1. We are multiplying 4-digit numbers by 1-digit numbers using the grid method. 2. For each problem, multiply each place value by the single-digit multiplier and then sum all the partial products. 3. Problem 1: $6586 \times 5$ - $6000 \times 5 = 30000$ - $500 \times 5 = 2500$ - $80 \times 5 = 400$ - $6 \times 5 = 30$ - Sum: $30000 + 2500 + 400 + 30 = 32930$ 4. Problem 2: $6682 \times 9$ - $6000 \times 9 = 54000$ - $600 \times 9 = 5400$ - $80 \times 9 = 720$ - $2 \times 9 = 18$ - Sum: $54000 + 5400 + 720 + 18 = 60138$ 5. Problem 3: $9870 \times 4$ - $9000 \times 4 = 36000$ - $800 \times 4 = 3200$ - $70 \times 4 = 280$ - $0 \times 4 = 0$ - Sum: $36000 + 3200 + 280 + 0 = 39480$ 6. Problem 4: $1476 \times 4$ - $1000 \times 4 = 4000$ - $400 \times 4 = 1600$ - $70 \times 4 = 280$ - $6 \times 4 = 24$ - Sum: $4000 + 1600 + 280 + 24 = 5904$ 7. Problem 5: $4217 \times 7$ - $4000 \times 7 = 28000$ - $200 \times 7 = 1400$ - $10 \times 7 = 70$ - $7 \times 7 = 49$ - Sum: $28000 + 1400 + 70 + 49 = 29519$ 8. Problem 6: $1815 \times 6$ - $1000 \times 6 = 6000$ - $800 \times 6 = 4800$ - $10 \times 6 = 60$ - $5 \times 6 = 30$ - Sum: $6000 + 4800 + 60 + 30 = 10890$ 9. Problem 7: $8292 \times 8$ - $8000 \times 8 = 64000$ - $200 \times 8 = 1600$ - $90 \times 8 = 720$ - $2 \times 8 = 16$ - Sum: $64000 + 1600 + 720 + 16 = 66336$ 10. Problem 8: $8940 \times 8$ - $8000 \times 8 = 64000$ - $900 \times 8 = 7200$ - $40 \times 8 = 320$ - $0 \times 8 = 0$ - Sum: $64000 + 7200 + 320 + 0 = 71520$ 11. Problem 9: $5512 \times 5$ - $5000 \times 5 = 25000$ - $500 \times 5 = 2500$ - $10 \times 5 = 50$ - $2 \times 5 = 10$ - Sum: $25000 + 2500 + 50 + 10 = 27560$ 12. Problem 10: $9706 \times 8$ - $9000 \times 8 = 72000$ - $700 \times 8 = 5600$ - $0 \times 8 = 0$ - $6 \times 8 = 48$ - Sum: $72000 + 5600 + 0 + 48 = 77648$ Final answers: 1. 32930 2. 60138 3. 39480 4. 5904 5. 29519 6. 10890 7. 66336 8. 71520 9. 27560 10. 77648