Tenths and Hundredths: Adding Fractions with Denominators 10 and 100

One tenth is ten hundredths

Take a bar and cut it into $10$ equal parts. Each part is one tenth.

Now cut every one of those parts into $10$ smaller ones. The bar is in $100$ parts, and each of them is one hundredth.

Count the small parts inside a single tenth. There are $10$ of them.

$$\frac{1}{10} = \frac{10}{100}$$

Three of the ten parts are shaded. Split every part into ten and each shaded part becomes $10$ small ones.

$3$ shaded parts turn into $30$ shaded small parts. The shading itself has not moved.

$$\frac{3}{10} = \frac{30}{100}$$

Same shading. Two ways of counting it.

The rewrite in numbers

The picture cut each part into $10$, which does two things at once. There are $10$ times as many parts, and $10$ times as many of them are shaded.

$$\frac{3}{10} = \frac{3 \times 10}{10 \times 10} = \frac{30}{100}$$

Multiplying the top and the bottom by the same number is the equivalent fraction rule. Here the number is always $10$.

Tenths	Hundredths
$\tfrac{1}{10}$	$\tfrac{10}{100}$
$\tfrac{4}{10}$	$\tfrac{40}{100}$
$\tfrac{7}{10}$	$\tfrac{70}{100}$
$\tfrac{10}{10}$	$\tfrac{100}{100}$

Write a zero after the top number. That is all the rewrite amounts to, and the split into ten is the reason it works.

Why tenths and hundredths cannot be added as they stand

$$\frac{3}{10} + \frac{25}{100}$$

Adding the top numbers gives $28$. Of what? The two fractions count parts of different sizes, so $28$ describes nothing.

This is the same reason you cannot add $3$ meters to $25$ centimeters and call the answer $28$. Two counts only add when they count the same unit.

Fix it by rewriting one of them:

$$\frac{3}{10} = \frac{30}{100}$$ $$\frac{30}{100} + \frac{25}{100} = \frac{55}{100}$$

Now both count hundredths, so the tops add.

Always convert the tenths up to hundredths, not the other way. Going down would need $25$ hundredths written as $2.5$ tenths, and a fraction wants whole numbers on top.

The same idea as decimal places

A decimal point does this conversion without being asked.

Fraction	Decimal
$\tfrac{3}{10}$	$0.3$
$\tfrac{30}{100}$	$0.30$
$\tfrac{25}{100}$	$0.25$
$\tfrac{55}{100}$	$0.55$

$0.3$ and $0.30$ are the same number. That extra zero is exactly the rewrite from $\tfrac{3}{10}$ to $\tfrac{30}{100}$.

$$0.3 + 0.25 = 0.55$$

Lining up the decimal points does the converting for you. That is what column addition of decimals relies on.

Worked examples

Common mistakes

Practice problems