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Theory: Adding three-digit numbers

Task

Find the sum:

\(\displaystyle +\)\(\displaystyle 5\)\(\displaystyle 3\)\(\displaystyle 7\)
\(\displaystyle 1\)\(\displaystyle 7\)\(\displaystyle 6\)
 
Solution

1. First, we add the ones of the number \(\displaystyle 53\underline{7}\) (this is the number \(\displaystyle 7\)) with the ones of the number \(\displaystyle 17\underline{6}\) (this is the number \(\displaystyle 6\)):

\(\displaystyle 7 + 6 = 13{\small . }\)

We write \(\displaystyle 3\) (ones of the number \(\displaystyle 1\underline{3}\)), and \(\displaystyle 1\) (tens of \(\displaystyle \underline{1}3\)) moves to the tens place for further addition.

   \(\displaystyle {\small 1}\)
\(\displaystyle +\)\(\displaystyle 5\)\(\displaystyle 3\)\(\displaystyle \color{red}{7}\)
\(\displaystyle 1\)\(\displaystyle 7\)\(\displaystyle \color{red}{6}\)
 \(\displaystyle ?\)\(\displaystyle ?\)\(\displaystyle \color{red}{\bf3}\)

2. Add tens \(\displaystyle 3 + 7\) and add \(\displaystyle 1\) from the previous step:

\(\displaystyle 3 + 7 + 1 = 11{\small . }\)

 

We write \(\displaystyle 1\) (ones of the number \(\displaystyle 1\underline{1}\)), and \(\displaystyle 1\) (tens of the number \(\displaystyle \underline{1}1\)) moves to the hundreds place for further addition.

 \(\displaystyle \)\(\displaystyle {\small 1}\)\(\displaystyle \color{red}{\small +1}\)
\(\displaystyle +\)\(\displaystyle 5\)\(\displaystyle \color{red}{3}\)\(\displaystyle 7\)
\(\displaystyle 1\)\(\displaystyle \color{red}{7}\)\(\displaystyle 6\)
 \(\displaystyle ?\)\(\displaystyle \color{red}{\bf1}\)\(\displaystyle 3\)

3. We add hundreds \(\displaystyle 5 + 1\) and add \(\displaystyle 1\) from the previous step:

\(\displaystyle 5+1+1=7{\small . }\)

 

  \(\displaystyle \color{red}{\bf{\small +1}}\)\(\displaystyle \)\(\displaystyle \)
\(\displaystyle +\)\(\displaystyle \color{red}{5}\)\(\displaystyle 3\)\(\displaystyle 7\)
\(\displaystyle \color{red}{1}\)\(\displaystyle 7\)\(\displaystyle 6\)
 \(\displaystyle \color{red}{\bf7}\)\(\displaystyle 1\)\(\displaystyle 3\)

Therefore,

\(\displaystyle 537 + 176 = 713{\small . }\)

Answer: \(\displaystyle 713{\small . }\)