Real Numbers, Percentages and Financial Maths

An irrational number cannot be written as a fraction. $\sqrt{2} = 1.41421\ldots$ continues without repeating, so it is irrational. ($\sqrt{4}=2$ is rational, and $0.333\ldots = \dfrac{1}{3}$ is a recurring decimal, which is rational.)

A recurring decimal repeats a fixed block, so it can always be written as a fraction (here $\dfrac{18}{99} = \dfrac{2}{11}$), making it rational. An irrational number such as $\sqrt{2}$ never ends and never repeats.

Let $x = 0.444\ldots$, then $10x = 4.444\ldots$. Subtracting, $9x = 4$, so $x = \dfrac{4}{9}$. (Check: $\dfrac{4}{9} = 0.444\ldots$)

$0.6 = \dfrac{6}{10}$, and dividing the top and bottom by 2 gives $\dfrac{3}{5}$.

Multiply the amount by the exchange rate: $200 \times 0.65 = 130$ US dollars. ($307.69$ came from dividing by the rate, which converts the other way (US dollars back to Australian dollars).)

Percentage change is always measured against the original amount: $\dfrac{50 - 40}{40}\times 100 = \dfrac{10}{40}\times 100 = 25\%$. Dividing by the new price ($50) is the common error that gives 20%.

$\dfrac{3}{5}\times 100 = 60\%$. (35% concatenates the digits; multiplying 3 and 5 gives the wrong 15%.)

Her base pay is $22 \times 7.5 = \$165$. The 10% bonus adds $0.10 \times 165 = \$16.50$, so the total is $165 + 16.50 = \$181.50$. ($\$165$ is the pay before the bonus is added.)

The tip makes the total $80 \times 1.10 = \$88$. Split between 4 people: $\dfrac{88}{4} = \$22$ each. ($\$20$ split the bill before adding the tip.)

First find the floor area: $4 \times 3 = 12$ m². Then the cost is $12 \times \$35 = \$420$. ($\$245$ added the side lengths $(4 + 3 = 7)$ instead of multiplying them to get the area.)

The decrease is $40 - 30 = 10$ dollars. As a percentage of the original price: $\dfrac{10}{40}\times 100 = 25\%$. (Dividing by the new price of 30 gives the wrong 33.33%.)

$20\%$ of 45 is 9, so the sale price is $45 - 9 = 36$ dollars. (Or $80\%$ of 45 = 36. $9 gives only the discount.)

After a 30% discount the sale price is 70% of the original, so $0.70\times \text{original} = 420$, giving $\text{original} = \dfrac{420}{0.70} = 600$ dollars. (Check: 30% of 600 = 180, and 600 − 180 = 420.)

Cost per book $= \dfrac{60}{5} = 12$ dollars; profit per book $= 18 - 12 = 6$ dollars. Percentage profit on cost $= \dfrac{6}{12}\times 100 = 50\%$. (Dividing by the 18 selling price gives the wrong 33.33%.)

Loss $= 80 - 60 = 20$ dollars. As a percentage of the cost price: $\dfrac{20}{80}\times 100 = 25\%$. (Dividing by the 60 selling price gives the wrong 33.33%.)

Mark-up $= 30\%$ of 40 = 12, so the selling price is $40 + 12 = 52$ dollars. (Or $130\%$ of 40 = 52. $70 treats 30% as 30 dollars.)

GST $= 10\%$ of 60 = 6, so the inclusive price is $60 + 6 = 66$ dollars. (Or $1.10\times 60 = 66$.)

The inclusive price is 110% of the original, so the pre-GST price is $\dfrac{110}{1.10} = 100$ dollars. (The trap is taking 10% off 110 to get 99, but the 10% should be of the pre-GST price.)

Use $I = \dfrac{PRT}{100}$ with $P=2000,\ R=5,\ T=3$: $I = \dfrac{2000\times 5\times 3}{100} = \dfrac{30\,000}{100} = 300$ dollars. ($30,000 forgets to divide by 100; $100 is one year only; $2300 is the total amount.)

Rearrange to $R = \dfrac{I\times 100}{P\times T} = \dfrac{1200\times 100}{5000\times 4} = \dfrac{120\,000}{20\,000} = 6\%$. (Check: 6% of 5000 = 300 per year, times 4 years = 1200. Forgetting to divide by the years gives 24%.)

Compare the price per gram. Brand A: $\dfrac{5}{400} = 0.0125$ per gram; Brand B: $\dfrac{7.20}{600} = 0.012$ per gram. Brand B is cheaper per gram, so it is the better buy. (Choosing Brand A for its lower total ignores value for money per gram.)

Work backwards: $\text{original}\times 0.80\times 1.10 = 880$, so $\text{original}\times 0.88 = 880$ and $\text{original} = \dfrac{880}{0.88} = 1000$ dollars. (Check: 1000 × 0.80 = 800, then × 1.10 = 880. $800 only undoes the GST; $1100 only undoes the discount.)

Mark-up per book $= 25\%$ of 12 = 3, so the profit per book is 3 dollars. Total profit for 100 books $= 100\times 3 = 300$ dollars. ($1200 is the cost of buying them; $1500 is the revenue from selling them.)

Savings $= 15\%$ of 600 = $\$90$; rent $= 25\%$ of 600 = $\$150$. What is left is $600 - 90 - 150 = \$360$. (Together savings and rent take $40\%$, leaving $60\%$ of 600, which is also 360.)

Let the original be $P$. A 20% rise gives $1.20P$; a 20% fall then gives $1.20P\times 0.80 = 0.96P$. So $0.96P = 96$ and $P = 100$. (The two 20% changes do not cancel, because the fall is calculated from a larger base.)