Math / Numbers & Operations

QuestionAdd or subtract in the indicated base. [4.8]
46. $101_{2}+1101_{2}$
47. $110_{2}-101_{2}$
48. $413_{5}-324_{5}$

Studdy Solution

STEP 1

1. We are working with numbers in different bases: binary (base 2) and quinary (base 5).
2. We need to perform addition and subtraction in these bases without converting to base 10.
3. We will handle each problem separately.

STEP 2

1. Add the binary numbers $101_2$ and $1101_2$ .
2. Subtract the binary numbers $110_2$ and $101_2$ .
3. Subtract the quinary numbers $413_5$ and $324_5$ .

STEP 3

Align the binary numbers $101_2$ and $1101_2$ for addition. Add from right to left, carrying over when necessary.
$\begin{array}{c} \phantom{0}1101_2 \\ + \phantom{00}101_2 \\ \hline \end{array}$

STEP 4

Add the digits column by column:
- Rightmost column: $1 + 0 = 1$ - Next column: $0 + 0 = 0$ - Next column: $1 + 1 = 10_2$ (write 0, carry 1) - Leftmost column: $1 + 1 = 10_2$ (write 0, carry 1)
Write the result:
$\begin{array}{c} \phantom{0}1101_2 \\ + \phantom{00}101_2 \\ \hline 1010_2 \\ \end{array}$

STEP 5

Align the binary numbers $110_2$ and $101_2$ for subtraction. Subtract from right to left, borrowing when necessary.
$\begin{array}{c} 110_2 \\ - 101_2 \\ \hline \end{array}$

STEP 6

Subtract the digits column by column:
- Rightmost column: $0 - 1$ (borrow from next column, becomes $10_2 - 1 = 1$ ) - Next column: $0 - 0 = 0$ - Leftmost column: $1 - 1 = 0$
Write the result:
$\begin{array}{c} 110_2 \\ - 101_2 \\ \hline 001_2 \\ \end{array}$

STEP 7

Align the quinary numbers $413_5$ and $324_5$ for subtraction. Subtract from right to left, borrowing when necessary.
$\begin{array}{c} 413_5 \\ - 324_5 \\ \hline \end{array}$

STEP 8

Subtract the digits column by column:
- Rightmost column: $3 - 4$ (borrow from next column, becomes $13_5 - 4 = 4_5$ ) - Next column: $0 - 2$ (borrow from next column, becomes $10_5 - 2 = 3_5$ ) - Leftmost column: $3 - 3 = 0$
Write the result:
$\begin{array}{c} 413_5 \\ - 324_5 \\ \hline 034_5 \\ \end{array}$
The results are:
1. $101_2 + 1101_2 = 1010_2$
2. $110_2 - 101_2 = 001_2$
3. $413_5 - 324_5 = 034_5$

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