Directions: Answer the questions based on the information given below.
There are three types of coins i.e. 'Gold', 'Silver', and 'Copper' in each of three bags A, B, and C. The total number of coins in all the bags is 1510. The number of coins in bag 'A' is 580 and the total number of coins in bag 'B' is 510. The ratio between the number of copper coins in bag 'A', the number of copper coins in bag 'B', and the number of 'Gold' coins in bag 'C' is 3 : 2 : 4 respectively. The number of silver coins in bag 'B' is 60 more than the number of copper coins in bag 'B'. The number of 'Gold' coins in bag 'B' is 25% less than the number of silver coins in bag 'A'. The total number of 'gold' coins in all three bags is 570. The ratio of the total number of copper coins in all three bags and the total number of silver coins in all three bags is 20 : 27.
Find the total number of silver coins in bag 'B' and bag 'C' together?
Explanation:
Let's break down the information:
- Total coins: 1510
- Bag A: 580 coins
- Bag B: 510 coins
- Bag C: 1510 - 580 - 510 = 420 coins
- Copper coins in A : Copper in B : Gold in C = 3 : 2 : 4
- Let them be 3x, 2x, 4x
- Silver in B = Copper in B + 60 = 2x + 60
- Gold in B is 25% less than silver in A. If silver in A = y, then Gold in B = y - y/4 = 0.75y
- Total gold coins = 570
- Ratio total copper : total silver = 20 : 27
Let’s solve step-by-step:
Step 1: Find x (using copper ratio and gold in C)
- 3x (A copper), 2x (B copper), 4x (C gold)
Step 2: Express everything in terms of x and y
- A: Gold (A), Silver (A=y), Copper (A=3x), Total = 580
So Gold (A) = 580 - y - 3x
- B: Gold (B=0.75y), Silver (B=2x+60), Copper (B=2x), Total = 510
So: 0.75y + (2x+60) + 2x = 510
=> 0.75y + 4x + 60 = 510
=> 0.75y + 4x = 450 ........(1)
- C: Gold (C=4x), Silver (C=z), Copper (C), Total = 420
So: 4x + z + (copper C) = 420
Step 3: Total gold coins = 570
- Gold A + Gold B + Gold C = 570
- Gold A = 580 - y - 3x
So: (580 - y - 3x) + 0.75y + 4x = 570
=> 580 - y - 3x + 0.75y + 4x = 570
=> 580 - y + 0.75y - 3x + 4x = 570
=> 580 - 0.25y + x = 570
=> x - 0.25y = 570 - 580 = -10
=> x = 0.25y - 10.............(2)
Step 4: Solve for x and y using equations (1) and (2)
Substitute x = 0.25y - 10 into equation (1):
0.75y + 4x = 450
0.75y + 4*(0.25y - 10) = 450
0.75y + y - 40 = 450
1.75y = 450 + 40 = 490
y = 490/1.75 = 280
So y = 280
Now x = 0.25*280 - 10 = 70 - 10 = 60
So,
A copper = 3x = 180
B copper = 2x = 120
C gold = 4x = 240
Silver in B = 2x + 60 = 120 + 60 = 180
B gold = 0.75y = 0.75*280 = 210
Step 5: Find silver in C (z)
Total coins in C = 420
C gold = 240
Let C silver = z
C copper = 420 - 240 - z = 180 - z
Step 6: Total silvers calculation
A silver = y = 280
B silver = 180
C silver = z
Total silver = 280 + 180 + z = 460 + z
Step 7: Total copper calculation
A copper = 180
B copper = 120
C copper = 180 - z
Total copper = 180 + 120 + (180 - z) = 480 - z
Step 8: Ratio total copper: silver = 20:27
(480 - z)/(460 + z) = 20/27
Cross-multiplied:
27(480 - z) = 20(460 + z)
12960 - 27z = 9200 + 20z
12960 - 9200 = 27z + 20z
3760 = 47z
z = 3760/47 = 80
Step 9: Silver in B + C = 180 (B) + 80 (C) = 260
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## Summary of key points:
- The number of coins in each bag, as well as copper, silver, and gold coins, were deduced step by step.
- Using the ratios and constraints, each bag’s coin types were expressed algebraically and solved.
- The sum of silver coins in B and C = 180 + 80 = 260.
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Correct Option: 2 (260)