Arithmetic & Operations — Read the numbers before you crunch them.

First, see what a number is made of

Before you crunch anything, look at how a number is built. 34 is not a single thing — it is 3 tens and 4 ones. Picture a ten as a tall rod of ten cubes and a one as a single cube:

Three ten-rods and four single cubes. That is what ‘34’ means.

Each spot to the left is worth ten times the spot on its right: ones, tens, hundreds, thousands. So 506 is 5 hundreds, 0 tens, 6 ones — the 0 is holding the tens column open so the 5 stays in the hundreds. This is place value, and it is why ‘carrying’ works.

‘Carry’ is bundling ten ones into one ten

Add 27 + 5. The ones are 7 + 5 = 12 ones. But you are not allowed to keep 12 ones — the ones column only holds 0 through 9. So you bundle ten of them into one ten-rod and slide it left, leaving 2 ones behind. That slid rod is the little ‘1’ you write above the tens column.

Ten loose cubes snap into one rod and move to the tens column. That snap IS the carry.

Subtraction with ‘borrowing’ is the same trick in reverse: you break one ten-rod back into ten ones when the ones column runs short. Carry bundles up; borrow breaks down. Same ten, different direction.

Make a ten — the fastest way to add in your head

Adding 8 + 5, your brain does not count ‘9, 10, 11, 12, 13.’ The quick way: steal 2 from the 5 to finish the 8 into a clean 10, then add the 3 that is left. 8 + 5 = (8 + 2) + 3 = 10 + 3 = 13. Tens are easy to add; you are routing every sum through the nearest ten.

Jump to 10 first, then jump the leftover. Two easy hops beat counting one at a time.

Subtraction works as number-line hops too. 23 − 8? Hop back 3 to land on the clean 20, then back 5 more: 20 − 5 = 15. Always aim for the nearest ten — it is the rest stop your brain handles instantly.

Now: read in tiers

Read this out loud the way you would read a sentence: 2 + 3 × 4. Most people say ‘two plus three is five, times four is twenty.’ That answer is wrong. The real answer is 14. Arithmetic is not read straight across like English.

Watch what these have in common:

• 2 + 3 × 4 = 2 + 12 = 14 (not 20)
• 10 − 2 × 3 = 10 − 6 = 4 (not 24)
• 1 + 2 × 5 = 1 + 10 = 11 (not 15)

In every line the multiplication happened first, even though the plus or minus came earlier on the page. That is the pattern. Multiplication is repeated addition, so it has to settle before any plain addition does. We read a calculation in tiers, top tier first.

THE MOVE: read it in tiers, top down. Find the grouping and exponents, settle them, then sweep the multiplications and divisions left to right, then the pluses and minuses left to right.

Watch the order in action

One tier per row. The highlighted piece gets computed; its value drops to the next row.

Add these the slow way and you will burn a minute: 4 + 7 + 10 + 13 + 16. Now try Gauss's trick. Fold the list in half and add the outer pair: 4 + 16 = 20. The next pair in: 7 + 13 = 20. The lonely middle, 10, is half of 20. Every pair balances to the same total.

(4, 16) sums to 20. (7, 13) sums to 20. The middle 10 is half of 20. Average = 10.

So the total is 5 × 10 = 50 — count times average, no slogging. This works for any list that climbs by the same step each time. Such a list is an arithmetic series, and it has a gorgeous property:

The balance is the reason. For every number above the middle there is one the same distance below, so they cancel and the whole list averages to its center. This is what Gauss saw with 1 + 2 + … + 100: fifty pairs of 101 give 5050.

THE MOVE: never add an evenly-spaced list term by term. Pair from the ends, or use (first + last) ÷ 2 × count.

‘How many more?’ is a gap, not a pile

When a question asks how many more Ana has than Ben, or how much taller one bar is, you are not adding their piles — you are measuring the gap between them on a number line. Ana has 32 stickers, Ben has 19. Mark both and read the jump from one to the other:

‘How many more’ is the distance between the two marks — one subtraction, never an addition.

So ‘how many more / fewer’ is always the difference of the two amounts: bigger minus smaller. The trap is reading the bigger number alone, or adding the two by reflex. Find the two marks, measure the jump between them.

Fence-post counting — count the gaps, not the posts

Here is a trap that catches almost everyone. How many whole numbers run from 7 to 23? Your instinct is 23 − 7 = 16. Resist it. Count on your fingers: 7, 8, 9 — that is three numbers, yet 9 − 7 = 2. Subtraction gives the number of steps, not the number of terms. The true count is 23 − 7 + 1 = 17.

Why ‘fence-post.’ Build a straight fence 4 sections long. How many posts? Not 4 — 5. There is a post on each end plus the ones between. Posts = gaps + 1. When a problem talks about posts, trees, stripes, or chimes, decide first whether it wants the things (posts) or the spaces between them (gaps).

5 posts, but only 4 gaps between them. Posts = gaps + 1.

Look at 17 × 6 + 17 × 4. The slow road computes 102 + 68 = 170 — two multiplications and an addition. But both terms carry the same 17. Pull it out front: 17 × (6 + 4) = 17 × 10 = 170. One multiplication, and the 6 + 4 collapses to a round 10.

Try a few more and feel the pattern:

• 8 × 12 + 8 × 8 = 8 × 20 = 160
• 25 × 7 + 25 × 3 = 25 × 10 = 250
• 6 × 99 + 6 × 1 = 6 × 100 = 600

Every time, the shared number stepped outside and the leftovers added to something friendly. That is the distributive property, read in reverse.

Two blocks of height 17 glue into one strip 10 wide. Same area, one multiplication.

The area model — why ‘multiply’ is really ‘find the rectangle’

Here is the same idea, run the other way, and it is the secret behind multiplying two-digit numbers in your head. Picture 12 × 14 as the area of a rectangle 12 tall and 14 wide. Break the 12 into 10 + 2 and the 14 into 10 + 4. Two cuts split the rectangle into four tidy pieces, and the whole product is the four areas added up.

One big square, two skinny strips, one little corner. Add the four areas: 168.

No carrying, no column tricks — you chopped a hard rectangle into four easy ones. The big 10 × 10 does most of the work; the two strips and the tiny corner finish it. This is exactly what the distributive property does on paper: (10 + 2)(10 + 4) = 100 + 40 + 20 + 8. Every term is one of the four pieces.

One law, two directions — and the minus sign obeys it too

Pulling the common factor out and handing the factor in are not two rules — they are the same law read in opposite directions. Reading a × (b + c) as ab + ac is called expanding; reading ab + ac back as a × (b + c) is called factoring. Same equation, you just choose which side helps. Contests hide the factored side and reward you for reading it; algebra later will lean on the expanding side constantly.

Here is the part most kids never connect. The ‘minus in front of parentheses’ trap from the first chapter is also the distributive property — because a leading minus is secretly ‘times −1.’

−(b + c) = (−1)(b + c) = (−1)b + (−1)c = −b − c

That is why a minus sign flips every term inside: the −1 distributes to each one. So 20 − (3 + 4) = 20 − 3 − 4 = 13, never 20 − 3 + 4. Distribution and the sign-flip are one idea wearing two hats.

Framing inspired by AoPS Prealgebra.

THE MOVE: see the same number more than once in a sum? Factor it out, add the leftovers, multiply once at the end.

When a problem says closest to or approximately, stop and look at the answer choices before you touch a number. They are usually far apart — 10s, 100s, even 1000s. That spread is a gift: it tells you exactly how sloppy you are allowed to be.

If the choices sit 100 apart, you can be off by nearly 50 and still land in the right box. So round each number to something easy and multiply the clean versions.

• 2.46 × 8.16 ≈ 2.5 × 8 = 20 — no ugly decimals.
• $1.98 + $5.04 + $9.89 ≈ $2 + $5 + $10 = $17.
• 0.0003 × 8{,}000{,}000 = 2400 — messy made tidy.

This is safe because multiplication is gentle: a 2% nudge in one factor moves the product by about 2%. Round each factor by 5% and you are still within 10–15% of the truth — almost always well inside the answer-choice gap.

THE MOVE: read the spread of the choices first. The wider the gap, the harder you are allowed to round.

Your instincts about numbers were trained on numbers bigger than 1. Squaring makes things bigger. A reciprocal makes things smaller. Negatives feel ‘small.’ All three of those instincts break the moment a number is negative or sits between 0 and 1 — and contests live in exactly those gaps.

Watch the instincts fail:

• Square a fraction and it gets smaller: (1/2)² = 1/4.
• Take the reciprocal of a fraction and it gets bigger: 1 ÷ (1/2) = 2.
• Flip the sign of a negative and it becomes a big positive: −(−3) = +3.
• The reciprocal of −2 is −1/2, which is actually bigger than −2.

−1/2 sits to the right of −2 on the line — so −2 is less than its own reciprocal.

Trying to argue your way through this in your head is how you lose points. There is a faster, safer path.

Some problems hand you the end of a story and ask for the start. ‘I doubled a number, added 4, and got 20 — what was the number?’ You could guess and check. The fast way is to walk the story backward, undoing each step with its opposite.

Run it in reverse: end at 20, undo the ‘add 4’ by subtracting (20 − 4 = 16), undo the ‘double’ by halving (16 ÷ 2 = 8). Start was 8. Forward to check: 8 × 2 + 4 = 20. ✓

Top arrows are the story; bottom arrows undo it from the right end, each flipped to its opposite.

THE MOVE: when a problem gives the result and hides the start, do not guess — reverse the steps and undo each one.

A team's three games averaged 20 points. How many did they score in all? You did not need the individual games — 20 × 3 = 60. That tiny step is the secret to every average problem: an average is a disguise for a total. Strip the disguise and you can do plain arithmetic.

See it: an average is a leveling-off

Picture four stacks of blocks of heights 3, 7, 5, 1. The average is the height they would all be if you knocked the tall ones down and used the blocks to fill the short ones — pour the towers into one flat row. The blocks never leave; they spread out evenly.

Total blocks = 3+7+5+1 = 16, shared among 4 stacks → height 4 each. That flat height is the average.

That is why average × count = total: the leveled height times how many stacks gives back every block you started with. The average is the total, spread flat.

THE MOVE: turn every average into a total before you do anything. Add, subtract, or combine totals, and divide back to an average only at the very end.

This single move kills the most popular average trap on the planet: never average two averages. A class of 20 kids averages 80; a class of 10 averages 70. The combined average is not 75. Go to totals:

20 × 80 + 10 × 70 = 1600 + 700 = 2300, over 20 + 10 = 30 kids, gives 2300 ÷ 30 ≈ 76.7.

Why 76.7 and not the tidy 75 halfway between? Picture a seesaw with one average sitting at 70 and the other at 80. The 70-end carries only 10 kids; the 80-end carries 20. The heavier crowd drags the balance point toward itself. The combined average always lands nearer the bigger group — here, closer to 80.

The fulcrum sits closer to the heavier side. More kids = more pull.

That is weighted averaging, and every weighted-average problem on these contests is the same: go to totals, do plain arithmetic, divide once at the end.

Pictured-intuition adapted from Competition Math for Middle School (AoPS).

The seesaw trick — ups and downs must cancel

Here is a beautiful fact: the amounts numbers sit ABOVE the mean exactly cancel the amounts they sit BELOW it. Picture a seesaw balanced at the mean. Every number is a kid at some distance from the center. The board is level, so the two sides must match.

Below: −10 + (−5) = −15. Above: +5 + 10 = +15. They cancel.

So if four of five scores around a mean of 80 are 70, 75, 88, 92 (deviations −10, −5, +8, +12, totaling +5), the fifth must contribute −5 to rebalance — it is 75. Often faster than ‘total = mean × count, missing = total − sum so far.’

Shift the whole list — the mean rides along

Add the same amount to every value, and you do not need to re-add anything. If five candy bags average 11 pieces and you drop 23 more into each bag, the new average is just 11 + 23 = 34. The total grew by 5 × 23, but it is split among the same 5 bags, so the per-bag share climbs by exactly 23.

This is the engine behind ‘everyone gets older’ and ‘everyone gets a raise’ problems. You never need the individual values — only how much each one moved.

Swap a value — track only the change to the total

Adding or removing one value changes the total by exactly that value; replacing one value with another changes the total by the difference. Ten teammates average 13.5, so their ages total 10 × 13.5 = 135. A 15-year-old joins and an 11-year-old leaves: the total shifts by +15 − 11 = +4, to 139, over the same 10 people — new average 13.9. You touched two numbers, not eleven.

Framing inspired by AoPS Prealgebra.

Division leftovers — round up or round down?

Division is the same total-and-count machine run backward: total ÷ count. A clean problem wants the quotient and nothing more. The hard ones leave a leftover, and the trap is deciding what to do with it. The leftover never changes — the question does.

The bus picture. 95 kids, each bus holds 30. 95 ÷ 30 = 3 leftover 5. Full buses? 3 (round down). Buses to carry everyone? 4 — the last 5 kids still need a ride (round up). Same division, opposite answers.

Five people are in a room: four earn $30,000 and one earns $5,000,000. What is the ‘typical’ income? The mean says about $1,024,000 — a number nobody in the room is close to. The median says $30,000 — which actually describes the room. Same five people, two very different ‘middles.’ Contests love this gap.

The mean gets yanked around by extreme values; the median shrugs them off. Picture {1, 2, 3, 4, 100} on a line:

The lone 100 drags the mean far from the pack. The median stays planted in the middle of the sorted list.

All three on one picture

Take a friendly set with a repeat in it: {2, 4, 5, 5, 9}. Already sorted. Plot the five dots, stacking the repeat, and all three middles show up in one glance.

Mode = the tallest stack of dots. Median = the dot in the middle of the row. Mean = where the row would balance on a fulcrum.

Here all three happen to land on 5, but they answer different questions: mode is most common, median is middle of the line-up, mean is balance point of the whole row. Slide that lonely 9 out to 99 and only the mean lurches right — the median and mode do not budge. That is exactly the gap contests poke at.

THE MOVE: sort the list once, then read all three middles off the same sorted line. The mean wants the raw sum, but a sorted list hands you the median and mode at a glance.

To find the median of n values: sort, then take position (n+1)÷2 if n is odd, or average positions n÷2 and n÷2 + 1 if even. Do not compute anything you do not need — sort.

You just met the three middles. This chapter zooms in on the fight between two of them — mean versus median — because that fight is where the setters hide their traps. Watch it play out: five friends compare how many books they read last summer: four read 3 each, and one bookworm read 83. What is the ‘typical’ number? The mean says (3+3+3+3+83)÷5 = 95÷5 = 19 — yet not one friend read anywhere near 19. The median says 3, which describes four of the five exactly. Same five friends, two wildly different ‘typical’ numbers. Contests live in this gap.

The reason shows up the moment you picture the mean as a balance point on a seesaw. One value far out at the end yanks the balance toward itself, no matter how many ordinary values sit on the other side.

One value far from the pack hauls the mean toward it. The median — the middle of the sorted line — does not move.

The median ignores what happens at the edges

Take the sorted list 2, 4, 5, 7, 9. The median is the middle value, 5. Now push the top value way out — change 9 to 9000. Sorted, it is 2, 4, 5, 7, 9000. The middle value is still 5. The median did not flinch, even though the mean exploded.

So the two statistics answer different questions. The mean asks ‘if we shared everything equally, how much each?’ — it depends on the total, so every value counts, including monsters. The median asks ‘what is the middle of the line-up?’ — it ignores how extreme the edges are.

Framing inspired by AoPS Prealgebra.

THE MOVE: when a list has one value far from the rest, expect the mean and median to disagree — and the mean to sit closer to the outlier.

This is the problem that breaks strong students: the arithmetic is trivial, but they misread one bar, one axis label, one row — and lose the whole point. The math is not the hard part. The reading is.

THE MOVE: name what you need, read only those numbers, then ignore the rest. A graph problem is mostly scenery built to distract you. Find the two or three values the question actually asks about, pull them out, and do the small arithmetic.

Pie charts: a slice's angle ÷ 360° is its fraction of the whole; multiply that fraction by the given total to get the slice's value.

If a question asks the spaghetti-to-manicotti ratio, read only those two bars: 250 and 100. The rest is scenery.

Some problems hand you a horror like 0.000315 × 7{,}928{,}564 and then offer choices that are miles apart: 210, 240, 2100, 2400, 24000. They are not asking for the exact product. They are asking one thing: how big is the answer? Computing it exactly is a waste — smart estimation wins.

THE MOVE: rewrite each ugly number as (a small number) × (a power of 10). Multiply the small parts, ADD the exponents, count the zeros.

• 0.000315 ≈ 0.0003 = 3 × 10⁻⁴
• 7,928,564 ≈ 8,000,000 = 8 × 10⁶
• Product ≈ 3 × 8 × 10⁻⁴⁺⁶ = 24 × 10² = 2400.

One caution. All of this works because the choices are far apart. When the answer choices are close together — within 10% of each other — aggressive rounding can land you in the wrong box. Then you compute exactly and round only at the very end.

A power is shorthand for the same number multiplied over and over. 2⁵ means five 2s multiplied: 2 × 2 × 2 × 2 × 2 = 32. The big number is the base (what you multiply); the little raised number is the exponent (how many copies). Read it as ‘two to the fifth.’

Once you see a power as ‘a stack of copies,’ every exponent rule becomes obvious — you are only counting copies.

Multiplying same-base powers: add the exponents

What is 2³ × 2⁴? Do not reach for 8 and 16. Count copies instead: 2³ is three 2s, 2⁴ is four 2s, so together you have seven 2s — that is 2⁷.

Three 2s next to four 2s makes seven 2s. The exponents add because the copies pile up.

Dividing same-base powers: subtract; a power of a power: multiply

Division cancels copies. 9⁷ ÷ 9³ is seven 9s over three 9s; three cancel top and bottom, leaving 9⁴. So you subtract exponents. And a power of a power, like (7⁵)³, is three copies of (five 7s) — fifteen 7s in all, so you multiply: 7¹⁵.

The staircase: why \(a^0 = 1\) and negatives flip

Here is the prettiest pattern in arithmetic. Walk the powers of 2 downward. Each step down divides by 2:

Each step down halves. Past 2¹ = 2 comes 2⁰ = 1, then 1/2, then 1/4. The pattern never breaks.

The staircase forces two definitions on us. Step below 2¹ = 2 and you land on 2⁰ = 1. Keep going and the exponents go negative, giving fractions: 2⁻¹ = 1/2, 2⁻² = 1/4. A negative exponent is not a negative number — it means one over the positive power.

These are not random rules to memorize. The product rule already demands them: 2³ × 2⁰ should be 2³, which only works if 2⁰ = 1. And 2² × 2⁻² should be 2⁰ = 1, which only works if 2⁻² = 1/4. The staircase keeps the copy-counting honest in both directions.

Framing inspired by AoPS Prealgebra.

THE MOVE: same base, one operation up. Multiply → add exponents. Divide → subtract. Power of a power → multiply. Walk the staircase down for zero and negatives.

The big one: (a + b)² is NOT a² + b²

This is the single most common slip in all of arithmetic, and even adults make it. Test it once and never forget it: (3 + 4)² = 7² = 49, but 3² + 4² = 9 + 16 = 25. Not equal — off by 24. You cannot hand an exponent out across a plus sign the way you hand out a multiplication.

So what does (a + b)² equal? Square is area, so draw the square. A square with side a + b splits cleanly into four pieces: the big a² corner, the little b² corner, and two identical a×b strips. That second ab strip is exactly the piece the wrong answer forgets.

Two big corners (a², b²) AND two equal strips (ab each). The forgotten 2ab is why the shortcut fails.

The next-square shortcut — square big numbers in your head

Put b = 1 and the formula becomes a mental-math gem: (a + 1)² = a² + 2a + 1. To square a number that sits one above a round one, start from the round square and add (twice the round number) + 1. Squaring 21? Start at 20² = 400, add 2×20 + 1 = 41: 21² = 441. No long multiplication.

Framing inspired by AoPS Prealgebra.