It's time to plant a spring vegetable garden 1/3 will be root plants, 1/3 will be stalk plants, and 1/3 will be vine plants. 1/2 of the stalk and vine plants will be grown organically without fertilizer. What fraction of the garden will be grown organically?

1/2 * 2/3 = ?

To find the fraction of the garden that will be grown organically, we can break down the problem step by step.

First, let's determine the fraction of the garden that will be dedicated to each type of plant:

1/3 will be root plants
1/3 will be stalk plants
1/3 will be vine plants

Next, let's find out what fraction of stalk and vine plants will be grown organically:

1/2 of stalk plants will be grown organically
1/2 of vine plants will be grown organically

Now, let's calculate the organic fraction for each type of plant:

Organic fraction of stalk plants = 1/2
Organic fraction of vine plants = 1/2

Finally, let's calculate the overall fraction of the garden that will be grown organically:

Fraction of the garden dedicated to stalk plants = 1/3
Organic fraction of stalk plants = 1/2

Fraction of the garden dedicated to vine plants = 1/3
Organic fraction of vine plants = 1/2

To find the overall organic fraction, we multiply the fraction of each plant type with its respective organic fraction and sum them up:

Organic fraction of stalk plants = (1/3) * (1/2) = 1/6
Organic fraction of vine plants = (1/3) * (1/2) = 1/6

Overall organic fraction = Organic fraction of stalk plants + Organic fraction of vine plants
= 1/6 + 1/6
= 2/6

Simplifying the fraction, we get:

Overall organic fraction = 1/3

Therefore, 1/3 of the garden will be grown organically.