# March 2019 (CML)

I will post seven new problems every weekend, one for each day of the week. The idea is that you work on your own during the week and I will post answers the following Saturday. For those of you who practiced with me last year, the process is similar.

For instructions on submitting weekly answers, click on the “Logistics” tab.
For previously posted problems, click on the “Archive” tab.
For previous prizes, click on the “News” tab.

Week of March 24 – March 30.

March 24: A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden?

Answer: 400 square feet. The perimeter of the rectangular garden is 120 feet. When it is made into a square, the side length is 30 feet. Therefore, the area is enlarged from 500 square feet to 900.

March 25: Bo, Coe, Flo, Joe, and Moe have different amounts of money. Neither Joe nor Bo has as much money as Flo. Both Bo and Coe have more than Moe. Joe has more than Moe, but less than Bo. Who has the least amount of money?

Answer: J < F, B < F, M < B, M < C, M < J < B. So M has the least amount.

March 26: The third exit on a highway is located at milepost 40 and the tenth exit is at milepost 160. There is a service center on the highway located three-fourths of the way from the third exit to the tenth exit. At what milepost would you expect to find this service center?

Answer: (160 – 40) * 3/4 + 40 = 130

March 27: A complete cycle of a traffic light takes 60 seconds. During each cycle the light is green for 25 seconds, yellow for 5 seconds, and red for 30 seconds. At a randomly chosen time, what is the probability that the light will NOT be green?

Answer: The probability of green is 25/60 = 5/12, so the probability of not green is therefore 7/12.

March 28: The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is $11/4$. To the nearest whole percent, what percent of its games did the team lose?

Answer: 4/15 is the fraction of the games lost, which is about 27%.

March 29: Tori’s mathematics test had 75 problems: 10 arithmetic, 30 algebra, and 35 geometry problems. Although she answered 70% of the arithmetic, 40% of the algebra, and 60% of the geometry problems correctly, she did not pass the test because she got less than 60% of the problems right. How many more problems would she have needed to answer correctly to earn a 60% passing grade?

Answer: Tori got 7 arithmetic, 12 algebra and 21 geometry problems right, for a total of 40 problems. To get 60% correct, she would need 45 problems. Therefore, she needed 5 more problems.

March 30:  How many whole numbers lie in the interval between $\frac{5}{3}$ and $2\pi?$

Answer: 5 whole numbers: 2, 3, 4, 5, 6.

Week of March 17 – March 23.

March 17: $a\triangle b = {a \times b \over {a-b}}$. For example, $3\triangle 2 = {{3\times 2}\over {3-2}} = 6$. How much bigger is $16\triangle 8$ than $10 \triangle 5$?

March 18: During track practice, Al ran 3/5 of a mile, Bob ran 4/7 of a mile, Carl ran 1/2 a mile and Dan ran 2/3 of a mile. Who ran the furthest?

March 19: Tom was still 14 on Wed, Sept 9. Exactly 3 weeks ago his birthday was in 40 days. On what day of the week will Tom be 15 years old?

Answer: Tom’s birthday is 19 days from Sept 9. So it is Sept 28, which is a Monday.

March 20: Tom was still 14 on Wed, Sept 9. Exactly 3 weeks ago his birthday was in 40 days. On what day of the week will Tom be 16 years old? (Assume no leap year is involved.)

Answer: In the following year Sept 9 is Tues.

March 21: How many multiples of 7 are between 15 and 95?

Answer: $\lfloor 95/7 \rfloor - \lfloor 15/7 \rfloor = 11$

March 22: It takes 5 men 8 hours to build 10 sheds. It would take 9 men ___ hours to build 45 sheds.

Answer: It takes 1 man 8 hours to build 2 sheds, and 1 man 4 hours to build 1 shed. So it takes 9 men 4 hours to build 9 sheds, and 20 hours to build 45 sheds.

March 23: Which of the following is not the product of two prime numbers?
A) 55  B) 39  C) 91 D) 63  E) 65

Week of March 10 – March 16.

March 10: $\sqrt{a}+\sqrt{b}+\sqrt{c}=20$. If $a = 49$ and $b= 16$, what is $c$?

March 11: $\sqrt{{4\over 3}\cdot{5\over 4}\cdot {6\over 5}\cdot {7\over 6}\cdot {8\over 7}\cdot\dots \cdot {a\over{a-1}}}=2$. What is the value of $a$?

Answer: $\sqrt {a\over 3} = 2$. Therefore, a=12.

March 12: $0.004389\times\Box = 4389\times 39$. What is the number in the $\Box$?

March 13: If ${a\over 3} = 4$ and ${b\over 4} = 24$, what is $a\over b$?

Answer: a=12, b=96. a/b = 1/8.

March 14: What is ${{100!-98!}\over 99!}$? Recall $n! =n\cdot (n-1)\cdot (n-2)\dots 3\cdot 2\cdot 1$.

Answer: $100-{1\over 99}$

March 15: If ${2\over 5} x ={3\over 5} y$, then ${3\over 5} x = \_\_\_ y$.

March 16: If $a\cdot b\cdot 7 = x$ and $a\cdot b =y$, then $x-y = k\cdot a\cdot b$. What is $k$?

Week of March 3 – March 9.

March 3: If $5 \times 7 = 19 + \Box$, what is the number that belongs in the box?

March 4: If $4x^2 = 7$ and $5y^2=9$, what is $(10xy)^2$?

Answer: $(10xy)^2 = 5(4x^2)(5y^2)=5\cdot 7\cdot 9=315$.

March 5: 10 is 50% of 10% of what number?

March 6: If $10(1^3+2^3+3^3+4^3) = m^3$, what is $m$?

Answer: $m^3 = 10(1+8+27+64)=1000$. So m=10.

March 7: If $_a F_b$ is defined as $b^a-a^b$, how much larger is $_5F_3$ than $_3F_5$?

Answer: $_5F_3- _3F_5 =(3^5-5^3) - (5^3-3^5)= 236$.

March 8: The average of 7 and x is equal to the average of 5, 9 and x. What is the value of x?

Answer: (7+x)/2 = (5+9+x)/3. Therefore, x=7.

March 9: If ${5\over 6} \cdot {9 \over A} = {1\over 2}$, what is $A$?