Final answer:
To make the ratio of books 5:3, Kate should give Natasha 28 books.
Step-by-step explanation:
To find the number of books that Kate should give Natasha for the ratio to become 5:3, we first need to determine the number of books each person currently has. Let's assume that Natasha has 7x books and Kate has 5x books.
Given that Natasha has 28 more books than Kate, we can set up the equation 7x = 5x + 28 to solve for x. By subtracting 5x from both sides of the equation, we get 2x = 28. Dividing both sides by 2, we find that x = 14.
Now that we know the value of x, we can calculate the number of books each person has. Natasha has 7 * 14 = 98 books and Kate has 5 * 14 = 70 books. To find how many books Kate should give Natasha to achieve a 5:3 ratio, we can set up the equation (70 - y) / (98 - y) = 5/3, where y represents the number of books Kate gives to Natasha.
Simplifying the equation and cross-multiplying, we get 3(70 - y) = 5(98 - y). Expanding this equation and solving for y, we find that Kate should give Natasha 28 books for the ratio to become 5:3.