Multiplying two-digit numbers by three-digit numbers is a mathematical operation that involves multiplying a two-digit number by a three-digit number to find their product. The two-digit number is called the multiplicand, and the three-digit number is called the multiplier.

To solve a two-digit by three-digit multiplication problem, you can use the standard algorithm, which is a step-by-step process that involves multiplying the digits of the multiplier by the digits of the multiplicand, carrying over digits as needed, and adding the results together to find the product.

For example, to multiply 35 by 456, you would follow these steps:

1. Multiply the ones digit of the two-digit number (5) by the ones digit of the three-digit number (6). The result is 30, which you will write in the ones place of the answer.
2. Multiply the ones digit of the two-digit number (5) by the tens digit of the three-digit number (5). The result is 25, which you will write in the tens place of the answer.
3. Multiply the ones digit of the two-digit number (5) by the hundreds digit of the three-digit number (4). The result is 20, which you will write in the hundreds place of the answer.
4. Multiply the tens digit of the two-digit number (3) by the ones digit of the three-digit number (6). The result is 18, which you will add to the tens place of the answer (25) to get 43.
5. Multiply the tens digit of the two-digit number (3) by the tens digit of the three-digit number (5). The result is 15, which you will add to the hundreds place of the answer (20) to get 35.
6. Multiply the tens digit of the two-digit number (3) by the hundreds digit of the three-digit number (4). The result is 12, which you will write in the thousands place of the answer.

## Sample test questions

Sample test questions for two-digit by three-digit multiplication:

1. What is the product of 38 and 405?
2. What is the product of 59 and 726?
3. What is the product of 47 and 834?
4. What is the product of 25 and 609?

To solve these problems, you can use the standard algorithm for two-digit by three-digit multiplication, which involves multiplying the digits of the multiplier by the digits of the multiplicand, carrying over digits as needed, and adding the results together to find the product.

1. What is the product of 38 and 405?
1. Multiply the ones digit of the two-digit number (8) by the ones digit of the three-digit number (5). The result is 40, which you will write in the ones place of the answer.
2. Multiply the ones digit of the two-digit number (8) by the tens digit of the three-digit number (0). The result is 0, which you will write in the tens place of the answer.
3. Multiply the ones digit of the two-digit number (8) by the hundreds digit of the three-digit number (4). The result is 32, which you will write in the hundreds place of the answer.
4. Multiply the tens digit of the two-digit number (3) by the ones digit of the three-digit number (5). The result is 15, which you will add to the tens place of the answer (0) to get 15.
5. Multiply the tens digit of the two-digit number (3) by the tens digit of the three-digit number (0). The result is 0, which you will add to the hundreds place of the answer (32) to get 32.
6. Multiply the tens digit of the two-digit number (3) by the hundreds digit of the three-digit number (4). The result is 12, which you will write in the thousands place of the answer. The final answer is 15,320.
2. What is the product of 59 and 726?
1. Multiply the ones digit of the two-digit number (9) by the ones digit of the three-digit number (6). The result is 54, which you will write in the ones place of the answer.
2. Multiply the ones digit of the two-digit number (9) by the tens digit of the three-digit number (2). The result is 18, which you will write in the tens place of the answer.
3. Multiply the ones digit of the two-digit number (9) by the hundreds digit of the three-digit number (7). The result is 63, which you will write in the hundreds place of the answer.
4. Multiply the tens digit of the two-digit number (5) by the ones digit of the three-digit number (6). The result is 30, which you will add to the tens place of the answer (18) to get 48.
5. Multiply the tens digit of the two-digit number (5) by the tens digit of the three-digit number (2). The result is 10, which you will add to the hundreds place of the answer (63) to get 73.
6. Multiply the tens digit of the two-digit number (5) by the hundreds digit of the three-digit number (7). The result is 35, which you will write in the thousands place of the answer. The final answer is 43,854.
3. What is the product of 47 and 834?
1. Multiply the ones digit of the two-digit number (7) by the ones digit of the three-digit number (4). The result is 28, which you will write in the ones place of the answer.
2. Multiply the ones digit of the two-digit number (7) by the tens digit of the three-digit number (3). The result is 21, which you will write in the tens place of the answer.
3. Multiply the ones digit of the two-digit number (7) by the hundreds digit of the three-digit number (8). The result is 56, which you will write in the hundreds place of the answer.
4. Multiply the tens digit of the two-digit number (4) by the ones digit of the three-digit number (4). The result is 16, which you will add to the tens place of the answer (21) to get 37.
5. Multiply the tens digit of the two-digit number (4) by the tens digit of the three-digit number (3). The result is 12, which you will add to the hundreds place of the answer (56) to get 68.
6. Multiply the tens digit of the two-digit number (4) by the hundreds digit of the three-digit number (8). The result is 32, which you will write in the thousands place of the answer.The final answer is 37,689.
4. What is the product of 25 and 609?
1. Multiply the ones digit of the two-digit number (5) by the ones digit of the three-digit number (9). The result is 45, which you will write in the ones place of the answer.
2. Multiply the ones digit of the two-digit number (5) by the tens digit of the three-digit number (0). The result is 0, which you will write in the tens place of the answer.
3. Multiply the ones digit of the two-digit number (5) by the hundreds digit of the three-digit number (6). The result is 30, which you will write in the hundreds place of the answer.
4. Multiply the tens digit of the two-digit number (2) by the ones digit of the three-digit number (9). The result is 18, which you will add to the tens place of the answer (0) to get 18.
5. Multiply the tens digit of the two-digit number (2) by the tens digit of the three-digit number (0). The result is 0, which you will add to the hundreds place of the answer (30) to get 30.
6. Multiply the tens digit of the two-digit number (2) by the hundreds digit of the three-digit number (6). The result is 12, which you will write in the thousands place of the answer. The final answer is 15,545.