Percentages of quantities and percentage comparisons

Percentage

\(\displaystyle 25\%\)

Which value is equivalent to the percentage shown?

One hundred percent names the reference whole

A percentage is a proportion, not a fixed number of units. The reference whole determines the quantity it represents

Both calculations take one half, but the wholes are different

To find a percentage of a quantity, divide the percentage number by one hundred and multiply by the quantity

The result has the same unit as the reference quantity

Use one quarter as a shortcut

Find the exact quantity, not a percentage

Divide the reference quantity into four equal parts

One quarter of sixty is fifteen

Find one half of the reference quantity

Build a percentage from ten percent or one percent

Ten percent is one tenth, so divide the quantity by ten

One percent is one hundredth, so divide by one hundred

Use these pieces to build other percentages

For twenty percent, take twice the ten-percent amount

Find ten percent

Find the exact quantity, not a percentage

Identify the given quantity as the reference whole

Convert the percentage to a fraction of one whole

Multiply that fraction by the reference quantity

The percentage is below one hundred, so the result is smaller than the positive reference quantity

Check by comparing the result with the original quantity

Build twenty percent

Use the decimal multiplier

Find the exact quantity, not a percentage

Identify the given quantity as the reference whole

Convert the percentage to a fraction of one whole

Multiply that fraction by the reference quantity

The percentage is below one hundred, so the result is smaller than the positive reference quantity

Check by comparing the result with the original quantity

Find a non-benchmark percentage

The percentage number can have a fractional part

A percentage such as twelve point five percent still means out of one hundred

Taking one eighth may be faster than multiplying a decimal

Thirty-seven point five percent is three eighths

Do not round the percentage number before finding an exact quantity

Apply an exact fractional percentage

Find the exact quantity, not a percentage

Identify the given quantity as the reference whole

Convert the percentage to a fraction of one whole

Multiply that fraction by the reference quantity

The percentage is below one hundred, so the result is smaller than the positive reference quantity

Check by comparing the result with the original quantity

Use three eighths or an exact decimal

The reference quantity can be a decimal

Zero percent and one hundred percent

Zero percent takes none of the reference quantity

One hundred percent takes the entire reference quantity

More than one hundred percent is more than one reference whole

This is meaningful when comparing amounts or calculating a multiple of a reference, not when claiming that a subset contains more than every member of its whole

Take none of the quantity

Take the full quantity

The reference quantity stays at eighty. Change the percentage from zero through one hundred and beyond

What happens when the setting moves from fifty percent to one hundred percent?

Find more than one whole

Find the exact quantity, not a percentage

Identify the given quantity as the reference whole

Convert the percentage to a fraction of one whole

Multiply that fraction by the reference quantity

The percentage exceeds one hundred, so the result is larger than the reference quantity

Check by comparing the result with the original quantity

Take one and a half times the reference

Find the percentage when both quantities are known

To express one quantity as a percentage of another, the quantity named after of is the reference whole

Divide the compared quantity by the reference, then multiply the ratio by one hundred

For twenty-four out of sixty, the ratio is two fifths

Twenty-four is forty percent of sixty. The decimal ratio zero point four is not the percentage number

Convert a compared-to-reference ratio

Express the compared quantity as a percentage of the reference quantity. Both quantities use the same units

Use the stated reference quantity as one hundred percent

Divide the compared quantity by that reference, keeping both in the same units

Multiply the exact ratio by one hundred to obtain the percentage number

The compared quantity is smaller than the reference, so its percentage is below one hundred

Check that the percentage of the reference reproduces the compared quantity

Use the specified reference

An exact comparison can have a decimal percentage

Express the compared quantity as a percentage of the reference quantity. Both quantities use the same units

Use the stated reference quantity as one hundred percent

Divide the compared quantity by that reference, keeping both in the same units

Multiply the exact ratio by one hundred to obtain the percentage number

The compared quantity is smaller than the reference, so its percentage is below one hundred

Check that the percentage of the reference reproduces the compared quantity

Do not round an exact fractional percentage

Interpret the size of the compared quantity

Zero compared with a positive reference is zero percent

An amount equal to its reference is one hundred percent

A larger amount can exceed one hundred percent of its reference

Forty-five is one hundred and twenty-five percent of thirty-six. This asks for the whole compared amount, not just the excess

A reference of zero cannot be used in this division

Express zero relative to a positive reference

Compare two equal amounts

Compare an amount greater than its reference

Express the compared quantity as a percentage of the reference quantity. Both quantities use the same units

Use the stated reference quantity as one hundred percent

Divide the compared quantity by that reference, keeping both in the same units

Multiply the exact ratio by one hundred to obtain the percentage number

The compared quantity is larger than the reference, so its percentage is above one hundred

Check that the percentage of the reference reproduces the compared quantity

Express the entire compared amount

Reversing the reference changes the percentage

Thirty is fifty percent of sixty

But sixty is two hundred percent of thirty

The same two quantities give different comparisons because a different amount is named as one hundred percent

Compared quantity

\(\displaystyle 18\)

Reference quantity

\(\displaystyle 24\)

Which calculation expresses eighteen as a percentage of twenty-four?

Compare quantities in matching units

The numerator and denominator must use the same unit before their ratio is calculated

There are one hundred centimetres in one metre, so two metres is two hundred centimetres

The matching length units cancel in the ratio. The final comparison is twenty-five percent, not twenty-five centimetres

Convert the reference before comparing

Decide which quantity is missing

If a group has forty members and twenty-five percent attend a session, the missing value is the number attending

Ten members attend. The answer is a count

If ten out of forty members attend and the percentage is missing, divide by the group total and multiply by one hundred

The answer is twenty-five percent. These are related calculations, but they answer different questions

Find the number of participants

Find a percentage of all responses

Use a calculator without losing the meaning

A calculator can carry out the same percentage calculation. Enter the percentage number divided by one hundred, multiplied by the reference quantity

Avoid relying on a calculator percent key whose behaviour you have not checked

For a quick size check, twenty percent of two hundred and thirty-five is forty-seven, so eighteen percent should be a little smaller

Keep an exact terminating result rather than rounding it without a stated reason

Calculate and check an exact measurement

Keep amount, proportion and reference distinct

For a percentage of a quantity, convert the percentage to a multiplier and apply it to the reference whole

For one quantity as a percentage of another, divide by the stated reference and multiply by one hundred

Zero percent is none of the reference and one hundred percent is the full reference. A comparison can exceed one hundred percent when the compared amount is larger

Match units before comparing, and do not confuse a whole compared amount with just its excess

Interpret the answer as an amount or a percentage according to the question, then check it against the original quantities