Welcome to the Shopify Community! Hilary here, I work at Shopify. I'd be happy to help with this issue today.
From what you have described, it sounds like you have set a sale price of 50% off using the Compare at price feature, is that correct? When this feature is used, only the final price is shown on the cart page, and at the checkout.
It is possible to edit the cart page by editing the code on your theme. That being said, it is not possible to edit the code on your checkout, unless your store is on Shopify Plus. This is mentioned in our developers help article here. This means only the cart page can be changed. If you wish to change the cart page, there are a few options to do this:
An alternate option is to use an automatic discount, or discount code to show this information on the cart, and at the checkout. To do this, you would need to remove any sale prices you have set on your products. This means sale prices would no longer be displayed on each product page and are only shown once added to the cart, and at the checkout. Once all products are set to their original full price, you need to create the discount.
If you prefer this alternate option, I recommend using an announcement banner on your storefront to highlight the sale, since the sale will not be shown on the product pages. This banner is a great place to communicate important information like discounts and sales with your customers.
If all products are on sale, I recommend using an automatic discount so that your customers receive the discount automatically at checkout. This way, they will not miss out on the sale. To create an automatic discount, here are the steps:
Once the automatic discount has been created, you can preview your store, add a product to the cart, and see the result. You will see the original price, the discount amount, and the total price. When you continue to the checkout, you will also see all of this information. Here is an example from the checkout to show you how it will look:
Let me know what you think of these options, and what you decide is best for your business!