Percents, Hundred Grids, and Decimals

A rate tells how much there is of one quantity for each fixed amount of another. A percent is a rate per hundred. Per means for each, so a percent tells how many for each $100$.

A ratio compares two quantities. A percent is a special ratio. It compares an amount with a whole, and it always counts the whole as $100$ equal parts. So $37\%$ is the ratio $37$ to $100$. You can also write it as the fraction $\frac{37}{100}$, read "$37$ hundredths". One hundredth is $1$ of the $100$ equal parts of a whole.

For example, suppose $37$ of every $100$ seats in a hall are filled. Then $37\%$ of the seats are filled.

A percent does not tell you how big the whole is. It tells you how many of the whole's $100$ equal parts you have. So a percent has meaning only when you know which whole it describes.

A hundred grid is a square cut into $10$ rows and $10$ columns. That makes $10 \times 10 = 100$ equal small squares. The full grid is the whole. All $100$ of its squares together make $100\%$.

Each small square is $1$ of the $100$ equal parts. So one square is $\frac{1}{100}$ of the grid, which is $1\%$ of the grid.

In the figure, $37$ squares are shaded. That is $37$ for each $100$, so $37\%$ of the grid is shaded. Count in tens to check: the shaded squares make $3$ full groups of $10$ and $7$ more, and $3 \times 10 + 7 = 37$.

To model a whole-number percent from $0\%$ to $100\%$, shade that many squares of one hundred grid. A percent greater than $100\%$ needs more than one grid, as you will see later in this lesson.

A decimal can count hundredths too. Here is the place-value fact you need. In a decimal, the first place to the right of the decimal point counts tenths. The second place counts hundredths.

A tenth and a hundredth are parts of different sizes. One tenth of a whole is the same amount as $10$ hundredths of that whole: $\frac{1}{10} = \frac{10}{100}$. On a hundred grid, one full row of $10$ squares is one tenth of the grid, and so is one full column.

Read $0.37$ place by place: $3$ tenths and $7$ hundredths of one whole. To add them, count both in parts of the same size. Each tenth is $10$ hundredths, so $3$ tenths is $3 \times 10 = 30$ hundredths. Then $\frac{30}{100} + \frac{7}{100} = \frac{37}{100}$.

So $0.37$ is $37$ hundredths of the whole. That is exactly what $37\%$ means.

The key idea gives you a rule for each direction.

• Percent to decimal: divide the number in front of the $\%$ sign by $100$. For $37\%$, that is $37 \div 100 = 0.37$. This works because $\frac{37}{100}$ means $37 \div 100$. Dividing by $100$ makes each digit worth one hundredth as much: the $3$ tens become $3$ tenths, and the $7$ ones become $7$ hundredths.
• Decimal to percent: multiply the decimal by $100$. For $0.37$, that is $0.37 \times 100 = 37$, so $0.37 = 37\%$. Multiplying by $100$ counts how many hundredths the decimal holds.

Multiplying by $100$ and dividing by $100$ are inverse operations: each one undoes the other. So converting and then converting back returns the number you started with. The amount does not change. Only the way you write it changes.

Two conditions go with these rules. The percent and the decimal must describe the same whole. And every digit must stay in its own place, so use a $0$ to hold any place that has no other digit.

A percent can be greater than $100\%$. All $100$ squares of one hundred grid make $100\%$, which is one whole. To show more than that, you need another grid.

The extra grid must be the same size as the first and cut into the same $100$ equal squares. Then each grid is one equal-sized whole, and each small square is still $1\%$ of one grid.

The next figure shows $125\%$ on two equal-sized grids. In all, $125$ squares are shaded: $100$ fill one grid and $25$ are in the other.

Here is the idea to keep. A percent is a rate per hundred: it tells how many for each $100$ equal parts of a whole. On a hundred grid, each small square is $1\%$ of the grid, so the shaded squares show the percent. A percent counts hundredths, so you divide its number by $100$ to write a decimal, and you multiply a decimal by $100$ to write a percent. Keep every digit in its place, and make sure both forms describe the same whole.

Next, you will convert between percents and fractions and use benchmark percents with strip models.