This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the solutions to the multiplication problems using the lattice method. The lattice method involves: 1. Drawing a grid (lattice) based on the number of digits in the multiplicands. 2. Multiplying individual digits and placing the tens digit in the upper triangle and the units digit in the lower triangle of each cell. 3. Summing the numbers along the diagonals, starting from the bottom right, carrying over any tens digits to the next diagonal. 4. Reading the result from the top left to the bottom right. Since I cannot draw the lattice grids directly, I will describe the products within the cells and then show the diagonal sums. For a number like AB × C, the lattice would have cells for A × C and B × C. For ABC × DE, it would be a 3 × 2 grid. For single-digit multipliers, the lattice has one row. The cells will contain the products of each digit of the first number with the single digit of the second number. Example: 135 × 2 Cells: 1 × 2 = 02, 3 × 2 = 06, 5 × 2 = 10. Diagonals (from right to left): 1. Units: 0 (from 10) 2. Tens: 1 (from 10) + 6 (from 06) = 7 3. Hundreds: 0 (from 06) + 2 (from 02) = 2 4. Thousands: 0 (from 02) Result: 270 For multi-digit multipliers, the lattice has multiple rows. Example: 182 × 24 Cells (row 1 for multiplier 2, row 2 for multiplier 4): 1 × 2 = 02, 8 × 2 = 16, 2 × 2 = 04 1 × 4 = 04, 8 × 4 = 32, 2 × 4 = 08 Lattice (conceptual): ` 1 8 2 +---+---+---+ 2 |0/2|1/6|0/4| +---+---+---+ 4 |0/4|3/2|0/8| +---+---+---+ ` Diagonals (from bottom right to top left): 1. Units: 8 (from 0/8) 2. Tens: 4 (from 0/4) + 2 (from 3/2) + 4 (from 0/4) = 10. Write 0, carry 1. 3. Hundreds: 0 (from 0/4) + 6 (from 1/6) + 3 (from 3/2) + 0 (from 0/4) + (carry 1) = 10. Write 0, carry 1. 4. Thousands: 2 (from 0/2) + 1 (from 1/6) + (carry 1) = 4. 5. Ten Thousands: 0 (from 0/2) Result: 4368 --- 1. 135 × 2 Cells: 1 × 2 = 02, 3 × 2 = 06, 5 × 2 = 10 Diagonal sums: Units: 0 Tens: 1 + 6 = 7 Hundreds: 0 + 2 = 2 Thousands: 0 Result: 270 2. 182 × 24 Cells: Row 1 (for 2): 1 × 2 = 02, 8 × 2 = 16, 2 × 2 = 04 Row 2 (for 4): 1 × 4 = 04, 8 × 4 = 32, 2 × 4 = 08 Diagonal sums: Units: 8 Tens: 4 + 2 + 4 = 10. Write 0, carry 1. Hundreds: 0 + 6 + 3 + 0 + 1 (carry) = 10. Write 0, carry 1. Thousands: 2 + 1 + 1 (carry) = 4 Ten Thousands: 0 Result: 4368 3. 631 × 2 Cells: 6 × 2 = 12, 3 × 2 = 06, 1 × 2 = 02 Diagonal sums: Units: 2 Tens: 0 + 6 = 6 Hundreds: 0 + 2 = 2 Thousands: 1 Result: 1262 4. 832 × 3 Cells: 8 × 3 = 24, 3 × 3 = 09, 2 × 3 = 06 Diagonal sums: Units: 6 Tens: 0 + 9 = 9 Hundreds: 0 + 4 = 4 Thousands: 2 Result: 2496 5. 912 × 3 Cells: 9 × 3 = 27, 1 × 3 = 03, 2 × 3 = 06 Diagonal sums: Units: 6 Tens: 0 + 3 = 3 Hundreds: 0 + 7 = 7 Thousands: 2 Result: 2736 6. 452 × 3 Cells: 4 × 3 = 12, 5 × 3 = 15, 2 × 3 = 06 Diagonal sums: Units: 6 Tens: 0 + 5 = 5 Hundreds: 1 + 2 = 3 Thousands: 1 Result: 1356 7. 213 × 4 Cells: 2 × 4 = 08, 1 × 4 = 04, 3 × 4 = 12 Diagonal sums: Units: 2 Tens: 1 + 4 = 5 Hundreds: 0 + 8 = 8 Thousands: 0 Result: 852 8. 923 × 4 Cells: 9 × 4 = 36, 2 × 4 = 08, 3 × 4 = 12 Diagonal sums: Units: 2 Tens: 1 + 8 = 9 Hundreds: 0 + 6 = 6 Thousands: 3 Result: 3692 9. 324 × 5 Cells: 3 × 5 = 15, 2 × 5 = 10, 4 × 5 = 20 Diagonal sums: Units: 0 Tens: 2 + 0 = 2 Hundreds: 1 + 5 = 6 Thousands: 1 Result: 1620 10. 416 × 5 Cells: 4 × 5 = 20, 1 × 5 = 05, 6 × 5 = 30 Diagonal sums: Units: 0 Tens: 3 + 5 = 8 Hundreds: 0 + 0 = 0 Thousands: 2 Result: 2080 11. 805 × 5 Cells: 8 × 5 = 40, 0 × 5 = 00, 5 × 5 = 25 Diagonal sums: Units: 5 Tens: 2 + 0 = 2 Hundreds: 0 + 0 = 0 Thousands: 4 Result: 4025 12. 424 × 6 Cells: 4 × 6 = 24, 2 × 6 = 12, 4 × 6 = 24 Diagonal sums: Units: 4 Tens: 2 + 2 = 4 Hundreds: 1 + 4 = 5 Thousands: 2 Result: 2544 What's next? Send 'em! 📸