← 2022 Paper 2

UPSC 2022 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 $(1093.21875)_{10}$ into octal and $(1693.0628)_{10}$ into hexadecimal.

Technique

Integer-part conversion by repeated division; fractional-part conversion by repeated multiplication.

Solution

Part 1 — $(1093.21875)_{10}$ to octal

Integer part: 1093 to octal.

$1093\div 8=136$ rem $5$. $136\div 8=17$ rem $0$. $17\div 8=2$ rem $1$. $2\div 8=0$ rem $2$.

Read remainders bottom-up: $(2105)_8$.

Verify: $2\cdot 8^3+1\cdot 8^2+0\cdot 8+5=1024+64+0+5=1093$ ✓.

Fractional part: 0.21875 to octal.

$0.21875\times 8=1.75$ → integer 1. $0.75\times 8=6.0$ → integer 6. $0.0$ — done.

Fractional octal: $.16$.

Verify: $1/8+6/64=8/64+6/64=14/64=7/32=0.21875$ ✓.

Combined: $(1093.21875)_{10}=(2105.16)_8$.

Part 2 — $(1693.0628)_{10}$ to hexadecimal

Integer part: 1693 to hex.

$1693\div 16=105$ rem $13$ → D. $105\div 16=6$ rem $9$. $6\div 16=0$ rem $6$.

Read bottom-up: $(69D)_{16}$.

Verify: $6\cdot 256+9\cdot 16+13=1536+144+13=1693$ ✓.

Fractional part: 0.0628 to hex.

$0.0628\times 16=1.0048$ → integer 1. $0.0048\times 16=0.0768$ → integer 0. $0.0768\times 16=1.2288$ → integer 1. $0.2288\times 16=3.6608$ → integer 3. $0.6608\times 16=10.5728$ → integer A (10). $0.5728\times 16=9.1648$ → integer 9. … continues; truncate to a few places.

Fractional hex: $.10 13A9\ldots$ (to 6 places, approximately).

So $(1693.0628)_{10}\approx(69D.1013A9)_{16}$.

Summary

Answer