A has some coins. He gives half of the coins and 2 more to B. B gives half of the coins and 2 more to C. C gives half of the coins and 2 more to D. The number of coins D has now, is the smallest two-digit number. How many coins does A have in the beginning?
The correct answer is (C) 60.
Explanation:
Let us work backwards from D. The number of coins D has is the smallest two-digit number, which is 10. Let C have c coins. C gives half of the coins and 2 more to D, meaning D receives (c/2) + 2 = 10, which gives c/2 = 8, so c = 16. Working backwards for B with b coins: (b/2) + 2 = 16, so b/2 = 14, giving b = 28. Finally, working backwards for A with a coins: (a/2) + 2 = 28, so a/2 = 26, giving a = 52? Wait, let's re-verify: A gives half and 2 more. If A has 76, half is 38, plus 2 is 40 given to B. A is left with 36. B has 40. B gives half (20) plus 2 (22) to C, leaving B with 18. C receives 22. C gives half (11) plus 2 (13) to D, leaving C with 9. D receives 13? Let's check with 76: A starts with 76. Gives half (38) + 2 = 40 to B. B has 40. Gives half (20) + 2 = 22 to C. C has 22. Gives half (11) + 2 = 13 to D. The question says D's coins is smallest two-digit number (10). Wait, let's solve algebraically: ((a/2 - 2)/2 - 2)/2 - 2 = 10. Let's solve: ((a/2 - 2)/2 - 2)/2 = 12 ⇒ (a/2 - 2)/2 - 2 = 24 ⇒ (a/2 - 2)/2 = 26 ⇒ a/2 - 2 = 52 ⇒ a/2 = 54 ⇒ a = 108? Let's check the options given: 76, 68, 60, 52. Let's test 76: 76/2 = 38, 38+2=40 to B. Remaining with A = 36. B has 40. 40/2 = 20, 20+2=22 to C. Remaining with B = 18. C has 22. 22/2 = 11, 11+2=13 to D. D gets 13. Wait, option 76 gives 13. Let's check 68: 68/2=34, +2=36 to B. B has 36. 36/2=18, +2=20 to C. C has 20. 20/2=10, +2=12 to D. Let's check 60: 60/2=30, +2=32 to B. B has 32. 32/2=16, +2=18 to C. C has 18. 18/2=9, +2=11 to D. Let's check 52: 52/2=26, +2=28 to B. B has 28. 28/2=14, +2=16 to C. C has 16. 16/2=8, +2=10 to D. Exactly 10! Thus for 52, D gets 10. Wait, let's re-verify 76 or 52. Let's trace 52: A = 52. Gives half (26) + 2 = 28 to B. B has 28. B gives half (14) + 2 = 16 to C. C has 16. C gives half (8) + 2 = 10 to D. D has 10 coins, which is the smallest two-digit number. Hence, option (d) 52 is correct.
हिंदी में प्रश्न एवं आदर्श उत्तर
A के पास कुछ सिक्के हैं। वह उनमें से आधे सिक्कों में 2 और सिक्के मिलकर B को देता है। B उनमें से आधे सिक्कों में 2 और सिक्के मिलकर C को देता है। C उनमें से आधे सिक्कों में 2 और सिक्के मिलकर D को देता है। D के पास अब जितने सिक्के हैं, वह दो अंकों की लघुतम संख्या है। शुरु में A के पास कितने सिक्के हैं?
सही उत्तर (C) 60 है।
व्याख्या:
Let us work backwards from D. The number of coins D has is the smallest two-digit number, which is 10. Let C have c coins. C gives half of the coins and 2 more to D, meaning D receives (c/2) + 2 = 10, which gives c/2 = 8, so c = 16. Working backwards for B with b coins: (b/2) + 2 = 16, so b/2 = 14, giving b = 28. Finally, working backwards for A with a coins: (a/2) + 2 = 28, so a/2 = 26, giving a = 52? Wait, let's re-verify: A gives half and 2 more. If A has 76, half is 38, plus 2 is 40 given to B. A is left with 36. B has 40. B gives half (20) plus 2 (22) to C, leaving B with 18. C receives 22. C gives half (11) plus 2 (13) to D, leaving C with 9. D receives 13? Let's check with 76: A starts with 76. Gives half (38) + 2 = 40 to B. B has 40. Gives half (20) + 2 = 22 to C. C has 22. Gives half (11) + 2 = 13 to D. The question says D's coins is smallest two-digit number (10). Wait, let's solve algebraically: ((a/2 - 2)/2 - 2)/2 - 2 = 10. Let's solve: ((a/2 - 2)/2 - 2)/2 = 12 ⇒ (a/2 - 2)/2 - 2 = 24 ⇒ (a/2 - 2)/2 = 26 ⇒ a/2 - 2 = 52 ⇒ a/2 = 54 ⇒ a = 108? Let's check the options given: 76, 68, 60, 52. Let's test 76: 76/2 = 38, 38+2=40 to B. Remaining with A = 36. B has 40. 40/2 = 20, 20+2=22 to C. Remaining with B = 18. C has 22. 22/2 = 11, 11+2=13 to D. D gets 13. Wait, option 76 gives 13. Let's check 68: 68/2=34, +2=36 to B. B has 36. 36/2=18, +2=20 to C. C has 20. 20/2=10, +2=12 to D. Let's check 60: 60/2=30, +2=32 to B. B has 32. 32/2=16, +2=18 to C. C has 18. 18/2=9, +2=11 to D. Let's check 52: 52/2=26, +2=28 to B. B has 28. 28/2=14, +2=16 to C. C has 16. 16/2=8, +2=10 to D. Exactly 10! Thus for 52, D gets 10. Wait, let's re-verify 76 or 52. Let's trace 52: A = 52. Gives half (26) + 2 = 28 to B. B has 28. B gives half (14) + 2 = 16 to C. C has 16. C gives half (8) + 2 = 10 to D. D has 10 coins, which is the smallest two-digit number. Hence, option (d) 52 is correct.