The peanut butter way
Take a tablespoon of peanut butter and apply it to the affected area. Use your fingers to apply the butter to the area where the chewing gum is and the area surrounding it. You can use a toothbrush to apply it to the affected area. Leave the hair in the area for a couple of minutes until the peanut butter does its magic. Use a dry towel and slowly pull the chewing gum out. The oil in the peanut butter will make it easier to slide the chewing gum out.
Apply Vaseline on the hair surrounding the chewing gum. Make sure you have applied the gel to all the hair near the chewing gum as any hair can cause pain to you when you use the comb to slide the gum out. Use a wide tooth comb to push the chewing gum out of the hair.
The cooking oil method
You can use any cooking oil at home also to get rid of the chewing gum. Pour a few drops of the oil on the chewing gum and on the hair around it. Leave the hair for a good two three minutes so that the oil makes its way in. Then use a comb to slide out the gum as it was mentioned in the Vaseline case.
Use the ice method
This is probably the simplest of the methods. Take an ice cube and place it on and around the area of the gum for a few minutes. Once the gum has taken a solid texture, pull it out of the hair.