Chapter 7 · Topic 7.2 · Prelims + Mains2 of 3
Data Interpretation & Data Sufficiency (Class X level): Data sufficiency with numbers
Key points
- Data sufficiency with numbers
Prelims box: facts and traps
- I alone: sums below 23 from odd primes: 3 + 5 + 7 = 15, 3 + 5 + 11 = 19, 3 + 7 + 11 = 21, 3 + 5 + 13 = 21.
- II alone: 24 = 24 × 1, 12 × 2, 8 × 3, 6 × 4.
- II: 11(a + b) = 11/6 × 9(a − b) gives 2(a + b) = 3(a − b), so a = 5b: ages 51 and 15, difference 36 (whichever is older).
- II alone: 3, 5, 11 and 5, 7, 11 (sum 23) both work.
- Trap (UPSC Prelims 2024): “Which one of the following is correct in respect of the above Question and the Statements? other Statement alone ✓ either Statement alone” is incorrect.
- Trap: If a statement leaves more than one case, it is not enough.
Data sufficiency with numbers
For number DS, list the possible cases. If a statement leaves more than one case, it is not enough. If it leaves exactly one (or every case gives the same answer), it is enough. A firm “No” is also an answer.
CSAT 2024, Q64: A Question is given followed by two Statements I and II. Consider the Question and the Statements. Prelims 2024 I alone: x/y odd fits many pairs (3 and 1, 9 and 3,…). II alone: xy = 12 fits 3 and 4, 2 and 6, and so on. Together: of the pairs with product 12, only x = 6, y = 2 gives an odd whole number (3). Answer (c).
A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Question : What are the unique values of x and y, where x, y are distinct natural numbers?
Statement-I : x/y is odd.
Statement-II : xy = 12
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, Q59: A Question is given followed by two Statements I and II. Consider the Question and the Statements. Prelims 2024 I alone: m + n > mn only works when n = 1 (with n = 2, m = 3: 5 is not more than 6), but m can be anything. II alone: 24 = 24 × 1, 12 × 2, 8 × 3, 6 × 4. Together: n = 1 and m = 24. Answer (c).
A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Question : What are the values of m and n, where m and n are natural numbers?
Statement-I : m + n > mn and m > n.
Statement-II : The product of m and n is 24.
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, Q67: A Question is given followed by two Statements I and II. Consider the Question and the Statements. Prelims 2024 If 2 were one of the primes, the sum would be even and not prime, so all three are odd. I alone: sums below 23 from odd primes: 3 + 5 + 7 = 15, 3 + 5 + 11 = 19, 3 + 7 + 11 = 21, 3 + 5 + 13 = 21. Only 19 is prime: 3, 5, 11. II alone: 3, 5, 11 and 5, 7, 11 (sum 23) both work. Answer (a).
A Question is given followed by two Statements I and II. Consider the Question and the Statements.
There are three distinct prime numbers whose sum is a prime number.
Question : What are those three numbers?
Statement-I : Their sum is less than 23.
Statement-II : One of the numbers is 5.
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: (a) · UPSC official key
CSAT 2023, Q57: Question: Is (p + q – r) greater than (p – q + r), where p, q and r are integers? Prelims 2023 Subtract: (p + q − r) − (p − q + r) = 2(q − r). So the question is really: is q bigger than r?
I gives q < p; II gives p < r. Neither alone links q and r. Together q < p < r, so q < r and the answer is a firm No. Answer (c).
Question : Is (p + q – r) greater than (p – q + r), where p, q and r are integers ?
Statement-1: (p – q) is positive.
Statement-2: (p – r) is negative.
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, Q73: A Question is given followed by two Statements I and II. Consider the Question and the Statements. Prelims 2024 Ages 10a + b and 10b + a: difference 9 × (a − b), sum 11 × (a + b). II: 11(a + b) = 11/6 × 9(a − b) gives 2(a + b) = 3(a − b), so a = 5b: ages 51 and 15, difference 36 (whichever is older). I alone gives no number. Answer (a).
A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Age of each of P and Q is less than 100 years but more than 10 years. If you interchange the digits of the age of P, the number represents the age of Q.
Question : What is the difference of their ages?
Statement-I : The age of P is greater than the age of Q.
Statement-II : The sum of their ages is 11/6 times their difference.
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: (a) · UPSC official key
Mains box: analysis and answer points
- Together: of the pairs with product 12, only x = 6, y = 2 gives an odd whole number (3).
- Likely next angle: A firm No counts
Practice
Practice MCQs (8)
Question: Is the positive integer n even? Statements:
1. n is a multiple of 3.
2. n² is a multiple of 4.
Which one of the following is correct in respect of the question and the statements?
Answer (b) & explanation
Step 1: Statement 1 alone: multiples of 3 include 3 (odd) and 6 (even), so the answer can be yes or no. Not sufficient. Step 2: Statement 2 alone: if n were odd, n² would be odd and could not be a multiple of 4. So n must be even; and every even n has n² divisible by 4. The answer is a definite yes. Step 3: So Statement 2 alone is sufficient and Statement 1 alone is not.
Question: What is the value of x? Statements:
1. 3x + 5 = 20
2. x² = 25
Which one of the following is correct in respect of the question and the statements?
Answer (a) & explanation
Step 1: Statement 1 alone: 3x = 15, so x = 5. One value, so it is sufficient. Step 2: Statement 2 alone: x² = 25 gives x = 5 or x = −5. Two values, so it is not sufficient. Step 3: So Statement 1 alone answers the question and Statement 2 alone does not.
Question: What is the remainder when the positive integer N is divided by 6? Statements:
1. When N is divided by 12, the remainder is 5.
2. When N is divided by 3, the remainder is 2.
Which one of the following is correct in respect of the question and the statements?
Answer (a) & explanation
Step 1: Statement 1 alone: N = 12k + 5. Since 12 is a multiple of 6, 12k leaves remainder 0 on division by 6, so N leaves remainder 5. Sufficient. Step 2: Statement 2 alone: N = 3m + 2, so N could be 2 (remainder 2 on division by 6) or 5 (remainder 5). Two answers, not sufficient. Step 3: So Statement 1 alone is sufficient and Statement 2 alone is not.
On which day of the week did 1 March of a certain year fall?
1. 1 February of that year was a Sunday.
2. That year was not a leap year. Decide whether the data given in the statements are sufficient to answer the question.
Answer (c) & explanation
Step 1: Statement 1 alone: from 1 February to 1 March is 28 days in an ordinary year and 29 days in a leap year. 28 days is exactly 4 weeks, so 1 March would be a Sunday; 29 days would make it a Monday. Two possible answers, not sufficient. Step 2: Statement 2 alone tells us nothing about any weekday. Not sufficient. Step 3: Together: an ordinary year, so 28 days, exactly 4 weeks, and 1 March is a Sunday. Both together are sufficient.
Question: What is Ravi's present age? Statements:
1. Ravi is 4 years older than Sita.
2. The sum of the present ages of Ravi and Sita is 30 years.
Which one of the following is correct in respect of the question and the statements?
Answer (c) & explanation
Step 1: Statement 1 alone gives only the difference of ages: Ravi could be 10 (Sita 6) or 40 (Sita 36). Not sufficient. Step 2: Statement 2 alone gives only the sum: Ravi could be 20 (Sita 10) or 15 (Sita 15). Not sufficient. Step 3: Together: Ravi + Sita = 30 and Ravi − Sita = 4. Adding, 2 × Ravi = 34, so Ravi = 17. Both together are sufficient.
Question: What is the price of one pen? Statements:
1. Three pens and two pencils together cost ₹40.
2. Six pens and four pencils together cost ₹80.
Which one of the following is correct in respect of the question and the statements?
Answer (d) & explanation
Step 1: Statement 1 alone: 3 × pen + 2 × pencil = 40 has many solutions, e.g. pen ₹10 and pencil ₹5, or pen ₹8 and pencil ₹8. Not sufficient. Step 2: Statement 2 alone: 6 × pen + 4 × pencil = 80 is just Statement 1 doubled, so it has the same many solutions. Step 3: Together: there is still only one real equation, so the price of a pen is not fixed. Even both statements together are not sufficient.
Question: What is the two-digit number? Statements:
1. The sum of its digits is 9.
2. The number formed by reversing its digits is 27 more than the original number.
Which one of the following is correct in respect of the question and the statements?
Answer (c) & explanation
Step 1: Let the tens digit be a and the units digit b, so the number is 10a + b. Step 2: Statement 1 alone: a + b = 9 fits 18, 27, 36, 45, 54, 63, 72, 81 and 90. Not sufficient. Step 3: Statement 2 alone: (10b + a) − (10a + b) = 9(b − a) = 27, so b − a = 3: 14, 25, 36, 47, 58, 69. Not sufficient. Step 4: Together: a + b = 9 and b − a = 3 give b = 6, a = 3. The number is 36. Both together are sufficient.
Question: What is Meena's monthly salary? Statements:
1. Meena saves 20% of her monthly salary and spends the rest.
2. Meena's monthly expenditure is four times her monthly savings.
Which one of the following is correct in respect of the question and the statements?
Answer (d) & explanation
Step 1: Statement 1 alone gives a ratio, savings : salary = 1 : 5, but no amount. A salary of ₹20,000 or ₹50,000 fits equally. Not sufficient. Step 2: Statement 2 alone: expenditure = 4 × savings, so salary = 5 × savings. Again only a ratio. Not sufficient. Step 3: Together: saving 20% means spending 80%, which is exactly four times the savings. Statement 2 repeats Statement 1 in other words, and no rupee amount appears anywhere. Even together they cannot give the salary.
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In 150 words, explain the significance of Data Interpretation & Data Sufficiency with one recent example.
Flashcards (4)
Is a firm "No" a sufficient answer in DS?
Yes.
(p + q − r) vs (p − q + r) reduces to?
Is q bigger than r?
Three distinct primes with a prime sum can
include 2? No: the sum would be even.
Digits swapped ages: difference is a
multiple of? 9.
Next topic · 7.3 →Data Interpretation & Data Sufficiency (Class X level): Data sufficiency in word problems