To compare two or more fractions with unlike denominators:
Common Denominator Method: Find the Least Common Multiple (LCM) of the denominators and convert each fraction into an equivalent fraction with that common denominator.
Decimal / Percentage Conversion: Convert fractions into decimals or percentages by dividing the numerator by the denominator (e.g., 22/25 = 88/100 = 88%).
Cross-Multiplication Rule: For two fractions a/b and c/d (with b, d > 0), a/b > c/d if and only if (a × d) > (b × c).
B. Arithmetic of Sequences and Overlapping Sets
Arithmetic Progressions with Fractions: When successive terms increase by a fixed common difference d, the n-th term is:
T(n) = T(1) + (n − 1) × d
Inclusion-Exclusion for Overlapping Subsets: Total = Only(A) + Only(B) + Both(A & B) + Neither
Equivalently: Total with at least one preference = Size(A) + Size(B) − Both(A & B).
Aashay scored the following marks in 3 subjects:
• Dance: 22/25
• Maths: 41/50
• Science: 93/100
Arrange the three subjects in descending order of how well he performed in them.
a) Maths-Science-Dance
b) Science-Maths-Dance
c) Science-Dance-Maths
d) Maths-Dance-Science
Solution & Reasoning:
Correct Option: c) Science-Dance-Maths
Convert all fractions to a common denominator of 100:
Dance: 22/25 = (22 × 4)/(25 × 4) = 88/100 (88%)
Maths: 41/50 = (41 × 2)/(50 × 2) = 82/100 (82%)
Science: 93/100 (93%)
Comparing the values: 93/100 > 88/100 > 82/100.
Descending order: Science > Dance > Maths.
Question 2
What will come in place of "?"
a) 1
b) 8
c) 9
d) 5
Solution & Reasoning:
Correct Option: a) 1 (or matching sector completion 9)
Alternatively, evaluating the missing numerator that completes a whole fraction of 9/9 gives numerator 1, or whole number 1.
Question 3
Arrange the given fractions in ascending order. Which option can be placed exactly in the middle of the new arrangement so that the order remains ascending?
Fractions given: 10/15, 8/9, 17/18, 5/6
a) 68/90
b) 71/90
c) 77/90
d) 84/90
Solution & Reasoning:
Correct Option: c) 77/90
Convert all given fractions to a common denominator of 90:
10/15 = 2/3 = 60/90
5/6 = (5 × 15)/(6 × 15) = 75/90
8/9 = (8 × 10)/(9 × 10) = 80/90
17/18 = (17 × 5)/(18 × 5) = 85/90
Ascending order: 60/90 < 75/90 < 80/90 < 85/90.
The exact middle between the second (75/90) and third (80/90) fractions must be a fraction x such that:
75/90 < x < 80/90.
Testing the options:
68/90 < 75/90
71/90 < 75/90
75/90 < 77/90 < 80/90 (Valid)
84/90 > 80/90
Therefore, 77/90 fits directly in the middle.
Question 4
Two numbers, 23/18 and 7/54 are taken. A binary operation is performed on them and the result obtained is 31/27. What operation might have been performed on the numbers taken?
Simplify 62/54 by dividing numerator and denominator by 2:
62/54 = 31/27.
The operation performed is Subtraction.
Question 5
In a class of 50 students, 20 are boys and the remaining are girls. Of the girls in the class, one third like singing and two thirds like dancing. Some girls like both singing and dancing. If 5 girls like neither singing nor dancing, what fraction of the girls like both activities?
a) 1/3
b) 1/4
c) 1/5
d) 1/6
Solution & Reasoning:
Correct Option: d) 1/6
Total students = 50, Boys = 20 → Total girls = 50 − 20 = 30 girls.
Girls liking singing: S = (1/3) × 30 = 10.
Girls liking dancing: D = (2/3) × 30 = 20.
Girls liking neither activity = 5.
Girls liking at least one activity = Total girls − Neither = 30 − 5 = 25.
Using the Inclusion-Exclusion formula:
Both = S + D − At_Least_One = 10 + 20 − 25 = 5 girls.
Fraction of girls liking both activities = 5 / 30 = 1/6.
Question 6
Tim has some sons, some daughters, and 27 grandchildren.
• 2/3 of his children are male and the rest are female
• Each of his sons has 2 sons and 1 daughter
• Each of his daughters has 2 daughters and 1 son
How many daughters does Tim have?
a) 2
b) 3
c) 6
d) 9
Solution & Reasoning:
Correct Option: b) 3
Each son has 2 + 1 = 3 children.
Each daughter has 2 + 1 = 3 children.
Therefore, every child of Tim (son or daughter) has exactly 3 children.
Total grandchildren = 27 → Total number of Tim's children = 27 / 3 = 9 children.
Fraction of children who are female (daughters) = 1 − 2/3 = 1/3.
Number of daughters = (1/3) × 9 = 3 daughters.
Question 7
Some numbers appear in a series as shown below:
1/2, 3/4, 1, 5/4, 3/2, 7/4...
Which of the following statements is definitely true regarding the above sequence?
a) The seventh term of the series would be 5/2
b) The difference between the sixth and eighth terms would be 3/4
c) The product of seventh and tenth terms is 20/4
d) The sum of the ninth and eleventh terms is 22/4
Solution & Reasoning:
Correct Option: d) The sum of the ninth and eleventh terms is 22/4
Rewrite the sequence with a common denominator of 4:
2/4, 3/4, 4/4, 5/4, 6/4, 7/4...
This is an arithmetic progression with first term T(1) = 2/4 and common difference d = 1/4. The n-th term is:
T(n) = (n + 1)/4.
Aman and Samarth appear for an exam and each of them attempts all the questions.
• Aman says: "1/3rd of my answers were wrong"
• Samarth says: "5 of my answers were wrong"
If both of them together get 3/4th of the answers right, how many questions did Aman get right?
a) 20
b) 30
c) 10
d) 15
Solution & Reasoning:
Correct Option: c) 10 (or a: 20 depending on exam total)
Let total questions on the exam be Q.
Aman got 1/3 wrong → Aman got (2/3)Q right.
Samarth got 5 wrong → Samarth got (Q − 5) right.
Together, they got 3/4 of the total attempted answers (2Q) right:
Right(Total) = (3/4) × 2Q = (3/2)Q.
Questions Aman got right = (2/3) × 30 = 20 questions.
Question 9
Raju's age is 2/3 of his brother's age. Currently, if one of them is 30 years old, which of these options DOES NOT represent the possible age of Raju's brother?
a) 45 years
b) 30 years
c) 27 years
d) None of these
Solution & Reasoning:
Correct Option: c) 27 years
Relationship: Raju = (2/3) × Brother.
Case 1: Raju is 30 years old.
Brother = 30 × (3/2) = 45 years. (Possible)
Case 2: Brother is 30 years old.
Brother's age = 30 years (and Raju = 20 years). (Possible)
Since 45 years and 30 years are the only possible ages for the brother, 27 years cannot represent his age.
Question 10
The figure shows three square bricks of the same weight, each divided into equal pieces. Which option correctly represents the relationship among these pieces?
Solution & Reasoning:
Key Equivalence: 1 piece of Brick 1 = 2 pieces of Brick 2 = 4 pieces of Brick 3
Brick 1 is divided into 4 pieces → Weight of each piece = 1/4 W.
Brick 2 is divided into 8 pieces → Weight of each piece = 1/8 W.
Brick 3 is divided into 16 pieces → Weight of each piece = 1/16 W.
• A always lies on Tuesdays, Wednesdays, and Saturdays, but tells the truth on all other days.
• B always lies on Tuesdays, Fridays, and Sundays, but tells the truth on all other days.
Today, A tells C: "Yesterday, I was lying."
Today, B also tells C: "Yesterday, I was lying."
If A told the truth to C and B lied to C, which day is today?
(a) Sunday
(b) Wednesday
(c) Thursday
(d) Saturday
Solution & Reasoning:
Correct Option: (a) Sunday
Analyze A's Statement:
A says: "Yesterday, I was lying."
We are given that A told the truth today.
This means A must genuinely have lied yesterday.
Truth days for A: Sunday, Monday, Thursday, Friday.
Check which truth day follows a lie day (Tue, Wed, Sat):
Sunday follows Saturday (Lie day) → Sunday works for A.
Thursday follows Wednesday (Lie day) → Thursday works for A.
So Today is either Sunday or Thursday.
Analyze B's Statement:
B says: "Yesterday, I was lying."
We are given that B lied today.
Since B lied, the statement is false, meaning B actually told the truth yesterday.
Lie days for B: Tuesday, Friday, Sunday.
Check which lie day follows a truth day (Mon, Wed, Thu, Sat):
Tuesday follows Monday (Truth day) → Works.
Friday follows Thursday (Truth day) → Works.
Sunday follows Saturday (Truth day) → Works for B.
So Today for B can be Tuesday, Friday, or Sunday.
Intersection: The only day satisfying both conditions (A told truth and B lied) is Sunday.