C1.4, C1.5; E1.4, E1.5 - Fractions, decimals and percentages

Fractions as decimals

Which is larger: $\frac35$, $0.62$ or $58\%$? The largest-looking written number need not represent the largest amount. Converting to one common form will answer the question.

A fraction means division: $\frac38$ means $3\div8$. If its denominator can easily be made $10$, $100$ or $1000$, an equivalent fraction gives the decimal directly. For example, $\frac7{20}=\frac{35}{100}=0.35$. Otherwise, divide the numerator by the denominator, keeping track of remainders.

The conversions here use terminating decimals. Some decimal expansions continue forever; recognising their behaviour comes later.

Keep the denominator and change only the numerator. Seven eighths is close to one, which will help check the new decimal.

To go back from a terminating decimal to a fraction, use its place value, then simplify. For example,

Two decimal places mean hundredths, even when the first decimal digit is zero: $0.04=\frac4{100}=\frac1{25}$. For a value above one, the fraction can be improper: $1.25=\frac{125}{100}=\frac54$.

Percentages as hundredths

“Per cent” means “per hundred”. A percentage is another way to write a fraction with denominator one hundred: $37\%$ means $\frac{37}{100}$. The symbol is part of the value, so $37\%$ and $37$ are very different numbers.

Each grid below represents one whole split into one hundred equal cells. Thirty-seven shaded cells represent $37\%$. A full grid and another thirty-seven cells represent $137\%$.

[DIAGRAM: asset_name: Percentages below and above one whole; asset_slug: percentages_hundredths; caption: Thirty-seven hundredths is less than one whole; one hundred and thirty-seven hundredths is greater than one whole.; description: Two panels compare equal-size hundred-square grids. The first panel has one grid with thirty-seven blue cells labelled 37 percent equals 0.37. The second panel has two grids, the first entirely sage green and the second with thirty-seven sage green cells, labelled 137 percent equals 1.37. Each separate grid represents one whole.]

To convert a percentage to a decimal, divide the stated number by $100$. To convert a decimal to a percentage, multiply by $100$ and attach the percentage symbol. These operations undo each other: $0.6\times100=60$, so $0.6=60\%$.

Now increase only the percentage by one hundred percentage points. This adds exactly one whole.

Percentages may also contain decimals. For example, $2.5\%=\frac{2.5}{100}=0.025$. To write the last value as a fraction with whole-number numerator and denominator, use $0.025=\frac{25}{1000}=\frac1{40}$.

Converting and comparing representations

To compare different forms, convert them to a common form first. Decimals are convenient when they terminate; equivalent fractions with a common denominator are another valid choice. Keep exact values rather than rounding away a small difference.

You can also move directly from a fraction to a percentage by making the denominator $100$: $\frac35=\frac{60}{100}=60\%$. In reverse, $60\%=\frac{60}{100}=\frac35$. When a denominator does not fit neatly into $100$, divide to obtain a decimal, then multiply its value by $100$.

Change only $58\%$ to $68\%$. That value must move from the smallest position to the largest.

The opening comparison is now settled: $0.62$ is larger than both $\frac35=0.60$ and $58\%=0.58$. The written $58$ looked large only because its hundredths unit was hidden in the percentage symbol.

Quick Recap

• A fraction means numerator divided by denominator.
• A terminating decimal becomes a fraction over $10$, $100$, $1000$ or another suitable power of ten; then simplify.
• Percentage means per hundred. Divide by $100$ to obtain a decimal, or multiply a decimal by $100$ to obtain its percentage number.
• Percentages above $100\%$ represent values above one whole.
• Convert to a common exact form before comparing.

Of the three opening values, $0.62$ is greatest: $0.58<0.60<0.62$.