← 2021 Paper 2

UPSC 2021 Maths Optional Paper 2 Q5c-i — Step-by-Step Solution

5 marks · Section B

Number systems · Numerical Analysis · asked 7× in 13 yrs · Read the full method →

Question

Convert $(3798.3875)_{10}$ into octal and hexadecimal equivalents.

Technique

Standard base conversion: repeated division for integer part, repeated multiplication for fractional.

Solution

Integer part: 3798

Octal (base 8): $3798\div 8=474$ rem 6. $474\div 8=59$ rem 2. $59\div 8=7$ rem 3. $7\div 8=0$ rem 7.

Read bottom-up: $(7326)_8$.

Verify: $7\cdot 512+3\cdot 64+2\cdot 8+6=3584+192+16+6=3798$ ✓.

Hexadecimal (base 16): $3798\div 16=237$ rem 6. $237\div 16=14$ rem 13 → D. $14\div 16=0$ rem 14 → E.

Read bottom-up: $(ED6)_{16}$.

Verify: $14\cdot 256+13\cdot 16+6=3584+208+6=3798$ ✓.

Fractional part: 0.3875

Octal: $0.3875\times 8=3.1$ → integer 3. $0.1\times 8=0.8$ → integer 0. $0.8\times 8=6.4$ → integer 6. $0.4\times 8=3.2$ → integer 3. $0.2\times 8=1.6$ → integer 1. $0.6\times 8=4.8$ → integer 4. … continues (non-terminating).

Fractional octal: $\approx .306314\ldots$ (truncate as desired).

Hexadecimal: $0.3875\times 16=6.2$ → integer 6. $0.2\times 16=3.2$ → integer 3. $0.2\times 16=3.2$ → integer 3. … repeating.

Fractional hex: $\approx .633\overline{3}$ (repeating 3 after the initial 6).

Summary

Answer