Question: A tax table charges: first $\usd{}20\,000$ at $0\%$; next $\usd{}10\,000$ at $3.5\%$ (i.e. $\usd{}350$); so the first $\usd{}40\,000$ carries $\usd{}900$ tax; the next $\usd{}40\,000$ is at $8.5\%$. (a) Tom's chargeable income is $\usd{}27\,536$ — find his tax. (b) Jane's is $\usd{}64\,391$ — find hers.

Step 1. (a) Tom's income enters the second band. Only the part above $\usd{}20\,000$ is taxed:

Step 2. Tax that slice at its band rate:

Step 3. (b) Jane clears the first $\usd{}40\,000$, which costs the table's cumulative $\usd{}900$. The remainder:

Step 4.

⚠ Watch out: Progressive tax is charged slice by slice — Jane does not pay $8.5\%$ on her whole income (that would be $\usd{}5\,473$, nearly double). Use the table's cumulative column for the full bands, then the marginal rate only on the excess.

Question: A restaurant meal costs $\usd{}120$. A service charge of $10\%$ is added, and then GST of $7\%$ is charged on the total. Find the final bill.

Step 1. Order matters, and the convention is fixed: GST is charged after the service charge. Apply the service charge first:

Step 2. Now GST on that larger total:

Step 3. Or chain the multipliers in one line:

Step 4. Worth noticing: the combined multiplier is $1.10 \times 1.07 = 1.177$ — an effective $17.7\%$ on-cost, a little more than $10\% + 7\% = 17\%$, because the GST also taxes the service charge.

⚠ Watch out: The "++" on a menu means exactly this two-stage chain. Adding $17\%$ flat gives $\usd{}140.40$ — close, wrong, and specifically what the question is testing. The multiplier chain also reverses cleanly: a final bill $\div 1.07 \div 1.10$ recovers the menu price.

Question: In July 2019, Mrs Chong used $1156$ units of electricity at $23.07$ cents per unit, and $42.8$ m³ of water. Water: first $40$ m³ at tariff $\usd{}1.21$/m³ with $50\%$ water conservation tax (on the tariff) and waterborne fee $\usd{}0.92$/m³; above $40$ m³ the rates are $\usd{}1.52$, $65\%$ and $\usd{}1.18$. Find her total bill.

Step 1. Electricity — convert cents to dollars first:

Step 2. Water, first tier (the full $40$ m³):

tariff $40 \times 1.21 = \usd{}48.40$; conservation tax $50\% \times 48.40 = \usd{}24.20$; waterborne $40 \times 0.92 = \usd{}36.80$ — tier total $\usd{}109.40$.

Step 3. Water, second tier (the remaining $42.8 - 40 = 2.8$ m³):

tariff $2.8 \times 1.52 = \usd{}4.256$; tax $65\% \times 4.256 = \usd{}2.7664$; waterborne $2.8 \times 1.18 = \usd{}3.304$ — tier total $\usd{}10.3264$.

Step 4. Grand total:

⚠ Watch out: Tiered pricing never applies the higher rate to the whole usage — only the $2.8$ m³ above the threshold sees the $\usd{}1.52$ tier. The conservation tax is a percentage of the tariff only (not of tariff + waterborne fee), and $23.07$ cents entering as $\usd{}23.07$ inflates the bill a hundredfold — sanity-check the size.