Basic Mathematics for One-Paper Tests — Percentage, Average and Ratio with Solved Examples for PPSC, FPSC & CSS

Percentage, average and ratio solved step by step for PPSC, FPSC and CSS one-paper tests — formulas, 20 solved examples, exam traps and practice MCQs.

By Muhammad Younus · Updated 06 Nov 2026 · 16 min read

Basic mathematics is the most reliable mark-earner in a one-paper test. In a PPSC paper of 100 MCQs, in an FPSC screening test and in the CSS MPT, a small group of questions always comes from three topics: percentage, average, and ratio and proportion. These questions carry the same one mark as the hardest general-knowledge question, yet they take thirty seconds each once you know the method — and they punish only one thing: a shaky method under time pressure. This post teaches the three topics from zero, with every formula explained in plain words and twenty solved examples worked out step by step, exactly the way they appear in the paper.

This post is for candidates preparing for PPSC one-paper tests, FPSC screening tests, and the CSS MPT. You do not need to be good at mathematics in general. You need three methods, the habit of writing one line of working before choosing an option, and the traps listed at the end of this post. No calculator is allowed in the hall, so every example below is solved with mental arithmetic you can repeat in the paper.

How examiners ask mathematics questions

Mathematics questions in one-paper tests follow a small set of patterns, and knowing the pattern is half the preparation:

• Direct formula: "What is 25% of 360?" or "Find the average of 12, 18 and 24." One formula, one answer, thirty seconds.
• The everyday story: The same formula hidden inside a shop, a salary, a classroom or a cricket score. "A shopkeeper gives 20% discount" is a percentage question wearing a shopkeeper's clothes.
• The trap option: Three of the four options are the answers you get by making the classic mistake — adding instead of multiplying, using the wrong base, or averaging speeds directly. If your method is right, the trap options do not even tempt you.
• The two-step question: "The price rises by 20%, then falls by 20%. What is the net change?" Two operations, and the answer is never zero.

A useful rule for the whole section: write the formula first, then the numbers. Candidates who multiply before thinking pick the trap; candidates who write "increase = new − old, divided by old" get the mark. And with 0.25 negative marking in PPSC one-paper tests, a mathematics question you are unsure about is worth one careful attempt, not a blind guess — but a question whose method you know is a mark you must not leave blank.

Percentage — the formula and what it really means

Percent means "out of a hundred". 25% of a number is 25 parts out of every 100 parts of it, which is the same as one quarter of the number. Every percentage question in these papers is one of four forms, and the formulas below cover all of them:

• To find a percentage of a number: percentage × number ÷ 100. So 35% of 240 is 35 × 240 ÷ 100 = 84.
• To find what percentage one number is of another: (part ÷ whole) × 100. So 27 out of 45 is (27 ÷ 45) × 100 = 60%.
• Percentage increase or decrease: (change ÷ original) × 100. The original value — the value before the change — is always the base. This is the single most-tested rule in the topic.
• To find the original after a percentage change: new value ÷ (1 + rate) for an increase, or new value ÷ (1 − rate) for a decrease.

Learn the fraction equivalents by heart, because they turn most percentage questions into one-step division: 10% is 1/10, 20% is 1/5, 25% is 1/4, 50% is 1/2, 75% is 3/4, 12.5% is 1/8, 33⅓% is 1/3, 5% is 1/20, 40% is 2/5 and 60% is 3/5. When a question asks for 25% of 360, do not multiply by 25 and divide by 100 — take one quarter of 360, which is 90, and move on.

Percentage — solved examples

Work through each example with the formula beside it. Cover the answer, solve it yourself, then check.

1. What is 35% of 240? — 84. Method: 35 × 240 ÷ 100 = 8,400 ÷ 100 = 84. Or use 10% = 24, so 30% = 72 and 5% = 12, giving 72 + 12 = 84.
2. A shopkeeper marks a shirt at Rs. 1,500 and gives a 20% discount. What is the selling price? — Rs. 1,200. Method: discount = 20% of 1,500 = 300; selling price = 1,500 − 300 = 1,200. Faster: pay 80%, and 80% of 1,500 is 1,200.
3. A town's population is 40,000 and it grows by 10% every year. What is the population after 2 years? — 48,400. Method: after year one, 40,000 × 1.10 = 44,000; after year two, 44,000 × 1.10 = 48,400. Notice the second year's 10% is charged on 44,000, not on 40,000 — that is why the answer is more than 48,000.
4. A salary rises from Rs. 50,000 to Rs. 58,000. What is the percentage increase? — 16%. Method: increase = 8,000; (8,000 ÷ 50,000) × 100 = 16%. The base is the old salary, 50,000, never the new one.
5. In a class of 45 students, 60% passed the test. How many failed? — 18 failed. Method: 60% of 45 = 27 passed, so 45 − 27 = 18 failed. Or directly: 40% failed, and 40% of 45 = 18.
6. A shop gives a 20% discount and then a further 10% discount. What is the single equivalent discount? — 28%. Method: after the first discount you pay 80%; after the second you pay 90% of that 80%, which is 72%. So the total discount is 100% − 72% = 28%. Two discounts never simply add up to 30%.
7. A number increased by 25% becomes 200. What was the number? — 160. Method: 200 ÷ 1.25 = 160. Check: 25% of 160 is 40, and 160 + 40 = 200.
8. 15% of a number is 45. What is the number? — 300. Method: 45 ÷ 0.15 = 300. Or: if 15% is 45, then 5% is 15 and 100% is 15 × 20 = 300.

Average — the formula and what it really means

The average of a set of numbers is their total divided by how many there are. That is the whole topic in one sentence, and every average question is a rearrangement of it:

• Average = total ÷ number of items. So the average of 12, 18, 24, 30 and 36 is (12 + 18 + 24 + 30 + 36) ÷ 5 = 120 ÷ 5 = 24.
• Total = average × number of items. If the average of 8 numbers is 27, their total is 8 × 27 = 216. Examiners love asking for the total, or for a missing number, and this rearrangement is the key.
• Average of the first n natural numbers = (n + 1) ÷ 2. So the average of the numbers from 1 to 20 is 21 ÷ 2 = 10.5.
• Average speed = total distance ÷ total time. It is never the average of the two speeds unless the time spent at each speed is the same. This is the most common trap in the topic.

When a person joins or leaves a group, the total changes by that person's value and the count changes by one — recompute the total first, then divide. When a number is added to a set, add it to the total and add one to the count. Almost every "average" question in past papers is one of these two moves.

Average — solved examples

1. Find the average of 12, 18, 24, 30 and 36. — 24. Method: total = 120, count = 5, average = 120 ÷ 5 = 24.
2. What is the average of the first 20 natural numbers? — 10.5. Method: (20 + 1) ÷ 2 = 10.5. The average of 1 to 20 sits exactly in the middle, between 10 and 11.
3. The average age of 5 persons is 32 years. One person aged 40 leaves the group. What is the average age of the remaining 4? — 30 years. Method: total age = 5 × 32 = 160; after the person leaves, total = 160 − 40 = 120; average = 120 ÷ 4 = 30.
4. A car travels 60 km at 40 km/h and another 60 km at 60 km/h. What is the average speed for the whole journey? — 48 km/h. Method: time for the first part = 60 ÷ 40 = 1.5 hours; time for the second = 60 ÷ 60 = 1 hour; total distance 120 km over 2.5 hours = 48 km/h. The tempting answer 50 km/h is wrong because more time was spent at the slower speed.
5. The average of 8 numbers is 27. A ninth number, 63, is added. What is the new average? — 31. Method: total of the 8 numbers = 8 × 27 = 216; new total = 216 + 63 = 279; new average = 279 ÷ 9 = 31.
6. A batsman has an average of 40 runs after 10 innings. He scores 60 in his 11th innings. What is his new average? — About 41.8. Method: total after 10 innings = 400; after the 11th = 460; average = 460 ÷ 11 = 41.81. Cricket averages are the favourite story examiners use for this question, and the method never changes.
7. What is the average of four consecutive even numbers 12, 14, 16 and 18? — 15. Method: total = 60, count = 4, average = 15. For any run of consecutive even (or odd) numbers, the average is the middle value — halfway between the two middle numbers.

Ratio and proportion — the formula and what it really means

A ratio compares two quantities of the same kind. The ratio 2:3 means that for every 2 parts of the first quantity there are 3 parts of the second, and the whole is 5 parts. Every ratio question in these papers is one of four forms:

• To divide a quantity in a given ratio: add the parts, divide the quantity by the total parts to get one part, then multiply. To divide Rs. 900 in the ratio 2:3:5, the parts total 10, one part is Rs. 90, and the shares are Rs. 180, Rs. 270 and Rs. 450.
• To find a quantity from a ratio and a total: same method — one part = total ÷ total parts. If boys and girls are in the ratio 3:5 and the class has 40 students, one part is 40 ÷ 8 = 5, so there are 15 boys and 25 girls.
• To combine two ratios: make the common term equal. If A:B is 2:3 and B:C is 4:5, multiply the first ratio by 4 and the second by 3 so that B is 12 in both: A:B:C = 8:12:15.
• Proportion: if a:b = c:d, then a × d = b × c. If 5 pens cost Rs. 125, then 8 pens cost 125 ÷ 5 × 8 = Rs. 200. Proportion is simply the unitary method with a name.

Two warnings save marks here. First, always add the parts before dividing — candidates who divide by the first part instead of the total get a confident wrong answer. Second, keep the order: in the ratio A:B = 3:4, A is the smaller share, and swapping them is the trap the options are built around.

Ratio and proportion — solved examples

1. Divide Rs. 900 in the ratio 2:3:5. — Rs. 180, Rs. 270 and Rs. 450. Method: total parts = 10, one part = 900 ÷ 10 = 90; shares are 2 × 90, 3 × 90 and 5 × 90.
2. In a class, boys and girls are in the ratio 3:5 and there are 40 students. How many boys and how many girls? — 15 boys and 25 girls. Method: total parts = 8, one part = 40 ÷ 8 = 5; boys = 3 × 5 = 15, girls = 5 × 5 = 25.
3. If A:B = 2:3 and B:C = 4:5, find A:B:C. — 8:12:15. Method: make B equal in both ratios by multiplying the first by 4 (8:12) and the second by 3 (12:15), then combine.
4. Two numbers are in the ratio 3:4 and their sum is 49. Find the numbers. — 21 and 28. Method: total parts = 7, one part = 49 ÷ 7 = 7; numbers are 3 × 7 = 21 and 4 × 7 = 28.
5. A 35-litre mixture has milk and water in the ratio 5:2. How much milk is in it? — 25 litres. Method: total parts = 7, one part = 35 ÷ 7 = 5 litres; milk = 5 × 5 = 25 litres, water = 10 litres.
6. A father and son are in the ratio 5:2 and the sum of their ages is 56. Find their ages. — Father 40, son 16. Method: total parts = 7, one part = 56 ÷ 7 = 8; father = 5 × 8 = 40, son = 2 × 8 = 16.
7. If 5 pens cost Rs. 125, how much do 8 pens cost? — Rs. 200. Method: one pen costs 125 ÷ 5 = Rs. 25; 8 pens cost 8 × 25 = Rs. 200.
8. Simplify the ratio 45:60. — 3:4. Method: divide both sides by 15. Always simplify a ratio before using it — examiners often give the unsimplified form deliberately.

Speed tricks for the exam hall

These shortcuts are worth learning because they convert multi-step questions into one line:

• 10% of anything is the number with one zero removed (10% of 240 is 24). Build every other percentage from 10%: 20% is double it, 5% is half of it, 30% is three times it.
• 25% is a quarter, 50% is a half, 75% is three quarters. The fraction equivalents listed in the percentage section are the fastest tool in the whole topic.
• Successive percentage change: for a rise of a% followed by a fall of b%, the net change is a − b − (a × b ÷ 100). A 20% rise then a 20% fall is 20 − 20 − 4 = −4%, a net loss of 4%. The answer to "rise then fall by the same percentage" is never zero.
• Average of two numbers is their sum divided by two, and the average of consecutive numbers is the middle one. Average of 41 and 59 is 50 — do not add and divide the long way.
• Ratio questions: write the parts, add them, and divide once. One division gives you one part, and every share follows from it. If the total is not divisible by the parts, you have misread the question — recheck before calculating.
• Time check: a percentage question should take 30 seconds, an average question 45 seconds, a ratio question 45 seconds. If a question is taking two minutes, mark it, move on, and return at the end. Three easy questions elsewhere are worth more than one stubborn one.

Traps examiners set — and how to avoid them

• Percentage points are not percent. If a rate rises from 20% to 25%, it rose by 5 percentage points but by 25% in relative terms. Papers ask both versions.
• The wrong base. A percentage increase is always divided by the original value. Dividing by the new value is the most common error in the topic.
• Averaging speeds directly. Average speed is total distance over total time. Averaging 40 and 60 to get 50 is wrong unless the time at each speed is equal.
• Two discounts do not add. A 20% discount followed by a 10% discount is 28% in total, not 30%. The second discount applies to the already-reduced price.
• Ratio order. If the question says A:B = 3:4 and asks for B, the answer is the larger share. Read which letter is asked for before calculating.
• The missing number trap. If the average of 5 numbers is 20 and four of them are 10, 15, 25 and 30, the missing number is 100 − 80 = 20. Always recover the total first.
• Units. If one quantity is in kilometres and the other in metres, convert before forming the ratio. A ratio of 1 km to 500 m is 2:1, not 1:500.

Practice MCQs — test yourself

Cover the answers, attempt each question, and only then check. If you score below seven, revise the solved examples above once more before moving on.

1. What is 25% of 360? — 90
2. A number increased by 20% becomes 240. What was the original number? — 200
3. Find the average of 15, 25, 35 and 45. — 30
4. Divide Rs. 1,000 in the ratio 1:2:2. — Rs. 200, Rs. 400 and Rs. 400
5. A price rises from Rs. 80 to Rs. 100. What is the percentage increase? — 25%
6. What is the average of the first 10 odd numbers? — 10
7. Two numbers are in the ratio 2:3 and their sum is 75. What is the larger number? — 45
8. 40% of a number is 80. What is the number? — 200
9. A car travels 100 km at 50 km/h and another 100 km at 100 km/h. What is the average speed? — About 66.7 km/h
10. If A:B = 3:4 and B:C = 5:6, find A:C. — 5:8

Notice the pattern in these practice questions: the examiners' favourites are ordinary numbers with a twist — the twist is always the base, the total, or the order. That is good news. It means the mathematics section rewards a steady method, not talent, and not luck.

A realistic 7-day mathematics plan

• Day 1: Percentage formulas and the fraction equivalents. Solve the eight percentage examples above without looking at the answers.
• Day 2: Average — the total-first method, joining and leaving a group, and the average-speed trap.
• Day 3: Ratio and proportion — dividing quantities, combining ratios, and the unitary method.
• Day 4: Attempt one past paper's mathematics questions under time, 45 seconds per question, with the 0.25 negative-marking rule applied honestly.
• Day 5: Revise only the questions you got wrong on Day 4, and write each correct method in one line in your notebook.
• Day 6: Mixed revision — ten questions from each of the three topics, timed.
• Day 7: Full mixed practice of the ten practice MCQs above plus the solved examples, and nothing new. Stop new material the day before the test.

Seven days of this plan, at thirty minutes a day, will cover more mathematics marks than most candidates earn from months of casual reading. The three topics never change; only the numbers do.

FAQs

How many mathematics questions come in a PPSC one-paper test?

It varies by paper, but candidates should expect a small cluster of basic mathematics questions in every one-paper test — usually a handful out of the 100 MCQs, drawn from percentage, average, ratio, age problems and simple interest. Because each question carries the same mark as a hard general-knowledge question but takes far less time, mathematics is one of the best returns on study time in the whole paper.

Do I need a calculator for these questions?

No, and none is allowed in the hall. Every question in this post is designed to be solved with mental arithmetic — the fraction equivalents, the 10% trick and the one-part method for ratios are all mental tools. If a question seems to need long multiplication, you have probably missed the shortcut.

What is the difference between percentage and percentage points?

Percentage measures a change relative to its base; percentage points measure the simple difference between two percentages. A pass rate rising from 40% to 50% rose by 10 percentage points, but by 25% in relative terms (10 ÷ 40 × 100). Papers test both, so read the wording before choosing.

How do I handle a mathematics question when I do not know the method at all?

Eliminate first. In a ratio question, the shares must add up to the total — any option set that does not add up is wrong. In a percentage question, the answer must sit on the sensible side of the original number. If you still cannot decide, remember the 0.25 negative marking and skip rather than guess blindly — but never skip a question whose method you know.

Are percentage, average and ratio also tested in the CSS MPT and FPSC screening tests?

Yes. The MPT screens general ability including quantitative reasoning, and FPSC screening tests regularly include the same three topics. The methods in this post are identical across PPSC, FPSC and CSS — only the framing of the questions changes.

Basic mathematics rewards the candidate who learns three methods properly, not the candidate who memorises a hundred formulas. Learn percentage, average and ratio from this post, practise them on past papers, and the mathematics section will quietly become one of your safest sources of marks.