Applied Mental Arithmetic
Unlock high-speed mental multiplication, non-standard division, and rapid bounding heuristics. Replace pen-and-paper calculation with algebraic levers engineered for everyday quantitative decisions.
Like Duolingo, but for Applied Mental Arithmetic. Tomo turns the whole topic into a game you play five minutes a day, until it actually sticks.
A short one: 12 levels across 2 sections, about 24 minutes end to end, roughly 5 days at five minutes a day. It moves through Algebraic Levers for Mental Operations and Real-World Estimation and Bounding.
Free forever · No credit card · iPhone & Android
Key ideas in Applied Mental Arithmetic
- Scaling factors is not restricted to halving and doubling; tripling/thirding or multiplying by 2 and dividing by 2 can be chained repeatedly
- Doubling 35 produces 70, but doubling again yields 140 or multiplying by 2 while halving 48 yields 70 × 24, which re-doubled gives 140 × 12 = 1,680
- Targeting multiples of 10 or 100 eliminates mid-calculation digit carrying
- Difference of squares (a² - b²) eliminates standard cross-multiplication completely when factors share an even sum
- The midpoint of two factors equidistant from an anchor gives the base square, while the half-difference gives the value to square and subtract
- Recognizing how far each term deviates from the clean midpoint dictates the exact subtrahend
- Determine the algebraic deviations from the anchor base (100 - 96 = -4; 100 - 93 = -7)
- Adjust one factor by the other's signed offset to establish the hundreds place value (96 - 7 or 93 - 4 = 89 hundreds)
- Multiply the complementary offsets together to determine the final two digits ((-4) × (-7) = +28, yielding 8,928)
- Algebraic expansion of (10n + 5)² = 100n² + 100n + 25 = 100n(n + 1) + 25
- The coefficient 100 shifts the product of n(n + 1) precisely two decimal places, leaving the last two digits universally as 25
- The rule applies equally to multi-digit prefixes (e.g., 115² uses 11 × 12 = 132, yielding 13,225)
- Aliquot percentage fractions: 37.5% is equivalent to 3/8 of a whole
- Dividing by the aliquot denominator first (640 ÷ 8 = 80) followed by multiplying by the numerator (80 × 3 = 240) bypasses decimal multiplication
- Common commercial percentages (12.5%, 37.5%, 62.5%, 87.5%, 16.67%) correspond directly to unit and small-numerator fractions
- 1/14 is half of 1/7, which equates to multiplying by ~0.0714 or halving after dividing by 7
You've tried the other tabs
Thirty open tabs. Four facts you actually kept.
You watched. You nodded. By Sunday it was gone.
One answer, then back to scrolling.
Eight weeks. You meant to finish. You didn't.
Tomo gives Applied Mental Arithmetic the Duolingo treatment: levels, streaks, and quick quizzes that test what you just learned. That game loop is what the tabs above never had, so it's the one you actually finish.
Here's what playing it feels like
A real question from this course. Take your best guess.
You want to mentally solve 35 × 48 in two quick scaling steps. Which pair makes it simplest?
Get it right to open this lesson and 11 more in the app.
Where Applied Mental Arithmetic takes you
- 1
Algebraic Levers for Mental Operations
- Bilateral Balancing and Symmetric Multiplication
- Reciprocal Division and Modular Scaling
- 2
Real-World Estimation and Bounding
- Order-of-Magnitude Scaling and Fermi Bounds
- Dimensional Checking and Inversion Traps
2 sections · 4 units · 12 levels. Built to play, not to enroll.
Tomo is in ChatGPT
Tomo is also a ChatGPT plugin: the same playable courses, built and played inside your chat.
Add Tomo to ChatGPT- What to ask
- “Make me a Tomo course on black holes.” “Turn this chat into a Tomo course.”
- What you get
- Quick questions with instant feedback, hearts and stars, ready to play in about a minute.
- Where it goes
- No sign-up to start. Connect your Tomo account and your streak carries on in the app.
You pick the voice
Applied Mental Arithmetic is taught in the Professor style: clear, structured, thorough. Want a different feel? In the app you can spin up the same topic in any of Tomo's teaching styles. Same facts, totally different vibe.
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Start Applied Mental Arithmetic today.
Download Tomo, search Applied Mental Arithmetic, and play your first lesson in under a minute.