Answer: 40
Step-by-step explanation: To find the greatest number of green buttons in the jar, we need to follow these steps:
1. Let's assign variables to the number of pink, blue, and green buttons in the jar. Since we know the ratios, we can express them in terms of these variables.
Let the number of pink buttons be 2x, the number of blue buttons be 7x, and the number of green buttons be 4y (since the ratio of blue to green buttons is 3:4).
2. We are given that there are less than 170 buttons in the jar, so we can set up an inequality based on the total number of buttons:
2x + 7x + 4y < 170
3. Simplify the inequality:
9x + 4y < 170
4. To find the greatest number of green buttons, we need to maximize the value of y. Let's find the maximum value of y by assuming the other variables take the lowest possible values:
Let x = 1 (minimum value to keep it an integer)
Substitute x = 1 into the inequality:
9(1) + 4y < 170
9 + 4y < 170
4y < 161
y < 40.25
5. Since the number of buttons must be an integer, y can take the maximum value of 40 (since it's less than 40.25), which means the greatest number of green buttons in the jar is 40.
Therefore, the greatest number of green buttons in the jar can be 40.
