23. Percent Case Studies: Shopping, Grades, Budgeting, and Data

Percent questions become easier when you identify the base: what is the percent being taken of?

Case study 1 - shopping with discount and tax. A tablet costs 240 dollars. It is 15% off, then 6% sales tax is applied to the sale price.
\[ \text{sale price}=0.85(240)=204. \]
\[ \text{tax}=0.06(204)=12.24. \]
\[ \text{final price}=204+12.24=216.24. \]

Case study 2 - grades. A student earns 45 points out of 60:
\[ \frac{45}{60}=0.75=75\%. \]

Case study 3 - budgeting. A 2,400 dollar monthly income has 30% reserved for rent:
\[ 0.30(2400)=720. \]

Case study 4 - data comparison. Store A sells 45 of 60 items. Store B sells 72 of 90 items.
\[ \frac{45}{60}=75\%,\qquad \frac{72}{90}=80\%. \]
Store B has the higher sell-through rate, even though both sold different counts.

Case-study habit: write the base next to every percent. "15% off 240" and "6% tax on 204" use different bases.

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Deep reasoning: case studies may switch the reference whole

Multi-step percent problems often change bases between steps. A budget category may be a percent of total income, while an increase is a percent of last year's category amount. Write the base beside every rate before calculating.

Tables and graphs also require attention to what the displayed percent represents. Row percent, column percent, and percent of the grand total can answer different questions even when they use the same count.

GED® worked example

A store employee earns a base salary of ($1{,}600) plus 4% commission on ($12{,}500) in monthly sales. Twenty percent of total earnings is then set aside for taxes.

\[ \text{commission}=0.04(12{,}500)=500, \]

\[ \text{gross earnings}=1600+500=2100, \]

\[ \text{tax set-aside}=0.20(2100)=420, \]

\[ \text{remaining}=2100-420=1680. \]

The 4% rate uses sales as its base; the 20% rate uses total earnings. Applying both rates to the same number would misread the situation.

For data comparison, suppose 36 of 48 morning participants and 55 of 80 evening participants pass. Rates are 75% and 68.75%, so the morning group has the higher pass rate even though its pass count is smaller.

Error clinic

Do not choose a denominator merely because it is the largest number in the problem. Identify the logical whole for that particular percentage. Also distinguish a percent of total from a percent change over time.

Mastery summary

Draw a quantity map, attach each rate to its base, calculate one labeled step at a time, preserve full precision, and interpret the final result in context.