WASHINGTON » Apple will refund at least $32.5 million to consumers to settle a federal case involving purchases that kids made without their parents’ permission while playing on mobile apps, the government announced today.

The Federal Trade Commission said Apple will make full refunds for any such in-app purchases made by kids while playing on mobile phones and other devices, and incurring charges without parents’ knowledge or permission.

The commission said it had received tens of thousands of complaints about unauthorized charges.

Edith Ramirez, the agency’s head, said the settlement only involves children’s mobile apps and charges racked up when kids bought things such as virtual currency or dragon food. In some cases, Ramirez said, charges ran into the hundreds and even thousands of dollars.

The $32.5 million is the minimum that Apple would pay. As part of the settlement, Apple is required to pay full refunds to consumers for kids’ unauthorized purchases, so the number could go higher. If it doesn’t rise to $32.5 million, the difference would be paid to the FTC, Ramirez said.

The FTC said the apps included a 15-minute window in which passwords were not needed to make the purchases, but that Apple did not inform users about that.

Apple is the world’s most valued company as measured by market capitalization.