure
    Net Score in Dupe = 11 x 1 + 6 x (-1/3) + 18 x (-1/6) = 6.
    Number of failure in digit should be 6 and in dance 12.
    Digit, total problems 40 and net score is 11 and failure is 6.
    Now due to failure he loses score = 6 x (-1/3) = -2.
    Now score before = 11+2 = 13.
    Suppose out of 34, he successfully attempted B.
    B x 1 + (34 – B )(-1/6) =13.
    7B/6 = 13 + 34/6
    B = 16.
    Not attempted = 18.
    Dance,net score is 12.
    Failure 3.
    Due to failure scores lost = 3 x (-1/3) = -1.
    Scores before = 12 + 1 = 13.
    Hence suppose his successful attempts were C.
    C + (27-C)(-1/6) = 13.
    C= 15.
    Not attempted 12.
    Now problems attempted in dynasty was 39.where 30 successful and 9 failure.
    Net score = 26.
    Dice, net score = 12.

    GameProblemsAttemptedSuccessful attemptFailureNet score
    Dice3529171212
    Dance301815312
    Dupe35171166
    Digit402216611
    Dynasty453930926
    185125893667

    Set 2

    ln a wrestling charnpionship, five persons A, B, C, D and H -- secured the first five ranks, in the same order. The weight of each of the five persons is a distinct natural number.
    T he following information is known regarding their weights:

    1. C is the heaviest person and B is the second heaviest.
    2. A is heavier than D.
    3. The weight of the lightest person among the five is 54 kg and it is not H.
    4. The weight of the heaviest person is 87 kg.
    5. The weight of one of the persons is 19 kg more than that of H.
    6. The person whose weight is 82 kg secured a better rank than the person whose weight is 67 kg.

    Q1. What is the weight of A ?
    Q2. What is the weight of B ?
    Q3. What is total weight of all put together ?

    Solution :-

    From 3, weight of lightest person is 54 and it’s not H. One person’s weight should be more than H by 19 kg. 4 weights are given 54,67,82,87. H’s weight possible here is 67 only.
    One person’s weight shall be 86. A is heavier than D. It means A’s weight is 82 and D’s weight is 54.
    C’s weight is 87, B’s weight is 86.
    All answers can be given by above solution.
    Answer 1 :- 82
    Answer 2 :- 86.
    Answer 3 :- 87+86+82+67+54 = 376.

    Set 3

    The Indian Group of Industries owns four companies – A Ltd., J Ltd., P Ltd. and X Ltd. The Chartered Accountant of the group has reported Sales and Expenses figures to the Income Tax Department and submitted challans for the tax paid according to these figures. All the companies pay a 40% tax on profits, which are calculated as the difference between the Sales and Expenses. An Accountant with the Income Tax Department found the reported figures to be a bit odd and warded the following observations to an Auditor with the Department.

    1. The Profits of P Ltd. and X Ltd. were each equal to a fifth of the Sales of X Ltd.
    2. The Sales of J Ltd. and P Ltd. were each equal to four times the Profits of J Ltd..
    3. A Ltd. reported Profits as 15% of Sales, while P Ltd. reported Profits as a sixth of its Expenses.
    4. A Ltd. reported Expenses as 15% less than the Expenses of P Ltd and P Limited reported expenses as Rs. 432 crores.

    The Auditor scrutinized the books of all four companies and determined the following.

    1. The Sales of A Ltd. and J Ltd. were reported 10% less than actual while those of X Ltd. were reported 20% less than actual.
    2. The Expenses of A Ltd., J Ltd. and P Ltd. were reported 20% more than actual while those of X Ltd. were reported 25% more than actual. The Profits of P Ltd. were reported 20% less than actual.

    Based on this information, the Income Tax Department issued notices to all the companies stating that they would have to make good the difference in the actual tax and the amount of tax paid as well as pay a penalty amounting to ten times this difference.

    Q1. How much less Profits were reported by A Ltd. and J Ltd. together?
    Q2. What is the total penalty paid by J Ltd. and X Ltd. together?
    Q3. Which of the following represents the actual total Profits of the four companies as a percentage of their actual Sales?
    Q4. If Profitability is calculated as Profits as a percentage of Expenses, what is the difference between the actual Profitability of A Ltd. and X Ltd?

    Solution :-

    40% tax on profit.
    Profit = sales – Expense.
    Expense of P = 432 crore.
    The expenses of P Ltd. are Rs. 432 crores.
    So profits are 432/6 = Rs. 72 crore and sales are 432 + 72 = Rs. 504 crores.
    Since the profit of X Ltd. is also Rs. 72 crores, its sales are 72 × 5 = Rs. 360 crores and expenses are 360 – 72 = Rs. 288 crores.
    Since the sales of J Ltd. are also Rs. 504 crores, its profits are Rs. 126 crores and expenses are Rs. 378 crores. Expenses of A = 0.85 × 432 = Rs 367.2 crore. This must be 85% of the sales as profit is 15% of the sales. Hence sales = 367.2/0.85 = Rs 432 crores. Profits of A Ltd.,are 0.15 × 432 = Rs. 64.8 crores .
    According to the auditor, the sales of A Ltd. are only 90% of actual.
    So actual sales of A Ltd. are 432/0.9 = Rs. 480 crores.
    Actual expenses of A Ltd. are 367.2/1.2 = Rs. 306 crores and actual profits are 480 – 306 = Rs. 174 crores.
    Similarly, the actual sales of J Ltd. are 504/0.9 = Rs. 560 crores.
    The actual expenses of J Ltd. are 378/1.2 = Rs. 315 crores and its profits are 560 – 315 = Rs. 245 crores.
    The actual expenses of PQR Ltd. are 432/1.2 = Rs. 360 crores.
    The actual profits of PQR Ltd. are 72/0.8 = Rs. 90 crores and its actual sales are 360 + 90 = Rs. 450 crores.
    The actual sales of X Ltd. are 360/0.8 = Rs. 450 crores.
    The actual expenses of X Ltd. are 288/1.25 = Rs. 230.4 crores and its actual profits are 450 – 230.4 = Rs. 219.6 crores.

    Reported Values

    AJPX
    Sales432504504360
    Expenses367.2378432288
    Profit64.81267272
    Tax25.9250.428.828.8

    Actual values

    AJPX
    Sales480560450450
    Expenses306315360230.4
    Profit17424590220.4
    Tax69.6983687.84

    Answer 1) The total profits reported by A Ltd. and J Ltd. together are 64.8 + 126 = Rs. 190.8 crores while their actual profits are 174 + 245 = Rs. 419 crores. The required difference is Rs. 228.2 crores.
    Answer 2) J Ltd. and X Ltd. have together paid taxes of 50.4 + 28.8 = Rs. 79.2 crores while their actual taxes are 98 + 87.84 = Rs. 185.84 crores. The difference is Rs. 106.64 crores. Thus the penalty is Rs. 1066.4 crores.
    Answer 3) From the table, the actual total profits of the 4 companies are 174 + 245 + 90 + 219.6 = Rs. 728.6 crores and their actual total sales are 480 + 560 + 450 + 450 = Rs. 1940 crores. The required percentage is 37.55%.
    Answer 4) The actual profitability of ABC Ltd. is 174/306 = 56.86% while that of X Ltd. is 219.6/230.4 = 95.31%. The required difference is 38.45.


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