Question

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

Hint:

For ratio problems with transfers between two containers, first express initial quantities with a variable using the given ratio, then use the new ratio after transfer to form an equation. Finally, use total quantity (which stays constant) to handle any further equalisation like making the ratio 1:1.

Answer 1

Correct Answer: 272

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].

Answer 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.