Complete the equation to show how to break apart 316 using regrouping.
$316 = 200 + \square + 6$
The problem asks to find the missing value in the equation \(316 = 200 + \square + 6\). We can represent the missing value as \(x\). The goal is to express 316 as the sum of 200, an unknown value \(x\), and 6. This is a decomposition exercise, where 316 is broken down into parts that sum to the total value. In decimal notation, 316 is composed of 3 hundreds, 1 ten, and 6 ones, which is \(300 + 10 + 6\). The problem provides a modified version of this: \(200 + x + 6 = 316\).
To find the value of \(x\), we first simplify the right-hand side of the equation:
\[316 = 200 + x + 6\]
Group the known constant terms \(200\) and \(6\):
\[316 = (200 + 6) + x\]
\[316 = 206 + x\]
Now, we isolate \(x\) by subtracting 206 from both sides of the equation:
\[x = 316 - 206\]
Performing the subtraction:
\[x = 110\]
Thus, the missing value is 110. The original equation is completed as \(316 = 200 + 110 + 6\).
Understanding how to decompose numbers is a fundamental skill in arithmetic, especially when performing mental calculations. For example, if you are shopping and need to figure out how to pay \$316 using a mix of \$200 bills, \$110 in other denominations (like one \$100 bill and one \$10 bill), and \$6 in coins, this type of regrouping helps you manage your money effectively. Recognizing that numbers can be broken down into different, flexible parts allows for easier addition and subtraction in everyday financial and inventory tasks.