Fractions, Decimals & Percents — Three names for the same number.

A fraction like 3/5 means “3 out of 5 equal parts.” A decimal like 0.6 means “6 tenths.” A percent like 60% means “60 per hundred.”

They’re all the same number. Three costumes for one value. The kid who thinks in all three picks the easiest costume for each problem. Before any rules, look at the pictures — every rule below is a thing you can already SEE.

See it first: a fraction is a shaded part of a bar

Take a chocolate bar. Snap it into equal pieces. A fraction is “how many pieces you shade” over “how many equal pieces there are.” The bottom number tells you how many cuts; the top tells you how many you grabbed.

See it: equivalent fractions are the SAME shaded length

Here is the idea that unlocks everything else. Take the bar shaded to one-half. Now draw extra cut lines through it — you didn’t move any chocolate, you only chopped it finer. The shaded part is exactly as long as before, but now you can call it 2/4, or 3/6, or 50/100. Same length, more names.

That picture is the rule everyone memorizes: multiply top and bottom by the same number and the fraction does not change. Multiplying by the same number means cutting every piece into the same number of smaller pieces — nothing was added or removed. Running it backward (dividing top and bottom by a shared factor) is gluing pieces back together: that is “simplifying.”

See it: compare fractions by lining the bars up

Which is bigger, 2/3 or 3/5? Stack two equal-length bars, shade each, and look at which shaded edge sticks out farther. No arithmetic — your eyes do it.

The bars make the “common denominator” trick feel obvious. To compare with numbers, give both bars the same number of pieces (a common bottom): 2/3 = 10/15 and 3/5 = 9/15. Now it’s 10 pieces vs 9 pieces — 2/3 wins, exactly as the picture showed.

See it: a decimal is a 10×10 grid

Take a square and cut it into a 10×10 grid — 100 little cells, the whole square is “1.” Now each column is one tenth (0.1) and each tiny cell is one hundredth (0.01). A decimal is how many cells you shade.

Place value falls right out of the grid. The first decimal spot counts columns (tenths); the second counts single cells (hundredths). So 0.3 is three whole columns — 30 cells — which is why 0.3 = 0.30 = 30/100. Adding a zero on the end shades the same area, only counted in smaller cells.

See it: a percent is the SAME grid — one bar, three labels

Here is the punchline of the whole chapter. The 10×10 grid has exactly 100 cells. So “percent” — which means per hundred — is exactly how many of those 100 cells are shaded. The decimal grid and the percent grid are the same drawing. That is why one shaded bar can wear all three labels at once.

To turn any fraction into a percent, cut the bar into 100 pieces and count the shaded ones — that is exactly “scale the bottom to 100,” which you’ll use constantly below.

See it: fraction × fraction is the OVERLAP in a unit square

“Of” means multiply — but why does 1/2 × 1/3 come out so small, only 1/6? Watch it in a square that stands for one whole.

First take 1/3 of the square: shade a vertical strip one-third wide. Now take 1/2 of that strip: cut it in half the other way and shade the lower half. The piece you shaded twice — the overlap — is the answer. Count the little boxes: the square split into 2 rows and 3 columns has 6 equal boxes, and the overlap is exactly 1 of them.

The picture spells out the rule. The square is split into 2 × 3 = 6 equal boxes (rows times columns), and the overlap is only 1 box. That is exactly multiply the tops, multiply the bottoms: (1×1)/(2×3) = 1/6. Multiplying two fractions under 1 makes a smaller piece because you are taking a part of a part.

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

A worked example with the bar picture

Peter’s family orders a 12-slice pizza. Peter eats one slice, and splits a second slice equally with his brother. What fraction of the pizza did Peter eat?

Draw the pizza as a bar of 12 equal slices. Peter’s full slice is 1 slice. The shared slice splits in two, so Peter gets half a slice more. Peter’s total shaded length is 1 + ½ = 1½ slices out of 12.

That is 1.5/12. Double top and bottom to clear the half: 3/24 = 1/8 (answer C). The bar kept the “out of 12” honest — no guessing.

The six conversions, spelled out

Conversions to KNOW COLD

Repeating decimals — use the bar

Some fractions never stop when you divide them out (1/3, 1/6, 1/9, 1/99 above). Instead of rounding to “0.333” or “0.667,” write a bar over the block that repeats forever: 1/3 = 0.3,  2/3 = 0.6,  1/6 = 0.16,  1/99 = 0.01.

The pattern. One repeating digit over 9 is that digit as a ninth: 0.1 = 1/9,  0.4 = 4/9. Two repeating digits go over 99: 0.27 = 27/99 — which is why 1/99 = 0.01.

The trick: turn a repeating decimal into a fraction

Name the decimal, multiply by 10 (or 100…) to slide it one whole period, then subtract so the repeating tail cancels.

Same move for any repeater. x = 0.3 → 10x = 3.3 → 9x = 3 → x = 3/9 = 1/3. For a two-digit period, multiply by 100: x = 0.01 → 100x = 1.01 → 99x = 1 → x = 1/99.

When only PART of the decimal repeats

Trickier cousins like 0.28 (that is 0.2888…) have a non-repeating digit out front and only the 8 looping. The shift trick still works — you only have to line up the tails so they truly match before subtracting.

The watch-out. Do NOT write 10x − x = 0.8. The repeating 8-tails only cancel if the two numbers line up digit-for-digit past the decimal — 2.88 minus 0.28 leaves a clean 2.6, tails gone. Pick the power of 10 that lands the repeating block in the same spot in both numbers, then subtract.

See it: WHY some fractions stop and some repeat forever

1/4 stops dead at 0.25. 1/3 goes 0.3333… forever. What decides it? Not the size of the fraction — it’s the prime factors hiding in the bottom. Here is the whole secret in one idea.

A decimal that stops is really a fraction over 10, or 100, or 1000 — 0.25 = 25/100. And every power of ten is built from only two primes: 2 and 5 (10 = 2·5, 100 = 2·2·5·5). So a fraction can be rewritten over a power of ten exactly when its bottom is built from only 2s and 5s. If any other prime (3, 7, 11…) is stuck in the bottom, no power of ten will ever fit, and the long division never closes — it repeats.

The long-division loop for a repeater

Why does 1/7 repeat with a block exactly 6 long? Do the long division and watch the remainders. Dividing by 7, every remainder is one of 1, 2, 3, 4, 5, 6 (a remainder of 0 would mean it stopped). There are only 6 of those. So within 6 steps a remainder must repeat — and the instant a remainder comes back, the whole string of digits after it comes back too. The decimal is trapped in a loop.

That is the contest payoff: 1/7 = 0.142857, and the six sevenths are all the same six digits, only started at a different spot. The repeating block can never be longer than “denominator minus 1.”

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

Reading a shaded figure as a fraction

Contests love to draw a shape, shade part of it, and ask “what fraction is shaded?” (or “what percent?”). Your instinct is to measure areas with a ruler. Resist it. A fraction is parts out of equal parts — so the whole job is: cut the figure into pieces that are all the same size, then count.

The trick is choosing equal pieces. A grid hands them to you. Lines through a circle’s center make equal pie slices. A square cut into a clean number of smaller squares gives you a counting unit.

Worked example. Three circles share a center, and four straight lines run through that center; every other wedge is shaded. What percent of the figure is shaded?

Don’t fight the three rings. The four lines through the center cut the whole picture into 8 equal pie slices. The shading alternates — one slice on, the next off, all the way around — so exactly 4 of the 8 slices are shaded.

That holds in every ring at once, so the fraction is the same for the whole figure: 4/8 = 1/2 = 50% (answer E). No areas, no ruler — count equal slices.

Nobody at a bake sale asks for “seven-fourths of a pie.” They ask for “one and three-quarter pies.” Same amount, friendlier name. 7/4 is an improper fraction (top bigger than bottom); 1¾ is a mixed number — a whole number sitting next to a leftover fraction. They are two costumes for one value, and a contest will hand you whichever is more annoying.

Improper → mixed: it is just division with a leftover

To rename 23/5, ask the plain division question: how many whole 5s fit in 23, and what is left over? 23 ÷ 5 = 4 remainder 3. The quotient 4 is the whole part; the leftover 3 stays on top over the same bottom 5. So 23/5 = 4⅗. The remainder can never reach the bottom number — if it did, that would be one more whole.

Mixed → improper: glue the whole back on

Going back, 4⅗ means 4 + ⅗. Write the 4 as fifths too: 4 = 20/5, so 4⅗ = 20/5 + 3/5 = 23/5. The shortcut everyone memorizes — (whole × bottom + top) over the bottom — is exactly that: (4×5 + 3)/5 = 23/5.

Adding mixed numbers — watch the fraction part overflow

Add wholes to wholes and fractions to fractions, keeping the two stacks separate. The only twist: the fraction stack can pile up past 1, and then you regroup — trade the extra whole over to the whole stack.

Take 2⅔ + 1¾. Wholes: 2 + 1 = 3. Fractions: ⅔ + ¾ = 8/12 + 9/12 = 17/12 — that is more than a whole. Rename it: 17/12 = 1 and 5/12. Now carry that 1 over: 3 + 1 and 5/12 = 4 and 5/12.

Subtracting mixed numbers — borrowing a whole

Subtraction has the mirror twist. In 5¼ − 2¾ you cannot take ¾ from ¼ — the top fraction is too small. So borrow one whole and turn it into fourths, exactly like borrowing a ten in plain subtraction:

So 5¼ − 2¾ = 2½. (If borrowing feels slippery, the safe fallback is to flip BOTH numbers to improper fractions first: 21/4 − 11/4 = 10/4 = 2½ — same answer, no borrowing.)

Framing inspired by AoPS Prealgebra.

A $40 lunch, 8% tax. Most kids do two steps: find 8% of 40 (that’s $3.20), then add it on ($43.20). Watch a faster kid do it in one: $40 × 1.08 = $43.20. Done. No side calculation, no adding.

The difference is one habit. Stop reading a percent as a thing you find and add. Read it as a number you multiply by.

Build the multiplier straight from the words:

Once every percent is a multiplier, percent ‘arithmetic’ turns into plain multiplication. A $180 coat at 50% off is $180 × 0.5 = $90 — no ‘find half, then subtract’ two-step.

A stock goes up 25% on Monday and down 25% on Tuesday. Back to even, right? Wrong — and that gap is the most famous percent trap in contest math.

Start with $100. Up 25% → $125. Now drop that by 25%: $125 × 0.75 = $93.75. You ended below where you started, down $6.25.

Here’s why your gut was wrong: the 25% down is taken off the bigger number ($125), so it’s worth more dollars than the 25% up was. Same percent, different base.

The fix is to multiply the multipliers: 1.25 × 0.75 = 0.9375. The chain never adds.

Two important special cases:

• +25%, then −20%: ×1.25 × ×0.8 = ×1.00. Returns to start. Because 0.8 = 1/1.25.
• +10% four times: ×(1.1)⁴ = ×1.4641 — about 46.4%, NOT 40%.

Which is bigger, 4/9 or 17/35? Reach for a calculator and you’ve already lost the race. Both are a whisker under one-half — and a contest writes them that way on purpose, daring you to divide. You don’t have to. Three habits let you rank fractions by looking.

The number-line picture. Imagine a number line from 0 to 1, with ½ marked in the middle. Every fraction lives somewhere on this line. Comparing to ½ tells you which half it's in.

Four of these five fractions sit just below ½. Only 151/301 crosses over.

How to compare to ½ in your head. Double the top; if it's bigger than the bottom, the fraction beats ½.

• Is 151/301 bigger than ½? Double 151 = 302. Compare to 301: 302 > 301, so YES, 151/301 > ½.
• Is 100/201 bigger than ½? Double 100 = 200. Compare to 201: 200 < 201, so 100/201 < ½.

For five-way comparisons, use the landmark idea: compare each candidate to ½ first. Any below ½ is eliminated immediately if a candidate above ½ exists.

WHY cross-multiplying works (it is just a hidden common denominator)

The rule says a/b > c/d exactly when ad > bc. That can feel like a spell. It isn’t — it is the common-denominator method with the writing hidden.

To compare 3/5 and 4/7 the honest way, give both the same bottom, 5 × 7 = 35:

3/5 = (3×7)/35 = 21/35    and    4/7 = (4×5)/35 = 20/35.

Now both bottoms are 35, so the bigger fraction is just the bigger top: 21 vs 20. But look at where those tops came from — 3×7 and 4×5. Those are exactly the two cross-products! Cross-multiplying compares the numerators you would get over the common denominator bd — without writing the 35. Since 21 > 20, 3/5 > 4/7.

Framing inspired by AoPS Prealgebra.

Some problems look like a fraction that fell into a blender:

(1 − 1/3) ÷ (1 − 1/2)

That’s a complex fraction — fractions stacked inside fractions. Your eye wants to smash all four pieces under one giant common denominator at once. That’s where kids drop a sign or a factor. Don’t. Break it into THREE small, safe steps:

What division even MEANS: “how many fit?”

Before the flip rule, ask what dividing by a fraction is really asking. 6 ÷ 2 asks “how many 2s fit in 6?” (three). In the same way, 6 ÷ ¾ asks “how many ¾-cups fit in 6 cups?” The pieces are small, so a lot of them fit — that is why the answer comes out bigger than 6.

Counting confirms it: eight ¾-cups is 8 × ¾ = 6 cups exactly. And the flip rule gives the same thing in one line: 6 ÷ ¾ = 6 × 4/3 = 24/3 = 8. The reciprocal isn’t a magic trick — it is literally counting how many pieces fit.

Framing inspired by AoPS Prealgebra.

Why “flip the second” works

Dividing by 1/2 is the SAME as multiplying by 2. Because asking “how many halves fit?” doubles whatever you started with.

Picture a problem that asks you to multiply a hundred fractions in a row. Looks like a death march. It isn’t — if the fractions are built right, almost every number erases its neighbor and the whole line collapses to one tiny answer.

Start small. (2/3) × (3/4) × (4/5). The 3 on top of the second cancels the 3 on the bottom of the first; the 4 on top of the third cancels the 4 on the bottom of the second. All that survives is the very first top and the very last bottom: 2/5. Two cancellations, zero real multiplying. That domino effect is called telescoping.

Why? Look at consecutive factors: the 2 on top of 2/3 cancels the 2 on the bottom of 1/2. The 3 on top of 3/4 cancels the 3 on the bottom of 2/3. The 4 on top of 4/5 cancels the 4 on the bottom of 3/4. Each numerator (except the first) kills the denominator on its left, so everything in the middle cancels — leaving the very first numerator (1) over the very last denominator (N).

So a product of 1000 fractions of this kind collapses to a single fraction in one move.

Sums can telescope too

The same cancellation idea works when you’re adding fractions, not only multiplying them — you rewrite each fraction as a difference of two pieces first. Here’s the most useful identity:

Don’t take it on faith — compute a few:

• 1/(1·2) = 1/2 — and 1/1 − 1/2 = 1/2 ✓
• 1/(2·3) = 1/6 — and 1/2 − 1/3 = 1/6 ✓
• 1/(3·4) = 1/12 — and 1/3 − 1/4 = 1/12 ✓

So a sum like 1/(1·2) + 1/(2·3) + 1/(3·4) + … + 1/(99·100) rewrites as:

(1/1 − 1/2) + (1/2 − 1/3) + (1/3 − 1/4) + … + (1/99 − 1/100).

Now look closely: every term in the middle cancels. The +1/2 cancels the −1/2, the +1/3 cancels the −1/3, and so on, all the way down. The only survivors are the very first piece (1/1 = 1) and the very last piece (−1/100).

Adding 99 fractions becomes a single subtraction. Whenever you see a sum that looks like “one over (consecutive product),” split each piece and let the cancellation do the work.

Half the people in a room left. One third of those still there got up to dance. Twelve weren’t dancing. How many were in the room to start?

The number you want is hiding behind a chain of fractions-of-fractions. The slow road is to guess a starting number and grind forward. The fast road: notice you never need the in-between counts at all — you can multiply the fractions straight through.

The shaded strip is the not-dancers: ½ of the room, then ⅔ of that. Multiply the two keep-fractions and you have them as one slice of the whole: (½)(⅔) = ⅓ of N. Set that equal to 12 and N pops out: (⅓)N = 12 → N = 36.

Two clean routes, same answer:

• Forward (keep-fractions): at each step keep the fraction of the subgroup you care about, multiply them, then divide the final count by that product.
• Backward: start from 12 and undo each step. Not-dancers are ⅔ of the stayers, so stayers = 12 ÷ ⅔ = 18. Stayers are half the room, so room = 18 × 2 = 36.

Pick whichever has fewer steps for the numbers in front of you.

Class A (20 kids) averages 80. Class B (30 kids) averages 70. Put them together — what’s the average now? Almost everyone says 75, splitting the difference. Wrong. The classes aren’t the same size, so they don’t pull equally.

Think of a see-saw with 80 on one end and 70 on the other. The balance point sits over the middle only if both sides have equal weight. Class B has more kids, so it’s the heavier end — the average slides toward 70.

Forget the see-saw arithmetic for a second — the real rule is simpler than any formula:

Here: (20·80 + 30·70) / (20+30) = (1600 + 2100)/50 = 3700/50 = 74. As predicted — below 75, leaning toward the bigger class’s 70.

The combined average always lands between the two group averages, nearer the larger group. Sanity checks: it can’t be under 70 or over 80, and it should sit closer to whichever group is bigger.

One of the most consistent mid-contest traps is mixing up “A is what percent OF B” with “A is what percent MORE THAN B.” Two different sentences, two different formulas, two different answers.

Pictures kill the confusion. Take A = 50 and B = 40 (so A is bigger):

Phrase ↔ formula cheat-sheet