Missed ML Problems

Missed on Sprint 11813:
  • On Saturday, Wes rode his bike 15 miles in 70 minutes. On Sunday, Wes rode his bike 13 miles in 50 minutes. What was Wes’s average speed for his weekend of biking, in miles per hour?
    • 14 mph
      • 15+13=28
      • 70+50=120 = 2 hours
      • 28/2=14mph
    • I got this wrong since I did 13+15=18 ← a very silly mistake that I should have never done
    • On his first trip, Bob walked 12 miles in 100 minutes. On his second trip, he walked only 2 miles in 20 minutes. What is his average speed for the two trips?
      • 12+2=14
      • 100 + 20 = 120 = 2 hours
      • 14/2 = 7 mph
    • Going to work, Bob took a train that went 5 miles in 15 minutes. Coming back, he biked instead taking 30 minutes. What was his average speed for the two trips?
      • 5+5=10
      • 30+15=45=¾ hour
      • 10/(¾)= 40/3 mph
  • Cole scored 88, 91, and 94 on his first three exams, while Carly scored 93, 98, and 97 on her first three exams. They both took a fourth exam, and Carly’s average score on the four exams was 2 points higher than Cole’s average score on the four exams. What is the value of Cole’s score on the fourth exam minus Carly’s score on the fourth exam?
    • 7
      • Cole: 88+91+94=273
      • Carly: 93+98+97=288
      • So, (288+x)/4 = ((273+y)/4) + 2 where x = Carly’s score on the 4th exam and y = Cole’s score on the 4th exam
      • Multiplying both sides by 4: 288+x = 273+y+8, 288+x=281+y
      • We are trying to find y-x, and we get y - x =7
    • I didn’t use algebra to find it and rushed through the problem. I should be more careful and slow down a bit to be more precise
    • On the first four exams, Bob’s average score was 95 and Joe’s average score was 96. After the final exam, Joe’s average became 3 points lower than Bob’s. How many more points than Joe did Bob get on the final exam?
      • Bob’s total before = 95*4 = 360+20=380
      • Joe’s total before = 96*4 = 380+4=384
      • B = bob’s score on final, J = Joe’s score on final
      • (380+B)/5 = ((384+J)/5) + 3
      • Multiplying both sides by 5, 380+B=384+J+2, 380+B=386+J
      • We are trying to find B-J, and we get B-J=6
    • While playing a game where one can only score from 0-20 points, Bob played 5 rounds, earning an average of 12 points. His friend, Joe, played 5 rounds earning an average of 16 points. After both played another round, their average scores were equal. How many points did Bob score?
      • ****Had to change problem to get a whole number****
      • Bob total score = 5*12=60
      • Joe’s total score = 5*16=80
      • B = Bob’s score on that round, J = Joe’s score on that round.
      • (60+B)/6=(80+J)/6
      • Multiplying both sides by 6, we get: 60+B=80+J → B=20+J
      • ****This question was sort of lucky because the only possibility was J=0 and B=20 so I changed it from “how many points did Bob score more that Joe” to “how many points did Bob score”.****
      • ****Also, if it is a question that says their scores are equal, then just take the beginning totals and subtract.****
  • Two identical pizzas were given to two groups of students. The first group had four students; their pizza was cut into eight equal slices, and each student ate two slices of pizza. The second group had five students; their pizza was cut into ten equal slices, and each student ate two slices of pizza. What percent more pizza did a student in the first group receive than a student in the second group?
    • 25%
      • First group student = ¼= 25%
      • Second group student = ⅕ = 20%
      • **** “Percent more” is always: (Greater value - Lesser value)/(Lesser Value)****
      • Difference between them = 25-20=5
      • Ans = 5/20 = 25%
    • Instead of doing percent more, I put the difference because I got confused about the percent of pizza and percent more.
    • Bob ate ¼ of a cake and Joe ate ½ more than him. What percent more did Joe eat than Bob?
      • Bob=¼, Joe = ¾
      • Difference = ½
      • (½)/(¼) = 2 = 200%
    • Sam took 5 more candies than Joe. Joe took twice as many candies as Bob. Bob took 25 less candies than Sam. What percent more candies did Sam take than Bob?
      • B + 25 = S
      • 2B = J
      • J + 5 = S
      • 2B + 5 = S
      • 2B + 5 = B + 25
      • B = 20
      • J = 40
      • S = 45
      • Difference between S and B = 25
      • 25/20 = 5/4 = 125%
  • Each of the vertices of  smaller square lies on a side of a larger square, partitioning each side of the larger square into two segments in the ratio of 1:3. What is the ration of the area of the smaller square to the area of the larger square? Express you answer as a common fraction.
    • 5/8
      • We can assume it is a 4*4 square and subtract the 4 3*1 triangles
      • So 16-(4*3*1/2) = 10
      • 10/16=5/8
      • Or, we can see that it is a 1-3-√10 right triangle
    • I found the triangle areas and forgot to subtract it from the total!!! So I got 3/8
    • The square is instead split into the ratio of 2:3
      • Assume it is a 5*5 square and subtract 4 2*3 triangles
      • 25-(4*2*3/2) = 13
      • 13/25
    • An equilateral triangle is split into the ratio of 1:2
      • Assume the side length is 3 so its area is (9√3)/4
      • Then the side length of the smaller equilateral triangle is √3 so its area is (3√3)/4 and that is 1/3 of the original triangle
  • The sum of a positive number and three times its reciprocal is equal to two less than twice the number. What is the number?
    • 3
      • x+3/x=2x-2
      • We can get: x-2-3/x = 0 and multiplying by x, we get: x^2-2x-3=0
      • Completing the square: (x-1)^2=4
      • Square rooting both sides: x-1=±2
      • So x=-1, 3
      • The positive number = 3
      • ****You don't really need to complete the square since it is easy to see that the roots are -1 and 3 since x+y=2 and xy=-3****
    • I didn't read the question correctly and put -1
    • The sum of three times a positive number and six times its reciprocal is equal to the number and 8. What is the sum of all the possible solutions?
      • 3x + 6/x = x+8 🠂 x^2 - 4x + 3 = 0
      • We can easily see that the solutions are 1 and 3 because x+y=4 and xy=3
      • So 1+3=4
    • The sum of a negative number squared and -35 is equal to twice the number. What is that number cubed?
      • We get: x^2 - 35 = 2x 🠂 x^2 - 2x - 35 = 0
      • We see that x = -5, 7 since x+y=2 and xy = -35
      • So -5^3 = -125
  • A box with four sides and a bottom, but no top, has a base of 2 feet by 3 feet and a height of 3 inches. Including both the inside and outside, what is the total surface area of the box, in square feet? Note that there are 12 inches in 1 foot.
    • 17 sq ft
      • We can find the total surface area and then subtract the top and then multiply by 2
      • Dimensions are 2*3*1/4
      • So we get 2*3*2+2*1/4*2+3*1/4*2-2*3=12+1+3/2-6 = 8 1/2
      • To get both the inside and outside, we multiply by 2 to get 17 sq ft
    • I got this wrong because I didn't read it carefully and did height = 3 feet instead of 3 inches
    • The dimensions are 4 ft by 7 ft base and 6 inch height, inside and outside
      • 2(4*7+(4+7)*2*1/2) = 39*2 = 78 sq ft
    • The dimensions of a square pyramid is a 8*8 ft base and a 3 ft height. What is the total surface area in square yards?
      • The triangle areas are 5*8/2*4 = 80
      • The base = 8*8 = 64
      • 64+80 = 144
      • 144/9 = 16 sq yds
  • What is the value of 1^2 - 2^2 + 3^2 - 4^2 + 5^2 ... + 49^2 - 50^2?
    • -1275
      • First, we can compute the first few to get
      • -3 - 7 - 11 - 15 - 19.....
      • This is an arithmetic sequence with 25 terms and a difference of -4
      • So the total value is ((-99-3)/2) * 25 = -1275
    • I got this wrong since I didn't correctly get the sum of the arithmetic sequence
    • What is the value of 1^2 + 2^2 + 3^2 - 4^2 - 5^2 - 6^2 + 7 ^2 +8^2 + 9^2 - 10^2 - 11^2 - 12^2?
      • We can get the sum of square from 1 to 3 = 14, the sum from 1 to 6 - sum from 1 to 3 = 77, the sum from 1 to 9 - the sum from 1 to 6 = 194, the sum from 1 to 12 - sum from 1 to 9 = 365
      • We can compute 14-77+194-365 = -234
    • What is the value of 5 + 9 + 13 + 17 + 21 + 25 + 29 + 33 + 37 + 41 + 45 ..... + 85 + 89
      • The first and last terms of this sequence are 5 and 89
      • (5+89)/2 * 22 = 1034
  • Two equilateral triangles share the same center, but the area of one of the triangles is twice the area of the other. If the height of the smaller triangle is 3, what is the shortest possible distance between a vertex of the smaller triangle and a vertex of the larger triangle? Express you answer in simplest radical form.
    • 2√2-2
      • We can find that the side length of the smaller triangle is 2√3 because it is a 30-60-90 triangle
      • Then, we find that the side length of the larger triangle is √2 times larger so it is 2√6 and the height will be √18 = 3√2
      • The distance from the center to a corner is 2/3 of the height so the answer is 2√2-2
    • I got this wrong since I didn't know what the distance from the center to a vertex was.
    • The height of the smaller is still 3, but the larger triangle is 3 times larger.
      • 2√3*√3 = 6 so the height of the larger is 3√3
      • Then it is 2√3 - 2
    • The height is 6 and the larger triangle is 6 times larger
      • 6√6 is height of larger
      • 4√6-4
  • A sequence is generated by a set of three rules: Firstly, if the current term is odd and greater than 11, then the next term is found by subtracting 9 from the current term. Secondly, if the current term is even and greater than 11, then the next term is found by dividing the current term by 2. Finally, if the current term is less than 12, then the next term is found by multiplying the current term by 7. If the first term of the sequence is 2017, then what is the 2017th term?
    • 49
      • We can find the sequence is
      • 2017, 2008, 1004, 502, 251, 242, 121,112, 56, 28, 14, 7, 49, 40, 20, 10, 70, 35, 26, 13, 4, 28....
      • We see it repeats on 28
      • So 2017 - first few = 2017 - 9 = 2008
      • 2008 = 167*12 R4
      • So from 28, the 4th is 49
    • I didn't have time to write out the whole sequence to find it is cyclic
    • What is the 2017th term of the sequence 1, 2, 4, 7, 3, 13, 4123, 31, 134, 41, 645, 76, 75, 43, 34, 1, 2, 4, 7, 3, 13....
      • The sequence has 15 terms before it repeats, and 2017/15 = 134 R7
      • 7th term = 4123
    • What is the 2018th term of the sequence 13, 21, 17, 19, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, ...
      • There are 4 terms in front of the sequence, so it is 2018-4 = 2014
      • Then, we find that there are 13 in the repeating part
      • 2014/13 = 154 R 12
      • The 12th term is 14
  • The base of a semicircle of radius 6 is partitioned into three equal segments, and an isosceles triangle is constructed on each segment, with the vertex that lies between the two congruent sides of each triangle located on the semicircle. The area of the smallest triangle is N. Find N^2.
    • 80 512/9                
      • From the above diagram, we can solve for h to find the area of the smallest triangle.
      • \sqrt{16-h^2} + \sqrt{36-h^2} = 6 or \sqrt{36-h^2} = 6 - \sqrt{16-h^2}
      • Squaring, we get 36-h^2 = 36 + (16-h^2) - 12\sqrt{16-h^2}
      • So, 12\sqrt{16-h^2} = 16, which simplifies to h^2 = \frac{16*8}{9}
      • Thus, h = \frac{8\sqrt{2}}{3} and area N = \frac{1}{2}(4)(\frac{8\sqrt{2}}{3}) = \frac{16\sqrt{2}}{3}, N^2 = \frac{512}{3}.

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