1.2, 1.3 - Fractions, decimals and percentages

Decimal place value and ordering

A label says $0.35$, another says $\frac7{20}$, and a third says $35\%$. Do they describe different amounts? We will decide by writing them in the same form. First, make every decimal digit count.

The decimal point separates whole-number places from fractional places. Moving one place right divides the place value by ten.

Read $0.506$ as five tenths, no hundredths and six thousandths. The zero holds the hundredths place. A zero at the end of the fractional part adds no value: $0.56=0.560$. A zero inserted before a nonzero digit changes its place: $0.506\ne0.56$.

To order decimals, align the points and compare from left to right. The first unequal place decides the order. These are place-value facts, absent from both the Foundation and Higher formulae sheets.

Now change the number of decimal places. Trailing zeros let you compare the same sized parts.

Here is a surprising consequence: decimals have no “next value” after $0.56$. For example, $0.56<0.5601<0.561<0.57$. You can keep inserting a value between two different decimals by using a finer place. A longer decimal need not be larger.

For negative decimals, use position on the number line: $-0.65<-0.56$, because $-0.65$ is farther left. Keep each given number once when writing an ordered list.

Terminating decimals as exact fractions

A decimal's final place gives you a fraction denominator. Three decimal places mean thousandths, even when some of the digits are zero.

Write the digits as a whole-number numerator over the appropriate power of ten. Then cancel common factors. The result is exact, so use an equals sign. This place-value conversion is not in either tier's formulae sheet: learn it.

Now keep three decimal places but put a zero immediately after the point. That zero still occupies a place.

The same method works above one: $1.24=\frac{124}{100}=\frac{31}{25}$. An improper fraction is appropriate because the value exceeds one. Every terminating decimal is a fraction because its denominator can be a power of ten. Conversely, reduce a fraction first: its decimal terminates exactly when its denominator has no prime factors other than $2$ and $5$. Those are the prime factors of $10$, so the denominator can scale to a power of ten. For example, $\frac3{8}=\frac{375}{1000}$ terminates, while $\frac13$ cannot acquire a power-of-ten denominator and recurs. Denominator $1$ also works: it gives a whole number.

“Give your answer as a fraction in simplest form” asks for the final cancellation too. A rounded decimal cannot replace that exact fraction.

Converting fractions, decimals and percentages

A fraction tells you which division to do. A percentage tells you how many hundredths you have. These meanings connect all three forms.

To turn a fraction into a decimal, divide its numerator by its denominator. To turn a decimal into a percentage, multiply by $100$ and write $\%$. Reverse that operation to turn a percentage into a decimal.

A denominator that scales easily to $100$ gives a useful route without a calculator.

Now change only the fraction to one whose decimal does not terminate. The conversion still works, but we must show the recurring tail rather than round it away.

Dot notation marks a digit that repeats for ever. Converting a recurring decimal back to a fraction needs a further algebraic method; here we only convert the fraction forwards. For terminating decimals, both directions are available: $0.08=\frac8{100}=\frac2{25}=8\%$. Similarly, $12.5\%=\frac{12.5}{100}=\frac{125}{1000}=\frac18$.

The opening labels match: $\frac7{20}=0.35=35\%$. Writing quantities in the same form makes a comparison fair.

Quick Recap

• Compare decimals from the left, using equal place values.
• Trailing zeros do not change a decimal's value.
• Write a terminating decimal over a power of ten, then simplify.
• Divide for fraction-to-decimal conversion; multiply a decimal by $100$ for its percentage number.

The opening labels all describe the same amount: $\frac7{20}=0.35=35\%$.