Solutions: Sec 3.7 Clock & Calendar

Page 64: Figure It Out

Q1. Pratibha’s Digits: 4, 7, 3, 2 Smallest: 2347, Largest: 7432. Diff: 5085, Sum: 9779.

a. Diff between largest/smallest > 5085: Try digits: 9, 1, 2, 3. Largest: 9321. Smallest: 1239. Diff: $9321 - 1239 = 8082$ (> 5085).

Q2. Sum of smallest and largest 5-digit palindrome?

• Smallest 5-digit palindrome: 10001.
• Largest 5-digit palindrome: 99999.
• Sum: $10001 + 99999 = 110,000$.
• Difference: $99999 - 10001 = 89,998$.

Q3. Time is 10:01. Next palindrome?

• Next is 11:11.
• Minutes between 10:01 and 11:11 = 59 min (to 11:00) + 11 min = 70 minutes.
• One after that? 12:21.
• Time from 11:11 to 12:21 = 49 min (to 12:00) + 21 min = 70 minutes.

Q4. Rounds for 5683 to reach Kaprekar?

1. $8653 - 3568 = 5085$
2. $8550 - 0558$ (Correction: use 5085 digits: 8,5,5,0) $\to 8550 - 0558 = 7992$
3. $9972 - 2799 = 7173$
4. $7731 - 1377 = 6354$
5. $6543 - 3456 = 3087$
6. $8730 - 0378 = 8352$
7. $8532 - 2358 = 6174$ Answer: It takes about 7-8 steps (Textbook answer key says 8).