Three Ways to Say It

A fraction, a decimal and a percent are three languages for the same amount.

Like dog, perro and 狗. Three words. One dog.

14=0.25=25%

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

Cut the same pizza into 100

Your quarter would be 25 of those tiny pieces.

Percent means “out of 100”.

Money already does this

A dollar is one whole, already cut into 100 pennies.

Stuck on a question? Ask yourself: what would this be in money?

Learn these seven so well that you never have to think about them.

Fraction	Decimal	Percent	Think
1/2	0.5	50%	half
1/4	0.25	25%	a quarter
3/4	0.75	75%	three quarters
1/5	0.2	20%	20 out of 100
1/10	0.1	10%	10 out of 100
1/100	0.01	1%	one penny
1	1.0	100%	all of it

Now you get more for free

If 1/5 is 20%, just count in twenties.

Tenths work the same way. 1/10 is 10%, so 7/10 is 70%.

The first place after the point counts tenths.

So the 3 in 0.3 means 3 tenths.

Now cut each strip into 10

You get 100 little squares, and your 3 strips become 30 squares.

That is also why 0.3 and 0.30 are the same number.

Tidying a fraction up

If the top and the bottom both divide by the same number, you can shrink them.

Same amount, smaller numbers. That is called the simplest form.

6/10 is not wrong — it is the same pizza. But 3/5 is tidier, and it is what a test will ask for.

• 4/10both divide by 2 → 2/5
• 50/100both divide by 50 → 1/2
• 3/10nothing divides both → already simplest

• 0.1= 10/100 = 10%
• 0.6= 60/100 = 60%
• 0.9= 90/100 = 90%

Two digits after the point are hundredths — and percent means hundredths.

So the work is already done for you.

Decimal	Hundredths	Percent
0.45	45/100	45%
0.08	8/100	8%
0.60	60/100	60%
1.00	100/100	100%

Going backwards

A percent is already hundredths, so write it over 100.

Want the fraction too?

Divide the top and the bottom by the same number until you can’t any more.

Multiply the top and the bottom by the same number. The amount does not change.

Take 3 pieces out of 20. Cut every piece into 5. Now it is 15 out of 100.

Which bottoms can reach 100?

Only these. Learn the number you multiply by.

Bottom	×	Example
2	×50	1/2 = 50/100 = 50%
4	×25	3/4 = 75/100 = 75%
5	×20	2/5 = 40/100 = 40%
10	×10	7/10 = 70/100 = 70%
20	×5	9/20 = 45/100 = 45%
25	×4	7/25 = 28/100 = 28%
50	×2	17/50 = 34/100 = 34%

A fraction is a division. 3/8 means 3 ÷ 8.

Picture it: 3 chocolate bars shared between 8 kids. How much does each one get?

Eighths never land on 100, so sharing is the way.

8 into 3 wholes	none each, 3 left	0.
3 wholes = 30 tenths	3 each, 6 left	0.3
6 tenths = 60 hundredths	7 each, 4 left	0.37
4 hundredths = 40 thousandths	5 each, none left	0.375

The half in 37.5% is half a penny. That is what the last digit means.

The eighths family

Each eighth is 12.5%. Just keep adding 12.5.

Fraction	Decimal	Percent
1/8	0.125	12.5%
3/8	0.375	37.5%
5/8	0.625	62.5%
7/8	0.875	87.5%

Quick way: 1/8 is half of a quarter, and half of 25% is 12.5%.

Share 1 chocolate bar between 3 kids and the dividing never finishes.

10 tenths ÷ 3	3 each, 1 left	0.3
10 hundredths ÷ 3	3 each, 1 left	0.33
10 thousandths ÷ 3	3 each, 1 left	0.333
…forever	always 1 left	0.3333…

There is always 1 left over, so we round it and use ≈, which means “about”.

A puzzle

33.3% three times is 99.9%, but three thirds is a whole pizza. Where did the missing bit go? Into the rounding. That is what ≈ warns you about.

How to tell which ones stop

Look at the bottom number.

• stops2, 4, 5, 8, 10, 20, 25, 50, 100
• never3, 6, 7, 9, 12

Our numbers are built on 10, and 10 is 2 × 5. A 3 never fits.

In practice these are marked ≈, and 0.33 or 0.333 both count as right.

You have done all four of these by hand. Now take the shortcut.

1	Fraction → decimaldivide the top by the bottom	3/4 = 3 ÷ 4 = 0.75
2	Decimal → percentmultiply by 100	0.75 × 100 = 75%
3	Percent → decimaldivide by 100	40% ÷ 100 = 0.4
4	Percent → fractionput it over 100, then simplify	40% = 40/100 = 2/5

Which road do I take?

When you meet a fraction, ask one question.

• can it?Bottom is 2, 4, 5, 10, 20, 25 or 50 → make it 100. You can do this in your head.
• can’t?Bottom is 3, 6, 7, 8, 9 → divide, then ×100.

Today anything can come up, and there are no pictures. Just like the test.

You cannot add two different languages. Turn them into one first.

You can’t add 3 apples and 2 oranges until you call them both fruit.

For + and −, decimals are easiest

3/8 = 3 ÷ 8	day 6	0.375
0.2 becomes 0.200	so they line up	0.200
0.200 + 0.375	add as normal	0.575

In money: 20¢ + 37½¢ = 57½¢.

Unless it never ends

Thirds never finish as decimals, so use fractions and make the bottoms match.

Now they are both sixths: 3 + 2 = 5/6.

For × and ÷, turn everything into fractions.

Multiplying fractions is easy: top times top, bottom times bottom.

0.5 = 1/2	an anchor from day 2	1/2
1/2 × 2/7	tops, then bottoms	2/14
2/14, both ÷ 2	simplify	1/7

“Of” is the everyday one

• 25% of 60= 1/4 × 60 = 15
• 0.3 of 40= 30% of 40 = 12

Dividing: flip the second one

How many quarter pizzas are in 3 whole pizzas? Twelve.

Dividing asks “how many fit inside?”, so the answer usually gets bigger.

Check it: how many quarters fit in a half? Two.

Fifteen questions. No pictures, no hints, no second tries.

• 12+you pass, and you get the Graduate badge
• 15Perfect Paper
• againretake it as often as you like — your best score counts

If a question stalls, ask “how many out of 100?” and say it in money.