The problem provides the heights of Anas, Iman, and Muaz in different bases. Anas's height is $10100001_2$, Iman's height is $230_8$, and Muaz's height is $1123_5$. The problem asks us to: (a) State Muaz's height in base 10. (b) Find the difference between Anas's height and Iman's height and express the result in base 5.

1. Problem Description

The problem provides the heights of Anas, Iman, and Muaz in different bases. Anas's height is $10100001_2$, Iman's height is $230_8$, and Muaz's height is $1123_5$. The problem asks us to:

(a) State Muaz's height in base

1

0. (b) Find the difference between Anas's height and Iman's height and express the result in base

5.

2. Solution Steps

(a) Converting Muaz's height from base 5 to base 10:

$1123_5 = (1 \times 5^3) + (1 \times 5^2) + (2 \times 5^1) + (3 \times 5^0)$

$1123_5 = (1 \times 125) + (1 \times 25) + (2 \times 5) + (3 \times 1)$

$1123_5 = 125 + 25 + 10 + 3$

$1123_5 = 163_{10}$

(b) First, convert Anas's height from base 2 to base 10:

$10100001_2 = (1 \times 2^7) + (0 \times 2^6) + (1 \times 2^5) + (0 \times 2^4) + (0 \times 2^3) + (0 \times 2^2) + (0 \times 2^1) + (1 \times 2^0)$

$10100001_2 = (1 \times 128) + (0 \times 64) + (1 \times 32) + (0 \times 16) + (0 \times 8) + (0 \times 4) + (0 \times 2) + (1 \times 1)$

$10100001_2 = 128 + 0 + 32 + 0 + 0 + 0 + 0 + 1$

$10100001_2 = 161_{10}$

Next, convert Iman's height from base 8 to base 10:

$230_8 = (2 \times 8^2) + (3 \times 8^1) + (0 \times 8^0)$

$230_8 = (2 \times 64) + (3 \times 8) + (0 \times 1)$

$230_8 = 128 + 24 + 0$

$230_8 = 152_{10}$

Now, find the difference between Anas's height and Iman's height in base 10:

Difference $= 161 - 152 = 9_{10}$

Finally, convert the difference from base 10 to base 5:

$9 \div 5 = 1$ remainder $4$

$1 \div 5 = 0$ remainder $1$

So, $9_{10} = 14_5$

3. Final Answer

(a) $163_{10}$

(b) $14_5$