# Feb 2018 (CML)

SMC is now finished for the 2017-2018 season. I’ll be back in Sept!  Meanwhile, you can practice problems from the SMC Archive

This is the last week for the 2017-2018 season. I’ve picked out seven problems that had the most number of wrong answers in the submissions.  I hope they are easier this time!

Week of Feb 25 – Mar 3.

Feb 25: The reversal of a number reverses the ordering of the digits. For example, the reversal of 213 is 312.  Add 627 to its reversal, and call this sum A. Subtract A from the reversal of A. Call this difference B. Add B to its reversal. What is the answer?

Feb 26:  How many units away from the origin is the point (−3, −4)?

Feb 27: There are ___ 2-digit numbers such that the number is decreased by 27 when the 2 digits are swapped.

Answer: 6.  96-69, 85-58, 74-47, 63-36, 52-25, 41-14.

Feb 28: Stephanie was born on a Tuesday, June 27.  If Emily was born on Aug 13 of the same year, what day of the week was Emily born?

Mar 1: The original price of a coat is $500. Store A first reduced the price by 25% and then another 25% off the reduced price. Store B took off 50% off the original price of the coat. Mrs. X bought the coat from shop A; Mrs. Y bought it from shop B. Who paid more and by how much? Answer: Mrs. X paid$31.25 more. Mrs. X paid 500*0.75*0.75 = 281.25; Mrs.Y paid 250.

Mar 2: In rectangle ABCD, point P lies on side BC and point Q lies in the interior of the rectangle so that triangle APQ is equilateral. If the measure of angle PAB is 17 degrees, what is the measure of angle QPC?

Answer:  47 deg. Angle BPA = 73 deg; angle APQ = 60 deg. So angle QPC = 180 – angle BPA – angle APQ = 47.

Mar 3: A standard 52-card deck of playing cards includes four aces. What is the probability that two cards selected randomly, without replacement, will both be aces? Express your answer as a common fraction.

I noticed that some of you haven’t learned the Pythagorean Theorem, something really magical about right triangles.  It says that if $a$, $b$ and $c$ are the sides of a right triangle where $c$ is the longest side, then $a^2+b^2 = c^2$.  For example, 3, 4 and 5 form 3 sides of a right triangle. I include a video link from the Khan Academy here.  This week we will do a Pythagorean Stretch.

Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written $(a, b, c)$, and a well-known example is (3, 4, 5). If $(a, b, c)$ is a Pythagorean triple, so is $(ka, kb, kc)$ for any positive integer $k$.

Week of Feb 18 – Feb 24

Feb 18:  Complete the following Pythagorean triples:  (?, 4, 5), (6, 8, ?), (5, ?, 13)

Answer: (3, 4, 5), (6, 8, 10), (5, 12, 13)

Feb 19:  Complete the following Pythagorean triples:  (9, 40, ?), (?, 15, 17), (15, 36, ?)

Answer: (9, 40, 41), (8, 15, 17), (15, 36, 39)

Feb 20:  Complete the following Pythagorean triples:  (7, ?, 25) , (16, 30, ?), (?, 60, 61)

Answer: (7, 24, 35), (16, 30, 34), (11, 60, 61)

Feb 21: Complete the following Pythagorean triples:  (25, ?, 65), (?, 80, 82), (15, 20, ?)

Answer: (25, 60, 65), (18, 80, 82), (15, 20, 25)

Feb 22: In a right triangle ABC, AB=3, BC=4 and AC=5.  The height from B to the base AC is BH.  How long is BH?

Answer: BH = 12/5 = 2.4.  Two ways to compute the area of ABC, AB*BC*(1/2) = BC*BH*(1/2).

Feb 23: For the above problem, what is the area of triangle BHA?

Answer: BHA is a right triangle.  BH=2.4, AB=3, from Pythagorean AH=1.8.  Area of BHA=2.4*1.8*(1/2)=2.16

Feb 24: For the above problem, what is perimeter of triangle BHC?

Answer: BHC is a right triangle. BH=2.4, BC=4, CH=3.2.  Perimeter of BHC=9.6

Week of Feb 11 – Feb 17

Feb 11: When half of the students in Mr. Freeman’s class joined the National Math Club, there were 13 students who did not join. What is the total number of students in Mr. Freeman’s class?

Feb 12: Debbie is counting by fives. If the first number she says is 110, what is the fifth number Debbie says?

Feb 13: A carton of one dozen jumbo eggs has a total weight of 32 ounces. If the carton itself weighs 2 ounces, how much does each egg weigh, on average? Express your answer as a decimal to the nearest tenth.

Feb 14: Given that 3 feet = 1 yard and 240 yards = 1 cable, how many feet are equal to
1 cable?

Feb 15: Velma earned the following scores on seven science quizzes: 88, 95, 84, 100, 86, 99, 94. What is the median of Velma’s quiz scores?

Feb 16: What decimal is equivalent to ${1\over 2}$ percent?

Feb 17: What is the value of $100-$ ${10\over 0.1}$ ?

Week of Feb 4 – Feb 10

Feb 4: If ${c \over d} =4$, what is the value of $2d \over c$?

Feb 5 What is the value of (−20) + (−17) + (−14) + … + 13 + 16 + 19 + 22?

Feb 6: What is the closest integer to the real number x such that $2^x =1000$?

Feb 7: What is the value of x if ${1\over x} + {1\over {2x}} = {1\over 2}$?

Feb 8: If $45 + c = 49$, what is the value of $c^2$?
Feb 10: If ${3\over 4} = {a \over 36}= {36 \over b}$, what is the value of $a+b$?