The Central Idea
A percentage adjustment starts from a reference amount.
If the original amount is \(A\) and the rate is \(r\), then the percentage amount is
\[ \boxed{rA}. \]An increase by rate \(r\) produces
\[ \boxed{A(1+r)}, \]while a decrease by rate \(r\) produces
\[ \boxed{A(1-r)}. \]Percentages Must Be Written as Decimals
Convert the stated percentage before substitution:
\[ 8\%=0.08, \] \[ 15\%=0.15, \]and
\[ 2.5\%=0.025. \]Sales Tax
If an item has pre-tax price \(P\) and tax rate \(t\), then the tax amount is
\[ \boxed{T=tP}. \]The total price is
\[ P+T. \]Therefore,
\[ \boxed{ P_{\text{total}}=P(1+t) }. \]Example: Find Sales Tax
An item costs \(\$80\) before tax. The tax rate is \(7.5\%\).
\[ T=0.075(80). \] \[ T=6. \]Therefore,
\[ \boxed{\$6\text{ tax}}. \]Example: Find the Total after Tax
Continue the previous example.
\[ P_{\text{total}}=80(1.075). \] \[ P_{\text{total}}=86. \]Therefore,
\[ \boxed{\$86}. \]Reverse Tax Problems
Sometimes the after-tax price is known and the pre-tax price is unknown.
If
\[ A=P(1+t), \]then
\[ \boxed{ P=\frac{A}{1+t} }. \]Example: Recover the Pre-Tax Price
An after-tax total is \(\$107\) with a \(7\%\) tax rate.
\[ P=\frac{107}{1.07}. \] \[ P=100. \]Therefore,
\[ \boxed{\$100\text{ before tax}}. \]Discounts
If an item has original price \(P\) and discount rate \(d\), then the discount amount is
\[ \boxed{D=dP}. \]The sale price is
\[ P-D. \]Therefore,
\[ \boxed{ P_{\text{sale}}=P(1-d) }. \]Example: Find a Discount
A \(\$120\) item is discounted by \(25\%\).
\[ D=0.25(120)=30. \]Therefore, the discount is
\[ \boxed{\$30}. \]Example: Find the Sale Price
Using the multiplier directly,
\[ P_{\text{sale}}=120(0.75). \] \[ P_{\text{sale}}=90. \]Therefore,
\[ \boxed{\$90}. \]Reverse Discount Problems
If a sale price \(S\) results from discount rate \(d\), then
\[ S=P(1-d). \]Therefore,
\[ \boxed{ P=\frac{S}{1-d} }. \]Example: Recover the Original Price
After a \(20\%\) discount, an item costs \(\$64\).
\[ 64=P(0.80). \] \[ P=\frac{64}{0.80}. \] \[ P=80. \]Therefore,
\[ \boxed{\$80\text{ original price}}. \]Discount Followed by Tax
If a discount is applied first and tax is then applied to the discounted price, the multipliers are applied in sequence.
With original price \(P\), discount rate \(d\), and tax rate \(t\),
\[ \boxed{ P_{\text{final}} = P(1-d)(1+t) }. \]Example: Discount Then Tax
An item costs \(\$200\), is discounted \(15\%\), and then taxed at \(8\%\).
Discount multiplier:
\[ 0.85. \]Tax multiplier:
\[ 1.08. \]Therefore,
\[ P_{\text{final}}=200(0.85)(1.08). \] \[ P_{\text{final}}=183.60. \]Thus,
\[ \boxed{\$183.60}. \]Successive Discounts
Two discounts are applied multiplicatively, not additively.
If discounts \(d_1\) and \(d_2\) are applied in sequence,
\[ \boxed{ P_{\text{final}} = P(1-d_1)(1-d_2) }. \]Example: Two Discounts
A product receives a \(20\%\) discount followed by another \(10\%\) discount.
The final multiplier is
\[ 0.80(0.90)=0.72. \]Therefore, the final price is
\[ \boxed{72\%\text{ of the original}}. \]The equivalent single discount is
\[ \boxed{28\%}, \]not \(30\%\).
Equivalent Single Discount
If two discounts \(d_1\) and \(d_2\) are applied,
\[ 1-d_{\text{eq}} = (1-d_1)(1-d_2). \]Therefore,
\[ \boxed{ d_{\text{eq}} = 1-(1-d_1)(1-d_2) }. \]Successive Increases
Repeated increases also multiply.
If an amount increases by \(r_1\) and then by \(r_2\),
\[ \boxed{ A_{\text{final}} = A(1+r_1)(1+r_2) }. \]Equal Increase and Decrease Do Not Cancel
Suppose a price rises by \(20\%\) and then falls by \(20\%\).
The combined multiplier is
\[ 1.20(0.80)=0.96. \]Therefore, the final price is
\[ \boxed{96\%\text{ of the original}}. \]Commission
A commission is often calculated as a percentage of sales.
If sales are \(S\) and the commission rate is \(c\), then commission is
\[ \boxed{C=cS}. \]Example: Straight Commission
A salesperson earns \(6\%\) commission on \(\$18{,}000\) in sales.
\[ C=0.06(18000). \] \[ C=1080. \]Therefore,
\[ \boxed{\$1{,}080}. \]Base Pay plus Commission
Some compensation models combine fixed pay \(B\) with commission.
If sales are \(S\) and commission rate is \(c\),
\[ \boxed{ E=B+cS }. \]Example: Base plus Commission
A salesperson earns \(\$500\) base pay plus \(4\%\) of sales. If sales are \(\$20{,}000\),
\[ E=500+0.04(20000). \] \[ E=500+800. \] \[ E=1300. \]Therefore,
\[ \boxed{\$1{,}300}. \]Find Required Sales for a Target Commission
If
\[ C=cS, \]then
\[ \boxed{ S=\frac{C}{c} }. \]Example: Required Sales
A salesperson earns \(5\%\) commission and wants \(\$2{,}000\) in commission.
\[ S=\frac{2000}{0.05}. \] \[ S=40000. \]Therefore,
\[ \boxed{\$40{,}000\text{ in sales}}. \]Target Earnings with Base plus Commission
If
\[ E=B+cS, \]and target earnings are \(E_t\), then
\[ E_t=B+cS. \]Therefore,
\[ \boxed{ S=\frac{E_t-B}{c} }. \]Example: Required Sales for Target Earnings
A salesperson earns \(\$600\) base pay plus \(3\%\) commission. How much must be sold to earn \(\$1{,}500\)?
\[ 1500=600+0.03S. \] \[ 900=0.03S. \] \[ S=30000. \]Therefore,
\[ \boxed{\$30{,}000\text{ in sales}}. \]Tiered Commission
Some commission plans change rates after a threshold.
For example:
- \(4\%\) on the first \(\$10{,}000\) of sales;
- \(6\%\) on sales above \(\$10{,}000\).
Such models are piecewise rather than one single linear formula.
Example: Tiered Commission
Suppose sales are \(\$16{,}000\).
First tier:
\[ 0.04(10000)=400. \]Remaining sales:
\[ 16000-10000=6000. \]Second-tier commission:
\[ 0.06(6000)=360. \]Total commission:
\[ 400+360=760. \]Therefore,
\[ \boxed{\$760}. \]Markup
Markup is commonly defined as an amount added to cost.
If cost is \(C\) and markup rate on cost is \(m\), then markup amount is
\[ \boxed{M=mC}. \]Selling price is
\[ C+M. \]Therefore,
\[ \boxed{ S=C(1+m) }. \]Example: Markup on Cost
An item costs a retailer \(\$80\). It is marked up \(35\%\) on cost.
\[ S=80(1.35). \] \[ S=108. \]Therefore,
\[ \boxed{\$108}. \]Reverse Markup
If selling price \(S\) and markup rate \(m\) on cost are known,
\[ S=C(1+m). \]Therefore,
\[ \boxed{ C=\frac{S}{1+m} }. \]Example: Recover Cost from Markup
A product sells for \(\$156\) after a \(30\%\) markup on cost.
\[ C=\frac{156}{1.30}. \] \[ C=120. \]Therefore,
\[ \boxed{\$120\text{ cost}}. \]Markup Rate
If cost is \(C\) and selling price is \(S\), then markup amount is
\[ S-C. \]Markup rate on cost is
\[ \boxed{ m=\frac{S-C}{C} }. \]Example: Find Markup Rate
An item costs \(\$75\) and sells for \(\$96\).
Markup:
\[ 96-75=21. \]Markup rate:
\[ \frac{21}{75}=0.28. \]Therefore,
\[ \boxed{28\%\text{ markup on cost}}. \]Markup versus Profit Margin
Markup rate and profit margin use different reference amounts.
Markup on cost:
\[ \boxed{ \text{markup rate} = \frac{S-C}{C} }. \]Gross margin as a fraction of selling price:
\[ \boxed{ \text{margin rate} = \frac{S-C}{S} }. \]They are not generally equal.
Example: Markup versus Margin
An item costs \(\$80\) and sells for \(\$100\).
Difference:
\[ 100-80=20. \]Markup on cost:
\[ \frac{20}{80}=0.25. \]Therefore, markup is
\[ \boxed{25\%}. \]Margin on selling price:
\[ \frac{20}{100}=0.20. \]Therefore, margin is
\[ \boxed{20\%}. \]Convert Markup Rate to Margin Rate
If selling price is
\[ S=C(1+m), \]then gross profit is
\[ S-C=mC. \]Margin rate is
\[ \frac{mC}{C(1+m)}. \]Therefore,
\[ \boxed{ \text{margin} = \frac{m}{1+m} }. \]Convert Margin Rate to Markup Rate
If margin rate is \(g\), then
\[ g=\frac{S-C}{S}. \]The corresponding markup rate on cost is
\[ \boxed{ m=\frac{g}{1-g} }. \]Example: Margin to Markup
A seller wants a \(25\%\) gross margin on selling price.
Then
\[ m= \frac{0.25}{1-0.25}. \] \[ m= \frac{0.25}{0.75} = \frac13. \]Therefore, the equivalent markup on cost is
\[ \boxed{33\frac13\%}. \]Markdown after Markup
A retailer may first mark up cost and later discount the listed price.
If cost is \(C\), markup rate is \(m\), and later discount rate is \(d\), then
\[ \boxed{ S_{\text{final}} = C(1+m)(1-d) }. \]Example: Markup Then Discount
An item costs \(\$100\), is marked up \(50\%\), then discounted \(20\%\).
\[ S=100(1.50)(0.80). \] \[ S=120. \]Therefore,
\[ \boxed{\$120}. \]Even after the discount, the selling price remains above cost.
Net Percent Change from Multipliers
If the final multiplier is \(M\), then the net percent change relative to the original is
\[ \boxed{M-1}. \]If
\[ M>1, \]the result is a net increase.
If
\[ M<1, \]the result is a net decrease.
Example: Net Change after Markup and Discount
A \(40\%\) markup followed by a \(25\%\) discount gives multiplier
\[ 1.40(0.75)=1.05. \]Therefore, the final price is
\[ \boxed{5\%\text{ above cost}}. \]Reverse Successive-Change Problems
If the final amount \(F\) is known after several percentage adjustments, divide by the combined multiplier.
For discount \(d\) followed by tax \(t\),
\[ F=P(1-d)(1+t). \]Therefore,
\[ \boxed{ P= \frac{F}{ (1-d)(1+t) } }. \]Example: Reverse Discount and Tax
After a \(20\%\) discount and \(5\%\) tax, an item costs \(\$84\). Find the original price.
\[ 84=P(0.80)(1.05). \] \[ 84=0.84P. \] \[ P=100. \]Therefore,
\[ \boxed{\$100}. \]Reference Amount Matters
Percentages are always percentages of something.
Tax is usually a percentage of taxable price.
Discount is usually a percentage of the current listed price.
Commission is usually a percentage of sales.
Markup is often a percentage of cost.
Margin is a percentage of selling price.
Percent of Cost versus Percent of Selling Price
Suppose profit amount is
\[ S-C. \]Dividing by cost answers:
“What percent of cost is the profit?”
Dividing by selling price answers:
“What percent of selling price is the profit?”
Those are different questions.
Percentage Points versus Percent Change
Suppose a commission rate rises from \(4\%\) to \(5\%\).
The increase is
\[ 1 \]percentage point.
Relative to the original \(4\%\), the percent increase in the rate is
\[ \frac{5-4}{4}=0.25. \]Therefore, the commission rate increased by
\[ \boxed{25\%} \]relative to its original level.
Combining Flat Amounts with Percentages
Some problems combine percentage adjustments with fixed-dollar amounts.
For example, a bill may include a base fee \(F\) plus a percentage charge \(rA\):
\[ \boxed{ T=F+rA }. \]Do not force every applied problem into a pure multiplier model.
Example: Flat Fee plus Percentage
A service charges \(\$25\) plus \(3\%\) of transaction value.
For a \(\$2{,}000\) transaction,
\[ T=25+0.03(2000). \] \[ T=25+60. \] \[ T=85. \]Therefore,
\[ \boxed{\$85}. \]Piecewise Percentage Rules
Some tax, commission, fee, or discount systems use different rates on different ranges.
Such systems are modeled piecewise.
The amount in each tier is calculated separately, then the pieces are added.
This is the same interval logic used in staggered work-rate problems.
Checking a Tax Result
A positive tax should make the after-tax price greater than the pre-tax price:
\[ P(1+t)>P \]when
\[ t>0. \]Checking a Discount Result
A positive discount less than \(100\%\) should satisfy
\[ 0<P(1-d)<P. \]Checking a Commission Result
For straight commission with
\[ 0<c<1, \]commission should be less than total sales:
\[ 0<cS<S. \]Checking a Markup Result
With positive markup on cost,
\[ S=C(1+m) \]should satisfy
\[ S>C. \]Common Error: Using the Percent as a Whole Number
A \(7\%\) tax rate is
\[ 0.07, \]not
\[ 7. \]Common Error: Adding Successive Percentages
A \(20\%\) discount followed by a \(10\%\) discount does not usually equal a \(30\%\) discount.
Use multipliers:
\[ 0.80(0.90)=0.72. \]The equivalent discount is
\[ \boxed{28\%}. \]Common Error: Subtracting Tax after a Discount
Tax is an increase, so its multiplier is
\[ 1+t. \]A discount uses
\[ 1-d. \]Keep the signs conceptually tied to whether the adjustment raises or lowers the current amount.
Common Error: Using the Final Amount as the Original Base
If an item is discounted \(20\%\), the discount is \(20\%\) of the pre-discount price, not \(20\%\) of the sale price.
Reverse problems therefore require division by the multiplier.
Common Error: Confusing Markup with Margin
Markup uses cost in the denominator:
\[ \frac{S-C}{C}. \]Margin uses selling price:
\[ \frac{S-C}{S}. \]Always identify the stated reference amount.
Common Error: Applying a Tier Rate to the Entire Amount
In a tiered commission model, a higher rate may apply only to the amount above a threshold.
Do not apply the higher rate retroactively to the entire sales total unless the plan explicitly says so.
Common Error: Ignoring Order
When flat fees and percentage changes are mixed, order can matter.
For example,
\[ (P+F)(1+t) \]differs from
\[ P(1+t)+F. \]Translate the stated sequence carefully.
A Reliable Procedure for Taxes and Discounts
- Identify the reference price.
- Convert the percentage to decimal form.
- Use \(1+r\) for an increase.
- Use \(1-r\) for a decrease.
- Apply successive adjustments in the stated order.
- For reverse problems, divide by the multiplier.
- Check that the result moves in the expected direction.
A Reliable Procedure for Commission Problems
- Identify the sales amount.
- Identify the commission rate.
- Write \(C=cS\) for straight commission.
- Add base pay if the plan includes it.
- Split tiered plans into separate ranges.
- For target earnings, solve the resulting equation for sales.
A Reliable Procedure for Markup Problems
- Identify cost \(C\).
- Determine whether the stated percent is markup on cost or margin on selling price.
- For markup on cost, use \(S=C(1+m)\).
- For reverse markup, divide by \(1+m\).
- For margin, use \((S-C)/S\).
- Check that the selling price is consistent with the stated percentage basis.
Practice 1: Sales Tax
Find the sales tax on a \(\$240\) purchase at \(6.5\%\).
Practice 2: Total after Tax
Find the total price for Practice 1.
Practice 3: Reverse Tax
An after-tax total is \(\$216\) with an \(8\%\) tax rate. Find the pre-tax price.
Practice 4: Discount
A \(\$150\) item is discounted by \(30\%\). Find the sale price.
Practice 5: Reverse Discount
After a \(25\%\) discount, an item costs \(\$90\). Find the original price.
Practice 6: Discount Then Tax
A \(\$300\) item is discounted \(20\%\) and then taxed at \(7.5\%\). Find the final price.
Practice 7: Successive Discounts
An item receives a \(15\%\) discount and then another \(20\%\) discount. Find the equivalent single discount.
Practice 8: Straight Commission
Find a \(7\%\) commission on \(\$24{,}000\) in sales.
Practice 9: Base plus Commission
A salesperson earns \(\$700\) plus \(4\%\) of sales. Find total earnings on \(\$15{,}000\) in sales.
Practice 10: Target Earnings
A salesperson earns \(\$500\) plus \(5\%\) of sales. How much must be sold to earn \(\$1{,}750\)?
Practice 11: Markup
An item costs \(\$120\) and is marked up \(40\%\) on cost. Find the selling price.
Practice 12: Reverse Markup
An item sells for \(\$195\) after a \(30\%\) markup on cost. Find the cost.
Practice 13: Markup versus Margin
An item costs \(\$60\) and sells for \(\$75\). Find the markup rate on cost and the margin rate on selling price.
Practice 14: Markup Then Discount
An item costs \(\$200\), is marked up \(50\%\), then discounted \(10\%\). Find the final selling price and the net percent change from cost.
Practice 15: Identify the Error
A student says a \(20\%\) discount followed by a \(20\%\) tax always returns a price to its original value.
Explain the error and find the final multiplier.
Practice Answers
Practice 1
\[ T=0.065(240). \] \[ T=15.60. \]Therefore,
\[ \boxed{\$15.60}. \]Practice 2
\[ 240+15.60=255.60. \]Therefore,
\[ \boxed{\$255.60}. \]Practice 3
\[ P=\frac{216}{1.08}. \] \[ P=200. \]Therefore,
\[ \boxed{\$200}. \]Practice 4
\[ P_{\text{sale}}=150(0.70). \] \[ P_{\text{sale}}=105. \]Therefore,
\[ \boxed{\$105}. \]Practice 5
\[ 90=P(0.75). \] \[ P=\frac{90}{0.75}=120. \]Therefore,
\[ \boxed{\$120}. \]Practice 6
\[ P=300(0.80)(1.075). \] \[ P=258. \]Therefore,
\[ \boxed{\$258}. \]Practice 7
Final multiplier:
\[ 0.85(0.80)=0.68. \]Therefore, the equivalent single discount is
\[ 1-0.68=0.32. \] \[ \boxed{32\%}. \]Practice 8
\[ C=0.07(24000). \] \[ C=1680. \]Therefore,
\[ \boxed{\$1{,}680}. \]Practice 9
\[ E=700+0.04(15000). \] \[ E=700+600. \] \[ E=1300. \]Therefore,
\[ \boxed{\$1{,}300}. \]Practice 10
\[ 1750=500+0.05S. \] \[ 1250=0.05S. \] \[ S=25000. \]Therefore,
\[ \boxed{\$25{,}000\text{ in sales}}. \]Practice 11
\[ S=120(1.40). \] \[ S=168. \]Therefore,
\[ \boxed{\$168}. \]Practice 12
\[ C=\frac{195}{1.30}. \] \[ C=150. \]Therefore,
\[ \boxed{\$150}. \]Practice 13
Profit amount:
\[ 75-60=15. \]Markup rate:
\[ \frac{15}{60}=0.25. \] \[ \boxed{25\%\text{ markup}}. \]Margin rate:
\[ \frac{15}{75}=0.20. \] \[ \boxed{20\%\text{ margin}}. \]Practice 14
\[ S=200(1.50)(0.90). \] \[ S=270. \]Therefore,
\[ \boxed{\$270}. \]Relative to cost,
\[ \frac{270-200}{200}=0.35. \]Therefore, the net increase from cost is
\[ \boxed{35\%}. \]Practice 15
The tax is applied after the discount, so the two percentages use different reference amounts.
The final multiplier is
\[ 0.80(1.20)=0.96. \]Therefore, the final price is
\[ \boxed{96\%\text{ of the original price}}. \]What to Remember
An increase by rate \(r\) uses multiplier
\[ \boxed{1+r}. \]A decrease by rate \(r\) uses multiplier
\[ \boxed{1-r}. \]Sales tax:
\[ \boxed{ P_{\text{total}}=P(1+t) }. \]Discount:
\[ \boxed{ P_{\text{sale}}=P(1-d) }. \]Straight commission:
\[ \boxed{C=cS}. \]Markup on cost:
\[ \boxed{ S=C(1+m) }. \]The most important habit is to identify the correct reference amount before calculating a percentage. Successive percentage changes should usually be handled with multipliers rather than by adding or subtracting the rates.
Where We Go Next
Taxes, discounts, commissions, and markups apply algebra to familiar financial adjustments. The next family broadens applied algebra into scientific and engineering settings, where variables may represent physical quantities, scaling laws, units, and empirical relationships.
The next lesson, Scientific, Engineering, and Scaling Models, develops those applications.