2.03a, 2.03b - Percentages of quantities

Parts per hundred

Would $25\%$ of $80$ be larger than $80\%$ of $25$? We will return to this comparison after giving the percentage sign a precise meaning.

Imagine a square split into one hundred equal cells. Shading twenty-five cells shades $25\%$ of the square. The whole square is $100\%$; the percentage describes a share of that whole.

[DIAGRAM: asset_name: Twenty-five parts per hundred; asset_slug: percent-hundred-grid; caption: Twenty-five shaded cells out of one hundred represent one quarter of the whole.; description: A ten-by-ten square grid with exactly twenty-five pale blue cells. Labels identify 25 out of 100, 25 percent, 0.25 and one quarter as the same proportion. The whole grid represents 100 percent.]

The whole need not contain one hundred objects. $25\%$ of twenty counters is five counters, because a quarter of twenty is five. The percentage tells you the share; the amount depends on the size of the whole.

Percentages can exceed $100\%$. A jug holding $150\%$ of a cup's volume holds one and a half cups. This does not mean $150\%$ of the jug is full: always identify which amount represents $100\%$.

Check that you can distinguish the proportion from its amount.

Fractions, decimals and percentages

Fractions, decimals and percentages can name exactly the same proportion.

Since a percentage counts hundredths, multiply a decimal by $100$ to obtain its percentage number: $0.08=8\%$. Reverse this by dividing the percentage number by $100$: $8\%=0.08$. To write a percentage as a fraction, put its number over $100$, then simplify. Thus $8\%=\frac8{100}=\frac2{25}$.

Change only the numerator from $3$ to $7$. More than five fifths means more than one whole, so expect a percentage above $100\%$.

A fractional percentage is possible too: $12.5\%=0.125=\frac18$. The percentage sign already means “divide by one hundred”; do not divide twice.

Convert a larger number of fifths using the same method.

A percentage of an amount

To find a percentage of an amount, multiply the amount by the fraction or decimal represented by that percentage.

For friendly percentages, use familiar fractions or build from $10\%$, $5\%$ and $1\%$. For example, $15\%$ of $80$ is $8+4=12$. On a calculator, the direct calculation is $15\div100\times80$, which gives the same result.

The fraction method often makes a non-calculator calculation shorter.

Change only the percentage to $17.5\%$. The whole stays $64\,\mathrm{kg}$; a direct decimal multiplication is now convenient.

Now answer the opening question: $25\%$ of $80$ and $80\%$ of $25$ are both $20$. More generally, $\frac a{100}\times b=\frac b{100}\times a$: swapping the percentage number and the numerical amount leaves the product unchanged. It can make a calculation easier, such as $4\%$ of $75$ becoming $75\%$ of $4$, which is $3$.

Apply the same percentage to a new whole.

One amount as a percentage of another

To express one amount as a percentage of another, first decide which amount is the whole.

If you scored $18$ out of $30$, the whole is $30$, so your share is $\frac{18}{30}$. Expressing this share in hundredths gives $60\%$. This reverses “find a percentage of an amount”. Both amounts must use the same units; convert units first if necessary.

The denominator is a decision, not simply the larger of the two numbers.

Change only the reference amount, to $24\,\mathrm{kg}$. The compared amount remains $30\,\mathrm{kg}$, so its percentage must now exceed $100\%$.

Keep the “of” amount in view when choosing the denominator.

Compare a new amount with the same reference.

Quick Recap

• A percentage names parts per hundred of a chosen whole.
• Multiply a decimal by $100$ to obtain its percentage number; divide to reverse.
• Find a percentage of an amount by multiplying by its decimal or fractional form.
• Express one amount as a percentage of another by dividing by the reference amount, then multiplying by $100\%$.

The opening amounts are equal: $25\%$ of $80$ and $80\%$ of $25$ both give $20$.