Math Problem Monday - September 28th, 2020 | Mathnasium Livermore, CA

Lower Elementary:
Question: Zoey bought a candy bar for $1.85. She gave the cashier $2. The cashier gave Zoey 7 coins for change. What 7 coins did the cashier give Zoey?
Answer: 2 nickels and 5 pennies
Solution: Zoey gave the cashier $2 and spent $1.85. So she received 15¢ back in change (2.00 – 1.85 = 15¢). Now we need to figure out which 7 coins can make 15¢. The ways to make 15¢ are:
15 pennies
1 nickel, 10 pennies
2 nickels, 5 pennies
1 dime, 5 pennies
1 dime, 1 nickel
3 nickels
The only way to make 15¢ with 7 coins is 2 nickels and 5 pennies.

Upper Elementary:
Question: I am a 5 digit number that is divisible by 10. I have 0s for two of my digits and prime numbers for the rest of my digits. I am greater the 60,000. My thousands place is the same as my ones place. My hundreds digit is greater than my tens digit. My tens digit is the third prime number. What number am I?
Answer: 70,750
Solution: Since the digits are 0s prime numbers, the choices we have are 0, 2, 3, 5, and 7. The number is divisible by 10, so the ones place has to be a 0.
_ _ , _ _ 0
Since the thousands place and the ones place are the same, the thousands digit is 0.
_ 0 , _ _ 0
The number is greater than 60,000; since the only prime digit that is greater than 6 is 7, the ten thousands digit has to be 7.
7 0, _ _ 0
The tens digit is the third prime number, so the tens digit is 5.
7 0, _ 5 0
Since the hundreds place is larger than the tens place, the hundreds place has to be 7 because 7 is the only single-digit prime number greater than 5. So, the number is 70,750.

Middle School:
Question: A picture is 12 inches long and 8 inches wide. When framed, the uncovered part of the picture is 10 inches long and 6 inches wide. What is the area of the part of the picture that is covered by the frame?
Answer: 36 inches²
Solution: One way to solve this problem is to calculate the area of the whole picture and subtract the area of the uncovered part of the picture in the picture frame. The area of the picture is 12 × 8 = 96 inches². The area of the uncovered part of the picture is 10 × 6 = 60 inches². The area of the picture that is covered is 96 – 60 = 36 inches².

Algebra and Up:
Question: Cody is running up and down an escalator at a constant rate of 4 feet/second. The escalator is moving down. It takes Cody 10 seconds to run down the escalator and 30 seconds to run up the escalator. What is the speed of the escalator?
Answer: 2 feet/second
Solution: One way to solve this problem is to compare the equations of Cody running down the escalator and Cody running up the escalator. The formula for speed is:
Speed = Distance ÷ Time, which can be rewritten as Distance = Speed × Time
Let L = the length of the escalator, R_e be the speed of the escalator, R_c be Cody’s speed, T_d be the time it takes to run down the escalator, and let T_u be the time it takes to run up the escalator.
When going down the escalator, both the escalator and Cody are going in the same direction, so the rates are added together.
The formula for going down the escalator is L = (R_c + R_e)T_d.
When going up the escalator, Cody is running against the direction of the escalator. Therefore, we subtract the speed of the escalator from Cody’s running speed.
The formula for going up the escalator is L = (R_c – R_e)T_d.
Notice that in both equations, the length of the elevator is the same. So, we can set the equations equal to each other.
(R_c + R_e)T_d = (R_c – R_e)T_d
We know Cody’s speed, the time it takes to go down the escalator, and the time it takes to go up the escalator. Plug in each piece of information into our equation and we get:
(4 + R_e)10 = (4 – R_e)30
We need to figure out the speed of the escalator. If we solve for R_e, we will find the speed of the escalator.
Distribute each side.
40 + 10 R_e = 120 – 30 R_e
Combine like terms: put R_e on one side and the constants on the other.
40 R_e = 80
Divide by 40.
R_e = 2
The speed of the escalator is 2 feet/second.