Generate an image of a blank notebook with a mechanical pencil, an eraser, and a ruler placed beside it on a wooden table. The pencil's lead is in contact with the paper, set in position as if someone was about to solve a problem. On the table, there is a well-used math textbook lying open, the pages filled with arithmetic problems, numbers and geometric diagrams. The rays of the afternoon sun are peeking through a window nearby, illuminating the study materials and casting a soft, warm glow over the table.

The first term of an A.P is -8. The ratio of the 7th term to the 9th term is 5:8. Calculate the common difference of the progression

a = - 8

Ratio of the 7th and 9th term is a + 6d : a + 8d = 5:8
8(a+6d) = 5(a+8d)
8a+48d=5a+40d
48d - 40d = 5a + 8a
8d = -3a
d = -3a÷8
d = -3×(-8)÷8 = 3
Therefore common difference = 3

8 - 5 is 3 ie 3 is the common difference

Correction on the last part

R.H.S

5(d-1)=5d-5

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Where a=-8

(a+6d)/(a+8d)=5/8

(6d-8)/(8d-8)=5/8

It boils down too

8(3d-4)=5(4d-4)

2(3d-4)=5(d-1)

6d-8=5d-1

d=-1+8=7