Unit 1 · Foundations
Add within 20: Make ten first
Break one number so it fills up to ten, then add what is left.
Curriculum example
8 + 5 → 8 + 2 + 3 → 10 + 3 → 13
Free adult mental-math guide
How to Improve Mental Math as an Adult: 66 Methods + Practice
Mental arithmetic improves when you have a small move to try, not just an answer to memorize. Learn the method, follow the worked example, then practise eight representative moves on nearby numbers in your browser.
Start with the move, then build speed
This guide groups 66 practical methods into nine units. You do not need to complete it in order: begin with the kind of calculation you meet most often, try the 24-example sampler, and return to foundations whenever a later method feels crowded.
- 66
- methods
- 9
- units
- $0
- web access
Free method trainer
Practice the move, not just the answer
Choose one representative method from each unit. Try three nearby examples, check your answer, and reveal one clear route through the calculation.
No score · no saved result · no network request
8 starter methods · 24 hand-checked examples
Try it yourself
Example 1 of 3
7 + 6
One way to use this move
- 7 + 3 + 3
- 10 + 3
- 13
This is a fixed website-only sampler. Hone Math's generated practice, adaptation, progress, Daily Workout, and puzzles are separate app features.
Unit 1 · 2 methods
Foundations
Build two habits that make later arithmetic easier: filling a ten and finding a difference by counting up.
Method 01
Add within 20
Make ten first
Break one number so it fills up to ten, then add what is left.
Worked example
8 + 5 → 8 + 2 + 3 → 10 + 3 → 13
Method 02
Subtract within 20
Count up
Instead of taking away, count up from the smaller number.
Worked example
13 − 8 → 8 up to 13 is 5
Unit 2 · 11 methods
Times tables
Reconstruct multiplication facts from doubling, halving, and friendly multiples instead of relying on recall alone.
Method 03
The 2 times table
Double it
Two of a number is the number plus itself.
Worked example
2 × 7 → 7 + 7 → 14
Method 04
The 10 times table
Add a zero
Ten times a whole number just puts a zero on the end.
Worked example
10 × 7 → 70
Method 05
The 5 times table
Half of ten times
Multiply by 10, then halve it.
Worked example
5 × 8 → 80 ÷ 2 → 40
Method 06
The 4 times table
Double, double
Double the number, then double again.
Worked example
4 × 7 → 14 → 28
Method 07
The 3 times table
Double, plus one more
Double the number, then add it once more.
Worked example
3 × 6 → (6 + 6) + 6 → 18
Method 08
The 9 times table
Ten times, minus one
Multiply by 10, then subtract one copy of the number.
Worked example
9 × 7 → 70 − 7 → 63
Method 09
The 6 times table
Five times, plus one
Multiply by 5, then add the number once more.
Worked example
6 × 7 → 35 + 7 → 42
Method 10
The 8 times table
Double three times
Double the number three times over.
Worked example
8 × 6 → 12 → 24 → 48
Method 11
The 7 times table
Split it up
Break 7 into 5 + 2, multiply by each, then add.
Worked example
7 × 8 → (5 × 8) + (2 × 8) → 40 + 16 → 56
Method 12
The 11 times table
Ten times, plus one
Multiply by 10, then add one more copy.
Worked example
11 × 6 → 60 + 6 → 66
Method 13
The 12 times table
Ten times, plus two times
Multiply by 10 and by 2, then add them.
Worked example
12 × 7 → 70 + 14 → 84
Unit 3 · 6 methods
Mental addition
Turn awkward additions into round numbers, nearby doubles, or a left-to-right running total.
Method 14
Two-digit addition
Add tens, then ones
Add the tens together, add the ones, then combine.
Worked example
47 + 35 → 70 + 12 → 82
Method 15
Near doubles
Double, then add the gap
When two numbers are close, double the smaller one and add the small gap.
Worked example
34 + 36 → 34 + 34 → 68 → 68 + 2 → 70
Method 16
Round and adjust
Round up, then take back
Round the second number up to the nearest ten, add that, then take back what you added.
Worked example
47 + 38 → 47 + 40 → 87 → 87 − 2 → 85
Method 17
Give and take
Move a little across
Move just enough off the second number onto the first to make it a round ten, then add what is left.
Worked example
58 + 37 → move 2 across → 60 + 35 → 95
Method 18
Add a near hundred
Use the whole hundred
When the number you are adding sits just under a hundred, add the whole hundred instead, then take back the few extra.
Worked example
268 + 97 → 268 + 100 → 368 → 368 − 3 → 365
Method 19
Left-to-right addition
Biggest parts first
Add the hundreds, then the tens, then the ones, keeping one running total as you go.
Worked example
342 + 275 → 500 → 610 → 617
Unit 4 · 6 methods
Mental subtraction
Replace borrowing on paper with counting up, compensation, and same-difference moves you can hold in your head.
Method 20
Two-digit subtraction
Count up
Count up from the smaller number to the larger. No borrowing.
Worked example
63 − 28 → 28→30 is 2, 30→63 is 33 → 35
Method 21
Subtract from 100 or 1000
Nines, then ten
Take every digit from 9, and the last digit from 10.
Worked example
100 − 37 → 9 − 3 is 6, 10 − 7 is 3 → 63
Method 22
Subtract in parts
Tens first, then ones
Take away the tens, then take the ones off what is left.
Worked example
63 − 28 → 63 − 20 → 43 → 43 − 8 → 35
Method 23
Subtract a round ten
Take too much, then give back
Round the number you take away up to the nearest ten, subtract that, then add back the extra you took.
Worked example
83 − 29 → 83 − 30 → 53 → 53 + 1 → 54
Method 24
Subtract a near hundred
Take the whole hundred, then give back
When the number you take away sits just under a hundred, take the whole hundred, then give back the few extra you took.
Worked example
435 − 198 → 435 − 200 → 235 → 235 + 2 → 237
Method 25
Same difference
Move both numbers
Add the same amount to both numbers until the second one is a round ten. The gap between them never changes.
Worked example
142 − 68 → add 2 to each → 144 − 70 → 74
Unit 5 · 12 methods
Multiplication tricks
Use place value, factor moves, and numbers near 10 or 100 to simplify larger products.
Method 26
Multiply by 11
Ten times, plus one
Multiply by 10, then add one more copy.
Worked example
11 × 14 → 140 + 14 → 154
Method 27
Multiply by 9
Ten times, minus one copy
Multiply by 10, then subtract one copy of the number.
Worked example
9 × 47 → 470 − 47 → 423
Method 28
Two-digit × one-digit
Break it apart
Split the two-digit number into tens and ones, multiply each, then add.
Worked example
7 × 23 → 7×20 + 7×3 → 140 + 21 → 161
Method 29
Multiply by 15
Ten times, plus half of that
Multiply by 10, then add half of what you just got. 15 is 10 plus 5, and 5 is half of 10.
Worked example
15 × 24 → 240 + 120 → 360
Method 30
Multiply by 25
A hundred times, then quarter it
25 is a quarter of 100. Multiply by 100, then halve it twice.
Worked example
25 × 16 → 1600 → 800 → 400
Method 31
Double and halve
Halve one, double the other
Halve the even number and double the other one. The answer never changes, and doubling a number ending in 5 always lands on a round ten.
Worked example
14 × 35 → 7 × 70 → 490
Method 32
Multiply by 99
A hundred times, minus one copy
Multiply by 100, then subtract one copy of the number.
Worked example
99 × 34 → 3400 − 34 → 3366
Method 33
Multiply by 101
A hundred times, plus one copy
Multiply by 100, then add one more copy of the number.
Worked example
101 × 34 → 3400 + 34 → 3434
Method 34
Squares ending in 5
Next one up, then 25
Drop the 5 and multiply what is left by the next number up. Write 25 after it.
Worked example
35² → 3 × 4 → 12 → 1225
Method 35
Squares
Round, then correct
Go to the nearest ten. Move one factor onto it and the other the same distance the opposite way, then add the square of that distance.
Worked example
17² → 20 × 14 → 280 + 3² → 289
Method 36
Perfect square roots
Work backward from a square
A square root asks which number was multiplied by itself. Recall the square fact, then work backward.
Worked example
√144 → 12 × 12 = 144 → 12
Method 37
Two numbers near 100
Work with the gaps
See how far each number is from 100. Take one gap off the other number for the hundreds, then multiply the two gaps for the last two digits.
Worked example
97 × 96 → 97 − 4 → 93 → 9300 + 3 × 4 → 9312
Unit 6 · 6 methods
Division
Run multiplication facts backward, or replace one difficult division with repeated halves and friendlier operations.
Method 38
Division facts
The tables, backwards
Division undoes multiplication, so ask "what times the divisor gives this?"
Worked example
56 ÷ 7 → 7 × ? = 56 → 8
Method 39
Halving
Split, then halve
Break the number into two parts that are each easy to halve, halve them, then add.
Worked example
74 ÷ 2 → 60 + 14 → 30 + 7 → 37
Method 40
Divide by 4
Halve it, then halve again
Four is two twos, so halving twice divides by four.
Worked example
92 ÷ 4 → 46 → 23
Method 41
Divide by 8
Halve it three times
Eight is two, times two, times two, so halving three times divides by eight.
Worked example
184 ÷ 8 → 92 → 46 → 23
Method 42
Divide by 5
Double, then divide by ten
Five is half of ten. Double the number first, then just divide by ten.
Worked example
135 ÷ 5 → 270 ÷ 10 → 27
Method 43
Divide by 25
Times four, then divide by a hundred
Four 25s make 100. Multiply by 4 first, then just divide by 100.
Worked example
300 ÷ 25 → 300 × 4 → 1200 → 12
Unit 7 · 7 methods
Percentages
Build percentages from 10% and 1%, flip a calculation when useful, and work backward to an original price.
Method 44
Ten percent
Move the decimal
10% of a number is that number with the decimal moved one place left.
Worked example
10% of 80 → 8.0 → 8
Method 45
Common percentages
Build from 10%
Find 10% first. Half of that is 5%, double it is 20%. For 25%, take a quarter of the original number instead.
Worked example
20% of 60 → 10% is 6 → double → 12
Method 46
Bigger percentages
Count 10% blocks
Find 10%, then take as many of those blocks as you need. 70% is seven blocks, 90% is nine.
Worked example
70% of 120 → 10% is 12 → 7 × 12 → 84
Method 47
The one percent block
Find 1%, then scale it
One percent is the number with the decimal moved two places left. Find that, then take as many of them as the percent asks for.
Worked example
7% of 400 → 1% is 4 → 7 × 4 → 28
Method 48
Flip the percent
Turn it around
Swap the two numbers: 8% of 75 is the same as 75% of 8. Flip it so the friendly one becomes the percent.
Worked example
8% of 75 → 75% of 8 → three quarters of 8 → 6
Method 49
Percent off, percent on
Turn it into one percent
25% off means you pay 75%. A 20% tip means you pay 120%. Work out that single percent of the starting amount.
Worked example
25% off $60 → you pay 75% → 10% is $6 → $45
Method 50
Back to the original price
Work back to 100%
What you paid is 100% minus the discount. Work out how many 10% blocks that is, find one block, then take ten of them.
Worked example
20% off, you paid $48 → that is 80% → 10% is $6 → $60
Unit 8 · 8 methods
Everyday math
Apply the core methods to tips, discounts, bills, change, unit prices, budgets, and receipts.
Method 51
Leaving a tip
Build from easy parts
Find 10%. Halve it for 5%, add half for 15%, double for 20%, or take a quarter of the bill for 25%.
Worked example
15% tip on $80 → 10% is $8 + 5% is $4 → $12
Method 52
Sale discounts
The percent is your saving
Work out the percent of the price. That is how much comes off.
Worked example
25% off $80 → 25% of 80 → $20 off
Method 53
Splitting a bill
Divide by the people
Split the total evenly: divide the bill by the number of people.
Worked example
$60 split 4 ways → 60 ÷ 4 → $15 each
Method 54
Working out change
Count up to what you paid
Count up from the price to the amount you handed over.
Worked example
$8 item, paid $20 → 8 up to 20 → $12 change
Method 55
Price per item
Divide by how many you get
Divide the pack price by the number of items to get the price of one. That is what lets you compare two sizes.
Worked example
$12 for 4 → 12 ÷ 4 → $3 each
Method 56
Buying several
Price times how many
Multiply the price of one by how many you are buying. Round the price to a friendly number first if it helps, then adjust.
Worked example
6 at $7 → 6 × 7 → $42
Method 57
What's left
Add the spends, then take them off
Add up what you spent first, then subtract that one total from what you started with. One subtraction beats two.
Worked example
$50, spent $18 and $12 → 18 + 12 → 30 → 50 − 30 → $20 left
Method 58
Adding up a receipt
Pair them into tens
Look for two prices that make a round ten, add each pair first, then add the round numbers together.
Worked example
$13 + $7 + $12 + $8 → 20 + 20 → $40
Unit 9 · 8 methods
Test-style problems
Break number series, remainders, last-digit cycles, rates, and age problems into small arithmetic relationships.
Method 59
Steady-step series
Find the step
Take the gap between neighbours. If it never changes, the next term is just the last one plus that step.
Worked example
5, 8, 11, 14 → step 3 → 17
Method 60
Multiplier series
Find the multiplier
When the gaps grow fast, divide a term by the one before it. A constant multiplier means: multiply the last term once more.
Worked example
3, 6, 12, 24 → ×2 → 48
Method 61
Growing-gap series
Look at the gaps' gaps
Write down the gaps between terms. If the gaps themselves grow by a steady amount, extend the gaps first, then add.
Worked example
2, 5, 10, 17 → gaps 3, 5, 7 → next gap 9 → 26
Method 62
Two chains in one
Split it into two chains
Read every OTHER term: positions 1, 3, 5 make one simple series and positions 2, 4 make another. Extend the chain the next position belongs to.
Worked example
1, 10, 3, 20, 5 → chains 1, 3, 5 and 10, 20 → 30
Method 63
Remainders
Biggest multiple, then subtract
Find the largest multiple of the divisor that fits, then subtract it. What is left over is the remainder.
Worked example
87 ÷ 6 → 6 × 14 = 84 → 87 − 84 = 3
Method 64
Last digits of powers
Powers cycle
Last digits of powers repeat in a short cycle: for 3 it is 3, 9, 7, 1, over and over. Reduce the exponent by the cycle and you only compute one small power.
Worked example
23^14 → cycle of 4 → 14 leaves 2 → 3² → 9
Method 65
Rates
Amount is rate times time
Find what happens in ONE minute (or one hour, or one item), then multiply by how many of them there are.
Worked example
12 boxes a minute for 7 minutes → 12 × 7 → 84
Method 66
Age problems
One relationship at a time
Turn each sentence into arithmetic before touching the question. Multiply for 'times as old', then add the years that pass.
Worked example
Ben is 8, Maya is 3 times as old → 24; in 5 years → 29