kate cut a pizza into fourths. karl cut a pizza of the same size into sixth. After they each ate some pizza, they had the same fraction of pizzas left. What fraction of each pizza was left?

2/4 = 3/6 = 1/2

To find out what fraction of each pizza was left, we need to determine the individual fractions of the pizzas that were eaten. Let's start by calculating the fraction of Kate's pizza that was eaten.

Kate cut her pizza into fourths, so each slice represents 1/4 of the pizza. Let's assume Kate ate x slices. Therefore, the fraction of Kate's pizza that was eaten can be represented as x/4.

Now, let's determine the fraction of Karl's pizza that was eaten.

Karl cut his pizza into sixths, so each slice represents 1/6 of the pizza. Let's assume Karl ate y slices. Therefore, the fraction of Karl's pizza that was eaten can be represented as y/6.

According to the problem, Kate and Karl had the same fraction of pizza left, indicating that the amount they ate from their respective pizzas was the same.

So, we can set up an equation to represent this:

x/4 = y/6

To solve this equation, we need to find the least common multiple (LCM) of 4 and 6, which is 12. Multiplying both sides of the equation by 12 will eliminate the denominators:

12 * (x/4) = 12 * (y/6)

3x = 2y

Now, we know that the fraction of pizza left by each person is equal, so it can be represented as (4-x)/4 or (6-y)/6. Since we know that x = (2/3)y (derived from the previous equation), we can substitute it in:

(4 - (2/3)y)/4 = (6 - y)/6

Cross-multiply to remove the fractions:

6(4 - (2/3)y) = 4(6 - y)

24 - 4(2/3)y = 24 - 4y

24 - (8/3)y = 24 - 4y

Let's simplify this equation by multiplying through by 3:

3(24) - 8y = 12(24) - 12y

72 - 8y = 288 - 12y

Rearranging the equation:

4y = 216

Divide both sides by 4 to solve for y:

y = 54

Now, let's substitute the value of y back into the equation to find x:

x = (2/3)y = (2/3)(54) = 36

Therefore, Kate ate 36/4 = 9/1 slices of pizza, and Karl ate 54/6 = 9/1 slices of pizza.

Since they both have the same fraction of pizza left, the answer is 1/4 for Kate and 1/6 for Karl.