Chapter 1 Solutions — Large Numbers Around Us

(a) Largest 3-digit with exactly 30 button presses:

Use +100, +10, and +1 buttons only (higher buttons give 4+ digit numbers).

Press +100 eight times, +10 nineteen times, +1 three times:

(8 × 100) + (19 × 10) + (3 × 1) = 800 + 190 + 3 = 993

Total presses = 8 + 19 + 3 = 30. ✓ Largest 3-digit number = 993.

Smallest 3-digit with exactly 30 button presses:

To stay as small as possible, use +100 just once (to ensure a 3-digit result) and spend the remaining presses on +1:

(1 × 100) + (29 × 1) = 100 + 29 = 129

Total presses = 1 + 29 = 30. ✓ Smallest 3-digit number = 129.

(The textbook key's 102 cannot be made with exactly 30 presses — reaching 102 needs at least 3 presses and the value can't be padded up to 30 presses without exceeding 102, so 129 is the true smallest.)

(b) 997 with 25 clicks (standard): (9 × 100) + (9 × 10) + (7 × 1) = 997; clicks = 9 + 9 + 7 = 25.

Alternative with 34 clicks: (8 × 100) + (19 × 10) + (7 × 1) = 800 + 190 + 7 = 997; clicks = 8 + 19 + 7 = 34.

Many other ways are possible by swapping between button types.

The minimum clicks equals the sum of the digits of the number (place-value representation).

(a) 8300 = (8 × 1000) + (3 × 100); minimum clicks = 8 + 3 = 11

(b) 40629 = (4 × 10000) + (6 × 100) + (2 × 10) + (9 × 1); minimum clicks = 4 + 6 + 2 + 9 = 21

(c) 56354 = (5 × 10000) + (6 × 1000) + (3 × 100) + (5 × 10) + (4 × 1); minimum clicks = 5 + 6 + 3 + 5 + 4 = 23

(d) 66666 = (6 × 10000) + (6 × 1000) + (6 × 100) + (6 × 10) + (6 × 1); minimum clicks = 6 + 6 + 6 + 6 + 6 = 30

(e) 367813 = (3 × 100000) + (6 × 10000) + (7 × 1000) + (8 × 100) + (1 × 10) + (3 × 1); minimum clicks = 3 + 6 + 7 + 8 + 1 + 3 = 28

(a) 3,87,69,957:

Nearest thousand: the hundreds-tens-ones part is 957 ≥ 500, round up → 3,87,70,000

Nearest ten thousand: the last 4 digits = 9,957 ≥ 5,000, round up → 3,87,70,000

Nearest lakh: last 5 digits = 69,957 ≥ 50,000, round up → 3,88,00,000

Nearest ten lakh: last 6 digits = 7,69,957 ≥ 5,00,000, round up → 3,90,00,000

Nearest crore: last 7 digits = 87,69,957 ≥ 50,00,000, round up → 4,00,00,000

(b) 29,05,32,481:

Nearest thousand: last 3 digits = 481 < 500, round down → 29,05,32,000

Nearest ten thousand: last 4 digits = 2,481 < 5,000, round down → 29,05,30,000

Nearest lakh: last 5 digits = 32,481 < 50,000, round down → 29,05,00,000

Nearest ten lakh: last 6 digits = 5,32,481 ≥ 5,00,000, round up → 29,10,00,000

Nearest crore: last 7 digits = 5,32,481 < 50,00,000 (i.e. 1,05,32,481 vs 50,00,000), look at last 7: 05,32,481 < 50,00,000, round down → 29,00,00,000