Math Problem Monday - December 14th, 2020 | Mathnasium Livermore, CA

Lower Elementary:
Question: A group of pirates found a treasure map that will lead them to buried treasure. Following the map, it took the pirates 3 days to get to the first island and another 2 days to get to the second island. Then, there was a storm that lasted for 2 days, so they stayed on the island until the storm passed. Afterward, it then took the pirates 4 days to get to the final island. They explored the last island for 2 days before they found the treasure. If they found the treasure on Monday, what day did they start their adventure?
Answer: Tuesday
Solution: First, let’s see how long the entire adventure took. 3 + 2 + 2 + 4 + 2 = 13. The entire adventure took 13 days. Count backwards 13 days from Monday and we have that they started on Tuesday. Alternatively, there are 14 days in 2 weeks. Since the adventure took 13 days, it would be 1 day less than 2 weeks. 2 weeks would be Monday, so one day less would be Tuesday.

Upper Elementary:
Question: Albert made a pizza. Half of the pizza was cheese and the other half was pepperoni. Albert also wanted 1/3 of the pizza to have mushrooms, but he did not want any mushrooms to be on the cheese half of the pizza. What fractional part of the pizza was pepperoni only?
Answer: 1/6
Solution: Half of the pizza is cheese and the other half is pepperoni. Since mushrooms were not added to the cheese half of the pizza, they must have been added to the pepperoni half of the pizza. To find what part of the pizza is just pepperoni, subtract the fractional part that is pepperoni and mushrooms from the fractional part that is pepperoni. Pepperoni is half of the pizza and mushrooms were added to 1/3 of the pizza. 1/2 – 1/3. In order to subtract these fractions, they must have the same denominator, or same name. If we convert the fractions to sixths, we have 3/6 – 2/6 = 1/6. 1/6 of the pizza is pepperoni only.

Middle School:
Question: How much interest will be earned if $10,000 is put into an account that is compounded annually at 10% for 2 years?
Answer: $2,100
Solution: The formula for compounded interest is A = P(1 + r/n)^nt where A is the amount in the account after applying the compound interest, P is the principle, r is the rate, t is the time in years, and n is the number of times compounded a year. With the given information, we have that P = 10,000, r = 10% = 0.1, t = 2, and n = 1. Plugging in these values into the formula, we have:
A = 10,000(1 + (0.1)/(1))⁽¹⁾⁽²⁾
A = 10,000(1 + 0.1)²
A = 10,000(1.1)²
A = 10,000(1.21)
A = 12,100
Thus the total amount in the account after 2 years is $12,100. Since the account started with $10,000, the amount of interest that was earned was 12,100 – 10,000 = $2,100.

Algebra and Up:
Question: Find the value of x: 2^{(x + 2)} = 4^{(x – 3)}
Answer: x = 8
Solution: One way to solve this problem is to rewrite the bases so that they are the same base. We can write 4 as 2². Since an exponent raised to another power is the same as multiplying the exponents together, we can rewrite the problem as:
2^{(x + 2)} = 2^{2(x – 3)}
Since the bases are the same, that means the exponents must be the same.
x + 2 = 2(x – 3)
Distribute the 2.
x + 2 = 2x – 6
Subtract an x from both sides and add a 6 to both sides.
8 = x
Thus, x = 8.