FN12, FR10, HN12, HR10 - Percentages as operators

FN12, FR10, HN12, HR10 - Percentages as operators

Read percentages as parts per hundred, convert them to fractions and decimals, and use them to calculate amounts. FN12 and HN12 explain percentage operators; FR10 and HR10 connect those operators with percentages above one hundred and comparisons between quantities.

Parts per hundred

Two students each score 1818 marks. Have they scored the same proportion of the available marks? Percentages let us compare with a common reference of one hundred, even when the actual totals differ.

Percentage

A percentage expresses a quantity as a number of parts per hundred. The symbol %\% means “per hundred”, so p%=p100p\%=\frac p{100}.

Thus 100%=1100\%=1 whole, 50%=1250\%=\frac12, and 0%=00\%=0. Divide the percentage number by 100100 to get a decimal; multiply a decimal by 100100 to get the percentage number.

Worked example 1: percentage to decimal and fraction

Question. Write 72%72\% as a decimal and as a fraction in simplest form.

Plan. Divide by 100100, then simplify the fraction.

72%=72100=0.7272100=1825divide numerator and denominator by 4\begin{aligned} 72\%&=\frac{72}{100}=0.72\\ \frac{72}{100}&=\frac{18}{25} &&\text{divide numerator and denominator by }4 \end{aligned}

Answer. 0.720.72 and 1825\frac{18}{25}.

Check. 0.72×100=720.72\times100=72, and 1825×44=72100\frac{18}{25}\times\frac44=\frac{72}{100}.

Add one whole hundred to the percentage. It now represents more than one whole.

Worked example 2: more than one hundred percent

Question. Write 172%172\% as a decimal and as a fraction in simplest form.

Plan. Use the same division by 100100, retaining the whole-number part.

172%=172100=1.72172100=4325divide both by 4\begin{aligned} 172\%&=\frac{172}{100}=1.72\\ \frac{172}{100}&=\frac{43}{25} &&\text{divide both by }4 \end{aligned}

Answer. 1.721.72 and 4325\frac{43}{25}.

Check. 172%=100%+72%=1+0.72172\%=100\%+72\%=1+0.72, agreeing with 1.721.72.

Small percentages need place-value care too: 6.5%=0.0656.5\%=0.065, while 0.7%=0.0070.7\%=0.007. To change a fraction to a percentage, either make the denominator 100100 or divide and multiply by 100100. For example, 38=0.375=37.5%\frac38=0.375=37.5\%.

Watch out

0.70.7 is 70%70\%, not 0.7%0.7\%. The percent sign itself means division by 100100; do not simply attach it to an unchanged decimal.

Convert a fresh percentage above one whole.

Percentage of an amount

To find a percentage of an amount, multiply by the corresponding fraction or decimal. The amount is the whole to which the percentage applies.

Percentage of an amount

p% of A=p100×Ap\%\text{ of }A=\frac p{100}\times A

Here AA is the reference amount. For example, 25%25\% of AA is 14A\frac14 A.

Not in the formula booklet: learn it.

Without a calculator, build useful percentages from 10%10\%, 5%5\% or 1%1\%. For instance, 10%10\% of 250250 is 2525, 5%5\% is 12.512.5, and 1%1\% is 2.52.5.

Worked example 3: a percentage below one hundred

Question. Work out 18%18\% of 250250.

Plan. Multiply 250250 by 18/10018/100.

18% of 250=18100×250=0.18×250=45\begin{aligned} 18\%\text{ of }250&=\frac{18}{100}\times250\\ &=0.18\times250=45 \end{aligned}

Answer. 4545.

Check. 10%+5%+3%=18%10\%+5\%+3\%=18\%, and 25+12.5+7.5=4525+12.5+7.5=45, using 3%=3×2.53\%=3\times2.5.

Change only the percentage by adding a whole hundred percent. The result now includes the original amount as well.

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