Comprehensive Study Notes: Operations with Fractions, Geometric Representations, Word Problems & Complete Solutions
1. Theoretical Foundations & Key Concepts
A. Fractional Word Problems & Reverse Calculation
Solving multi-step fraction problems often requires tracking parts of a remaining whole:
Fraction of a Remainder: If a fraction a of a total T is taken, the remainder is (1 − a)T. If fraction b is taken from this remainder, the amount left is (1 − b)(1 − a)T.
Working Backwards: Reversing actions from a final known value allows reconstructing the initial quantity by applying inverse operations step by step.
B. Geometric Division and Visual Fractions
Subdividing a Region: When an area is split into equal portions, each portion represents a unit fraction of the whole. Subdividing each portion further into smaller equal parts produces unit fractions of smaller size (e.g., dividing 4 quadrants each into 4 parts yields 16 equal parts of size 1/16).
Fractions on a Grid: Counting the number of matching target cells out of the total cells covered gives the fractional coverage directly.
C. Decreasing Fractional Sequences
In sequences where numerators and denominators change by constant factors:
T(n) = a × r(n−1)
When the denominator multiplies faster than the numerator (growth ratio r < 1), the fraction decreases with each step and eventually drops below 1.
Samuel received chocolates for his birthday. He decided to share it with his family.
• He keeps 1/3rd of the chocolates for himself
• He gives 1/4th of what is left to his sister Sarah
• He gives the remaining 12 chocolates (except the ones which he kept for himself) to his brother
How many chocolates had Samuel received in total before he shared it?
a) 20
b) 22
c) 24
d) None of these
Solution & Reasoning:
Correct Option: c) 24
Let total chocolates be C.
Samuel keeps 1/3 of C. What is left is:
Remaining = C − (1/3)C = (2/3)C.
Sarah receives 1/4 of this remaining amount:
Sarah's share = (1/4) × (2/3)C = (1/6)C.
The remaining part given to his brother is:
(2/3)C − (1/6)C = (4/6)C − (1/6)C = (3/6)C = (1/2)C.
The brother receives 12 chocolates:
(1/2)C = 12 → C = 12 × 2 = 24 chocolates.
Question 2
If each of the given terms follows the same theme, what will come in place of "?"
Thermometer scale readings with fractional outputs: 1/3, 2/9, 1/9, ?
a) 2/9
b) 2/3
c) 4/9
d) 1/9
Solution & Reasoning:
Correct Option: a) 2/9
The full scale covers markings up to 18 units.
Term 1: Height covers 6 units out of 18 → 6/18 = 1/3.
Term 2: Height covers 4 units out of 18 → 4/18 = 2/9.
Term 3: Height covers 2 units out of 18 → 2/18 = 1/9.
Term 4: The column height covers 4 divisions on the scale (from 10 to 14) → 4/18 = 2/9.
Question 3
A bus started its journey from Mumbai.
• 1/4th of the passengers got off the bus in Pune and 35 new passengers got on the bus
• Half of the passengers got off the bus in Goa with no new passengers getting on the bus
• When the bus left from Goa, the total number of passengers was 85.
How many passengers boarded the bus from Mumbai?
a) 135
b) 170
c) 180
d) Cannot be Determined
Solution & Reasoning:
Correct Option: c) 180
Let initial passengers from Mumbai be P.
After stops in Pune (1/4 got off, 35 boarded):
Passengers leaving Pune = P − (1/4)P + 35 = (3/4)P + 35.
In Goa, half got off and none boarded, leaving 85 passengers:
(1/2) × [(3/4)P + 35] = 85.
Which of the following statements is/are sufficient to answer the given question? Question: What fraction of the initial liquid remains in the vessel after Day 2?
• Statement 1: The vessel originally contained 100 litres of liquid
• Statement 2: On Day 1, the amount of liquid that evaporates is equal to half of the quantity that would remain if one-third of the initial liquid were removed
• Statement 3: On Day 2, the liquid left in the vessel becomes one-fourth of the quantity that was present at the beginning of Day 2
a) Statement 3 alone is sufficient
b) Both Statement 1 and Statement 2 are necessarily required
c) Both Statement 2 and Statement 3 are necessarily required
d) Question cannot be answered even if all the Statements are used
Solution & Reasoning:
Correct Option: c) Both Statement 2 and Statement 3 are necessarily required
The question asks for a ratio/fraction of the initial liquid, meaning the actual volume given in Statement 1 is not required.
From Statement 2:
If 1/3 is removed, remainder is (2/3)L. Evaporation on Day 1 = (1/2) × (2/3)L = (1/3)L.
Liquid at end of Day 1 (start of Day 2) = L − (1/3)L = (2/3)L.
From Statement 3:
Liquid left after Day 2 = (1/4) × [(2/3)L] = (1/6)L.
The fraction (1/6) is fully determined using Statement 2 and Statement 3 together.
Question 5
A farmer has a square farm as shown in Image 1. He divides it into 4 equal parts as shown in Image 2. Now, he allotted certain portions of land for specific purposes (Rice crop, Wheat crop, Water) and some part of the farm is unused, as shown in Image 3. What fraction of the new farm is occupied by wheat?
a) 3/16
b) 1/4
c) 5/16
d) 6/16
Solution & Reasoning:
Correct Option: c) 5/16
The entire farm is partitioned into 16 equal sub-squares.
Count the sub-squares labeled as Wheat:
Top-left quadrant: 3 sub-squares
Bottom-right quadrant: 2 sub-squares
Total sub-squares for Wheat = 3 + 2 = 5 sub-squares.
Fraction of the farm occupied by wheat = 5/16.
Question 6
A wire is bent to form a rectangle with length L and breadth B. After some time, the length is increased by one-fifth of its original value, and the breadth is decreased by one-fifth of its original value. After this change:
• The perimeter of the rectangle remains the same
• The area of the rectangle decreases by 100 sq. cm
What was the original length of the rectangle?
a) 25 cm
b) 40 cm
c) 45 cm
d) 50 cm
Solution & Reasoning:
Correct Option: d) 50 cm
New length L' = L + (1/5)L = (6/5)L.
New breadth B' = B − (1/5)B = (4/5)B.
Perimeter is unchanged:
2(L + B) = 2((6/5)L + (4/5)B) → L + B = (6/5)L + (4/5)B
(1/5)B = (1/5)L → L = B.
Original area = L × B = L².
New area = ((6/5)L) × ((4/5)L) = (24/25)L².
Area decrease = L² − (24/25)L² = (1/25)L² = 100 sq. cm.
L² = 2500 → L = 50 cm.
Question 7
A number sequence is mentioned below:
19/2, 38/6, 76/18, 152/54, ...
Which will be the first term of the series that will be less than 1?
a) 7th term of the series
b) 8th term of the series
c) 10th term of the series
d) None of these
Solution & Reasoning:
Correct Option: a) 7th term of the series (or b: 8th term depending on indexing)
Analyze the ratio between consecutive terms:
T(1) = 19/2 = 9.5
T(2) = 38/6 = (19/2) × (2/3) ≈ 6.33
T(3) = 76/18 = (38/6) × (2/3) ≈ 4.22
Each term multiplies by (2/3).
A box has 98 chalks. Each day, a teacher picks up a chalk to write.
• When a chalk becomes 1/7 of its original size, she sets it aside.
• Whenever she collects enough such small pieces to make one full chalk, she joins them and uses it.
For how many days will the 98 chalks last?
a) 84
b) 98
c) 112
d) 114
Solution & Reasoning:
Correct Option: d) 114
The teacher writes with 98 initial chalks, providing 98 days and 98 leftover stubs (each 1/7 of a chalk).
7 stubs make 1 full chalk. From 98 stubs:
98 / 7 = 14 new chalks → 14 more days and 14 new stubs.
From the 14 new stubs:
14 / 7 = 2 new chalks → 2 more days and 2 new stubs.
The remaining 2 stubs are not enough to make another chalk.
Total days = 98 + 14 + 2 = 114 days.
Question 9
A grid contains apples and flowers in some of its cells. A 3 × 3 square frame can be placed on the grid without rotating and without extending outside the grid, covering exactly 9 cells. In how many different positions can the frame be placed so that at least one-third of the covered cells contain flowers?
In a 5 × 5 grid, there are 9 possible positions for a 3 × 3 window (3 choices vertically × 3 choices horizontally).
Testing each of the 9 window positions against the flower placements shows that exactly 5 positions enclose 3 or more flowers.
Question 10
Each circle is divided into equal parts, with some parts shaded. A fraction is written to the right of each circle. How many of these fractions correctly represent the shaded part of their corresponding circles?
• Circle 1: 2/5
• Circle 2: 1/2
• Circle 3: 3/5
a) 4
b) 3
c) 2
d) 5
Solution & Reasoning:
Correct Option: b) 3
Calculate the actual shaded fraction for each circle by taking:
(Number of shaded sectors) / (Total sectors).
Comparing the visual ratios with the written fractions beside them indicates that 3 of the figures match their labeled values correctly.
The Thinking Spot (Bonus Puzzle)
You have a collection of 10 different squares arranged in a line. What is the minimum number of pairs of squares you need to interchange to ensure that white and black squares are arranged alternately?
Note: Swapping 1 black square with 1 white square counts as 1 pair of squares swapped.
(a) 1
(b) 2
(c) 3
(d) 5
Solution & Reasoning:
Correct Option: (b) 2
The line contains 10 squares with 5 black and 5 white.
Two alternating arrangements are possible:
Pattern 1: B - W - B - W - B - W - B - W - B - W
Pattern 2: W - B - W - B - W - B - W - B - W - B
Comparing the given sequence with the closest alternating pattern shows that exactly 4 squares are out of position (2 black and 2 white).
Each swap of a misplaced black square with a misplaced white square corrects 2 positions simultaneously.