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save up (for something)
To put aside money in small increments in order to accumulate savings (typically to be able to buy something). We'll have to save up if we want to take that vacation to Greece next summer. I've stopped eating out at restaurants so I can save up for a new car.
save something up
to save something; to accumulate something. I'm saving up cans for recycling. If you'd only save your money up, you could buy anything you want.
Accumulate something for a particular purpose, as in Jan had been saving up her allowance for a new bicycle. [First half of 1800s]
To collect something for future use; accumulate something: I haven't saved up enough money yet to buy a new TV. You should save your money up for college. We've been saving up to buy a new car.