Mental Addition and Subtraction Tips — Math Tricks with Arthur Benjamin

Overview

Arthur Benjamin teaches a left-to-right mental arithmetic method for addition and subtraction that scales from one-digit problems to three-digit calculations by breaking numbers into hundreds, tens, and ones. He emphasizes using complements and smart rewrites—such as turning 766 + 489 into 766 + 500 - 11—to convert carry-heavy additions into simpler subtractions and borrowing-heavy subtractions into easier additions, including practical applications like making change from $10, $20, and $100.

Key takeaways

Arthur Benjamin's core strategy is left-to-right arithmetic: add or subtract the highest place values first, then handle the remaining digits one chunk at a time.
Carry-heavy addition can often be simplified by rounding to a nearby multiple of 10 or 100 and then correcting with a small subtraction, such as 766 + 489 = 766 + 500 - 11.
Borrowing-heavy subtraction can be transformed into addition by using complements, such as 835 - 467 = 835 - 500 + 33.
The complement to 100 is easy to compute mentally because the digits pair as 9s and 10s: 67 complements to 33, while 80 complements to 20 as a special trailing-zero case.
Three-digit complements extend naturally to money math, making cash-change calculations from $10, $20, or $100 fast without paper.
Benjamin argues that large-number mental arithmetic becomes limited more by working memory than by computation, which is why estimation and simplification matter for four-digit and larger problems.

Chapters

0:00 Left-to-Right Mental Addition with One- and Two-Digit Numbers

Mental addition begins by working one digit at a time from left to right, starting with simple examples like 52 + 4 = 56.
Two-digit sums are handled by adding tens first, then ones, as in 62 + 24 = 86 and 73 + 16 = 89.
When carries appear, the method still works cleanly: 23 + 58 becomes 73 + 8 = 81, and 84 + 87 becomes 164 + 7 = 171.

4:15 Scaling Up to Three-Digit Addition with Place-Value Steps

Three-digit addition starts with easy cases like multiples of 100: 500 + 200 = 700 and 700 + 600 = 1300.
A number with trailing zeros simplifies the process, such as 400 + 567 = 967 and 235 + 800 = 1035.
When both numbers have tens and ones, the approach is still left to right: 450 + 320 = 770 and 410 + 360 = 770.
Benjamin notes that memory becomes the bottleneck as numbers grow, especially when several carries are involved.

9:50 Turning Hard Addition into Easier Subtraction

A difficult sum like 766 + 489 can be rewritten as 766 + 500 - 11, which is easier to compute mentally.
The key idea is that carry-heavy addition can often be replaced by a round-number addition followed by a small subtraction.
This rewrite sets up the transition to subtraction, where the same left-to-right strategy applies in reverse.

11:15 Two-Digit Subtraction: Standard Method and the 'Subtract 30, Add 1' Trick

Two-digit subtraction is done from left to right, as in 93 - 41 = 52 by subtracting 40, then 1.
For 74 - 29, one method subtracts 20 then 9, but Benjamin prefers 30 - 1, so 74 - 29 becomes 74 - 30 + 1 = 45.
The same idea works for 82 - 48 by converting 48 into 50 - 2, giving 82 - 50 + 2 = 34.
For 121 - 57, the method becomes 121 - 60 + 3 = 64.

14:54 Three-Digit Subtraction Without Borrowing and with Complements

When each digit of the minuend is at least as large as the corresponding digit of the subtrahend, subtraction is straightforward: 846 - 225 = 621.
Most three-digit problems require borrowing, so Benjamin introduces complements as a way to avoid it.
The complement of a number to 100 is found digit-wise: 75 ↔ 25, 49 ↔ 51, 67 ↔ 33, and 80 ↔ 20.
For 835 - 467, Benjamin subtracts 500 and adds back the complement of 67, yielding 835 - 500 + 33 = 368.

21:00 Using Complements for Change-Making from 10, 20, and 100

Three-digit complements generalize to money: $6.75 from $10 gives $3.25, and $8.49 from $10 gives $1.51.
For change from $100, the dollars complement to 99 and the cents complement to 100, so $23.58 returns $76.42.
For change from $20, the dollars complement to 19 and the cents still complement to 100, so $13.57 returns $6.43.
Benjamin stresses that these complement patterns are practical mental-math tools, not just abstract number tricks.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, The Great Courses.