Commission is a slice of the selling price; VAT rides on top of the base. Both are one multiplication — the skill is multiplying by the right thing.
The problem
An agent earns 4% commission on every sale. A house sells for ₱2,500,000. How much is the commission?
Commission = rate × selling price = 0.04 × 2,500,000 = ₱100,000. A and C are decimal-place slips — 0.004 and 0.4 instead of 0.04. Writing the percent as a decimal before multiplying kills this whole family of errors.
commission = 0.04 × 2,500,000
the percent, as a decimal, times the price
= ₱100,000
what the agent takes home
seller keeps 96%: ₱2,400,000
one number splits into two shares
Backwards for the rate: divide the commission by the price. ₱24,000 earned on a ₱600,000 sale is 24,000 ÷ 600,000 = 0.04, a 4% rate.
A 12% tax on top of a base price means the shelf price is 112% of the base.
price with VAT = base × 1.12
100% of the base plus 12% of it
₱500 × 1.12 = ₱560
₱60 of that is the tax
base = price with VAT1.12
undo a multiplication by dividing
₱560 ÷ 1.12 = ₱500 ✓
lands exactly back on the base
₱560 × 0.88 = ₱492.80 ✗
the trap — 12% of the WRONG number
Why the trap fails: the 12% was computed on the ₱500 base, but "take 12% off" computes it on the ₱560 total — a bigger number, so too much comes off. Percent on, percent off, are not inverses unless the base is the same.
commission, then 10% tax on it
two percents in a row
take-home = price × 0.04 × 0.90
each step multiplies — never add the percents
(a) Salary ₱15,000 a month plus 6% on sales above ₱80,000; sales were ₱200,000. (b) 3% on the first ₱500,000 of sales and 5% on the rest; sales were ₱1,200,000.
(a) 200,000 − 80,000 = 120,000
only the part above the quota earns
0.06 × 120,000 = 7,200 → 15,000 + 7,200 = ₱22,200
then add the salary
(b) 0.03 × 500,000 = 15,000
first tier
0.05 × 700,000 = 35,000
only the 700,000 above the tier line
15,000 + 35,000 = ₱50,000
add the tiers
6% of all ₱200,000 (₱12,000) ignores the quota; 5% of all ₱1,200,000 (₱60,000) ignores the lower tier. Both wrong totals will be among the choices.
Only the tax. A shelf price of ₱3,360 includes 12% VAT. How much of it is VAT?
base = 3,360 ÷ 1.12 = 3,000
remove VAT the right way
VAT = 3,360 − 3,000 = ₱360
the difference
shortcut: 3,360 × 12112 = 3,360 × 328 = 360
VAT is 12 parts of every 112
What is different: the answer is the tax, not the base. 12% of ₱3,360 (₱403.20) uses the wrong base again.
A discount on the price without VAT. A senior citizen buys a meal whose shelf price, VAT included, is ₱1,120. Seniors pay no VAT and get 20% off the VAT-free price. How much does the senior pay?
1,120 ÷ 1.12 = 1,000
strip the VAT first
1,000 × 0.80 = ₱800
then take 20% off
What is different: two changes on different bases. Taking 20% off the shelf price (₱896) leaves the VAT in.
₱1,120 × 0.80 = ₱896 charges the senior VAT they do not owe. Divide by 1.12 first, then apply the discount to what is left.
Commission on a discounted price. A lot listed at ₱2,000,000 sells after a 10% discount. The agent earns 5% of the actual selling price. Find the commission.
selling price = 2,000,000 × 0.90 = 1,800,000
the money that changed hands
commission = 1,800,000 × 0.05 = ₱90,000
or 2,000,000 × 0.90 × 0.05 in one line
What is different: the list price is not the selling price. 5% of the list price (₱100,000) pays the agent on money nobody paid.
(a) An agent earns 3% commission, and 10% of it is withheld; she takes home ₱81,000. (b) A seller receives ₱2,350,000 after paying a 6% commission. Find each selling price.
(a) commission = 81,000 ÷ 0.90 = 90,000
undo the last step first
price = 90,000 ÷ 0.03 = ₱3,000,000
then undo the commission rate
(b) seller keeps 94% → price = 2,350,000 ÷ 0.94
divide by the share kept
= ₱2,500,000
check: 6% of it is 150,000 ✓
Adding 6% of ₱2,350,000 back (₱2,491,000) uses the wrong base — the same trap as taking 12% off a VAT-inclusive price.
Expect VAT removed from a shelf price, commission above a quota or in tiers, withholding on a commission, and the backwards versions. Write the whole chain as multipliers (price × 0.03 × 0.90 = take-home), then multiply forward or divide back. Always check by running your answer forward to the given number.
The problem
A gadget's shelf price is ₱4,480 with 12% VAT already included. What was the price before VAT?
The shelf price is 112% of the base, so divide: 4,480 ÷ 1.12 = ₱4,000 — and forward-checking, 4,000 × 1.12 = 4,480 ✓. A is the trap: 4,480 × 0.88 takes 12% of the tax-inclusive price, which was never the base of the 12%.
