Given: The diagram shows a grid made of vertical and horizontal lines. The question asks which part was cut from the grid.
Explanation: To determine the missing part of the grid, we need to compare each option (A to E) with the grid’s structure. The key is to observe the pattern of lines in the grid and identify which option corresponds to the removed section.
Option (D) shows a section where both vertical and horizontal lines match the positions and counts in the original grid, indicating that it fits perfectly into the missing space without altering the overall pattern of the grid.
Correct Answer: (D)
The method used in the problem is called geometric reasoning or geometric analysis. This method involves analyzing the properties and characteristics of geometric shapes to determine the outcome of certain operations or transformations on those shapes.
Given: Identify which of the following shapes cannot be cut into two trapeziums with one single straight line. The shapes given are a triangle, rectangle, trapezium, regular hexagon, and square.
Explanation: A trapezium (or trapezoid) has one pair of parallel sides. To form two trapeziums from a given shape, the shape must allow for a straight cut that creates two new shapes, each with at least one pair of parallel sides.
Triangle: No pair of parallel sides means a straight cut can’t create two trapeziums.
Rectangle: Can be cut diagonally to form two trapeziums.
Trapezium: Cutting parallel to the bases can form two trapeziums.
Regular Hexagon: A vertical or horizontal cut can create two trapeziums.
Square: A diagonal cut can form two trapeziums.
The triangle is the shape that cannot be divided into two trapeziums with one straight line.
Correct Answer: (A) Triangle
The mathematical method used to solve this problem involves modular arithmetic, which handles the circular sequence of numbers on the watch dial, and spatial reasoning, which helps visualize and understand the fixed distance relationship between the holes on the disc.
Given: A dark disc with two holes is placed over a watch dial, showing the number 8 through one of the holes. The question asks which numbers could be seen through the other hole now.
We know that if one of the holes shows 1 the other shows 5, meaning that the difference between these 2 holes is 4.
Explanation: To solve this, visualize the watch dial with numbers arranged from 1 to 12. The disc has two holes fixed in a specific arrangement. When the number 8 is visible through one hole, the other hole will show a number that is a fixed distance from 8 on the dial.
By examining the dial, if one hole shows 8, the other hole could reveal numbers that are the same distance apart as 8 is from those numbers. So we have to find out what we would get if we subtract 8 from 4 and if we added 8 from 4. And you’d realize that we would get 4,12 thus making A the answer.
Correct Answer: (A) 4 and 12
Given: John throws 150 coins, and 60 of them show heads while the others show tails. He wants an equal number of heads and tails. The question asks how many heads need to be turned to achieve this balance.
Explanation: John has 60 heads and 90 tails. To balance the number of heads and tails, he needs the same number of heads as tails.
If he turns 15 tails into heads :
He will have 60 + 15 = 75 heads.
He will have 90 – 15 = 75 tails.
Therefore, flipping 15 heads to tails results in 75 heads and 75 tails, achieving the desired balance.
Correct Answer: (B) 15
Given: Kristina has a piece of see-through foil with points and lines drawn, folded along a dotted line. The question asks what configuration can be seen now.
Explanation: To determine the new configuration, visualize the points and lines on the foil before and after folding along the dotted line. The fold changes the alignment of the lines and points. By following the fold, the new arrangement of lines and points forms the pattern shown in option (C).
Correct Answer: (C) 5 : 6 : 9
Given: A grid needs to be divided along the black lines into identical shapes with no leftover pieces. This problem is solved using a mathematical technique known as tessellation. Tessellation involves covering a surface with a repeating pattern of shapes that fit together perfectly without overlapping or leaving any gaps.
Explanation: The task is to identify the shape that cannot uniformly fit into the grid. The total area of the grid is 4×6=24. Therefore, we need to check if the number of squares in each shape can evenly divide 24.
A) Shape A consists of 3 squares. Since 24 is divisible by 3, shape A can perfectly tessellate the grid 8 times.
B) Shape B consists of 4 squares. Since 24 divided by 4 equals 6, shape B can also tessellate the grid without any leftover pieces.
C) Shape C consists of 5 squares. Since 24 divided by 5 equals 4.8, which is not an integer, shape C cannot tessellate the grid without leaving leftover pieces. Therefore, it’s impossible to cut the grid into shape C.
D) Shape D consists of 6 squares. Since 24 divided by 6 equals 4, shape D can tessellate the grid without any leftover pieces.
E) Shape E consists of 4 squares. Since 24 divided by 4 equals 6, shape E can also tessellate the grid without any leftover pieces.
Conclusion: The correct answer is C, as shape C cannot be used to tessellate the grid evenly.
The method used is “vector analysis” or “collision detection in kinematics,” involving tracking each car’s motion vector and determining the intersection points to predict collisions. This approach uses kinematic equations to analyze positions and detect when two paths will intersect.
Given: The diagram shows the starting position, direction, and distance covered within 5 seconds by four bumper cars. The question asks which two cars will first crash into each other.
Explanation: To determine which cars will first crash, observe the direction and distance each car travels:
Car A moves right.
Car B moves down.
Car C moves left.
Car D moves up.
By plotting their paths on the grid, you can see that Car A and Car C are on a collision course. Car A moves horizontally towards the right, and Car C moves vertically downwards, intersecting at the same point.
Correct Answer: (D) A and C
Given: Werner wants to label each side and each corner point of the rhombus with one number. The number on each side should equal the sum of the numbers on the corner points of that side. The numbers 8, 9, 13, and ? are given.
Explanation: Let’s label the corners of the rhombus as A, B, C, and D. The sides should equal the sum of the two adjacent corners.
Using the given numbers:
Side AB = A + B = 8
Side BC = B + C = ?
Side CD = C + D = 13
Side DA = D + A = 9
Now lets simply guess and check:
Side AB = 4 + 4 = 8
Side BC = 4 + 8 = ?
Side CD = 8 + 5 = 13
Side DA = 4 + 5 = 9
IT WORKS!!! All the equations are satisfied which means side BC is equal to 12!
The balance equations confirm the Correct Answer: (B) 12
To solve the problem, let’s break down the steps:
Given:
A 2×4 grid with numbers 1 to 8.
The sum of the numbers in each row and each column should be the same.
Numbers 3, 4, and 8 are already placed in the grid as shown.
Steps:
Sum of All Numbers:
The sum of numbers from 1 to 8 is 1+2+3+4+5+6+7+8=36
Sum of Rows and Columns:
The grid is 2 rows by 4 columns. The sum in each row (and column) should be equal.
Total sum is 36, so the sum of numbers in each row is 36/2=18
Identify the Numbers:
The grid has numbers 3, 4, and 8 already placed. We need to find the remaining numbers.
Step-by-Step Calculation:
Top Row (First Row):
The sum of the first row should be 18.
Current numbers in the first row: 4 (in the second column) and 8 (in the fourth column).
The sum so far is 4+8=12
The remaining sum needed in the first row is 18−12=6
There are two empty spots in the first row (first and third columns). The two numbers that sum to 6 and fit are 2 and 1.
Bottom Row (Second Row):
The sum of the second row should also be 18.
Current numbers in the second row: 3 (in the first column) and 7 (in the third column).
The sum so far is 3+7=10
The remaining sum needed in the second row is 18−10=8.
There are two empty spots in the second row (second and fourth columns). The two numbers that sum to 8 and fit are 6 and 2.
Final Placement:
The correct number to place in the dark field (third column, top row) is 2.
So the answer is (B) 2.
Let’s break down the given problem and analyze it step by step.
Understanding the Symbols
We are given three consecutive natural numbers in increasing order, represented by symbols as follows:
⬛ ◆ ♦
♡ △ △
♡ △ ⬛
We need to identify what numbers these symbols represent and find the next consecutive number.
Step 1: Analyzing the First and Third Numbers
You pointed out that the hundreds place of the first number (⬛) changes in the third number (♡). This suggests that the first two numbers are close to moving to the next hundred.
From this, we can infer that:
The diamond (◆) represents 9, as this would allow the transition from something like 199 to 200 in the next number.
Therefore, ⬛ (square) must be 1, since we are looking at numbers just before 200. The first number could be 199.
Step 2: Interpreting the Second Number
Let’s follow the reasoning further. If the first number is 199:
The second number (♡ △ △) must be 200 because it immediately follows 199.
From this, we can deduce the following:
♡ (heart) represents 2, since the number jumps to 200, with the hundreds place changing to 2.
△ (triangle) must represent 0, as in 200, both the tens and ones digits are zero.
Step 3: Completing the Third Number
Now let’s look at the third number: ♡ △ ⬛.
We’ve already established that ♡ = 2 and △ = 0.
Since ⬛ = 1 (as deduced earlier), this makes the third number 201.
Step 4: Finding the Next Number
The three numbers we now have are:
199 (⬛ ◆ ♦)
200 (♡ △ △)
201 (♡ △ ⬛)
The next number in this sequence would be 202. To express 202 in symbols:
The hundreds place stays the same (♡ = 2).
The tens place is still 0 (△ = 0).
The ones place moves to 2 (♡ = 2).
Thus, the next number, 202, would be written as: ♡ △ ♡.
Conclusion:
The next number in Dorli’s notation is ♡ △ ♡, which corresponds to option (E).
A cube has 12 edges and 6 faces. Each face is a square with 4 edges.
Objective:
We need to color the minimum number of edges such that every face has at least one red edge.
Strategy:
The cube’s faces share edges with neighboring faces. This means that coloring an edge that touches two faces counts for both.
The key is to color edges that maximize coverage across multiple faces.
Solution:
You can achieve this by coloring three edges that meet at one vertex. This ensures that all adjacent faces to that vertex will have at least one red edge.
Conclusion:
The minimum number of red edges required is 3.
Answer: (A) 3
We need to find how many different numbers can be formed using exactly 6 matchsticks. Here’s the breakdown of matchstick counts for each digit:
0 → 6 matchsticks
1 → 2 matchsticks
2 → 5 matchsticks
3 → 5 matchsticks
4 → 4 matchsticks
5 → 5 matchsticks
6 → 6 matchsticks
7 → 3 matchsticks
8 → 7 matchsticks
9 → 6 matchsticks
Since we need exactly 6 matchsticks, we can immediately eliminate 2, 3, 5, and 8, as they either use too many or too few matchsticks.
Possible Numbers:
Single-digit numbers:
0, 6, 9 (all use 6 matchsticks)
Two-digit numbers:
Combine digits with matchsticks adding to 6:
14 (1 uses 2 + 4 uses 4 = 6 matchsticks)
41 (4 uses 4 + 1 uses 2 = 6 matchsticks)
77 (7 uses 3 + 7 uses 3 = 6 matchsticks)
Conclusion:
The possible numbers are: 0, 6, 9, 14, 41, and 77.
Thus, the total number of different positive numbers that can be formed is 6.
Answer: (E) 6.
Key Conditions:
Three beavers are in the circle, and no beavers stand next to each other.
There is exactly one group of three kangaroos standing next to each other.
Solution Approach:
We need to place three beavers and some number of kangaroos in a circle, keeping in mind that no two beavers can be adjacent and we need exactly one group of three adjacent kangaroos.
Step-by-step arrangement:
Placing the three beavers: To avoid having beavers next to each other, there must be at least one kangaroo between any two beavers. This means each beaver will have a kangaroo between them.
So the arrangement starts like this:
B – K – B – K – B – K (where B = beaver, K = kangaroo).
Inserting three adjacent kangaroos: We need to place a group of three adjacent kangaroos somewhere in the circle. The best place to do this is by replacing one of the single kangaroos in the arrangement with the three adjacent kangaroos.
So, for example, we can extend one of the Ks into a group of three, like this: B – K – K – K – B – K – B.
In this configuration, we now have:
Three adjacent kangaroos (K – K – K).
Beavers are not adjacent to each other.
The arrangement maximizes the number of kangaroos.
Count of kangaroos:
In this configuration, there are 5 kangaroos in total.
Thus, the biggest possible number of kangaroos in the circle is 5.
Therefore, the correct answer is (B) 5.
Let’s start by defining an algebraic expression for each of them:
If falling on the outer ring is represented by xxx,
The middle ring is yyy,
And the center ring is zzz,
Then the expressions are:
Tom: 2x+y+3z=46
John: 2x+3y+z=34
Lily: 2x+2y+2z
Let’s add Tom’s and John’s points together to find their total sum:
2x+y+3z=46 + 2x + 3y + z = 34
Adding these: (2x+y+3z)+(2x+3y+z)=4x+4y+4z=80
We can observe that this total is exactly half of Lily’s expression, which is 2x+2y+2z.
Since 4x+4y+4z is double 2x+2y+2z so we can get the answer by dividing 80 by 2 which gives us 40. Thus, the answer is 40.
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