Brianna and Gavin used a method known as "partial quotients" to divide 552 by 23, and while both of their methods yield the correct quotient, they approached the problem differently. Let's break down their methods and relate them to the division process:
Brianna's Work:
-
Step 1: Subtract \(20 \times 23\) from 552:
\(552 - 460 = 92\)
(This is correct because \(20 \times 23 = 460\)) -
Step 2: Subtract \(4 \times 23\) from 92:
\(92 - 92 = 0\)
(Since \(4 \times 23 = 92\), this is also correct) -
Step 3: Add the partial quotients:
\(20 + 4 = 24\)
Gavin's Work:
-
Step 1: Subtract \(10 \times 23\) from 552:
\(552 - 230 = 322\)
(Correct, as \(10 \times 23 = 230\)) -
Step 2: Subtract \(10 \times 23\) from 322:
\(322 - 230 = 92\)
(Again correct) -
Step 3: Subtract \(2 \times 23\) from 92:
\(92 - 46 = 46\)
(Correct, as \(2 \times 23 = 46\)) -
Step 4: Subtract \(2 \times 23\) from 46:
\(46 - 46 = 0\)
(Again, correct) -
Step 5: Add the partial quotients:
\(10 + 10 + 2 + 2 = 24\)
Conclusion:
Both students arrived at the correct answer for \(552 \div 23\), which is 24. The process of partial quotients allows for flexibility in choosing how much to subtract (the partial quotients) at each step.
- Brianna effectively broke down the problem more quickly, using larger partial quotients initially (20 and 4).
- Gavin took a more detailed approach by breaking it into smaller parts (two subtractions of 10 and then smaller numbers).
Both methods are valid and lead to the same final result. This shows that division can be approached from multiple angles, and understanding the underlying principles is crucial for flexible problem solving.
Ultimately, the answer is \(24\), and both methods exemplify effective strategies using the partial quotients technique.