Base, rate and percentage — three names, one equation, and the word “of” tells you which is which.
The problem
15% of what number is 48?
0.15 × n = 48, so n = 48 ÷ 0.15 = 320. B multiplies instead of dividing — that gives 15% of 48. The phrase "of what number" means the whole is unknown, and finding a whole means dividing.
1
One equation covers every percentage question.
P =
R × B
base — B
the whole
follows the word of
percentage — P
the part
follows the word is
rate — R
the percent
carries the % sign
Two words do all the work
Whatever comes after of is the base. Whatever comes after
is is the percentage. Label those two and the third is what
you solve for.
Finding the part is one multiplication: 45% of 260 = 0.45 × 260 = 117. When the
question asks about the rest, use 100% minus the rate: if 72% of 25 seats are taken,
28% are empty, and 0.28 × 25 = 7 seats.
2
Finding the base.
“15% of what number is 48?” — the base is the unknown.
rate
15%
percentage
48
base
?
48 = 0.15 × B
15% becomes 0.15
B = 48 ÷ 0.15
÷ 0.15 both sides
B = 320
the whole
Divide, do not multiply
15% of 48 is 7.2 — a completely different question. If the unknown is the base,
the part gets divided by the rate.
3
Finding the rate.
“What percent of 80 is 28?” — now the rate is missing.
28 = R × 80
80 follows “of”
R = 28 ÷ 80 = 0.35
a decimal, not yet a percent
R = 35%
× 100
320 · 35%
Part ÷ whole always gives the rate. Forgetting the × 100 leaves you with 0.35%.
4
Percent of change.
A price rises from ₱250 to ₱310. By what percent?
percent change =
changeoriginal
× 100
change = 310 − 250 = 60
the rise
60 ÷ 250 = 0.24
divide by the old price
= 24% increase
× 100
The base is always the original
Dividing by 310 gives 19.4%, which answers a question nobody asked. Change is
always measured against where you started.
5
Two phrases that are not the same.
20% of 50
10
a part of it
20% more than 50
60
the original plus the part
“More than” and “less than” mean 100% ± the rate — so multiply by 1.20 or 0.80.
Wording
Multiplier
On 50
20% of
0.20
10
increased by 20%
1.20
60
decreased by 20%
0.80
40
is 20% of
÷ 0.20
250
6
Backwards from a sale price, and changes in a row.
After a 25% discount a bag sells for ₱360. What was the original price?
₱360 is 75% of the original
100% − 25% is what is left
original = 360 ÷ 0.75 = ₱480
finding the base: divide
check: 25% of 480 = 120, 480 − 120 = 360 ✓
run it forward
A price rises 20%, then the new price falls 20%. What is the net change?
1.20 × 0.80 = 0.96
multiply the multipliers
0.96 = 96% of the original → a 4% decrease
on ₱100: 120, then 96
Percents of different bases do not add
Adding 25% back to ₱360 gives ₱450 — wrong, because the 25% was taken from the original.
And +20% then −20% is not “no change”: the second 20% is taken from a bigger number.
7
Variations: more than vs less than, a rate above 100%, and undoing a rise.
The same gap, two answers: 50 is what percent more than 40? And 40 is what percent less than 50?
gap = 50 − 40 = 10
the change is the same both ways
more than 40: 10 ÷ 40 = 25%
the base follows “than”
less than 50: 10 ÷ 50 = 20%
a different base, a different rate
What is different: the word after “than” is the base, so swapping the two numbers changes the answer.
A part bigger than the whole: 52 is what percent of 40?
R = 52 ÷ 40 = 1.3
part ÷ base, as always
= 130%
more than the whole, so past 100%
What is different: the “part” is larger than the base. Dividing the other way, 40 ÷ 52 ≈ 77%, answers the reverse question.
Backwards from an increase: after an 8% rise a fee is ₱1,350. What was it before?
₱1,350 is 108% of the old fee
100% + 8%
old = 1350 ÷ 1.08 = ₱1,250
check: 8% of 1,250 = 100, 1,250 + 100 = 1,350 ✓
What is different: the same move as working back from a sale price, but the multiplier is above 1.
Not ₱1,242
Taking 8% off the new fee, 1350 × 0.92 = 1,242, uses the new fee as the base. The 8% was
worked out on the old fee, so we divide by 1.08.
8
Combining groups, and adding to a mixture.
A student scores 70% on a 40-item test and 90% on a 60-item test. What is the overall percentage?
0.70 × 40 = 28 and 0.90 × 60 = 54
turn each percent into a count
82 out of 100 = 82%
totals over totals
30 L of a solution is 20% acid. How much pure acid must be added to make it 40% acid?
acid now: 0.20 × 30 = 6 L
the part
6 + x30 + x = 0.40
the added acid joins the part and the whole
6 + x = 12 + 0.4x → 0.6x = 6 → x = 10 L
check: 16 of 40 L = 40% ✓
Averaging the percents
(70% + 90%) ÷ 2 = 80% treats the two tests as equal in size. With groups of different
sizes — tests, classes, boys and girls — always go through the counts. For the acid,
keeping the whole at 30 L gives 6 L, which forgets that the tank grows.
On the exam
Percent items reward knowing which number is the base: the original for a change, the
original again when working back from a sale price, and the combined total for a whole
group. Use multipliers (1.20, 0.75) for chains, and check every answer by running it forward.
✓
Check: run the answers forwards.
base0.15 × 320 = 48matches ✓
rate0.35 × 80 = 28matches ✓
sensible?28 is over a third of 8035% fits ✓
Remember the flow
After “of” · After “is” · Solve for the third
Change is measured against the original.
Now you try
Practice problem
The problem
24 is 30% of what number?
0.30 × n = 24, so n = 24 ÷ 0.3 = 80. C computes 30% of 24 instead. Quick check: 30% is less than a third, so the whole has to be comfortably more than 24.
Every lesson in the UPCAT Math Review has one of these. 30 questions, marked as you answer, with the working shown for each.
Question 1 of 30
What is 35% of 240?
0.35 × 240 = 84. (10% is 24, so 30% is 72 and 5% is 12; 72 + 12 = 84.)
Question 2 of 30
18 is what percent of 45?
Rate = part ÷ whole = 18 ÷ 45 = 0.4 = 40%. Dividing the other way gives 250%.
Question 3 of 30
12% of what number is 54?
0.12 × B = 54, so B = 54 ÷ 0.12 = 450. Multiplying instead gives 6.48, which is 12% of 54.
Question 4 of 30
A store buys an item for ₱400 and sells it for ₱520. What is the percent markup based on the cost?
The markup is 520 − 400 = ₱120, measured against the cost: 120 ÷ 400 = 0.30 = 30%. Dividing by the selling price gives about 23%.
Question 5 of 30
In a class of 40 students, 65% passed the test. How many students failed?
If 65% passed, 35% failed: 0.35 × 40 = 14. Answering 26 gives the number who passed.
Question 6 of 30
The population of a town grew from 25,000 to 28,000. What was the percent increase?
Change = 3,000, measured against the original: 3,000 ÷ 25,000 = 0.12 = 12%. Dividing by 28,000 gives about 10.7%.
Question 7 of 30
After a 15% discount, a pair of shoes was sold for ₱2,125. What was the original price?
The sale price is 85% of the original: 0.85 × P = 2,125, so P = 2,125 ÷ 0.85 = ₱2,500. Adding 15% of 2,125 gives 2,443.75, which uses the wrong base.
Question 8 of 30
A price is increased by 25% and the new price is then decreased by 20%. What is the net change from the original price?
Multiply the factors: 1.25 × 0.80 = 1.00, so the price returns exactly to where it started. Percent changes do not simply add (25 − 20 = 5 is the trap).
Question 9 of 30
A 40 L solution is 30% acid. How many litres of pure acid must be added to make the solution 50% acid?
There are 0.30 × 40 = 12 L of acid. Adding x litres of pure acid: (12 + x)/(40 + x) = 1/2 → 24 + 2x = 40 + x → x = 16 L. Check: 28/56 = 50%.
Question 10 of 30
A student scored 60% on a 50-item test and 80% on a 30-item test. What was the student's overall percentage on the two tests combined?
Correct items: 0.6(50) + 0.8(30) = 30 + 24 = 54 out of 80. Then 54 ÷ 80 = 0.675 = 67.5%. Averaging the two percentages (70%) ignores the different test sizes.
Question 11 of 30
What is 62% of 450?
0.62 × 450 = 279. 171 is the other 38%; 725.8 divides by 0.62 instead of multiplying.
Question 12 of 30
In a barangay of 1,200 households, 35% have no internet connection. How many households have internet?
The rest is 100% − 35% = 65%, and 0.65 × 1,200 = 780. 420 is the households without internet.
Question 13 of 30
54 is what percent of 72?
R = 54 ÷ 72 = 0.75 = 75%. 133⅓% divides 72 by 54, the wrong way round; 0.75% forgets the × 100.
Question 14 of 30
30% of what number is 87?
87 = 0.30 × B, so B = 87 ÷ 0.30 = 290. 26.1 is 30% of 87, a different question.
Question 15 of 30
A tricycle fare went up from ₱20 to ₱25. By what percent did it increase?
Change = 5, and 5 ÷ 20 = 25%. 20% divides by the new fare, ₱25; 125% is the new fare as a percent of the old.
Question 16 of 30
A ₱650 school bag goes up in price by 18%. What is the new price?
18% more means 118%: 1.18 × 650 = ₱767. ₱117 is only the increase; ₱533 takes 18% off.
Question 17 of 30
₱84 is what percent of ₱60?
Part ÷ base = 84 ÷ 60 = 1.4 = 140%; the part is bigger than the base, so the rate passes 100%. 71.4% divides 60 by 84; 40% is only the excess.
Question 18 of 30
After a 30% discount a rice cooker sells for ₱1,820. What was its original price?
₱1,820 is 70% of the original, so original = 1,820 ÷ 0.70 = ₱2,600. ₱2,366 adds 30% back to the sale price, but the 30% was taken from the original.
Question 19 of 30
A number is increased by 40%, and the result is then decreased by 25%. The final number is what percent of the original?
1.40 × 0.75 = 1.05, so the final number is 105% of the original. 115% adds +40% and −25% as if they were on the same base.
Question 20 of 30
In a survey, 48 of 300 respondents chose sinigang as their favourite dish. What percent did NOT choose sinigang?
252 of 300 did not, and 252 ÷ 300 = 0.84 = 84%. 16% is the share who did choose sinigang.
Question 21 of 30
60 is what percent less than 80?
Gap = 20, and the base follows “than”: 20 ÷ 80 = 25%. 33⅓% divides by 60, which answers “80 is what percent more than 60”.
Question 22 of 30
After a 15% raise, a worker's monthly salary is ₱23,000. What was it before the raise?
₱23,000 is 115% of the old salary: 23,000 ÷ 1.15 = ₱20,000. ₱19,550 takes 15% off the new salary, but the 15% was worked out on the old one.
Question 23 of 30
A 50 L solution is 10% alcohol. How many litres of pure alcohol must be added to make it 28% alcohol?
Alcohol now = 5 L. (5 + x) ÷ (50 + x) = 0.28 → 5 + x = 14 + 0.28x → 0.72x = 9 → x = 12.5 L. 9 L keeps the whole at 50 L, forgetting that the tank grows.
Question 24 of 30
In Section A, 40% of 35 students passed a test; in Section B, 70% of 40 students passed. What percent of the 75 students passed?
Counts: 0.40 × 35 = 14 and 0.70 × 40 = 28, so 42 of 75 = 56%. 55% averages the two percents, treating the sections as the same size.
Question 25 of 30
The price of a kilo of pork rose 25% in March and then fell 12% in April. What is the net change from the original price?
1.25 × 0.88 = 1.10, so the price is 110% of the original — a 10% increase. A 13% increase subtracts 12 from 25, but the 12% is taken from the bigger price.
Question 26 of 30
A blender is discounted 20%, and then a further 10% is taken off the discounted price. It now costs ₱2,160. What was its original price?
The multipliers give 0.80 × 0.90 = 0.72, so original = 2,160 ÷ 0.72 = ₱3,000. ₱3,085.71 treats the two discounts as a single 30% (2,160 ÷ 0.70).
Question 27 of 30
Lito's allowance is 25% more than Bea's. By what percent is Bea's allowance less than Lito's?
Let Bea = 100, so Lito = 125. The gap is 25, and “less than Lito's” puts 125 underneath: 25 ÷ 125 = 20%. 25% uses Bea's allowance as the base again.
Question 28 of 30
In a class, 3/5 of the students are boys. If 30% of the boys and 55% of the girls joined the choir, what percent of the whole class joined?
Boys: 0.30 × 0.60 = 0.18 of the class; girls: 0.55 × 0.40 = 0.22. Total 0.40 = 40%. 42.5% averages 30% and 55% as if the groups were equal.
Question 29 of 30
This year's enrolment at a school is 135% of last year's. If 1,620 students enrolled this year, how many more students enrolled than last year?
Last year = 1,620 ÷ 1.35 = 1,200, so the increase is 1,620 − 1,200 = 420. 567 takes 35% of this year's figure, using the wrong base; 1,200 is last year's enrolment.
Question 30 of 30
A shop raised a price by 20%. By what percent must the new price be reduced to bring it back to the original?
On ₱100: up to ₱120, and the ₱20 drop is measured against ₱120: 20 ÷ 120 = 16⅔%. 20% would take the price to ₱96, because the cut is from the bigger number.
End of the free lesson
Every lesson like this one is free.
All 176 lessons of the UPCAT Math Review are open to everyone — no account needed. Each comes with its 5-question check. Make a free account and your progress is saved.
The practice set for every lesson and the 7 full-length mock exams come with the UPCAT Math Review — one payment of ₱200, no expiry.
