AD Practice DS SET – 3
DS: Age & Digits – Level 3
1.
What is the two-digit number \( N \)?
(1) When the digits are reversed, the new number is 9 less than \( N \).
(2) The sum of the digits is 15.
(1) When the digits are reversed, the new number is 9 less than \( N \).
(2) The sum of the digits is 15.
Solution (C):
Let \( N = 10t + u \). Reverse \( M = 10u + t \).
(1) \( M = N – 9 \implies (10u + t) = (10t + u) – 9 \implies 9t – 9u = 9 \implies t – u = 1 \).
Pairs \((t,u)\) with diff 1: 21, 32, 43, 54… Not sufficient.
(2) \( t + u = 15 \). Pairs: 69, 78, 87, 96. Not sufficient.
Combined: \( t – u = 1 \) and \( t + u = 15 \).
Adding: \( 2t = 16 \implies t = 8 \). \( u = 7 \). Number is 87. Sufficient.
2.
What is the present age of A?
(1) 10 years ago, A was half as old as B.
(2) 10 years from now, A will be \(3/5\) as old as B.
(1) 10 years ago, A was half as old as B.
(2) 10 years from now, A will be \(3/5\) as old as B.
Solution (C):
(1) \( A-10 = 0.5(B-10) \). One equation, two unknowns. Not sufficient.
(2) \( A+10 = 0.6(B+10) \). One equation, two unknowns. Not sufficient.
Combined: System of two linear equations with two variables.
\( 2A – 20 = B – 10 \implies B = 2A – 10 \).
Substitute into (2): \( A+10 = 0.6(2A – 10 + 10) = 1.2A \).
\( 10 = 0.2A \implies A = 50 \). Sufficient.
3.
Is the three-digit number \( n \) divisible by 4?
(1) The number formed by the last two digits of \( n \) is divisible by 4.
(2) The sum of the digits of \( n \) is 12.
(1) The number formed by the last two digits of \( n \) is divisible by 4.
(2) The sum of the digits of \( n \) is 12.
Solution (A):
(1) The divisibility rule for 4 is exactly this: if the last two digits form a number divisible by 4, the whole number is divisible by 4. Sufficient (Yes).
(2) Sum of digits = 12. Example: 39 (not 3 digit), 129 (No), 138 (No), 132 (Yes). Not sufficient.
4.
What is the average age of a group of people?
(1) The sum of their ages is 250.
(2) There are 10 people in the group.
(1) The sum of their ages is 250.
(2) There are 10 people in the group.
Solution (C):
Average = Sum / Count.
(1) Gives Sum. No Count. Not sufficient.
(2) Gives Count. No Sum. Not sufficient.
Combined: \( 250 / 10 = 25 \). Sufficient.
5.
If \( N = 10a + b \) is a two-digit number, what is the value of \( N \)?
(1) \( b = 3a \).
(2) \( 10b + a = N + 18 \).
(1) \( b = 3a \).
(2) \( 10b + a = N + 18 \).
Solution (C):
(1) \( b = 3a \). Pairs \((a,b)\): (1,3), (2,6), (3,9). Numbers: 13, 26, 39. Not sufficient.
(2) Reverse \( M = N + 18 \implies 10b + a = 10a + b + 18 \implies 9b – 9a = 18 \implies b – a = 2 \).
Pairs with diff 2: 13, 24, 35, 46… Not sufficient.
Combined: \( b = 3a \) and \( b – a = 2 \).
\( 3a – a = 2 \implies 2a = 2 \implies a = 1 \).
\( b = 3 \). Number is 13. Sufficient.
Score: 0 / 0