6.5 Percent Applications: Tax, Tip, Discount, Markup & Percent Change

6.5 Percent Applications: Tax, Tip, Discount, Markup & Percent Change

Quick Answer: Percent means per hundred, so $37% = \frac{37}{100} = 0.37$. Every percent question is one of three types: find the part, find the percent, or find the whole. The fastest reliable technique is the single multiplier - a 15% increase is $\times 1.15$ and a 15% decrease is $\times 0.85$, done in one step. Percent change is always $\frac{\text{new} - \text{original}}{\text{original}}$, with the original in the denominator. Successive changes multiply, so a 20% markup followed by a 20% discount lands below where it started.

---

Where This Sits in the Blueprint

Percent is one of the highest-yield topics in middle school mathematics because it recurs in data analysis (a category band reported as a percentage), in proportional reasoning, and in nearly every financial word problem.

---

The Conversion Triangle

Fractions, decimals, and percents describe the same quantity three ways. Fluent movement among them is a prerequisite for everything else.

                        PERCENT
                       /        \
      (divide by 100) /          \ (multiply by 100)
                     v            v
              DECIMAL <--------> FRACTION
                    (place value / simplify)

[!IMPORTANT] Percents can exceed 100 and can fall below 1. 150% of 40 is 60 - more than the whole, which is exactly what a markup does. 0.5% of 40 is 0.2 - and 0.5% is $0.005$, not $0.5$. Misplacing that decimal by a factor of 100 is the most expensive error in this topic.

---

The Three-Part Percent Relationship

$\text{part} = \text{percent} \times \text{whole}$

Every basic percent item supplies two of the three and asks for the third.

The proportion form works identically and is what many Florida classrooms teach:

$\frac{\text{part}}{\text{whole}} = \frac{\text{percent}}{100}$

Use whichever you can execute without hesitation. Speed matters more than elegance on a timed adaptive test.

---

The Single Multiplier

Two-step methods - find the tax, then add it - double the number of chances to slip. One multiplication does the same work.

Worked example. A jacket lists at $$68.00$, is discounted 25%, and then carries 7% Florida sales tax.

$\$68.00 \times 0.75 \times 1.07 = \$51.00 \times 1.07 = \$54.57$

Two multiplications, no intermediate addition, no rounding drift.

[!IMPORTANT] Order matters less than students fear - but only for multiplication. Applying tax then discount gives $$68.00 \times 1.07 \times 0.75 = $54.57$, the same result, because multiplication is commutative. What is not interchangeable is a percent change and a flat dollar coupon. A $$5$ coupon before a 25% discount is not the same as after it.

---

Percent Increase and Decrease

$\text{percent change} = \frac{\text{new value} - \text{original value}}{\text{original value}} \times 100$

The original value is always the denominator. A positive result is an increase; a negative result is a decrease.

Example. A club grows from 40 members to 52.

$\frac{52 - 40}{40} = \frac{12}{40} = 0.30 = 30\% \text{ increase}$

Example. The same club falls from 52 back to 40.

$\frac{40 - 52}{52} = \frac{-12}{52} \approx -0.2308 \approx 23.1\% \text{ decrease}$

Notice the asymmetry: up 30%, then down 23.1%, returns to the start. The percentages differ because the denominators differ. This is not a rounding artifact - it is the whole point.

---

Successive Percent Changes Multiply

A store marks an item up 20%, then runs a 20% off sale. Is the price back to the original?

$P \times 1.20 \times 0.80 = P \times 0.96$

No. The final price is 96% of the original - a net 4% decrease. The markup was 20% of the smaller original; the discount was 20% of the larger marked-up price, so the discount removed more dollars than the markup added.

[!WARNING] Never add successive percent changes. "30% off, then an extra 20% off" is not 50% off. It is $0.70 \times 0.80 = 0.56$, which is 44% off. Retail advertising depends on shoppers adding; FAST items depend on students not.

---

Working Backward from a Final Amount

This is the item type that separates procedural fluency from real understanding. A final price is given and the original is unknown.

Example. After a 7% sales tax, a customer pays $$85.60$. What was the pre-tax price?

The wrong move is to take 7% of $$85.60$ and subtract it, because the tax was 7% of the original, not of the total. The right move divides by the multiplier:

$\text{original} = \frac{\$85.60}{1.07} = \$80.00$

Check forward: $$80.00 \times 1.07 = $85.60$. Correct.

Example. A coat costs $$91$ after a 35% discount. What was the original price?

$\text{original} = \frac{\$91}{0.65} = \$140$

Check: $$140 \times 0.65 = $91$. Correct. Subtracting 35% of $$91$ would have given $$59.15$ - not even in the right direction, since the original must be larger than the discounted price.

   FORWARD:   original  --  x multiplier  -->  final
   BACKWARD:  original  <--  / multiplier  --  final

   ALWAYS check by running the forward direction on your answer.

---

Simple Interest and Percent Error

Simple interest: $I = Prt$, where $P$ is principal, $r$ is the annual rate as a decimal, and $t$ is time in years. On $$1{,}200$ at 4% for 3 years: $I = 1200 \times 0.04 \times 3 = $144$, so the balance is $$1{,}344$. Note that $t$ must be in years - 18 months is $t = 1.5$, not 18.

Percent error: $\frac{|\text{measured} - \text{actual}|}{\text{actual}} \times 100$. A student measures 47 cm when the true length is 50 cm: $\frac{3}{50} = 6%$ error. The actual value is the denominator, mirroring percent change.

---

Common Exam Traps & Misconceptions

[!WARNING]

Trap 1: Putting the new value in the denominator

Percent change divides by the original. Going from 80 to 100 is a 25% increase ($20/80$), not a 20% increase ($20/100$). Ask which number came first in time.

[!WARNING]

Trap 2: Subtracting the percent from the final amount

To undo a 7% tax you divide by 1.07. Taking 7% off the total gives the wrong answer because the tax was computed on a smaller base.

[!WARNING]

Trap 3: Adding stacked discounts

Stacked percent changes multiply. 30% then 20% is 44% off, not 50% off.

[!WARNING]

Trap 4: Misplacing the decimal on percents under 1

A 0.5% fee on $$4{,}000$ is $$20$, not $$2{,}000$. Convert first and check the size: 0.5% is half of one percent, so the answer must be less than 1% of the amount.

[!WARNING]

Trap 5: Answering the tax instead of the total

Distractors regularly include the tax amount, the discount amount, or the pre-tax subtotal. Re-read the final sentence of the problem before selecting.