In a logistics network, the numbers of parcels delivered respectively by Hub X and Hub Y are in the ratio 13 : 21. If Hub X delivers 160 more parcels and Hub Y delivers 240 more parcels, the ratio becomes 15 : 23. How many parcels were originally delivered by Hub X?
65
Let the original numbers of parcels delivered by Hub X and Hub Y be 13k and 21k, since they are in the ratio 13 : 21.
After the extra deliveries, \(\frac{13k + 160}{21k + 240} = \frac{15}{23}\), so 23(13k + 160) = 15(21k + 240).
Expanding, 299k + 3680 = 315k + 3600, so 16k = 80 and k = 5.
Therefore Hub X originally delivered 13 × 5 = 65 parcels (and Hub Y delivered 21 × 5 = 105 parcels).
Check: 65 + 160 = 225 and 105 + 240 = 345, and 225 : 345 = 15 : 23 on dividing both by 15; also 65 : 105 = 13 : 21.
Hence, 65.
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