Question 21. There are three candidates for one position as class president, and 130 students are voting. Suhami has 24 votes so far, while Khairul has 29, and Akmal has 37. How many more votes does Akmal need in order to be elected? Now, if we add up all the three total votes, we get a total of 90, so we know that 90 students have voted so far. Now, if Akmal gets 17 more votes, he will have 54 total votes, and with this number, it will be impossible for any of the other students, Khairul or Suhami, to be elected, because there will be 23 votes left, and even if Khairul gets those 23 votes, they will only have 52 total votes, therefore losing. So the correct answer is E, 17. Question 22. The diagram shows the net of an unfolded rectangular box. What is the volume of the box? So we see the box unfolded, and we have some different measures of 10 centimeters, 7 centimeters, and 26 centimeters. I like to draw a box to help me visualize, and we can write down height, width, and length, and if we transfer this over to the net, we have length, width, and height for its corresponding sides. Now, if we look at this as a formula, we can add together, there's one width and another width, plus a length and a length gives us 26 centimeters. A height and a length will give us 10 centimeters, and a height and a width gives us 7. So doing this, we know that one width is equal to 7 minus the height, and one length is 10 minus the height. So if you put this into the formula of width plus length plus width plus length equals 26, we get something that looks like this. To simplify, we get 34 minus 4h height equals 26. Therefore, negative 4 height is equal to negative 8. So if we divide by negative 4, height is 2. Now we can plug this into the other formulas. 2 plus length equals 10. Length is 8, and 2 plus width is 7. So width equals 5. Now, we were originally asked to find the volume of the box. To find the volume of a box, all we must do is do length times width times height, and that gives us the volume. 2 times 5 times 8 will give us a total of 80. Therefore, the correct answer is C, 80 centimeters cubed. Question 23. Ria wants to write a number in every cell on the border of the 6 by 5 table. In each cell, the number she writes is equal to the sum of the two numbers in the cells with which this cell shares an edge. Two of the numbers are given in the diagram. What number will she write in the cell marked X? Take a closer look at the picture. We start off by labeling each box with a variable letter, such as A, B, C, D, E, F, G, H, I, and J. This will help us find the solution. We know that F plus D is equal to 3. Since the cell that shares two boxes, its total will be equal to the two neighboring boxes. We also know that 3 plus C equals D. Now we also know that D plus B is equal to C. C plus A is equal to B, and B plus 10 is equal to A. 10 plus G is equal to E. E plus H is equal to G. G plus I is equal to H. H plus X is equal to I. Now if we start looking at this algebraically, we can notice that C is equal to B plus C plus 3. Since we know that D is equal to C plus 3, and that D plus B equals C, we can substitute the D for C plus 3, getting C equals B plus C plus 3. We simplify that, we get the answer of B is equal to negative 3. Knowing this, we can substitute the B, and the B plus 10 equals A, with a negative 3, getting A equals 7. Following the same logic, we get to C plus 7 equals negative 3. C is negative 10. D minus 3 equals negative 10. D is negative 7. A plus E is equal to 10. 7 plus E is equal to 10, since we know the value of A. So E is 3. Then we move on. 10 plus G is equal to 3, since the E is a 3. So G is negative 7. 3 plus H is equal to G, or negative 7. So H is negative 10. Negative 7 plus I is equal to negative 10. So I is a negative 3. Now we can look at the final formula. Negative 10 plus X is equal to negative 3, since H is negative 10 and I is negative 3. And we get X equals 7. Therefore, the correct solution is B, 7. Question 24. Simon and Ian decide to have a race. Simon runs around the perimeter of the pool shown in the diagram, while Ian swims lengths of the pool. Simon runs three times faster than Ian swims. Ian swam six lengths of the pool in the same time Simon ran around the pool five times. How wide is the pool? We know that the pool has a length of 50 meters, and since Ian swam it six times, that is a total of 5 times 6, or 300 meters. Now Simon's total length is 5 times, since he ran around the perimeter of the pool five times. And the perimeter of the pool is 50 plus 50, plus the width, plus the width, also known as 2W width, plus 100. Now we also know that we have to divide by their speeds. Now we know that Simon can run three times faster, so we will represent running speed as R. So if we do 300 divided by R, or running speed, is equal to 5 times 2W plus 100, and the total of that divided by 3R, since he runs three times faster, we have an equality. We multiply both sides by 3R, we multiply both sides by 3R, we get 900 is equal to 5 times 2W plus 5 times 100. Now this can be simplified by dividing 5 from both sides. So 180 equals 2W plus 100. So 80 is equal to 2W, and if we divide both sides by 2, we get W equals 40. So the correct answer is B, 40 meters. Question 25. Freda's Flying Club designed a flag with a flying dove on the graph paper as shown. The area of the dove is 192 centimeters. All parts of the perimeter of the dove are either parts of a circle or straight lines. What are the dimensions of the flag? Look at the flag more closely. We notice that parts of it can be moved around to get a simpler view of the flag. So if we move this side, like so, over, then we notice that it fits in perfectly with the rest of it. And if we look at the bottom triangle, remove it, and replace it over here, we see that we get a nice rectangle. Since we know the area has remained unchanged, the area is 192. We count off all the squares, all the red squares, we get 12. 192 divided by 12 gives us 16. Knowing this, we can figure out that there are 24 by 16. So the correct answer is D, 24 centimeters by 16 centimeters. Question 26. Domino tiles are said to be arranged correctly if the number of dots at the ends that touch are the same. Polyus laid six dominoes in a line as shown in the diagram. He can make a move by either swapping the position of any two dominoes or by rotating one domino. There's a smaller number of moves he needs to make to arrange all the tiles correctly. Look at all six dominoes. We notice that there are three 4s, three 6s, two 3s, two 2s, and two 1s. Therefore, since there is an odd number of 4s and 6s, we know that 4s and 6s will have to remain on the ends, since there will be one 4 without a pair and one 6 without a pair. And if we look at the dominoes, the way they are already arranged, there is a domino with a 4 on the end of the left side, and there is a domino with a 6 on the end of the right side. So it is wisest to leave those unmoved. Now, the first move we must make is to switch these two dominoes. Therefore, the 6 matches with the left 6. Next, we must move these two dominoes to match the 2 and the 4. And the only thing we have left is to rotate the third domino. This gives us a total of three moves, so the correct answer is C, 3. Question 27. Points N, M, and L lie on the sides of the equilateral triangle ABC, such that segment NM is perpendicular to segment BC, segment ML is perpendicular to AB, and segment LN is perpendicular to segment AC, as shown in the diagram. The area of the triangle ABC is 36. What is the area of triangle LMN? It is important to note that triangle ABC is an equilateral triangle. Therefore, all angles will be 60 degrees. A special and key feature of equilateral triangles helps us determine the solution to this problem. Now, if we also look, we can see that there are right angles in the smaller white triangles, which means they are 90 degrees since they are perpendicular. Now, we can figure out that, for example, triangle CMN is a 30-60-90 triangle. Same goes for ALN and BML. And therefore, we know that 30-60-90 triangles have a property of having one side equal to x, the hypotenuse equal to 2x, and the other longer leg, root 3x. This is true for all three of these triangles. If we look at the area formula for an equilateral triangle, which is side length times root 3 times side length divided by 4, we can figure out the solution to our problem. If we look at the side lengths of the larger triangle ABC, we notice they are 3x. So we can substitute that in and simplify the problem. 9 times x times root 3 times x divided by 4. Now, if we look at the smaller shaded end triangle, we can do the same thing and get 3 times x times root 3 times x times 4. The only difference between the area of the larger ABC triangle and the smaller LMN triangle is the 9 and the 3. So therefore, we know that the larger triangle has 3 times the area that the shaded triangle has. The problem stated that the area of triangle ABC is 36. When we divide 36 by 3, we get 12. So the answer is B, 12. Question 28. Azmi, Burhan, and Chu went shopping. Burhan spent only 15% of what Chu spent. However, Azmi spent 60% more than Chu. Together, they spent $55. How many dollars did Azmi spend? Now, if we look at this as a formula, we put Chu as a C, Burhan as B, and Azmi as A, we can figure out that B is equal to 0.15 C, A is equal to 1.6 C, and A plus B plus C is equal to 55. Now, since C is the common factor in all these, we can get C plus 0.15 C plus 1.6 C is 55, or 2.75 C is equal to 55. If we divide this, we get C equals 20. So with this, we know that Chu spent $20. The question asks us how much Azmi spent. So we must put in 20 for the formula A equals 1.6 C, and we get A or Azmi is equal to 1.6 times 20, which is 32. So the correct answer is E, 32. Question 29. Viola is practicing the long jump. The average distance she has jumped so far today is 3.8 meters. On her next jump, she jumped 3.99 meters, and her average increased to 3.81 meters. What distance must she jump with her next jump to increase her average to 3.82 meters? Look at the formula for finding an average. We can see that 3.8, the current average, times X, the amount of times she's jumped, which is unknown, plus 3.99, the next jump she has made, divided by X, the amount of times she's jumped, plus 1, since we are adding the 3.99 meter jump, is equal to 3.81. This formula will help us find how many times Viola has jumped. Simplify this to 3.8X plus 3.99 is equal to 3.81X plus 3.81. With this, we see 0.18 equals 0.01X. With that, X equals 18. Now, if we look at the same formula, adding another variable of Y, since this is the distance she must jump to get an average of 3.82, and X is still the amount of times she has jumped, we know that X is 18. So with this, 68.4 plus 3.99 plus Y, which is the distance she must jump to get an average of 3.82, equals 3.82X plus 7.64. Simplifying this further, 72.39 plus Y is equal to 68.76 plus 7.64. With that, we can move all the numbers we know to one side and leave the Y alone. So we know that Y is equal to 68.76 plus 7.64 minus 72.39. We simplify this, we get Y equals 4.01. The correct solution is C, 4.01 meters. Question 30. In isosceles triangle ABC, points K and L are marked on sides AB and BC respectively, so that AK equals KL equals LB and KB equals AC. What is the measure of angle ABC? Now, since we know that this is an isosceles triangle, and there are two pairs that are adjacent and same length, we can determine that AKLC is a kite. Angle A and angle C are both equal. Using the properties of a kite, we can determine that angle KLC to be equal to angle A and angle C as well. Let angle A, angle C, and angle KLC each be represented by X. The measures of the four angles of a quadrilateral add up to 360 degrees. And knowing that angle A, angle C, and angle KLC are equal, we can determine that angle AKL is equal to 360 degrees minus 3X, like so. Now, in triangle BKL, segments BL and KL are equal, meaning that angle B and angle K are equal. Let angle B and angle K be represented using Y. Since angle BLK and KLC make up a linear pair, we subtract X from 180 to find angle BLK. Angle BLK and AKL also make up a linear pair, so the measure of angle BKL can be written as 180 degrees minus 360 degrees minus 3X, which means Y equals 180 degrees plus 3X. We know that triangle BKL and all angles angle B, angle BKL, and angle BLK add up to 180. This can be represented using Y plus Y plus 180 degrees minus X equals 80. If we solve this, then we get Y equals X divided by 2. Since we have already determined that Y is equal to 180 degrees plus 3X, we can set X divided by 2 and 180 degrees plus 3X equal to each other. Now, if we just solve for X, we will give us the measure 72 degrees. And since angle ABC is represented by Y, that's the angle we are looking for, we can plug in 72 for X into Y equals X divided by 2. To find our solution, all we must do is divide 72 by X. So the correct answer is C, 36 degrees.

math problems

class president election

rectangular box volume

domino arrangement

inscribed triangle area

isosceles triangle angle