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Chapter 1: Problem 37

Divide and then check by multiplying. $$305 \div 8$$

Short Answer

Expert verified
305 divided by 8 is 38 R1, and verified by multiplication.

Step by step solution

01

Perform Division

First, divide 305 by 8. Start by seeing how many times 8 fits into the first few digits of 305. 1. 8 goes into 30 (from 305) three times, as 8 x 3 = 24. Write 3 above the division bar. 2. Subtract 24 from 30 to get 6. 3. Bring down the next digit (5) from 305 to get 65. 4. 8 goes into 65 eight times, as 8 x 8 = 64. Write 8 above the division bar next to the 3. 5. Subtract 64 from 65 to get 1. The quotient is 38 with a remainder of 1, so we can write: 305 梅 8 = 38 R1.
02

Verify by Multiplying

To check your division, multiply the quotient by the divisor and add the remainder.1. Multiply the quotient (38) by the divisor (8): \[38 \times 8 = 304\]2. Add the remainder (1) to this result: \[304 + 1 = 305\]Since 305 matches the original dividend, the division is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiplication
Multiplication is one of the four basic arithmetic operations, and it can be thought of as repeated addition. For example, when you multiply 38 by 8, you are essentially adding 38 to itself 8 times. This can be written as:
  • \[38 \times 8 = 38 + 38 + 38 + 38 + 38 + 38 + 38 + 38\]
When multiplying, it is helpful to remember the multiplication tables, which can quickly help you calculate products like \(8 \times 3\) or \(8 \times 8\), as seen in the division example you worked on. The multiplication step in verifying division helps confirm that your division is correct. If the product, along with the remainder, correctly reconstructs the original number, you have a solid check on your calculation.
Quotient and Remainder
In division, the result is often given as a quotient with a remainder. The quotient tells you how many times the divisor fits into the dividend, whereas the remainder is what is left over when the division doesn't divide evenly. In our example, dividing 305 by 8 gives:
  • Quotient: 38 (which means 8 goes into 305, 38 full times)
  • Remainder: 1 (what is left after 38 full divisions of 8)
The remainder is an important part of the division process because it completes the full picture of the division. If you think of division as splitting into groups, the remainder is what remains ungrouped and can't fully form another group of your divisor's size.
Verification of Division
Verifying your division helps ensure accuracy and builds confidence in your calculations. To verify, you multiply your quotient by the divisor and then add the remainder:
  • First, calculate: \(38 \times 8 = 304\)
  • Then add the remainder: \(304 + 1 = 305\)
If the result equals your original dividend (305 in this case), your division was performed correctly. This step acts like a double-check, reminding you of the connection between multiplication and division: they are inverse operations. By verifying, you ensure that everything in your work adds up properly, literally and figuratively.

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