To find the ratio between the number of unemployed males and the number of female engineers, we need to calculate these two values separately based on the information given about the summer festival in Shimla.

Calculating Unemployed Males

The total number of visitors at the festival was 4000. We are told that 25% of the total visitors were unemployed.

Number of unemployed visitors = $25\% \text{ of } 4000$

Number of unemployed visitors = $\frac{25}{100} \times 4000 = 0.25 \times 4000 = 1000$

Out of these unemployed visitors, 20% were females.

Number of unemployed females = $20\% \text{ of } 1000$

Number of unemployed females = $\frac{20}{100} \times 1000 = 0.20 \times 1000 = 200$

The number of unemployed males is the total number of unemployed visitors minus the number of unemployed females.

Number of unemployed males = Number of unemployed visitors - Number of unemployed females

Number of unemployed males = $1000 - 200 = 800$

Calculating Female Engineers

There were 1500 engineers present at the festival. The ratio of males to females among these engineers was 8 : 7.

Ratio of male engineers : female engineers = 8 : 7

The sum of the parts in the ratio is $8 + 7 = 15$.

To find the number of female engineers, we calculate the fraction of females from the total number of engineers.

Number of female engineers = $\frac{7}{15} \times \text{Total number of engineers}$

Number of female engineers = $\frac{7}{15} \times 1500$

Number of female engineers = $7 \times \frac{1500}{15} = 7 \times 100 = 700$

Determining the Ratio

We need to find the ratio between the number of unemployed males and the number of female engineers.

Ratio = Number of unemployed males : Number of female engineers

Ratio = $800 : 700$

To simplify the ratio, we divide both numbers by their greatest common divisor, which is 100.

Simplified Ratio = $\frac{800}{100} : \frac{700}{100}$

Simplified Ratio = $8 : 7$

Therefore, the ratio between the number of unemployed males and the number of female engineers is 8 : 7.