An ant crawls along three connected semicircular arcs from point A to point B as shown in the diagram. The straight-line distance from A to B is 12 cm. What is the total distance traveled by the ant along the curved path?
Explanation: The total straight line length of 12 cm is divided into 3 equal diameters of 4 cm each (since 12 ÷ 3 = 4 cm).
The perimeter (arc length) of one semicircle is ½ × π × d = ½ × π × 4 = 2π cm.
Since there are 3 identical semicircles, total distance = 3 × 2π = 6π cm.
Trick/Concept: Even though the curved path wraps around, the total length of smaller identical semicircles on a straight segment equals the length of one single big semicircle spanning the entire distance (12 × π/2 = 6π).
An arched pedestrian bridge spans across a river that is 20 meters wide. A straight distance cable connects point P to point Q directly along the water level. If an engineer walks from P to Q over the arch of the bridge, and a fish swims underwater in a direct straight line from P to Q, which statement is ALWAYS true?
Explanation: The shortest distance between any two points in space is always a straight line segment. The fish travels along the straight displacement vector (20 m), whereas the engineer follows the curved arch of the bridge, making the total distance traveled strictly greater than 20 m.
A car enters a circular roundabout of radius R = 14 meters at point A and travels along the roundabout to point B, which is directly opposite to point A (halfway around). Compare the Distance (D) traveled by the car with its Displacement (S) (the direct straight line distance from start to end). (Take π = 22/7.)
Explanation:
• Displacement (S): The straight line from A to B across the center is the diameter = 2 × R = 2 × 14 = 28 m.
• Distance (D): Half the circumference of the circle = π × R = (22/7) × 14 = 44 m (taking π = 22/7).
Notice that Distance > Displacement because the path traveled is curved!
A robot moves from point X to point Y. Path 1 goes straight up a ramp along the diagonal line. Path 2 goes along a precise staircase pattern made of 5 identical vertical and 5 identical horizontal steps. Which path is longer?
Explanation: In any right triangle formed by horizontal and vertical step components, the hypotenuse (Path 1 line) is strictly shorter than the sum of the two legs (base + height). According to the Triangle Inequality Theorem, the sum of all horizontal and vertical staircase lengths (Path 2) will always be strictly greater than the direct diagonal straight line distance (Path 1).
A runner starts at marker P, completes 3 full laps around a circular track of radius 10 meters, and stops exactly back at marker P. What are the runner's total Distance Traveled and Net Displacement?
Explanation:
• Distance: Total length of actual ground covered. 1 lap = 2 × π × r = 2 × π × 10 = 20π m. For 3 laps, Total Distance = 3 × 20π = 60π m (≈ 188.5 m).
• Displacement: The shortest straight-line distance from the starting position to the final position. Since the runner ended up at the exact point P where they started, the net change in position is zero (0 m).
A giant wheel with a red cart on top takes 30 seconds to take a full turn. Another giant wheel with a yellow cart on top takes 20 seconds to take a full turn. If both start turning from the top position at the exact same time, how long will it take for both carts to be at the top position together again?
Explanation: To find when both events coincide again, we need the Least Common Multiple (LCM) of their rotation times.
Multiples of 30: 30, 60, 90, 120...
Multiples of 20: 20, 40, 60, 80...
The smallest common multiple is 60 seconds (1 minute).
An architect wants to tile a floor that is 84 cm long and 60 cm wide using the largest possible identical square tiles without cutting any tile. How many total square tiles will be needed to cover the entire floor?
Explanation:
1. To find the largest square tile size, find the HCF of 84 and 60.
Factors of 84 = 2 × 2 × 3 × 7
Factors of 60 = 2 × 2 × 3 × 5
HCF(84, 60) = 2 × 2 × 3 = 12 cm. (Each tile is 12 cm × 12 cm).
2. Tiles along length = 84 ÷ 12 = 7 tiles.
3. Tiles along width = 60 ÷ 12 = 5 tiles.
4. Total tiles needed = 7 × 5 = 35 tiles.
Trick Alert: Options often include 12, which is the side of the tile, not the count of tiles!
Three colored LED light strips flash at regular intervals: Red flashes every 6 seconds, Green every 8 seconds, and Blue every 12 seconds. If all three flash together at exactly 8:00 AM, how many times will they flash together between 8:00 AM and 8:05 AM (inclusive of both 8:00 AM and 8:05 AM)?
Explanation:
1. Find the LCM of 6, 8, and 12 to determine how often they flash together.
LCM(6, 8, 12) = 24 seconds.
2. Total time duration from 8:00 AM to 8:05 AM = 5 minutes = 300 seconds.
3. Number of intervals = 300 ÷ 24 = 12.5 → 12 full intervals.
4. Since they flash together at time = 0s (8:00 AM), total flashes = 12 + 1 = 13 times.
Trick Alert: Forgetting to add the initial flash at 8:00 AM (0 seconds) is a common mistake!
A teacher has 48 chocolate bars and 72 juice boxes to distribute equally into gift bags for a class party. Every bag must have the exact same combination of items with none left over. What is the maximum number of gift bags she can make, and how many total items will be in each bag?
Explanation:
1. To find the maximum number of bags, calculate the HCF of 48 and 72.
Factors of 48 = 24 × 3
Factors of 72 = 23 × 32
HCF(48, 72) = 23 × 3 = 24 bags.
2. Chocolates per bag = 48 ÷ 24 = 2.
3. Juice boxes per bag = 72 ÷ 24 = 3.
4. Total items per bag = 2 + 3 = 5 items.
Since both A and C express the true count (24 bags with 5 items total / 2 chocolates + 3 juices), option D is correct.
What is the smallest number which, when divided by 12, 15, and 18, leaves a remainder of 4 in each case?
Explanation:
1. First, find the smallest number divisible by 12, 15, and 18 without any remainder, which is their LCM.
12 = 22 × 3
15 = 3 × 5
18 = 2 × 32
LCM(12, 15, 18) = 22 × 32 × 5 = 4 × 9 × 5 = 180.
2. To leave a remainder of 4 in each case, add 4 to the LCM:
Required Number = 180 + 4 = 184.
Check: 184 ÷ 12 = 15 (rem 4), 184 ÷ 15 = 12 (rem 4), 184 ÷ 18 = 10 (rem 4).
Look at the given polyomino shape below. Which of the following block choices (A, B, C, or D) can be added to complete a full 3×3 square grid? (Note: Options can be rotated!)
Explanation:
1. A 3×3 square contains 9 unit squares in total.
2. The given figure has 5 filled unit squares, leaving 9 - 5 = 4 empty squares.
3. The 4 empty squares form a column of 3 unit squares on the right side plus 1 square sticking out to the left from the middle row. That is a T-shaped block of 4 squares.
4. Option B is a T-shape, so it fits the missing region perfectly. (A is an L-shape, C is a zig-zag and D is a straight line, so none of them fit.)
The net below can be folded to form a 3D cube. Which symbol will be on the face opposite to the Star (★)?
Explanation:
In a standard cube net, faces that are separated by one intervening face in a straight row or column fold up to become opposite faces.
• Triangle (Δ) is opposite Circle Dot (•)
• Star (★) is opposite Square (□)
• Heart (♥) is opposite Sun (☀)
A solid 3D structure is built using identical 1×1×1 unit cubes on a table, as shown below. What is the total number of unit cubes used to build this structure (including the hidden cubes supporting the top ones)?
Explanation:
Count the height of each vertical column from back to front:
• Back column = 3 cubes high
• Left column = 2 cubes high
• Right column = 1 cube high
• Front column = 1 cube high
Total Cubes = 3 + 2 + 1 + 1 = 7 cubes.
A large rectangle measures 10 cm in length and 6 cm in width. Inside it, two identical unshaded right triangles with base 4 cm and height 3 cm are removed. What is the area of the shaded region?
Explanation:
1. Total Area of Outer Rectangle = Length × Width = 10 cm × 6 cm = 60 cm².
2. Area of one unshaded triangle = ½ × Base × Height = ½ × 4 × 3 = 6 cm².
3. Area of both unshaded triangles = 2 × 6 cm² = 12 cm².
4. Area of Shaded Region = 60 cm² - 12 cm² = 48 cm².
A clock face showing the time 3:40 is reflected in a plane mirror placed vertically to its right. What time will the clock show in the mirror image?
Explanation:
A quick standard rule for finding mirror time on a 12-hour clock is to subtract the real time from 11:60 (which equals 12:00):
11 : 60
- 03 : 40
-----------
08 : 20
The mirror image displays 8:20.
On a safari park scale, the total weight of a Lion and a Deer is 300 kg. When a blue Whale gets onto a giant underwater scale along with the Lion and the Deer, their combined weight is 840 kg. What is the weight of the Whale?
Explanation:
1. We know: Lion + Deer = 300 kg.
2. We also know: Lion + Deer + Whale = 840 kg.
3. Replace (Lion + Deer) with 300 kg in the second equation:
300 kg + Whale = 840 kg
4. Whale = 840 kg - 300 kg = 540 kg.
Sammy the snail crawls at a steady speed of 3 cm every second. How far will Sammy travel in 45 seconds?
Explanation:
Distance = Speed × Time
Distance = 3 cm/s × 45 seconds = 135 cm.
A farmer wants to build a fence around an L-shaped garden plot. Given the side lengths shown in the diagram, what is the total perimeter of the garden?
Explanation:
1. Horizontal sides: top = 12 m and bottom = 6 m, so the missing horizontal step = 12 − 6 = 6 m.
2. Vertical sides: left = 10 m and right = 6 m, so the missing vertical step = 10 − 6 = 4 m.
3. Perimeter = 12 + 6 + 6 + 4 + 6 + 10 = 44 m.
Shortcut: an L-shape has the same perimeter as its bounding rectangle: 2 × (12 + 10) = 44 m.
Maya has 60 colorful marbles in a jar. 1/3 of the marbles are blue, 2/5 are red, and the remaining marbles are green. How many green marbles does Maya have?
Explanation:
1. Blue marbles = 1/3 of 60 = 60 ÷ 3 = 20 marbles.
2. Red marbles = 2/5 of 60 = (60 ÷ 5) × 2 = 12 × 2 = 24 marbles.
3. Total Blue + Red = 20 + 24 = 44 marbles.
4. Green marbles = Total - (Blue + Red) = 60 - 44 = 16 marbles.
In this magic number pyramid, each upper block is created by adding the two blocks directly underneath it. What is the value of the missing top block marked with X?
Explanation:
1. Bottom row check:
6 + 8 = 14 (Left middle block)
8 + 10 = 18 (Right middle block)
2. To find X, add the two middle blocks together:
X = 14 + 18 = 32.
Study the balanced scales below:
What is the combined weight of 1 Circle and 2 Triangles?
Explanation:
1. Subtract the second equation from the first:
(2 Circles + 1 Triangle) - (1 Circle + 1 Triangle) = 14 kg - 9 kg
1 Circle = 5 kg.
2. Find Triangle weight:
1 Circle + 1 Triangle = 9 kg → 5 kg + 1 Triangle = 9 kg → 1 Triangle = 4 kg.
3. Calculate target weight:
1 Circle + 2 Triangles = 5 + (2 × 4) = 5 + 8 = 13 kg.
A boy builds towers using identical wooden blocks. Tower 1 has 3 blocks, Tower 2 has 6 blocks, and Tower 3 has 10 blocks. Following this triangular growth pattern, how many blocks will be needed to build Tower 6?
Explanation:
The pattern follows triangular numbers where Tower n has 1 + 2 + 3 + ... + (n + 1) blocks:
• Tower 1: 1 + 2 = 3
• Tower 2: 1 + 2 + 3 = 6
• Tower 3: 1 + 2 + 3 + 4 = 10
• Tower 4: 10 + 5 = 15
• Tower 5: 15 + 6 = 21
• Tower 6: 21 + 7 = 28 blocks.
A shaded L-shaped figure is placed on a grid of identical squares, each with an area of 9 cm². What is the total perimeter of the shaded figure?
Explanation:
1. Area = side × side = 9 cm², so each grid square has side 3 cm.
2. Count the unit segments around the outline: top 1 + right step 2 + inner horizontal 3 + inner vertical 1 + bottom 4 + left 3 = 14 units.
3. Perimeter = 14 × 3 cm = 42 cm.
Two robotic bugs, Bug A and Bug B, start crawling from the same point around a track. Bug A crawls at 4 cm/s and Bug B crawls at 7 cm/s in the same direction. How far apart will they be after 45 seconds?
A parking lot has 22 vehicles parked in total. It has two types of vehicles: 2-wheeled bicycles and 4-wheeled cars. If there are a total of 64 wheels in the parking lot, how many cars are parked there?
Explanation:
1. Suppose all 22 vehicles were 2-wheeled bicycles.
2. Total wheels would be: 22 × 2 = 44 wheels.
3. But the question says there are 64 wheels in total. The extra wheels are: 64 − 44 = 20 wheels.
4. Each car has 2 more wheels than a bicycle (4 − 2 = 2).
5. Number of cars = 20 ÷ 2 = 10 cars.
Check: 10 cars (40 wheels) + 12 bicycles (24 wheels) = 22 vehicles and 64 wheels.
Farmer Joe has ducks and cows on his farm. Counting their heads, there are 15 animals in total. Counting their legs, there are 46 legs. How many ducks are on the farm?
Explanation:
1. If all 15 animals were ducks, there would be: 15 × 2 = 30 legs.
2. The difference in leg count is: 46 − 30 = 16 extra legs.
3. Each cow adds 2 extra legs (4 − 2 = 2).
4. Number of cows = 16 ÷ 2 = 8 cows.
5. Number of ducks = Total animals − Cows = 15 − 8 = 7 ducks.
Check: (7 ducks × 2 legs) + (8 cows × 4 legs) = 14 + 32 = 46 legs.
A school auditorium has 80 seats in total. It has two-seater benches and three-seater benches. There are 10 two-seater benches in the room. How many three-seater benches are there?
Explanation:
1. Calculate seats taken by two-seater benches: 10 benches × 2 seats = 20 seats.
2. Subtract these seats from the total seats available: 80 seats − 20 seats = 60 seats remaining.
3. Divide the remaining seats by 3 to find the number of three-seater benches: 60 ÷ 3 = 20 benches.
On a balanced scale, 2 identical toy robots and 1 block weigh 11 kg. On another balanced scale, 1 toy robot and 3 identical blocks weigh 13 kg. If all blocks are identical, how much does 1 toy robot weigh?
Explanation:
Let R = weight of 1 Robot, B = weight of 1 Block.
Scale 1: 2R + 1B = 11 kg → 1B = 11 − 2R.
Scale 2: 1R + 3B = 13 kg.
Substitute 1B into Scale 2: 1R + 3 × (11 − 2R) = 13
1R + 33 − 6R = 13 → 33 − 5R = 13 → 5R = 20 → R = 4 kg.
(Block weight = 11 − 2(4) = 3 kg).
Check: Scale 1 = 4+4+3 = 11 kg. Scale 2 = 4+3+3+3 = 13 kg.
Siddharth has 30 coins in his piggy bank consisting only of 5-rupee coins and 10-rupee coins. The total value of all coins is 220 rupees. How many 10-rupee coins does Siddharth have?
Explanation:
1. If all 30 coins were 5-rupee coins, the total would be 30 × 5 = 150 rupees.
2. The actual total is 220 rupees, a difference of 220 − 150 = 70 rupees.
3. Replacing one 5-rupee coin with a 10-rupee coin adds 5 rupees.
4. Number of 10-rupee coins = 70 ÷ 5 = 14 coins.
Check: 14 coins of ₹10 (140) + 16 coins of ₹5 (80) = 30 coins totalling ₹220.
Sam liked to do experiments. He painted some shapes on half of a circular paper as shown below. While the color was wet, he gently folded the paper along the dotted center line and pressed it. How will the picture look when he unfolds the paper?
Explanation: Folding paper with wet paint along a line creates a reflectional (mirror) symmetry about the dotted folding line. Each shape on the left side gets copied identically to the right side at the exact same distance and orientation relative to the center fold line.
A regular hexagon with total area of 120 cm² is divided into 6 equal triangular sections. Two sections are fully shaded blue, and one section is half-shaded yellow. What is the total shaded area of the figure?
Explanation:
1. Area of 1 full triangular sector = Total Area ÷ 6 = 120 ÷ 6 = 20 cm².
2. Two fully shaded blue sectors = 2 × 20 cm² = 40 cm².
3. One half-shaded yellow sector = ½ × 20 cm² = 10 cm².
4. Total Shaded Area = 40 + 10 = 50 cm².
Look at the two balanced scales below. How many strawberries are needed to perfectly balance one pineapple on Scale 3?
Explanation:
From Scale 1: 1 Pineapple = 2 Apples.
From Scale 2: 1 Apple = 3 Strawberries.
Therefore, 2 Apples = 2 × 3 = 6 Strawberries.
So, 1 Pineapple = 6 Strawberries.
A clock currently shows 3:00 PM. If the hour hand rotates clockwise through an angle of 150°, what time will the clock show?
Explanation:
A full circle on a clock is 360°, divided into 12 hours.
Angle per hour = 360° ÷ 12 = 30° per hour.
Number of hours traveled = 150° ÷ 30° = 5 hours.
Starting at 3:00 PM + 5 hours = 8:00 PM.
Which of the closed 3D cubes can be formed by folding the flat net shown on the left?
Explanation:
In a standard cube net, faces separated by one square in a line fold to become opposite faces:
• ⭐ (Star) and 🔺 (Triangle) are opposite each other.
• 🟢 (Circle) and 🟦 (Square) are opposite each other.
• ❤️ (Heart) and 🔷 (Diamond) fold on top/bottom, so they are opposite each other.
A swimmer aims to swim directly across a river from A to B. However, the current pushes the swimmer downstream to point C. Which statement correctly describes the distance traveled and displacement?
Explanation: The actual straight path traveled through water from start point A to end point C is line AC. Thus, both actual distance and direct displacement equal the length of AC.
Two runners run from point A to B. Runner 1 takes the inner track arc of radius 10 m, while Runner 2 takes the outer track arc of radius 20 m. How much further does Runner 2 run?
Explanation: Inner arc length = π × 10 = 10π m. Outer arc length = π × 20 = 20π m. Difference = 20π - 10π = 10π m.
A farmer walks along the perimeter of a rectangular field (40 m × 30 m) from corner A to diagonally opposite corner C. What is the ratio of Distance to Displacement?
Explanation: Distance along perimeter = 40 + 30 = 70 m. Diagonal Displacement = √(40² + 30²) = 50 m. Ratio = 70 : 50 = 7 : 5.
A beetle crawls along two consecutive semicircular arcs, each with diameter 6 cm. What is the ratio of total path distance to net displacement?
Explanation: Displacement = 6 + 6 = 12 cm. Distance for two semicircles = 2 × (½ × π × 6) = 6π cm. Ratio = 6π : 12 = π : 2.
An object travels along 3 connected semicircles of increasing diameters: 2 cm, 4 cm, and 6 cm. What is the total distance traveled?
Explanation: Arc length = ½πd. Total distance = ½π(2) + ½π(4) + ½π(6) = π + 2π + 3π = 6π cm.
A pendulum bob swings back and forth along a curved arc of length 15 cm. If it swings 4 times complete (back and forth), what is its net displacement from its starting equilibrium point?
Explanation: After completing full back-and-forth swing cycles, the bob returns to its initial starting position, making net displacement equal to 0 cm.
A person walks 80 m North, then 60 m East. How much distance is saved by going straight from the starting point to the final point instead?
Explanation: Original path distance = 80 + 60 = 140 m. Straight diagonal distance = √(80² + 60²) = 100 m. Distance saved = 140 - 100 = 40 m.
A driver travels three-quarters of the way around a circular track of radius 7 m. What is the magnitude of the driver's displacement?
Explanation: The start and end positions form a right-angled triangle with radius sides 7 m and 7 m. Displacement = √(7² + 7²) = 7√2 m (≈ 9.9 m).
A drone flies along a figure-8 path consisting of two identical touching circles of radius 5 m. What is the total distance traveled after 1 full figure-8 loop?
Explanation: Circumference of 1 circle = 2 × π × 5 = 10π m. Two circles = 2 × 10π = 20π m.
A ball is thrown vertically upwards to a height of 15 meters and caught back at the same release point. What are the total Distance and Displacement?
Explanation: Total distance covered = 15 m up + 15 m down = 30 m. Since initial and final points are identical, Net Displacement = 0 m.
Three runners take 12, 18, and 24 minutes to complete one lap around a track. If they start together at 9:00 AM, at what time will they meet again at the starting point?
Explanation: LCM(12, 18, 24) = 72 minutes = 1 hour 12 minutes. 9:00 AM + 1 hr 12 min = 10:12 AM.
Three ribbons of lengths 45 cm, 60 cm, and 75 cm are to be cut into equal pieces without any leftover ribbon. What is the maximum possible length of each piece?
Explanation: Maximum equal piece length = HCF(45, 60, 75) = 15 cm.
Bells A, B, and C toll at intervals of 15, 20, and 30 minutes respectively. If they toll together at 12:00 PM, when will they toll together next?
Explanation: LCM(15, 20, 30) = 60 minutes = 1 hour. Next toll = 12:00 PM + 1 hour = 1:00 PM.
The HCF of two numbers is 16 and their product is 3072. What is their LCM?
Explanation: Formula: HCF × LCM = Product of two numbers. LCM = 3072 ÷ 16 = 192.
Two milk tankers contain 850 liters and 680 liters of milk. What is the maximum capacity of a container that can measure the milk of both tankers an exact number of times?
Explanation: HCF(850, 680) = 170 liters.
What is the smallest 4-digit number that is exactly divisible by 12, 18, and 21?
Explanation: LCM(12, 18, 21) = 252. Smallest 4-digit multiple = 252 × 4 = 1008.
If two numbers x and y are coprime (share no common factors other than 1), what is their HCF and LCM?
Explanation: By definition, coprime numbers have HCF = 1, which makes their LCM equal to their direct product (x × y).
A school librarian wants to arrange 120 Science books and 144 Math books into equal stacks containing only one subject each. What is the maximum number of books per stack?
Explanation: Maximum stack height = HCF(120, 144) = 24 books.
Three consecutive traffic lights change color every 48s, 72s, and 108s. If they change simultaneously at 7:00:00 AM, at what time will they change together next?
Explanation: LCM(48, 72, 108) = 432 seconds = 7 minutes 12 seconds. Time = 7:00:00 AM + 7m 12s = 7:07:12 AM.
What is the Highest Common Factor (HCF) of the fractions 2/3, 4/9, and 8/15?
Explanation: Formula: HCF of Fractions = HCF(Numerators) / LCM(Denominators). HCF(2, 4, 8) = 2. LCM(3, 9, 15) = 45. Result = 2/45.
What is the place value of the digit 4 in the number 245,318?
Explanation: The digit 4 is in the ten-thousands place, so its value is 4 × 10,000 = 40,000.
Using each of the digits 3, 0, 7, 5 and 1 exactly once, what is the smallest 5-digit number you can make?
Explanation: A number cannot start with 0, so the smallest non-zero digit, 1, goes first. Then put the rest in increasing order: 0, 3, 5, 7 → 10357. (01357 is not a 5-digit number.)
What is the sum 1 + 2 + 3 + … + 20?
Explanation: Pair the numbers: (1 + 20), (2 + 19), … gives 10 pairs, each adding to 21. So 10 × 21 = 210.
How many 2-digit numbers are exactly divisible by 7?
Explanation: The 2-digit multiples of 7 run from 14 (7 × 2) to 98 (7 × 14). That is 14 − 2 + 1 = 13 numbers.
How many factors does 36 have?
Explanation: 36 = 2² × 3², so the number of factors is (2 + 1) × (2 + 1) = 9. They are 1, 2, 3, 4, 6, 9, 12, 18, 36.
How many prime numbers are there between 20 and 40?
Explanation: The primes between 20 and 40 are 23, 29, 31 and 37, so there are 4.
What is the largest 3-digit number that is divisible by 8?
Explanation: 999 ÷ 8 leaves remainder 7, so subtract 7: 999 − 7 = 992 (= 8 × 124).
The 4-digit number 347_ is divisible by 9. What is the missing digit?
Explanation: A number is divisible by 9 when its digit sum is. 3 + 4 + 7 = 14, so the missing digit must make 18: 18 − 14 = 4. The number is 3474.
Find the next number: 2, 6, 12, 20, 30, ?
Explanation: The differences are 4, 6, 8, 10, so the next difference is 12: 30 + 12 = 42.
Find the missing number: 3, 6, 11, 18, 27, ?
Explanation: The differences are 3, 5, 7, 9, so the next difference is 11: 27 + 11 = 38.
What comes next: 2, 3, 5, 8, 13, 21, ?
Explanation: Each number is the sum of the two before it: 13 + 21 = 34.
The average of five numbers is 18. The sum of four of them is 70. What is the fifth number?
Explanation: Sum of all five = 5 × 18 = 90. Fifth number = 90 − 70 = 20.
A student's average score in 3 tests is 72. What must the student score in the 4th test to make the average 75?
Explanation: Total needed = 4 × 75 = 300. Total so far = 3 × 72 = 216. The 4th test score = 300 − 216 = 84.
Which of these fractions is the greatest? 3/4, 5/8, 7/10, 2/3
Explanation: As decimals: 0.75, 0.625, 0.7 and 0.667. The greatest is 3/4.
What is 3/8 of 96?
Explanation: 96 ÷ 8 = 12, and 12 × 3 = 36.
Ravi has ₹240. He spends 1/4 of it on books and then 1/3 of the remaining money on toys. How much money does he have left?
Explanation: Books: 240 ÷ 4 = ₹60, leaving ₹180. Toys: 180 ÷ 3 = ₹60. Money left = 180 − 60 = ₹120.
Find the value of 4.5 + 3.75 − 2.6.
Explanation: 4.5 + 3.75 = 8.25, and 8.25 − 2.6 = 5.65.
A book marked at ₹250 is sold at a 20% discount. What is the selling price?
Explanation: Discount = 20% of 250 = ₹50. Selling price = 250 − 50 = ₹200.
In a class of 40 students, the ratio of boys to girls is 3 : 2. How many girls are there?
Explanation: 3 + 2 = 5 parts, so 1 part = 40 ÷ 5 = 8. Girls = 2 × 8 = 16.
₹360 is shared among three friends in the ratio 2 : 3 : 4. What is the largest share?
Explanation: 2 + 3 + 4 = 9 parts, so 1 part = 360 ÷ 9 = ₹40. Largest share = 4 × 40 = ₹160.
The sum of the ages of a father and his son is 50 years. The father is 4 times as old as the son. How old is the son?
Explanation: Son + father = 1 part + 4 parts = 5 parts = 50, so 1 part = 10. The son is 10 years old (and the father is 40).
Riya is 6 years older than her brother. After 4 years, the sum of their ages will be 32 years. How old is Riya now?
Explanation: Today the sum is 32 − (4 + 4) = 24. If the brother is x, then x + (x + 6) = 24, so x = 9. Riya is 9 + 6 = 15.
A train leaves at 9:45 AM and arrives at 2:20 PM on the same day. How long is the journey?
Explanation: 9:45 AM to 2:00 PM is 4 h 15 min, and 2:00 PM to 2:20 PM is 20 min more. Total = 4 h 35 min.
If the 5th of a month is a Monday, what day of the week is the 26th of the same month?
Explanation: 26 − 5 = 21 days, which is exactly 3 weeks, so the day of the week is the same: Monday.
1 March is a Thursday. What day of the week is 1 April of the same year? (March has 31 days.)
Explanation: 31 days = 4 weeks + 3 days. Thursday + 3 days = Sunday.
A pen costs ₹12 and a notebook costs ₹35. Meena buys 3 pens and 4 notebooks and pays with a ₹200 note. How much change does she get?
Explanation: 3 × 12 = 36 and 4 × 35 = 140, so the bill is 36 + 140 = ₹176. Change = 200 − 176 = ₹24.
A car travels 240 km in 4 hours at a steady speed. How far will it travel in 7 hours at the same speed?
Explanation: Speed = 240 ÷ 4 = 60 km/h. Distance in 7 hours = 60 × 7 = 420 km.
Rahul cycles at 12 km/h. How long will he take to cover 30 km?
Explanation: Time = 30 ÷ 12 = 2.5 hours = 2 h 30 min.
A square has an area of 81 cm². What is its perimeter?
Explanation: Side = √81 = 9 cm. Perimeter = 4 × 9 = 36 cm.
A rectangle has a perimeter of 40 cm and a length of 12 cm. What is its area?
Explanation: Length + width = 40 ÷ 2 = 20, so width = 20 − 12 = 8 cm. Area = 12 × 8 = 96 cm².
Two angles of a triangle are 65° and 45°. What is the third angle?
Explanation: The angles of a triangle add up to 180°. Third angle = 180 − (65 + 45) = 70°.
What is the angle between the hour hand and the minute hand at 3:30?
Explanation: At 3:30 the minute hand points at 6, which is 180° from 12. The hour hand has moved 3.5 × 30° = 105°. The angle between them = 180° − 105° = 75°.
For a cube, what is (faces) + (vertices) − (edges)?
Explanation: A cube has 6 faces, 8 vertices and 12 edges. 6 + 8 − 12 = 2.
A 3 × 3 × 3 cube is painted on all outer faces and then cut into 27 small cubes. How many small cubes have exactly one painted face?
Explanation: Only the centre cube of each of the 6 faces has exactly one painted face, so the answer is 6.
A 4 × 4 × 4 cube is painted on all outer faces and then cut into 64 small cubes. How many small cubes have no paint at all?
Explanation: The unpainted cubes form the inner 2 × 2 × 2 block: 2 × 2 × 2 = 8.
A square garden of side 20 m has a 2 m wide path running all around inside its edge. What is the area of the path?
Explanation: The inner square has side 20 − 2 − 2 = 16 m, so its area is 256 m². Path area = 400 − 256 = 144 m².
How many rectangles of all sizes are there in a grid of 2 rows and 3 columns of equal squares?
Explanation: A rectangle is chosen by picking 2 of the 3 horizontal lines (3 ways) and 2 of the 4 vertical lines (6 ways): 3 × 6 = 18.
How many squares of all sizes are there in a 4 × 4 grid of small squares?
Explanation: 1 × 1 squares: 16; 2 × 2: 9; 3 × 3: 4; 4 × 4: 1. Total = 16 + 9 + 4 + 1 = 30.
At a party, 8 friends each shake hands exactly once with every other friend. How many handshakes take place?
Explanation: Each friend shakes hands with 7 others: 8 × 7 = 56. Each handshake was counted twice, so 56 ÷ 2 = 28.
Tina has 3 shirts, 4 pairs of trousers and 2 caps. How many different outfits (one of each item) can she make?
Explanation: 3 × 4 × 2 = 24.
In a row of 25 students, Ram is 8th from the left end. What is his position from the right end?
Explanation: Position from the right = 25 − 8 + 1 = 18th.
Sita is 7th from the front and 12th from the back in a queue. How many people are in the queue?
Explanation: Total = 7 + 12 − 1 = 18, because Sita is counted in both positions.
Trees are planted every 5 m along a straight road 100 m long, with a tree at both ends. How many trees are planted?
Explanation: 100 ÷ 5 = 20 gaps. Number of trees = gaps + 1 = 21.
How many times does the digit 7 appear when writing all the numbers from 1 to 50?
Explanation: The digit 7 appears in 7, 17, 27, 37 and 47, so 5 times. (70 to 79 are not included.)
I think of a number, multiply it by 6, subtract 8 and get 40. What is my number?
Explanation: Work backwards: 40 + 8 = 48, then 48 ÷ 6 = 8.
A book has 120 pages. How many digits are used to number all the pages?
Explanation: Pages 1–9: 9 × 1 = 9 digits. Pages 10–99: 90 × 2 = 180 digits. Pages 100–120: 21 × 3 = 63 digits. Total = 9 + 180 + 63 = 252.