Question

complete the equation to show how to break apart 316 using regrouping.
$316 = 200 + \square + 6$

Explanation:

Step1: Analyze the original number breakdown

We know that \(316\) can be broken down, and from the given intermediate step \(316 = 200 + 100 + 10 + 6\) (from the first part of the problem's illustration), and now we have \(316=200 +?+ 6\).

Step2: Solve for the unknown

We can use the equation \(316=200 + x+ 6\) (let \(x\) be the unknown). Rearranging to solve for \(x\), we get \(x = 316-(200 + 6)\). First, calculate \(200+6 = 206\), then \(316 - 206=110\)? Wait, no, wait. Wait, from the first breakdown, \(316=300 + 10+ 6\), and then \(300 = 200+100\), so \(316 = 200+100 + 10+6\). So in the equation \(316=200 +?+ 6\), the? should be \(100 + 10=110\)? Wait, no, wait the equation is \(316 = 200+? + 6\). Let's sum \(200 + 6=206\), then \(316-206 = 110\)? Wait, no, wait the first breakdown was \(316=300 + 10+ 6\), then \(300 = 200 + 100\), so \(316=200+100 + 10+6\). So when we have \(316=200+? + 6\), the? is \(100 + 10=110\)? Wait, no, maybe I made a mistake. Wait, let's do it step by step. The number \(316\): the hundreds place is \(3\) (which is \(300\)), tens place is \(1\) (which is \(10\)), ones place is \(6\) (which is \(6\)). But in the regrouping, \(300\) is split into \(200\) and \(100\), so \(316 = 200+100+10 + 6\). Now the equation is \(316=200+? + 6\). So we can think of \(316 - 200 - 6=\)?. Calculate \(316-200 = 116\), then \(116 - 6=110\)? No, wait \(200 +?+ 6=316\). So \(?=316 - 200 - 6\). \(316-200 = 116\), \(116 - 6 = 110\)? Wait, but from the first breakdown, it's \(200+100+10 + 6\), so \(100 + 10=110\). Yes, so the unknown is \(110\)? Wait, no, wait the equation is \(316=200+? + 6\). Let's check: \(200+110 + 6=316\), yes, \(200+110=310\), \(310 + 6=316\). Alternatively, from the first part, \(316=300+10 + 6\), and \(300 = 200+100\), so \(316=200+100+10 + 6\). So in the equation \(316=200+? + 6\), the? is \(100 + 10 = 110\). Wait, but maybe the problem is simpler. Wait, the original breakdown was \(316 = 300+10 + 6\), then \(300\) is split into \(200\) and \(100\), so \(316=200+100+10 + 6\). Now the equation is \(316=200+? + 6\), so if we group \(100 + 10\) together, that's \(110\). So the unknown is \(110\). Wait, but let's do it with the equation. Let \(x\) be the unknown. Then \(200 + x+ 6=316\). Combine like terms: \(206 + x=316\). Subtract \(206\) from both sides: \(x = 316 - 206=110\).

Answer:

\(110\)