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Suppose she travelled x hours on foot.

Total time = 10 hours so she travelled (10-x) hours on horseback.

Therefore, on foot she travelled 3*x miles

and on horseback she went 7*(10-x) miles.

In total, that is 3x + 70 - 7x = 70 - 4x

The total distance travelled was 46 miles so that 70 - 4x = 46

Therefore 4x = 70-46 = 24 and so x = 24/4 = 6 hours.

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