# Fastest way to do large additions?

In our book, in the statistics chapter, we have to add a large amount of numbers, especially for finding the mean of some given numbers. Most of the students in our class can do it easily in <30 seconds, but I am a little bit slow with calculations, and I take about >1 minute to do these.

Right now I use two methods to solve these:

1. Divide the numbers into pairs, add these pairs, then add them, etc. E.g.,

 01 |- 35
34 |  |
113
22 |  |
56 |- 78
+
---
113
---

2. Add the numbers that can be added easily first, e.g, $4+6$, $3+7$, etc.

 01--|->23
34-----|
22--|  |->90
56-----|
+
---
113
---


The first method is more difficult, takes more time, but is more reliable. The second method is faster, but sometimes gives the wrong answer (because I sometimes forget to add some numbers).

I can use calculators now but I cannot use calculators in the exam.

So the question is that are there any other easy, fast, and reliable methods to solve these questions?

• Yes, calculators. – naslundx Apr 17 '14 at 14:24
• @naslundx I am using a calculator now but what will I do in the exam? – Kartik Apr 17 '14 at 14:26
• Just do it in the manner you consider to be the safest (which seems to be the first one). Try also to train yourself gradually with mental calculations. – Claude Leibovici Apr 17 '14 at 14:38

Do the addition in columns, but first cross out groups of digits that add up to 10. For example, suppose you are adding:

$$\begin{array}{rrr} 1&3&2\\ 1&1&1\\ 1&7&2\\ && 5\\ &9&5\\ &1&7\\ &8&6\\ 1&0&2\\ &8&9\\ & 1&9\\ \hline \end{array}$$

In the units' column we have 9+1, 6+2+2, 5+5, which is three tens:

$$\begin{array}{rrr} \require{cancel}\def\x#1{\rlap{\!\cancel{#1}}\hphantom{#1}} \def\b#1{\color{red}{\x#1}} & \color{blue}{3} & \\\hline 1&3&\b2\\ 1&1&\b1\\ 1&7&2\\ && \b5\\ &9&\b5\\ &1&7\\ &8&\b6\\ 1&0&\b2\\ &8&9\\ & 1&\b9\\ \hline \end{array}$$

In the units' column this leaves 7+2+9 = 18: $$\begin{array}{rrr} &{{3^{\color{blue}{1}} }} & \\\hline 1&3&\x2\\ 1&1&\x1\\ 1&7&2\\ && \x5\\ &9&\x5\\ &1&7\\ &8&\x6\\ 1&0&\x2\\ &8&9\\ & 1&\x9\\ \hline & & \color{blue}{8} \end{array}$$

Now find digits in the tens' column that add up to 10. Look at the big digits first, and then see if there are small digits that match. We match the 7 with the 3, the 9 with the 1, and the 8 with the 1 + 1. Cross out the digits as you match them so that you don't accidentally use a digit more than once:

$$\begin{array}{rrr} \color{blue}{3}&\rlap{3^1} \hphantom{0} & \\\hline 1&\b3&\x2\\ 1&\b1&\x1\\ 1&\b7&2\\ && \x5\\ &\b9&\x5\\ &\b1&7\\ &8&\x6\\ 1&0&\x2\\ &\b8&9\\ & \b1&\x9\\ \hline & & 8 \end{array}$$

The remaining digit in the tens' column is an 8, plus the 3+1 tens we carried from units' column:

$$\begin{array}{rrr} \rlap{3^{\color{blue}{1}}} \hphantom{0}&\rlap{3^1} \hphantom{0} & \\\hline 1&\x3&\x2\\ 1&\x1&\x1\\ 1&\x7&2\\ && \x5\\ &\x9&\x5\\ &\x1&7\\ &8&\x6\\ 1&0&\x2\\ &\x8&9\\ & \x1&\x9\\ \hline & \color{ blue}{2} & 8 \end{array}$$

Then the hundreds' column is easy:

$$\begin{array}{rrr} \rlap{3^1} \hphantom{0}&\rlap{3^1} \hphantom{0} & \\\hline 1&\x3&\x2\\ 1&\x1&\x1\\ 1&\x7&2\\ && \x5\\ &\x9&\x5\\ &\x1&7\\ &8&\x6\\ 1&0&\x2\\ &\x8&9\\ & \x1&\x9\\ \hline \color{blue}{8} & 2 & 8 \end{array}$$

(I generated this example at random, so it may or may not be typical. Sometimes you may need to match two large digits with a small digit, for example $8+8+4$. Sometimes you may want to match a large digit with one of the carried digits. There may be other tricks I didn't think of.)

I normally group them so that the addition gives round figures and then add the remaining....Also I try adding the more significant digits first. I dont think it is very safe though...