1. **State the problem:** We have 832 people surveyed about their birth month. \n\n2. **Given:** \n- $\frac{3}{13}$ of the people were born in May. \n- $\frac{5}{8}$ of the people were born in June. \n\n3. **Find:** How many people were born in neither May nor June? \n\n4. **Calculate the number of people born in May:** \n$$\text{May} = \frac{3}{13} \times 832 = \frac{3 \times 832}{13}$$\nCalculate numerator: $3 \times 832 = 2496$\n$$\text{May} = \frac{2496}{13} = 192$$\n\n5. **Calculate the number of people born in June:** \n$$\text{June} = \frac{5}{8} \times 832 = \frac{5 \times 832}{8}$$\nCalculate numerator: $5 \times 832 = 4160$\n$$\text{June} = \frac{4160}{8} = 520$$\n\n6. **Calculate total born in May or June:** \n$$192 + 520 = 712$$\n\n7. **Calculate number born in neither May nor June:** \n$$832 - 712 = 120$$\n\n**Final answer:** 120 people were born in neither May nor June.