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

Approach: Two ratio snapshots fix the actual counts. Use the first ratio to set variables, the second (after a known transfer) to solve for the unit, then count what's needed to level the boxes.

Step 1: Let A start with \(17k\) coins and B with \(7k.\) After moving \(108\) from A to B:
\[\frac{17k-108}{7k+108}=\frac{37}{20}.\]

Step 2: Cross-multiply: \(20(17k-108)=37(7k+108)\Rightarrow 340k-2160=259k+3996.\) So \(81k=6156\Rightarrow k=76.\)

Step 3: Initial counts: A \(=17(76)=1292,\ B=7(76)=532.\) After the \(108\)-coin shift: A \(=1184,\ B=640.\)

Step 4: The total \(1184+640=1824\) is fixed. For a \(1:1\) split each box must hold \(912.\) A currently has \(1184,\) so it must give away \(1184-912=272\) more coins.

Final answer: \(272\) coins.

Answer 2

Alternate approach — using the invariant total:
Shifting coins between A and B never changes the combined total. Let the initial counts be \( 17k \) and \( 7k \). After 108 coins move from A to B, the ratio becomes \( 37:20 \), so:
\[ \frac{17k-108}{7k+108} = \frac{37}{20} \]
Cross-multiplying: \( 20(17k-108) = 37(7k+108) \implies 340k - 2160 = 259k + 3996 \implies 81k = 6156 \implies k = 76 \).
So A had \( 17(76) = 1292 \) and B had \( 7(76) = 532 \), total \( 1824 \). After the first shift, A has \( 1292-108=1184 \) and B has \( 640 \).
Since the total \( 1824 \) never changes, an equal split needs \( 912 \) coins each. A must lose \( 1184 - 912 = \) 272 more coins to B.