Jane had $240. After Katherine bought a bag, Jane : Katherine = 6 : 7.

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• Jane = 6 units = $240 → 1 unit = $40
• Katherine after = 7 units = 7 × $40 = $280
• Given cost of bag = $20
• Katherine at first = $280 + $20 = $300
• Check: Jane : Katherine at first = 240 : 300 = 4 : 5

The cost of the bag was $20.
Check: Jane $240, Katherine $300. After spending $20: 240 : 280 = 6 : 7. ✓