Elena completed this division problem:

Question

A long division problem shows 4,297 under the division symbol. 38 is outside to the left of the symbol. The digit 1 is located above the 2 in the hundreds place of the dividend. The digit 0 is located above the 9 in the tens place of the dividend. The digit 7 is located above the 7 in the ones place of the dividend. There is a R to the right of the quotient with the remainder 31 following it. The number 38 is subtracted from the 4 in the thousands place and the 2 in the hundreds place. The digit 2 is below the 8 in the 38 being subtracted. The digit 9 is next to the 2, below the 9 in the tens place of the dividend. The digit 7 is next to the 9, below the 7 in the ones place of the dividend. 266 is subtracted from 297. The difference is 31.

4.) Explain the mistake Elena made and find the correct quotient.

Answer 1

Let's analyze Elena's long division problem step by step based on the details provided and identify the mistake.

1. Division Problem Setup:

   • Dividend: 4,297
   • Divisor: 38

2. Quotient Analysis:

   • Elena wrote down the digits of the quotient as follows:
     • 1 above the hundreds place (2): means \( 38 \times 1 = 38 \).
     • 0 above the tens place (9): means \( 38 \times 0 = 0 \).
     • 7 above the ones place (7): means \( 38 \times 7 = 266 \).

3. Calculating Step by Step:

   • The first digit of the quotient:
     • \( 38 \) (the divisor) goes into \( 42 \) (the first two digits of \( 4,297 \)), which goes 1 time \( (1 \times 38 = 38) \).
     • Subtract \( 38 \) from \( 42 \): \( 42 - 38 = 4 \).
   • Bring down the next digit (9). Now we have \( 49 \).
   • The next digit of the quotient:
     • \( 38 \) goes into \( 49 \) 1 time \( (1 \times 38 = 38) \).
     • Subtract \( 38 \) from \( 49 \): \( 49 - 38 = 11 \).
   • Bring down the next digit (7). Now we have \( 117 \).
   • The next digit of the quotient:
     • \( 38 \) goes into \( 117 \) 3 times \( (3 \times 38 = 114) \).
     • Subtract \( 114 \) from \( 117 \): \( 117 - 114 = 3 \).

4. Final Results:

   • The complete quotient is \( 113 \) with a remainder of \( 3 \).

Identifying the Mistake:

Elena made a mistake in her division when:

• She incorrectly added an additional 7 as part of her quotient instead of finding how many times 38 goes into the last number formed by bringing down all the digits correctly. Specifically, after bringing down the last digit, she should have calculated how many times 38 fits into 117, which is 3, not 7.

Conclusion:

The correct quotient of \( 4,297 \div 38 \) is \( 113 \) with a remainder of \( 3 \). The error in her quotient led her to an incorrect calculation of the remainder, which should actually have been 3 instead of 31.