8.3 Percentages, Percentage Change & Financial Problems

Percentages, Percentage Change & Financial Problems

Core Concept: Percentage questions assess more than basic fractions out of 100. High-scoring candidates treat percentages as algebraic multipliers. By viewing a $15\%$ increase as a multiplication by $1.15$ and a $20\%$ discount as a multiplication by $0.80$, you eliminate cumbersome intermediate steps, bypass common reverse-percentage traps, and effortlessly solve multi-step compounding financial problems.

Financial mathematics questions on Australian scholarship and selective entry tests frequently simulate real-world transactions: retail markups, promotional sales discounts, Australian Goods and Services Tax (GST), and term deposit simple interest. Understanding the underlying base of every percentage and mastering decimal multipliers is the key to solving these questions within the 30 to 45 seconds available per question.

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Percentages as Multipliers: The Mental Speed Advantage

A percentage is simply a rational fraction with a denominator of 100 ($r\% = \frac{r}{100}$). Instead of calculating the percentage amount and then adding or subtracting it in two distinct steps, apply a single decimal or fractional multiplier directly to the principal quantity.

Percentage Increase (+r%):   New Value = Original × (1 + r/100)
Percentage Decrease (-r%):   New Value = Original × (1 - r/100)

The Multiplier Dictionary

• Increase by 5%: $\times 1.05$
• Increase by 15%: $\times 1.15$
• Increase by 25%: $\times 1.25$ or $\times \frac{5}{4}$
• Decrease by 10%: $\times 0.90$ or $\times \frac{9}{10}$
• Decrease by 20%: $\times 0.80$ or $\times \frac{4}{5}$
• Decrease by 35%: $\times 0.65$ or $\times \frac{13}{20}$
• Decrease by 12.5%: $\times 0.875$ or $\times \frac{7}{8}$

High-Yield Benchmark Equivalence Table

Memorizing these fraction-decimal-percentage conversions allows you to bypass decimal long multiplication entirely:

Worked Example: What is $37.5\%$ of $480? $480 \times \frac{3}{8} = (480 \div 8) \times 3 = 60 \times 3 = 180$

The result is $180; Solved mentally in under 5 seconds.

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Percentage of an Amount & Finding the Whole

1. Decomposition Method (10% and 1% Building Blocks)

When calculating an unfamiliar percentage mentally, break it into $10\%$, $5\%$, and $1\%$ components:

• To calculate 17% of $350:
  • 10% = $35.00
  • 5% = $17.50 (half of 10%)
  • 1% = $3.50
  • 2% = $7.00
  • 17% = 10% + 5% + 2% = $35 + $17.50 + $7 = $59.50

2. The Reversibility Law ($x\%$ of $y = y\%$ of $x$)

Because multiplication is commutative:

$x\% \times y = \frac{x \times y}{100} = y\% \times x$

When asked to compute an awkward percentage of a friendly number, flip the terms:

• Evaluate: $16\%$ of $75$
• Flip: $75\%$ of $16 = \frac{3}{4} \times 16 = 12$. Solved instantly.
• Evaluate: $84\%$ of $25$
• Flip: $25\%$ of $84 = \frac{1}{4} \times 84 = 21$.

3. Finding the Whole (Unitary & Multiplier Methods)

If a question gives a partial amount and asks for the total quantity:

• Problem: If $15\%$ of an unknown budget is $45, find the total budget.

• Unitary Method: If 15% is $45, then 1% is $\frac{45}{15} = 3$ ($3), and 100% is $3 \times 100 = 300$ ($300).

• Multiplier Method: $\text{Total} = \frac{\text{Part}}{\text{Rate}} = \frac{45}{0.15} = \frac{4,500}{15} = 300$

  The total budget is $300.

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Reverse Percentage Problems: The #1 Scholarship Trap

Reverse percentage problems are a classic scholarship-test trap. They describe a price after an increase or discount has already been applied, and ask you to calculate the original pre-change price.

The Fundamental Fallacy

Critical Warning: If a coat is sold for $80 after a $20\%$ discount, the original price is NOT $96. Adding $20\%$ back onto $80 ($80 + $16 = $96) is completely wrong because the original $20\%$ discount was calculated on the larger original price, not on the discounted sale price!

The Reverse Calculation Formula

Always set up the relationship algebraically using the multiplier:

$\text{Original Value} = \frac{\text{Resulting Value}}{\text{Multiplier}} = \frac{\text{Resulting Value}}{1 \pm r}$

Worked Example: Original Price After Discount

A bicycle is purchased on sale for $360 after a $25\%$ discount. What was the original retail price?

• Step 1 (Identify multiplier): A $25\%$ discount means the customer paid $100\% - 25\% = 75\% = 0.75 = \frac{3}{4}$ of the original price.

• Step 2 (Set up equation): $\text{Sale Price} = \text{Original} \times 0.75$ $360 = \text{Original} \times \frac{3}{4}$

• Step 3 (Solve for original): $\text{Original} = 360 \div \frac{3}{4} = 360 \times \frac{4}{3} = 120 \times 4 = 480$

  The original retail price was $480.

Worked Example: Original Price Before GST / Markup

A smartphone sells for $1,035 including a $15\%$ luxury import levy. What was the pre-levy cost? $\text{Original Cost} = \frac{1,035}{1.15} = \frac{103,500}{115} = 900$

The pre-levy cost was $900.

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Successive Percentage Changes & Compounding Effects

When a price or quantity undergoes multiple consecutive percentage adjustments, percentages can never be directly added or subtracted.

$\text{Total Multiplier } M_{\text{total}} = M_1 \times M_2 \times M_3 \dots$

1. The Non-Additive Discount Fallacy

"A department store offers a 20% discount, followed by an additional 10% loyalty card discount at the register. What is the total effective discount?"

• Untrained candidates add: $20\% + 10\% = 30\%$ (Wrong).
• Correct compounding multiplier: $M_{\text{total}} = (1 - 0.20) \times (1 - 0.10) = 0.80 \times 0.90 = 0.72$
• Because the customer pays $72\%$ of the original price, the total effective discount is: $100\% - 72\% = 28\%$

2. Symmetrical Increase and Decrease ($+x\%$, then $-x\%$)

A classic scholarship question asks: "A share price increases by $20\%$ on Monday and decreases by $20\%$ on Tuesday. What is the net change from its original value?"

The Symmetric Loss Law: When an amount increases by $x\%$ and then decreases by the same $x\%$, the net result is always a decrease of $\left(\frac{x}{10}\right)^2\%$.

• Compound multiplier: $(1 + 0.20)(1 - 0.20) = 1.20 \times 0.80 = 0.96$
• Net result: $0.96 - 1.00 = -0.04 \implies$ 4% net decrease.
• Shortcut verification: $\left(\frac{20}{10}\right)^2 = 2^2 = 4\%$ loss.
• If $x = 30\%$: $(1.3)(0.7) = 0.91 \implies 9\%$ loss ($\left(\frac{30}{10}\right)^2 = 3^2 = 9\%$).

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Financial Mathematics: Profit, Loss, GST & Simple Interest

1. Profit & Loss on Cost Price

• Cost Price (CP): The expense incurred by the seller to acquire or manufacture the item.
• Selling Price (SP): The price at which the item is sold to the consumer.
• Profit: $\text{Profit} = SP - CP$ (when $SP > CP$)
• Loss: $\text{Loss} = CP - SP$ (when $SP < CP$)

Fundamental Rule: Percentage profit and percentage loss are always calculated as a percentage of the Cost Price (CP) unless the question explicitly specifies otherwise:

$\text{Percentage Profit} = \frac{SP - CP}{CP} \times 100\%$ $\text{Percentage Loss} = \frac{CP - SP}{CP} \times 100\%$

Worked Example: A merchant buys an antique clock for $250 and sells it for $325. What is the percentage profit? $\text{Profit} = 325 - 250 = 75 \quad (\text{Profit of } \text{\textdollar}75)$

$\text{Percentage Profit} = \frac{75}{250} \times 100\% = \frac{3}{10} \times 100\% = 30\%$

2. Australian Goods and Services Tax (GST)

In Australia, the Goods and Services Tax (GST) is a flat 10% tax levied on most goods and services.

• Adding GST: $\text{Price with GST} = \text{Price without GST} \times 1.10$
• Extracting GST from Shelf Price: $\text{GST Component} = \frac{\text{Shelf Price (incl. GST)}}{11}$ $\text{Price without GST} = \text{Shelf Price} \times \frac{10}{11}$

Example: An invoice shows a total of $242 including GST.

• The GST included is: $\frac{242}{11} = 22$ (giving $22).
• The pre-tax price is: $242 - $22 = $220. (Trap: Taking $10\%$ of $242 gives $24.20, which is completely incorrect!)

3. Simple Interest ($I = Prt$)

Simple interest is calculated exclusively on the original principal across the entire loan or investment duration:

$I = Prt$ $A = P + I = P(1 + rt)$

Where:

• $I$ = Total interest earned or paid
• $P$ = Principal (initial capital invested or borrowed)
• $r$ = Annual interest rate (expressed as a decimal or fraction: $6\% = 0.06 = \frac{6}{100}$)
• $t$ = Time duration expressed in years

Time Conversion Rules for Simple Interest

Examiners regularly test non-integer year durations. You must convert months or days into fractional years before substituting into the formula:

• Months: $t = \frac{\text{number of months}}{12}$ (e.g., $9 \text{ months} = \frac{9}{12} = \frac{3}{4} = 0.75 \text{ years}$)
• Days: $t = \frac{\text{number of days}}{365}$ (or $\frac{\text{days}}{360}$ if standard commercial bank year is specified)

Worked Example: Simple Interest Over Fractional Years

Amira deposits $4,800 into a fixed deposit account earning $5.5\%$ per annum simple interest for a period of 9 months. What is the total interest earned, and what is the final balance?

• Step 1 (Extract parameters): $P = 4,800, \quad r = 0.055 = \frac{11}{200}, \quad t = \frac{9}{12} = \frac{3}{4} \text{ years}$

• Step 2 (Calculate interest $I$): $I = 4800 \times \frac{11}{200} \times \frac{3}{4}$ $I = \left(\frac{4800}{200}\right) \times 11 \times \frac{3}{4} = 24 \times 11 \times \frac{3}{4} = 6 \times 11 \times 3 = 198$

  The total interest earned is $198.

• Step 3 (Calculate final balance $A$): $A = P + I = 4,800 + 198 = 4,998$

  The final balance is $4,998.

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High-Yield Traps in Percentages & Financial Math

1. The Base Shift Trap (Dividing by New Instead of Original)

A stock rises from $50 to $80. A student calculates the percentage gain as: $\frac{80 - 50}{80} = \frac{30}{80} = 37.5\%$. This is wrong. The base must always be the original starting value: $\frac{30}{50} = 60\%$.

2. The Successive Linear Addition Trap

Assuming an item increased by $10\%$ and then $15\%$ underwent a $25\%$ total increase. The true multiplier is $1.10 \times 1.15 = 1.265$, representing a $26.5\%$ increase.

3. The 10% Deduction Fallacy on Australian GST

Attempting to find the pre-GST cost of a $330 item by subtracting 10%: $330 - $33 = $297. This is false. The correct pre-GST cost is $330 / 1.1 = $300, with exactly $30 GST.