Chapter 7 · Topic 7.3 · Prelims + Mains3 of 3
Data Interpretation & Data Sufficiency (Class X level): Data sufficiency in word problems
Key points
- Data sufficiency (DS): you don't solve fully; you only decide whether the given facts are enough.
- Test Statement I alone, then II alone, then both together.
- Write each statement as a small equation and ask what it fixes.
- Watch for traps: a statement that is always true, or a missing unit or count.
Prelims box: facts and traps
- The four options: (a) one statement alone works; (b) either alone works; (c) both needed together; (d) even both are not enough.
- CSAT 2024: Q60 (c), Q65 (c), Q66 (d), Q69 (c). CSAT 2026 Q1: (a), provisional key.
Data sufficiency in word problems
Like a detective deciding whether the clues are enough to name the thief, without making the arrest.
The four answer choices, in plain words. (a) one statement alone works, the other doesn’t. (b) either statement alone works. (c) you need both together. (d) even both together are not enough.
Word DS often hides a trap: a statement that sounds useful but is always true, or a missing unit or count. Write each statement as a small equation and ask what it fixes.
CSAT 2024, Q60 (download time). Time = size ÷ rate. I gives only the size; II gives only the rate. Together: 12 megabytes ÷ 2.4 kilobytes per second gives the time. Answer (c).
A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Question : What is the time required to download the software?
Statement-I : The size of the software is 12 megabytes.
Statement-II : The transfer rate is 2·4 kilobytes per second.
Which one of the following is correct in respect of the above Question and the Statements?
- The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
- The Question can be answered by using either Statement alone
- The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
- The Question cannot be answered even by using both the Statements together
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Answer: (c) · UPSC official key
CSAT 2024, Q65 (who got the least). Think in “parts” of the total. I: X = 4/5 of (Y + Z), so X is 4 of 9 parts of the total. II: Y = 2/7 of (X + Z), so Y is 2 of 9 parts. Alone, each leaves the other two unknown. Together X = 4, Y = 2, Z = 3 parts: Y gets the least. Answer (c).
A Question is given followed by two Statements I and II. Consider the Question and the Statements.
A certain amount was distributed among X, Y and Z.
Question : Who received the least amount?
Statement-I : X received 4/5 of what Y and Z together received.
Statement-II : Y received 2/7 of what X and Z together received.
Which one of the following is correct in respect of the above Question and the Statements?
- The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
- The Question can be answered by using either Statement alone
- The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
- The Question cannot be answered even by using both the Statements together
Show answer
Answer: (c) · UPSC official key
CSAT 2024, Q66 (number of students). This is the “always true” trap. II is true for any class whose highest and lowest marks average 60, and I says they are 70 and 50, which average 60. Neither says how many students there are. Answer (d).
A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Question : If the average marks in a class are 60, then what is the number of students in the class?
Statement-I : The highest marks in the class are 70 and the lowest marks are 50.
Statement-II : Exclusion of highest and lowest marks from the class does not change the average.
Which one of the following is correct in respect of the above Question and the Statements?
- The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
- The Question can be answered by using either Statement alone
- The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
- The Question cannot be answered even by using both the Statements together
Show answer
Answer: (d) · UPSC official key
CSAT 2024, Q69 (price of p). p + r = 50 − 16 = 34, and both are more than 16 (q is the cheapest). Prices need not be whole rupees, so I alone (p not more than r) still allows 16.50, 16.80 and so on. II alone is similar. Together p = r = 17. Answer (c).
A Question is given followed by two Statements I and II. Consider the Question and the Statements.
A person buys three articles p, q and r for ₹ 50. The price of the article q is ₹ 16 which is the least.
Question : What is the price of the article p?
Statement-I : The cost of p is not more than that of r.
Statement-II : The cost of r is not more than that of p.
Which one of the following is correct in respect of the above Question and the Statements?
- The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
- The Question can be answered by using either Statement alone
- The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
- The Question cannot be answered even by using both the Statements together
Show answer
Answer: (c) · UPSC official key
CSAT 2026, Q1 (coins). X gets three different coins from 1, 2, 5, 10 and 20, totalling m. Does X have a 5? List all ten totals of three different coins. With a 5: 8, 16, 17, 26, 27, 35. Without a 5: 13, 23, 31, 32. I alone: 8, 16, 26, 27, 35 and 32 are not prime, so it could go either way. II alone: digit sums above 5 are 8, 16, 17, 26, 27, 35, all with a 5; the others (13, 23, 31, 32) have digit sum 4 or 5. So II alone answers Yes. Answer (a) (2026 answer provisional).
Question : X receives three coins of different denominations : 1, 2, 5, 10 and 20. If the total amount received by X is m, does X receive a coin of denomination 5?
Statement I : m is not a prime number.
Statement II : The sum of the digits of m is greater than 5.
- Select this option if the question can be answered using one of these statements alone, but cannot be answered using other statement
- Select this option if the question can be answered using either statement alone
- Select this option if the question can be answered using both the statements together, but cannot be answered using either statement alone
- Select this option if the question cannot be answered even using any of the statements
Show answer
Answer: (a) · UPSC provisional key (final key awaited)
What UPSC asks: write each statement as an equation; watch hidden units and “always true” statements. Next likely angle: test alone, then together.
Practice
Practice MCQs (6)
A pie chart shows how a family's total monthly income of ₹48,000 is distributed. The share of each head, as a percentage of the total, is given below. Head | Share (%) Food | 30 Rent | 25 Education | 15 Transport | 10 Savings | 12 Others | 8 What is the central angle of the sector representing Education in the pie chart?
Answer (c) & explanation
Step 1: A full pie is 360°, which stands for 100%. Step 2: Each 1% therefore takes 3.6°. Step 3: Education is 15%, so its angle = 15 × 3.6 = 54°.
Four consecutive even integers are written in increasing order. Which one of the following, if known, would by itself be sufficient to determine all four integers?
Answer (b) & explanation
Step 1: Call the numbers a, a + 2, a + 4 and a + 6. Knowing them means knowing a. Step 2: (a) 'All positive' allows 2, 4, 6, 8 or 10, 12, 14, 16 and so on: not enough. Step 3: (b) Sum = 4a + 12 = 60 gives a = 12: the numbers are 12, 14, 16, 18. Enough. Step 4: (c) Among four consecutive even numbers, two are always multiples of 4; the largest is one of them in many cases (e.g. 2, 4, 6, 8 and 10, 12, 14, 16): not enough. Step 5: (d) The product of four consecutive even numbers is always divisible by 16, so it tells us nothing.
x and y are integers. Are the facts below enough to decide whether x exceeds y?
1. x + y > 0
2. x − y = 3
Choose the correct option.
Answer (b) & explanation
Step 1: Statement 1 alone: x = 1, y = 0 gives x + y > 0 with x > y, but x = 0, y = 1 also gives x + y > 0 with x y always. A definite yes. Step 3: Statement 2 alone is sufficient; Statement 1 alone is not.
A bar chart shows the number of students (in thousands) enrolled in five streams of a State's colleges in 2024 and 2025. The bar heights are as follows. Stream | 2024 | 2025 Arts | 50 | 45 Science | 40 | 47 Commerce | 30 | 36 Engineering | 35 | 31 Medicine | 15 | 17
Which stream recorded the highest percentage increase in enrolment from 2024 to 2025?
Answer (b) & explanation
Step 1: Only Science, Commerce and Medicine grew; Arts and Engineering fell. Step 2: Percentage increase = rise ÷ 2024 value × 100. Science: 7/40 = 17.5%; Commerce: 6/30 = 20%; Medicine: 2/15 ≈ 13.3%. Step 3: Commerce has the highest percentage increase, though Science has the largest absolute rise (7 thousand).
A pie chart shows how a family's total monthly income of ₹48,000 is distributed. The share of each head, as a percentage of the total, is given below. Head | Share (%) Food | 30 Rent | 25 Education | 15 Transport | 10 Savings | 12 Others | 8 Next month, the family's expenditure on Food rises by 20%, the rupee amounts under all other heads stay the same, and the total rises accordingly. What will be the approximate share of Food in the new total?
Answer (a) & explanation
Step 1: Food now = 30% of ₹48,000 = ₹14,400. Step 2: After a 20% rise, Food = ₹17,280, an increase of ₹2,880. Step 3: New total = 48,000 + 2,880 = ₹50,880. Step 4: New share = 17,280 ÷ 50,880 × 100 ≈ 34.0%.
x and y are positive integers such that x + y = 10. Which one of the following additional facts is NOT sufficient by itself to determine the values of x and y?
Answer (a) & explanation
Step 1: (a) xy = 21 with x + y = 10: the pair of numbers is 3 and 7, but which is x? Both (3, 7) and (7, 3) work, so x and y are not fixed. Step 2: (b) x − y = 4 with x + y = 10 gives x = 7, y = 3. Fixed. Step 3: (c) x² − y² = (x + y)(x − y) = 10(x − y) = 40, so x − y = 4, again x = 7, y = 3. Fixed. Step 4: (d) x = 4y gives 5y = 10, so y = 2, x = 8. Fixed. Step 5: Only (a) fails to determine x and y.
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In 150 words, explain the significance of Data Interpretation & Data Sufficiency with one recent example.
Flashcards (3)
Download time needs?
Size and transfer rate.
Highest 70, lowest 50, average 60: class
size? Cannot be found.
Why does p ≤ r alone not fix p = 17 in 2024
Q69? Prices need not be whole rupees.
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