Chapter 6 · Topic 6.1 · Prelims + Mains1 of 3
Basic Numeracy (Class X level): Numbers and percentages
Key points
- Numbers, divisibility and digit counting.
- Percentages, profit and interest.
- Time and work.
Prelims box: facts and traps
| CSAT year to 7) | Type |
|---|---|
| 2011 to 2013 9, 2, 12 | Count from the CSAT PYQ book topic tags |
| 2014 to 2016 13, 17, 26 | Same count |
| 2017 to 2019 23, 17, 24 | Same count |
| 2020 to 2022 36, 28, 27 | Same count |
| 2023 to 2025 24, 31, 31 | Same count |
| 2026 28 | Same count; 2026 answers provisional. |
- Three habits solve most of them: take an easy number such as 100, try the options, and count in blocks (1-digit, 2-digit, 3-digit numbers).
- CSAT 2019, Q75: An 8-digit number 4252746B leaves remainder 0 when divided by 3.
- New price ₹ 99. ₹ 12,375 ÷ 99 = 125 litres = 125 × 16 = 2,000 km.
- Trap (UPSC Prelims 2019): “How many values of B are possible?” is incorrect.
- Trap: Step 5: 151 is not divisible by 2, 3, 5, 7 or 11 (151 = 7 × 21 + 4 = 11 × 13 + 8), so it is prime.
Numbers, divisibility and digit counting
Numeracy is the largest scoring block of CSAT after comprehension. The questions use Class X arithmetic; what they test is a calm method. Three habits solve most of them: take an easy number such as 100, try the options, and count in blocks (1-digit, 2-digit, 3-digit numbers).
| CSAT year | Numeracy questions (book topics 3 to 7) | Type |
|---|---|---|
| 2011 to 2013 | 9, 2, 12 | Count from the CSAT PYQ book topic tags |
| 2014 to 2016 | 13, 17, 26 | Same count |
| 2017 to 2019 | 23, 17, 24 | Same count |
| 2020 to 2022 | 36, 28, 27 | Same count |
| 2023 to 2025 | 24, 31, 31 | Same count |
| 2026 | 28 | Same count; 2026 answers provisional |
Battle IAS count of the CSAT PYQ book topic tags (numbers, percentages, ratio and averages, time and work, geometry); out of 80 questions a year. Includes clock and calendar items, which are taught in General Mental Ability.
Divisibility rules worth knowing: by 3 or 9 if the digit sum is; by 5 if the last digit is 0 or 5; by 11 if the alternate digit sums differ by 0 or 11. A number written as abcabc is abc × 1001, and 1001 = 7 × 11 × 13.
CSAT 2025, Q20: What is the unit digit in the multiplication of 1 × 3 × 5 × 7 × 9 ×… × 999? Prelims 2025 Every number in the product is odd, so the product is odd. One of the numbers is 5, so the product is a multiple of 5. An odd multiple of 5 ends in 5. Answer (c).
What is the unit digit in the multiplication of 1 × 3 × 5 × 7 × 9 × . . . × 999?
- 1
- 3
- 5
- 9
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Answer: (c) · UPSC official key
CSAT 2023, Q17: For any choices of values of X, Y and Z, the 6-digit number of the form XYZXYZ is divisible by: Prelims 2023 XYZXYZ = XYZ × 1001, for example 123123 = 123 × 1001. 1001 = 7 × 11 × 13, so it is always divisible by all three. Answer (d).
For any choices of values of X, Y and Z, the 6-digit number of the form XYZXYZ is divisible by :
- 7 and 11 only
- 11 and 13 only
- 7 and 13 only
- 7, 11 and 13
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Answer: (d) · UPSC official key
CSAT 2019, Q75: An 8-digit number 4252746B leaves remainder 0 when divided by 3. How many values of B are possible? Prelims 2019 Add the known digits: 4 + 2 + 5 + 2 + 7 + 4 + 6 = 30, already a multiple of 3. So B itself must be a multiple of 3: 0, 3, 6 or 9. Four values. Answer (c).
An 8-digit number 4252746B leaves remainder 0 when divided by 3. How many values of B are possible?
- 2
- 3
- 4
- 6
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Answer: (c) · UPSC official key
CSAT 2015, Q57: If ABC × DEED = ABCABC; where A, B, C, D and E are different digits, what are the values of D and E? Prelims 2015 ABCABC = ABC × 1001. So DEED = 1001: D = 1, E = 0. Answer (c).
If ABC × DEED = ABCABC; where A, B, C, D and E are different digits, what are the values of D and E?
- D = 2, E = 0
- D = 0, E = 1
- D = 1, E = 0
- D = 1, E = 2
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Answer: (c) · UPSC official key
CSAT 2017, Q80: What is the total number of digits printed, if a book containing 150 pages is to be numbered from 1 to 150? Prelims 2017 Count in blocks. Pages 1 to 9: 9 digits. Pages 10 to 99: 90 pages × 2 = 180. Pages 100 to 150: 51 pages × 3 = 153. Total 9 + 180 + 153 = 342. Answer (b).
What is the total number of digits printed, if a book containing 150 pages is to be numbered from 1 to 150?
- 262
- 342
- 360
- 450
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Answer: (b) · UPSC official key
Percentages, profit and interest
Start with 100. A 20% fall from 100 is 80; a 20% rise from 80 is 96, not 100, because the second change is on a new base. Profit and loss are percentages of the cost price; interest is a percentage of the principal.
CSAT 2024, Q26: P’s salary is 20% lower than Q’s salary which is 20% lower than R’s salary. By how much percent is R’s salary more than P’s salary? Prelims 2024 Take R = 100. Q is 20% lower: 80. P is 20% lower than 80: 64. R is more than P by 36, and 36 out of 64 = 56.25%. Answer (b).
P’s salary is 20% lower than Q’s salary which is 20% lower than R’s salary. By how much percent is R’s salary more than P’s salary?
- 48·75%
- 56·25%
- 60·50%
- 62·25%
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Answer: (b) · UPSC official key
CSAT 2024, Q47: A person buys three articles P, Q and R for ₹ 3,330. If P costs 25% more than R and R costs 20% more than Q, then what is the cost of P? Prelims 2024 Take Q = 100. R = 120, P = 25% more than 120 = 150. Total 370 units = ₹ 3,330, so 1 unit = ₹ 9. P = 150 × 9 = ₹ 1,350. Answer (d).
A person buys three articles P, Q and R for ₹ 3,330. If P costs 25% more than R and R costs 20% more than Q, then what is the cost of P?
- ₹ 1,000
- ₹ 1,200
- ₹ 1,250
- ₹ 1,350
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Answer: (d) · UPSC official key
CSAT 2026, Q43: A person saves 10% of his salary every month. If his salary increases by 12% and the expenditure increases by 10%, then what will be the change in his saving per month? Prelims 2026 Take salary 100: saving 10, spending 90. New salary 112, new spending 99, new saving 13. Saving rose from 10 to 13, a 30% increase. Answer (b) (2026 answer provisional).
A person saves 10% of his salary every month. If his salary increases by 12% and the expenditure increases by 10%, then what will be the change in his saving per month?
- 20% increase
- 30% increase
- 03% decrease
- 02% decrease
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Answer: (b) · UPSC provisional key (final key awaited)
CSAT 2025, Q17: The petrol price shot up by 10% as a result of the hike in crude oil prices. The price of petrol before the hike was ₹ 90 per litre. A person travels 2200 km every month and his car gives a mileage of 16 km per litre. By how many km should he reduce his travel if he wants to maintain his expenditure at the previous level? Prelims 2025 He now uses 2,200 ÷ 16 = 137.5 litres × ₹ 90 = ₹ 12,375. New price ₹ 99. ₹ 12,375 ÷ 99 = 125 litres = 125 × 16 = 2,000 km. Reduce by 200 km. Answer (b).
The petrol price shot up by 10% as a result of the hike in crude oil prices. The price of petrol before the hike was ₹ 90 per litre. A person travels 2200 km every month and his car gives a mileage of 16 km per litre. By how many km should he reduce his travel if he wants to maintain his expenditure at the previous level?
- 180 km
- 200 km
- 220 km
- 240 km
Show answer
Answer: (b) · UPSC official key
CSAT 2023, Q44: A principal P becomes Q in 1 year when compounded half-yearly with R% annual rate of interest. If the same principal P becomes Q in 1 year when compounded annually with S% annual rate of interest, then which one of the following is correct? Prelims 2023 Half-yearly compounding earns interest on interest within the year, so it grows faster at the same rate. To reach the same Q, the half-yearly rate R must be smaller than the yearly rate S. Answer (c).
A principal P becomes Q in 1 year when compounded half-yearly with R% annual rate of interest. If the same principal P becomes Q in 1 year when compounded annually with S% annual rate of interest, then which one of the following is correct ?
- R = S
- R > S
- R < S
- R ≤ S
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Answer: (c) · UPSC official key
Mains box: analysis and answer points
- A 20% fall from 100 is 80; a 20% rise from 80 is 96, not 100, because the second change is on a new base.
- Profit and loss are percentages of the cost price; interest is a percentage of the principal.
- CSAT 2023, Q44: A principal P becomes Q in 1 year when compounded half-yearly with R% annual rate of interest.
- Likely next angle: abcabc = abc × 1001
Practice
Practice MCQs (8)
Meera's salary is 25% more than Kabir's. By what percentage is Kabir's salary less than Meera's?
Answer (a) & explanation
Step 1: Take Kabir's salary as 100. Then Meera's salary is 125. Step 2: The gap is 25, but now it must be compared with Meera's salary (125), because the question asks 'less than Meera's'. Step 3: 25 ÷ 125 × 100 = 20%.
If 20% of A is equal to 30% of B, then A is how much percent more than B?
Answer (d) & explanation
Step 1: 20% of A = 30% of B means 0.2A = 0.3B, so A = 1.5 B. Step 2: A is more than B by 0.5B. Step 3: Compared with B (the base after 'than'), 0.5B is 50% of B.
What is the largest number that always divides the product of any four consecutive positive integers?
Answer (d) & explanation
Step 1: Among any four consecutive integers there are two even numbers, and one of them is a multiple of 4, so the product has at least 2 × 4 = 8 as a factor. Step 2: Among any three consecutive integers there is a multiple of 3, so 3 is a factor too. Step 3: 8 × 3 = 24 always divides the product. Step 4: It cannot be larger, because 1 × 2 × 3 × 4 = 24 itself.
What is the sum of all the digits used in writing the integers from 1 to 99?
Answer (c) & explanation
Step 1: Treat 1 to 9 as 01 to 09; adding a leading zero does not change any digit sum. Step 2: Units place: across 00 to 99 each digit 0-9 appears 10 times, so the units digits add to 10 × (0 + 1 + … + 9) = 10 × 45 = 450. Step 3: Tens place: the same, another 450. Step 4: Total = 450 + 450 = 900.
The rate of interest on a loan rises from 8% per annum to 10% per annum. Consider the following statements:
1. The rate has risen by 2 percentage points.
2. The rate has risen by 25%.
3. The rate has risen by 20%.
Which of the statements given above is/are correct?
Answer (b) & explanation
Step 1 (Statement 1): the difference between two rates is measured in percentage points: 10 − 8 = 2 percentage points. Correct. Step 2 (Statement 2): the relative rise is measured on the old rate: 2 ÷ 8 × 100 = 25%. Correct. Step 3 (Statement 3): 20% is 2 ÷ 10, which uses the new rate as the base. Wrong.
Arjun's income is 20% more than Bela's, and Bela's income is 25% less than Chitra's. Arjun's income is what percentage of Chitra's income?
Answer (b) & explanation
Step 1: Start with the person at the end of the chain. Take Chitra's income as 100. Step 2: Bela earns 25% less than Chitra: 75. Step 3: Arjun earns 20% more than Bela: 20% of 75 = 15, so Arjun earns 90. Step 4: 90 out of 100 = 90%.
Consider the following numbers:
1. 91
2. 119
3. 143
4. 151
How many of the numbers given above are prime?
Answer (a) & explanation
Step 1: To test a number below 169, it is enough to try the primes 2, 3, 5, 7, 11, because 13 × 13 = 169. Step 2: 91 = 7 × 13. Not prime. Step 3: 119 = 7 × 17. Not prime. Step 4: 143 = 11 × 13. Not prime. Step 5: 151 is not divisible by 2, 3, 5, 7 or 11 (151 = 7 × 21 + 4 = 11 × 13 + 8), so it is prime. Only one number is prime.
For every positive integer n, consider the following statements:
1. n² − n is even.
2. n³ − n is divisible by 6.
3. n² + n + 1 is odd.
4. n³ + n is divisible by 4.
How many of the statements given above are correct?
Answer (c) & explanation
Step 1 (Statement 1): n² − n = n(n − 1), a product of two consecutive integers; one of them is even, so it is always even. Correct. Step 2 (Statement 2): n³ − n = (n − 1) × n × (n + 1), three consecutive integers; among them there is always a multiple of 2 and a multiple of 3, so it is divisible by 6. Correct. Step 3 (Statement 3): n² + n = n(n + 1) is even, so adding 1 makes it odd. Correct. Step 4 (Statement 4): try n = 1: 1 + 1 = 2, not divisible by 4. Wrong. So three statements are correct.
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In 150 words, explain the significance of Basic Numeracy with one recent example.
Flashcards (8)
Unit digit of 1 × 3 × 5 × ... × 999?
5.
XYZXYZ is always divisible by?
7, 11 and 13 (it is XYZ × 1001).
Digits to number pages 1 to 150?
342.
Divisible by 3 when?
The digit sum is divisible by 3.
R = 100, Q 20% less, P 20% less than Q. R
exceeds P by? 56.25%.
Salary +12%, spending +10%, saving was
10%. Saving changes by? 30% increase.
Half-yearly vs yearly compounding to reach
the same amount? Half-yearly needs a lower rate.
Ratio 1/2 : 3/4 : 5/6 in whole numbers?
6 : 9 : 10.
Next topic · 6.2 →Basic Numeracy (Class X level): Averages, speed and work