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MathChapter 7: Percents

Notes

Percents and Equations Notes

Translate percent language into equations or proportions and solve for an unknown part, rate, or whole.

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Most ordinary percent questions use one relationship: a part equals a decimal rate times a whole. The wording determines which of those three quantities is unknown.

The part-rate-whole relationship

Master equation
\[P=rW\]

The part \(P\) equals the decimal rate \(r\) multiplied by the whole or base \(W\).

Three percent-equation problem types
Problem typeVerbal patternEquationProportion formUnknown
Finding the partWhat is \(p\%\) of \(W\)?\(P=rW\)\(p/100=P/W\)Part \(P\)
Finding the percentWhat percent of \(W\) is \(P\)?\(r=P/W\)\(p/100=P/W\)Rate \(r\) or \(p\)
Finding the whole\(P\) is \(p\%\) of what number?\(W=P/r\)\(p/100=P/W\)Whole \(W\)

Translate percent language

Three words reveal the equation

of

Translate ‘of’ as multiplication.

is

Translate ‘is’ as an equals sign.

what

Translate ‘what’ as the unknown quantity.

Worked example

Find a part

What is \(18\%\) of \(250\)?

  1. Convert

    \(18\%=0.18\).

  2. Translate

    \(P=0.18(250)\).

  3. Calculate

    \(P=45\).

The part is \(45\).
Worked example

Find a percent

What percent of \(64\) is \(14.4\)?

  1. Use the whole as the base

    \(r=14.4/64\).

  2. Calculate the decimal rate

    \(r=0.225\).

  3. Convert

    \(0.225=22.5\%\).

\(14.4\) is \(22.5\%\) of \(64\).
Worked example

Find a whole

\(7.2\) is \(0.6\%\) of what number?

  1. Convert carefully

    \(0.6\%=0.006\), not \(0.6\).

  2. Build the equation

    \(7.2=0.006W\).

  3. Solve

    \(W=7.2/0.006=1{,}200\).

The whole is \(1{,}200\).

Small, fractional, and large percentages

The same equation handles every rate. For \(0.5\%\), use \(0.005\). For \(2.5\%\), use \(0.025\). For \(125\%\), use \(1.25\). A rate above \(100\%\) can make the part larger than the whole.

Reverse percent problems

Recovering the base

  1. Name the base

    Let the unknown whole or original value be \(W\).

  2. Convert the rate

    Write the percent as a decimal.

  3. Write the relationship

    Use \(P=rW\) with the known part.

  4. Solve by division

    Use \(W=P/r\), then check \(rW=P\).

Worked example

A value larger than the base

A measurement of \(288\) is \(240\%\) of another measurement. Find the base.

  1. Convert

    \(240\%=2.40\).

  2. Equation

    \(288=2.40W\).

  3. Solve

    \(W=288/2.40=120\).

The base measurement is \(120\).

Common equation mistakes

  • Choosing the part as the whole or base.
  • Using a percent as a whole number instead of converting it to a decimal.
  • Treating \(0.5\%\) as \(0.5\) instead of \(0.005\).
  • Reversing part and whole in the proportion \(p/100=P/W\).
  • Translating ‘of’ as addition rather than multiplication.
  • Stopping with a decimal rate when the question requests a percent.
  • Using the percent-change formula for a simple percent-of problem.
Mini check

Check your understanding

In ‘\(24\) is \(15\%\) of what number,’ which equation is correct?

  1. \(24=0.15W\)
  2. \(W=0.15(24)\)
  3. \(24=15W\)
  4. \(0.15=24W\)
Show answer and explanation

Answer: \(24=0.15W\)

The known part is \(24\), the decimal rate is \(0.15\), and the unknown is the whole.

Key takeaways

What to remember

  • The reusable relationship is \(P=rW\), with \(r\) written as a decimal.
  • Use the wording to determine whether the unknown is the part, rate, or whole.
  • The proportion \(p/100=P/W\) is equivalent to the master equation.
  • Small percents require two decimal shifts: \(0.5\%=0.005\).
  • Check a reverse-percent answer by multiplying the recovered whole by the rate.
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.