19. Finding the Percent When You Know Part and Whole

Sometimes the question gives the part and whole, and asks for the percent.

Use:
\[ \text{percent}=\frac{\text{part}}{\text{whole}}\times100\%. \]

Example: 18 is what percent of 72?
\[ \frac{18}{72}=0.25=25\%. \]

Another example: A student got 42 out of 50 questions correct.
\[ \frac{42}{50}=0.84=84\%. \]

Case study - budget: A family spends 450 dollars on rent from a 1,800 dollar monthly budget.
\[ \frac{450}{1800}=0.25=25\%. \]
Rent is 25% of the budget.

Language clues:

• "What percent of" usually means part divided by whole.
• "is" often marks the part.
• "of" often points to the whole.

In "18 is what percent of 72," 18 is the part and 72 is the whole.

Common mistake: dividing whole by part. \(72\div18=4\), which would become 400%, not the correct 25%.

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Deep reasoning: the whole belongs in the denominator

When the part and whole are known, the rate is

\[ R=\frac{P}{W}. \]

The denominator must be the reference whole—the original group, total capacity, total attempts, or other complete amount. Convert the quotient to percent by multiplying by 100%.

GED® worked example

A student answers 54 of 72 questions correctly.

\[ R=\frac{54}{72}=\frac34=0.75=75\%. \]

If the same student misses 18 questions, the missed rate is \(\frac{18}{72}=25\%\). The correct and missed rates sum to 100%, providing a strong check.

Suppose a machine's output rises by 18 units from an original 120 units. If asked "18 is what percent of 120?", calculate

\[ \frac{18}{120}=0.15=15\%. \]

The wording identifies 18 as the part and 120 as the whole. A later percent-change session will emphasize why the original value serves as this base.

Error clinic

Reversing part and whole gives the reciprocal and often a percent above 100. In a survey where 45 of 60 agree, \(45\div60=75\%\); \(60\div45\approx133.3\%\) answers a different question.

Do not judge performance from the count alone. 48 correct out of 50 is a higher rate than 90 correct out of 100 even though 90 is the larger count.

Mastery summary

State the reference whole, form part divided by whole, convert the decimal to percent, label the rate, and use a complement or benchmark to check.