Everything hangs on one question: percent of what?
Find the original price
A pair of school shoes is on sale at 30% off. The sale price is ₱2,100. What was the price before the discount?
₱2,100 is not the whole — it is the 70% that is left after the discount. So 70% of the original is 2,100, and the original is 2,100 ÷ 0.70 = ₱3,000. B adds 30% to the sale price, which is the commonest slip: 30% of ₱2,100 is smaller than 30% of the original, so you never get back. A takes another 30% off, and D divides by 0.30 instead of 0.70.
1
Percent means "per hundred" — nothing more.
So a percent is a fraction whose denominator is already 100.
65%651000.65
Going from
to
do this
example
decimal
percent
× 100, add %
0.375 → 37.5%
percent
decimal
÷ 100, drop %
65% → 0.65
fraction
percent
divide, then × 100
3 ÷ 8 = 0.375 → 37.5%
percent
fraction
over 100, simplify
65% → 1320
Multiplying and dividing by 100 just slides the decimal point two places. You are not changing the number, only how it is dressed.
2
One equation covers every percent problem.
part = percent × whole
The word "of" tells you which number is the whole. "35% of 260" — 260 is the whole.
35% of 260
the question
0.35 × 260
percent → decimal first
= 91
the part
Check that it is sensible
35% is about a third, and a third of 260 is roughly 87. An answer of 91 is believable. An answer of 9,100 is not — that is what you get by forgetting to convert 35% to 0.35.
3
Rearrange the same equation for the other two questions.
Cover the quantity you want and read off what is left.
The question
Which is unknown
Do this
What is 18% of 350?
part
0.18 × 350 = 63
42 is what percent of 120?
percent
42 ÷ 120 = 0.35 → 35%
18 is 24% of what number?
whole
18 ÷ 0.24 = 75
Notice rows two and three both divide. Whenever the whole or the percent is missing, you divide the part by the other one.
4
Increase and decrease: divide by the original.
percent change = (change ÷ original) × 100
250 → 300
enrolment rose
change = 50
300 − 250
50 ÷ 250 = 0.20
divide by where it started
= 20% increase
× 100
The classic trap
A 25% rise followed by a 25% fall does not bring you back.
₱40 rises to ₱50, then 25% of ₱50 is ₱12.50, landing at ₱37.50. The second
percent is taken of a different whole.
5
Discount, VAT and tip in one multiplication.
Do not find the discount and subtract it — go straight to what is left.
30% off → × 0.7012% VAT → × 1.1210% tip → × 1.10
₱1,200 at 25% off
a dress on sale
100% − 25% = 75%
what you still pay
0.75 × 1,200 = ₱900
one step, no subtraction
₱900
The two-step route — 25% of 1,200 is 300, then 1,200 − 300 — gives the same ₱900.
The one-step route is worth learning because it is what makes step 6 possible.
6
Working backwards: the sale price is not the whole.
Ask what percent the number you were given represents.
0.70 × original = 2,100
₱2,100 is the 70% that survived
original = 2,100 ÷ 0.70
undo the multiplication
= ₱3,000
the answer to the opener
₱3,000
Check it30% of 3,000 = 900the discount
3,000 − 900 = 2,100the sale price ✓
The trap2,100 + 30% of 2,100 = 2,73030% of the wrong whole ✗
Remember the flow
Name the whole · Convert to a decimal · Multiply or divide · Check
If the number you were given came after a discount or a rise, it is not the whole.
7
Variations: more than 100%, less than 1%, and markup on cost.
Still part = percent × whole; these are the percents that look wrong but are not.
What is 150% of 64?
150% = 1.5
÷ 100, as always
1.5 × 64 = 96
the whole plus half again
A percent above 100 gives a part bigger than the whole, and that is correct.
A bank pays 0.5% interest a year. How much interest does ₱3,000 earn in one year?
0.5% = 0.005
slide the point two places: 0.5 → 0.005
0.005 × 3,000 = ₱15
check: 1% would be ₱30, so half of it is ₱15 ✓
Reading 0.5% as 0.5 gives ₱1,500, a hundred times too much.
A sari-sari store buys a bottle of vinegar for ₱40 and sells it for ₱52. What is the markup as a percent of the cost?
markup = 52 − 40 = ₱12
the change
12 ÷ 40 = 0.30 = 30%
"of the cost", so the cost is the whole
Markup is a percent increase in disguise: divide by what the store paid, not by the selling price (12 ÷ 52 is about 23%).
On the exam
Percent items are nearly always money: a discount, VAT, a markup, a percent increase, or the original price before a sale. The fastest route is one multiplication by what is left or what it becomes (× 0.75, × 1.12), and one division when the original is missing; the 10-second check is to run your answer forwards and see the given number come back.
Now you try
Practice problem
Find the original wage
After a 15% increase, a worker's daily wage is ₱690. What was the wage before the increase?
After a 15% rise the new wage is 115% of the old one, so 1.15 × old = 690 and old = 690 ÷ 1.15 = ₱600. Check: 15% of 600 is 90, and 600 + 90 = 690 ✓ A takes 15% off ₱690 and C adds 15% to it — both treat ₱690 as the original. D divides by 0.15 instead of 1.15.
Hint: this one is an increase, so the number you were given
is more than the whole, not less. That changes what you divide by — 1.15, not 0.85.
