Baby's nearly one year old now. We've worked out how to weigh her but nurse and I still have trouble measuring her height. She just will not stand up straight against our measuring chart. In fact she can't stand up at all yet! So we measure her upside down. Last year nurse held Baby's feet, keeping them level with the $140 \; \text{cm}$ mark, while I read off the mark level with the top of Baby's head: $97 \; \text{cm}$. This year it was my turn to hold the feet. Being taller than nurse, I held them against the $150 \; \text{cm}$ mark, while the nurse crawled on the floor to read the mark level with the top of Baby's head: $84 \; \text{cm}$. How many centimetres has Baby grown in her first year?

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This problem is taken from the UKMT Mathematical Challenges.
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