A Comprehensive Guide to Algebraic Expressions, Wages, Taxes, and Interest
Algebraic Expressions
Like Terms
You can only add and subtract like terms (terms that have the same letter or power). For example:
- 8 – 3w + 7 – 2w = 8 + 7 = 15 -3w – 2w = -5w = 15 – 5w
Multiplying and Dividing Like Terms
When multiplying or dividing like terms, multiply or divide the numbers first, then add the letters after. Letters must be in alphabetical order.
- Multiplication Example: 5w x 6v = 5 x 6 = 30 then add the letters in alphabetical order: 30vw
- Division Example: 10w / 2w = 10 / 2 = 5 then add the letters: 5w
Adding and Subtracting Algebraic Fractions
You can only add and subtract fractions if the denominator is the same. If they are not the same, find the Least Common Denominator (LCD).
- Example: k/2 + k/3. In this case, you will find the LCD, which is 6. Then, multiply the top and bottom by however many times the denominator goes into the LCD. In this case, 6. So for k/2, you will multiply the top and bottom by 3 because 2 goes into 6 three times. Then for k/3, you will multiply the top and bottom by 2 because 3 goes into 6 two times. This gives you: (2k + 3k) / 6
Multiplying and Dividing Algebraic Fractions
To multiply fractions, cancel any common factors, but it has to be the same factor on the bottom and top. You can’t cross out if they’re the same factors on the bottom. Then multiply the numerators together and the denominators together.
- Example: 3z/r x 2r/9dz. You would cross out the 3 and replace it with a 1 and cross out the 9 and replace it with 3. This is because 3 goes into itself 1 time and it goes into 9 three times. Then you would go ahead and cross out the same letters, in this case, r and z, so it should look like 2/3d.
When dividing, you flip the right side and just turn it into multiplication and do the same as explained above.
Expanding Expressions
When expanding expressions, you want to multiply the number outside the bracket by the numbers inside the bracket.
- Example: 2(x+3) – 3(x – 4) = 2x + 6 – 3x + 12. Then simplify so it looks like this: -x + 18. To simplify, just find the like terms and either add or subtract.
Factorising Expressions
Factorising expressions is the same as expanding expressions, just reversed, so you’re putting the brackets in. You find the Highest Common Factor (HCF) and put that outside the brackets.
Wages and Salaries
- 1 year = 12 months
- 1 fortnight = 2 weeks
- 1 year = 52 weeks for wage earners
- 1 year = 52.18 weeks for salary earners
Example: Thomas earns a salary of $70,400 p.a.
- Weekly income: $70,400 / 52.18 = $1349.18 (rounded to the nearest cent)
- Fortnightly income: 2 x $1349.18 = $2698.36
- Monthly income: $70,400 / 12 = $5866.67 (rounded to the nearest cent)
Commission, Piecework, and Leave Loading
Commission Example: Georgia is a real estate agent and is paid a commission of 2.5% on the value of the apartments she sells. She also receives a weekly retainer of $750. How much will Georgia earn if she sells an apartment for $152,500?
- Commission: 2.5% of $152,500 = $3812.50
- Total earnings: Commission + Retainer = $3812.50 + $750 = $4562.50
Piecework Example: Emad is a jewellery designer. He makes handmade jewellery and is paid at the following rates:
- $278 per necklace
- $62 per pair of earrings
- $95 per bracelet
This month, Emad made 23 necklaces, 7 pairs of earrings, and 19 bracelets. How much did he earn?
- Monthly earnings: (23 x $278) + (7 x $62) + (19 x $95) = $8633
Income Tax
Example: Sanjay’s annual salary is $70,590. For his Christmas holidays, he received 4 weeks’ normal pay plus 17.5% annual leave loading for the 4 weeks. Calculate Sanjay’s:
- Normal weekly pay: $70,590 / 52.18 = $1352.82 (rounded to the nearest cent)
- Annual leave loading: 17.5% x $1352.82 x 4 = $946.97 (rounded to the nearest cent)
- Total holiday pay: (4 x $1352.82) + $946.97 = $6358.25
PAYG Tax and Net Pay
Example: Sophia earned $62,348 last financial year and collected bank interest of $440.81. She had allowable deductions of $427.52 in work expenses and $110 in donations to charities.
- Taxable income: $62,348 + $440.81 – $427.52 – $110 = $62,251 (rounded down to the nearest dollar)
Assuming a hypothetical tax table where the tax bracket for $37,001 – $80,000 is $3572 plus 32.5% of the amount over $37,000:
- Income tax: $3572 + 0.325 x ($62,251 – $37,000) = $11,778.58 (rounded to the nearest cent)
Simple Interest
Example: Jay earns a gross pay of $2290.33 per fortnight. His deductions are for PAYG tax, $44.10 for private health insurance, and $55.82 for superannuation.
- Fortnightly PAYG tax: (Assuming a hypothetical tax bracket where the fortnightly PAYG tax for $2286 – $2291 is $454) = $454
- Net pay: $2290.33 – ($454 + $44.10 + $55.82) = $1736.41
- Total deductions: $454 + $44.10 + $55.82 = $553.92
- Deductions percentage: ($553.92 / $2290.33) x 100% = 24.2% (rounded to one decimal place)