The ratio of the number of coins in boxes A and B was 17:7. After 108 coins were shifted from box A to box B, this ratio became 37:20. The number of coins that needs to be shifted further from A to B, to make this ratio 1:1, is
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Correct Answer: 272
Approach Solution - 1
Let the initial number of coins in box A be 17x and in box B be 7x. When 108 coins are shifted from A to B, the new ratio becomes 37:20. The equation is:
\((17x-108)/(7x+108)=37/20\)
Cross-multiplying gives:
\(20(17x-108)=37(7x+108)\)
Expanding both sides:
\(340x-2160=259x+3996\)
Solving for x, we get:
\(81x=6156\)
\(x=76\)
Initial coins in box A = 17x = 1292; in box B = 7x = 532.
After shifting 108 coins, box A has 1292-108=1184 coins; box B has 532+108=640 coins.
To find further coins to shift for a 1:1 ratio:
Let z be the coins to shift further. Then:
\((1184-z)=(640+z)\)
Solving for z:
\(1184-z=640+z\)
\(1184-640=2z\)
\(544=2z\)
\(z=272\)
Therefore, 272 more coins need to be shifted from A to B to make the ratio 1:1.
The solution value 272 is within the given range [272, 272].
Approach Solution -2
Step 1: Define the initial quantities.
Let the number of coins in box A and box B be 17x and 7x respectively.
Initial coins in A = 17x,
Initial coins in B = 7x.
Step 2: After shifting 108 coins from A to B.
New number of coins:
A: 17x - 108,
B: 7x + 108.
Given that the new ratio is 37 : 20:
(17x - 108) / (7x + 108) = 37 / 20.
Step 3: Solve for x.
Cross-multiply:
20(17x - 108) = 37(7x + 108)
340x - 2160 = 259x + 3996
340x - 259x = 3996 + 2160
81x = 6156
x = 6156 / 81 = 76.
Step 4: Find the current number of coins in each box.
After the first shift:
Coins in A = 17 * 76 - 108 = 1292 - 108 = 1184,
Coins in B = 7 * 76 + 108 = 532 + 108 = 640.
Total coins:
1184 + 640 = 1824.
Step 5: Make the ratio 1:1.
For a 1:1 ratio, both boxes must have:
Target in each box = 1824 / 2 = 912.
Currently, box A has 1184 coins, so coins to be shifted from A to B:
Shift needed = 1184 - 912 = 272.
(Alternatively, let the further shift be y, then:
(1184 - y) / (640 + y) = 1
1184 - y = 640 + y
2y = 544
y = 272.
Thus, the number of coins that must be shifted further from A to B is 272.
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