Financial Mathematics

Simple interest $I = PRT = 2000 \times 0.05 \times 3 = \$300$. (Adding it to the principal, $2300, is the final balance, not the interest.)

A 25% discount leaves 75% of the price: $80 \times 0.75 = \$60$. ($20 is the amount of the discount, not the sale price.)

Compound interest: $A = P(1 + r)^n = 1000 \times 1.04^2 = 1000 \times 1.0816 = \$1081.60$. (Simple interest would give $1080; the extra 1.60 is interest earned on the first year's interest.)

Adding 10% GST multiplies the quote by 1.10: $600 \times 1.10 = \$660$. ($60 is the GST itself.)

Interest $= 500 \times 0.06 \times 4 = \$120$. Total amount $= 500 + 120 = \$620$. ($120 is only the interest, not the final balance.)

Straight-line depreciation subtracts the same amount each year: $10\,000 - 3 \times 1500 = 10\,000 - 4500 = \$5500$. ($4500 is the total loss in value, not the value remaining.)

Commission $= 8\% \text{ of } 15\,000 = 0.08 \times 15\,000 = \$1200$.

Final amount $A = 3000 \times 1.05^3 = 3000 \times 1.157625 = \$3472.88$. Interest earned $= 3472.88 - 3000 = \$472.88$. ($450 is the simple interest; $3472.88 is the final balance, not the interest.)

Simple interest $= 1000 \times 0.10 \times 2 = \$200$. Compound: $1000 \times 1.1^2 = \$1210$, so $210 interest. The difference is $210 - 200 = \$10$ (the interest earned on the first year's interest).

Compounded monthly, the rate per month is $\dfrac{12\%}{12} = 1\%$ over $12$ periods: $A = 2000 \times 1.01^{12} = 2000 \times 1.126825 = \$2253.65$. (Charging 12% just once gives $2240; monthly compounding earns more.)

Reducing balance keeps 85% of the value each year: $A = 20\,000 \times 0.85^2 = 20\,000 \times 0.7225 = \$14\,450$. ($14,000 wrongly subtracts a fixed 15% of the original, $3000, each year.)

Appreciation compounds like interest: $A = 500\,000 \times 1.06^3 = 500\,000 \times 1.191016 = \$595\,508$. ($590,000 adds a flat 6% of the original ($30,000) each year, missing the compounding.)

Apply the multipliers in turn: $1.20 \times 0.80 = 0.96$, which is a 4% decrease overall. (The percentages do not cancel, because the 20% fall acts on the larger, increased price.)

Rearrange $A = P(1+r)^n$: $P = \dfrac{A}{(1+r)^n} = \dfrac{2420}{1.1^2} = \dfrac{2420}{1.21} = \$2000$. (Taking 20% off $2420, giving $1936, is the common error.)

Rearrange $I = PRT$: $R = \dfrac{I}{PT} = \dfrac{600}{4000 \times 3} = \dfrac{600}{12\,000} = 0.05 = 5\%$ per year.

Rearrange $I = PRT$: $T = \dfrac{I}{PR} = \dfrac{320}{1000 \times 0.08} = \dfrac{320}{80} = 4$ years.

Try values of $1.06^n$ until it reaches 2: $1.06^{12} \approx 2.01$, so it takes about 12 years. (The "rule of 72" gives the same estimate: $72 \div 6 = 12$.)

The income over $45,000 is $60\,000 - 45\,000 = 15\,000$. Tax on that portion $= 0.325 \times 15\,000 = \$4875$. Total tax $= 5092 + 4875 = \$9967$. (Applying 32.5% to the whole income gives the wrong $19,500.)

Normal pay $= 38 \times 24 = \$912$. Overtime is 5 hours at $1.5 \times 24 = \$36$ per hour: $5 \times 36 = \$180$. Total $= 912 + 180 = \$1092$. (Paying all 43 hours at $24 gives $1032, missing the overtime loading.)

Compare the price per gram. 500 g pack: $\dfrac{320\text{c}}{500} = 0.64\text{ c/g}$. 750 g pack: $\dfrac{450\text{c}}{750} = 0.60\text{ c/g}$. The 750 g pack is cheaper per gram, so it is the better value.

Total repaid $= 250 \times 24 = \$6000$ over the 24 months. The interest is the extra paid above the amount borrowed: $6000 - 5000 = \$1000$. ($6000 is the total repaid, not the interest.)

Convert to US$: $800 \times 0.65 = \text{US}\$520$. After spending, $520 - 300 = \text{US}\$220$ is left. Convert back: $\dfrac{220}{0.65} = \text{AU}\$338.46$. (US$220 is the amount left in US dollars, before converting back.)

Compounded quarterly, the rate per quarter is $\dfrac{8\%}{4} = 2\%$ over $4$ periods: $A = 8000 \times 1.02^4 = 8000 \times 1.082432 = \$8659.46$. (Charging 8% once gives $8640; quarterly compounding earns a little more.)

The selling price is the cost price plus 20%, so $\text{cost} \times 1.20 = 360$, giving $\text{cost} = \dfrac{360}{1.20} = \$300$. (Taking 20% off $360, giving $288, is wrong because the profit is 20% of the cost, not of the selling price.)

Priya: $5000 + 5000 \times 0.04 \times 5 = 5000 + 1000 = \$6000$. Raj: $5000 \times 1.038^5 \approx 5000 \times 1.2050 = \$6025$. Even at a slightly lower rate, compounding edges ahead, so Raj has about $25 more.