Ms. Hernandez has 17 tomato plants that she wants to plant in rows. She will put 2 plants in some rows and 1 plant in others. How many different ways can she plant the tomato plants?
Answer (3)
You just neeed to figure out the number of ways you can form 17 with the numbers 2 and 1. The easiest way is to write 17 as a function of the number of rows. If there are x rows with two plants and y rows with 1 plant, this equation is valid: 17= 2x + y, from which you can solve for y, y = 17 - 2x. Now you can give different whole values to x, to find the possible solution: x = 1 => y = 17 - 2 = 15. That means 1 row with 2 plants ans 15 rows - with 1 plant; x = 2 => y = 17 - 4 = 13; x = 3 => y = 17 - 6 = 11; x= 4 => y = 17 - 8 = 9; x = 5 => y = 17 - 10 = 7; x = 6 => y = 17 - 12 = 5; x = 7 => y = 17 - 14 = 3; x = 8 => y = 17 - 16 = 1. You cannot use x = 9, because that gives 2x = 18 and there are only 17 plants. So you have 8 different ways to plant the tomato plants using only 2 plants sin som rows and 1 plant in others (x from 1 to 8).
The correct answer is:
8 ways.
Explanation :
She will have at least one row with 2 plants, so using 17 rows with 1 plant each will not work.
She can have 15 rows with 1 plant each and 1 row with 2 plants. She can have 13 rows with 1 plant each and 2 rows with 2 plants. She can have 11 rows with 1 plant each and 3 rows with 2 plants. She can have 9 rows with 1 plant each and 4 rows with 2 plants. She can have 7 rows with 1 plant each and 5 rows with 2 plants. She can have 5 rows with 1 plant each and 6 rows with 2 plants. She can have 3 rows with 1 plant each and 7 rows with 2 plants. She can have 1 row with 1 plant and 8 rows with 2 plants.
This makes 8 different ways to plant these tomatoes.
Ms. Hernandez has 9 different ways to plant her 17 tomato plants in rows, using combinations of rows with 1 or 2 plants. By solving the equation 2 x + y = 17 , we find the various possible combinations. Each valid whole number for x from 0 to 8 results in a different arrangement of plants in rows.
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