Option D:
is positive product.
Solution:
<em>If the negative sign is in even number of times then the product is positive.</em>
<em>If the negative sign is in odd number of times then the product is negative.</em>
To find which product is positive:
Option A:
![$\left(\frac{2}{5}\right)\left(-\frac{8}{9}\right)\left(-\frac{1}{3}\right)\left(-\frac{2}{7}\right)](https://tex.z-dn.net/?f=%24%5Cleft%28%5Cfrac%7B2%7D%7B5%7D%5Cright%29%5Cleft%28-%5Cfrac%7B8%7D%7B9%7D%5Cright%29%5Cleft%28-%5Cfrac%7B1%7D%7B3%7D%5Cright%29%5Cleft%28-%5Cfrac%7B2%7D%7B7%7D%5Cright%29)
Here, number of negative signs = 3
3 is an odd number.
So, the product is negative.
Option B:
![$\left(-\frac{2}{5}\right)\left(\frac{8}{9}\right)\left(-\frac{1}{3}\right)\left(-\frac{2}{7}\right)](https://tex.z-dn.net/?f=%24%5Cleft%28-%5Cfrac%7B2%7D%7B5%7D%5Cright%29%5Cleft%28%5Cfrac%7B8%7D%7B9%7D%5Cright%29%5Cleft%28-%5Cfrac%7B1%7D%7B3%7D%5Cright%29%5Cleft%28-%5Cfrac%7B2%7D%7B7%7D%5Cright%29)
Here, number of negative signs = 3
3 is an odd number.
So, the product is negative.
Option C:
![$\left(\frac{2}{5}\right)\left(\frac{8}{9}\right)\left(\frac{1}{3}\right)\left(-\frac{2}{7}\right)](https://tex.z-dn.net/?f=%24%5Cleft%28%5Cfrac%7B2%7D%7B5%7D%5Cright%29%5Cleft%28%5Cfrac%7B8%7D%7B9%7D%5Cright%29%5Cleft%28%5Cfrac%7B1%7D%7B3%7D%5Cright%29%5Cleft%28-%5Cfrac%7B2%7D%7B7%7D%5Cright%29)
Here, number of negative sign = 1
1 is an odd number.
So, the product is negative.
Option D:
![$\left(-\frac{2}{5}\right)\left(-\frac{8}{9}\right)\left(\frac{1}{3}\right)\left(\frac{2}{7}\right)](https://tex.z-dn.net/?f=%24%5Cleft%28-%5Cfrac%7B2%7D%7B5%7D%5Cright%29%5Cleft%28-%5Cfrac%7B8%7D%7B9%7D%5Cright%29%5Cleft%28%5Cfrac%7B1%7D%7B3%7D%5Cright%29%5Cleft%28%5Cfrac%7B2%7D%7B7%7D%5Cright%29)
Here, number of negative sign = 2
2 is an odd number.
So, the product is positive.
Hence option D is the correct answer.
is positive product.