Three Ways to Say It

A fraction, a decimal and a percent are three languages for saying the same amount. Nobody argues about whether a dog is a dog or un perro — it’s the same dog. Same here.

• 1/4“1 out of 4 equal pieces.”
• 0.25“25 hundredths of the whole.”
• 25%“25 out of every 100.”

Now cut the same pizza into 100

Imagine slicing that same pizza into 100 tiny equal pieces. Your quarter would be 25 of them.

Percent literally means “out of 100.” The little % sign is a squashed 100. So every percent question is really one question: how many hundredths?

Money is already in hundredths

You already know this topic — you just know it as money. One dollar is one whole, and it comes pre-cut into 100 pieces. That is why percent and money look so alike.

A penny is one hundredth of a dollar, so one penny is 1%. Four quarters, ten dimes, a hundred pennies — all of them are 100%.

Try it in your head

• 3 dimes$0.30  =  30/100  =  30%
• 7 pennies$0.07  =  7/100  =  7%
• 3 quarters$0.75  =  3/4  =  75%

When a question feels slippery, ask yourself: if this were money, what would it be?

These seven come up constantly. Learn them the way you know your own phone number — instantly, without working anything out. Everything harder gets built out of them.

Build on them

Once 1/5 = 20% is automatic, you get the whole family for free: 2/5 is two of those, so 40%. 3/5 is 60%. 4/5 is 80%. Same with tenths: 1/10 = 10%, so 7/10 = 70%.

The decimal point is a fence. To its left are whole things. To its right are pieces of one whole, and the rooms have names.

The first room after the point is tenths — the whole thing cut into 10 equal strips. So in 0.3, the 3 is sitting in the tenths room:

From tenths to hundredths

Percent wants hundredths, not tenths. So cut every strip into 10 little squares. Now there are 100 squares, and your 3 strips have become 30 squares.

That is also why 0.3 and 0.30 are the same number. The extra zero doesn’t add anything — it just says “and no spare hundredths.”

Say it out loud

• 0.1one tenth  =  10/100  =  10%
• 0.6six tenths  =  60/100  =  60%  =  3/5
• 0.9nine tenths  =  90/100  =  90%

The second room after the point is hundredths — pennies. And here is the lovely part: percent asks “how many hundredths?”, so a decimal with two digits after the point has already answered the question.

Read the two digits, add a % sign. That’s the whole job. No rule to memorise, because the digits were already counting hundredths.

Both directions

Going backwards is the same trip in reverse. A percent is already a number of hundredths, so write it over 100 and read it as a decimal:

People say “move the decimal point two places.” That’s fine as a shortcut, but the reason is the one you already know: you are switching between counting whole things and counting hundredths of them.

And the fraction?

Put it over 100 and simplify by dividing top and bottom by the same number:

So far every fraction has been easy to turn into hundredths. Today, the tool that does it: multiply the top and the bottom by the same number.

Why is that allowed? Because it is just re-cutting. Take 3 pieces out of 20 and slice every piece into 5 smaller ones. Now the whole thing is in 100 pieces, and your share is 15 of them. You didn’t gain or lose any pizza — the pieces are just smaller.

Which bottoms can reach 100?

Only the ones that divide into 100 exactly. Here they are, with the number you multiply by:

The same trick, backwards

Going from a percent to a tidy fraction is this in reverse: divide top and bottom by the same number until nothing else fits.

Some bottoms never land on 100. Try eighths: 8, 16, 24… 96, 104. It steps right over it. So we use the other meaning of a fraction.

A fraction is a division. 38 means “3 shared fairly between 8” — 3 chocolate bars, 8 children, how much does each child get?

3 ÷ 8, one room at a time

You can’t give each child a whole bar, so you start breaking them up. Each line below moves one room to the right: wholes, then tenths, then hundredths, then thousandths.

Where does the half in 37.5% come from? 0.375 is 37 hundredths and 5 thousandths — and 5 thousandths is half of a hundredth. So it is thirty-seven and a half out of a hundred. Half a penny.

The eighths family

Each eighth is 12.5%, so the whole family is just counting by 12.5:

Quick check: 1/8 is half of a quarter. Half of 25% is 12.5%. No division needed.

Try sharing 1 chocolate bar between 3 children, the same way as yesterday. Something strange happens.

The leftover never goes away, so the 3s never stop. We write it 0.333… with dots, or we round it and use the “about equal” sign:

Two thirds is two of those: 0.666…, which rounds up to 0.67, because the next digit is a 6 and 6 is past halfway.

A puzzle worth arguing about

33.3% + 33.3% + 33.3% = 99.9%. But three thirds is a whole pizza, which must be 100%. Where did the missing 0.1% go? Nowhere — we threw away a crumb each time we rounded. That’s exactly what ≈ is warning you about. (This is why bakers say a third of a pizza and never 33.3% of a pizza.)

Which ones stop and which ones run on

There is a real rule, and it is about the bottom number:

• stopsbottoms built only from 2s and 5s: 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 100
• runs onbottoms with a 3, 6, 7 or 9 in them: 3, 6, 7, 9, 12, 15

That makes sense: our whole number system is built on 10, and 10 is made of 2 × 5. Bottoms made of 2s and 5s fit neatly into tenths and hundredths. A 3 never does.

In practice these are marked with ≈, and any sensible rounding is accepted — 0.33 or 0.333 both count.

You have earned the shortcuts. Every one of them is something you already did by hand this week.

1. Fraction → decimal: top ÷ bottom
2. Decimal → percent: × 100
3. Percent → decimal: ÷ 100
4. Percent → fraction: put it over 100, then simplify

Which road do I take?

When you meet a fraction, ask one question first:

• Can it?Can the bottom become 100 (2, 4, 5, 10, 20, 25, 50)? → multiply top and bottom, read off the hundredths. Faster, and you can do it in your head.
• Can’t?Bottom is 3, 6, 7, 8, 9, 16… → divide top by bottom, then ×100. Round with ≈ if it never ends.

One worked example of each

Today is a mixed day: anything can come up, nothing is multiple choice, and there are no pictures. That is exactly what tomorrow’s test looks like.

You now know that 38, 0.375 and 37.5% are the same amount. Today that stops being a fact and starts being useful, because it lets you calculate with numbers that arrive in different languages.

Here is the whole idea: you cannot add two different languages. You can’t add 3 apples and 2 oranges until you agree to call them both fruit. In the same way 0.2 + 3/8 means nothing until both numbers are written the same way.

Which language should I translate into?

• + and −Decimals are usually easiest. Line up the points and go.
• unlessthe fractions are thirds or sixths — those never finish as decimals, so use fractions instead.

Worked example: 0.2 + 3/8

Money check: 20¢ plus 37½¢ is 57½¢. It works.

Worked example: 75% − 1/2

Three quarters of a pizza, minus half a pizza, leaves a quarter. The picture agrees with the arithmetic, which is how you know you did it right.

When decimals will not do: 1/2 + 1/3

A third never finishes, so rounding would lose a crumb. Use fractions, and use the tool from day 5: re-cut both so the bottoms match.

Now they are both sixths, so you can add them: 3 sixths + 2 sixths = 56, which is about 0.83, or 83.3%.

Same first move as yesterday — translate so both numbers speak one language — but today the best language flips.

• × and ÷Fractions are usually easiest, because multiplying fractions is simply top × top and bottom × bottom.

Worked example: 0.5 × 2/7

Sevenths never finish as decimals, which is exactly why fractions were the right language here. Trying it as 0.5 × 0.2857… would have been miserable.

“Of” is the everyday one

This is the version you meet in real life — sale prices, tips, a share of something.

• 25% of 60= 1/4 × 60 = 15
• 0.3 of 40= 30% of 40 = 12
• 3/4 of 20%= 3/4 × 1/5 = 3/20 = 0.15 = 15%

Dividing: flip the second one over

Start with a question you can picture: how many quarter pizzas are there in 3 whole pizzas? Twelve, obviously. And look:

Dividing by a fraction is the same as multiplying by that fraction turned upside down. Dividing asks “how many of these fit inside that?”, so the answer is usually bigger — the mirror image of yesterday’s surprise.

Worked example: 1/2 ÷ 0.25

Check it in your head: how many quarters fit in a half? Two. The picture and the arithmetic agree again.

Fifteen questions drawn from all ten lessons: hundredths, anchors, tenths, percents backwards, friendly fractions, eighths, thirds — and three where you actually have to calculate across the three languages. No pictures, no hints, no second tries — just you.

• 12+a pass, and the Graduate badge
• 15Perfect Paper
• againyou can retake it as many times as you like — your best score is the one that counts

Tip: if a question stalls, ask “how many out of 100?” and say it as money.