Q: 17. How many years are in 84 months?

The correct answer is: $7\text{ years}$ Explanation To convert months to years, divide the number of months by 12 (months per year). Steps: Formula: $$\text{years}=\frac{\text{months}}{12}$$ Substitute: $$\text{years}=\frac{84}{12}$$ Compute: $$\text{years}=7$$ Therefore, 84 months equals $7\text{ years}$.

Q: 1. Write 4.3125 as a fraction in simplest form.

The correct answer is: $69/16$ Explanation We convert the decimal part to a fraction and simplify. Steps: Separate the integer and decimal parts: $$4.3125 = 4 + 0.3125$$ Write the decimal as a fraction: $$0.3125 = \frac{3125}{10000}$$ Simplify the fraction by dividing numerator and denominator by $125$: $$\frac{3125}{10000} = \frac{25}{80} = \frac{5}{16}$$ Combine and convert to an improper fraction: $$4 + \frac{5}{16} = \frac{64}{16} + \frac{5}{16} = \frac{69}{16}$$ Therefore, $4.3125 = \frac{69}{16}$.

Q: Write and solve an equation that represents the following: Twenty more than a number is eighteen.

The correct answer is: $x = -2$ Explanation Let $x$ be the unknown number. "Twenty more than a number is eighteen" translates to the equation $x + 20 = 18$. Solve this equation to find $x$. Steps: Write the equation: $x + 20 = 18$ Subtract 20 from both sides: $x = 18 - 20$ Compute the result: $x = -2$ Therefore, the number is $-2$.

Q: We want to convert 54 degrees Celsius (°C) to degrees Fahrenheit (°F).

The correct answer is: $129.2^\circ\text{F}$ Explanation Use the Celsius-to-Fahrenheit conversion formula $F = C\times\frac{9}{5}+32$. For $C=54^\circ\text{C}$ substitute and compute. Steps: Formula: $$F = C\times\frac{9}{5}+32$$ Substitute: $$F = 54\times\frac{9}{5}+32$$ Calculate: $$F = 54\times1.8+32 = 97.2+32 = 129.2$$ Therefore, $54^\circ\text{C} = 129.2^\circ\text{F}$.

Q: How do you write 0.79 as a fraction?

The correct answer is: $79/100$ Explanation Write the decimal as a fraction by moving the decimal point two places (because there are two digits after the decimal) — this is equivalent to multiplying numerator and denominator by 100. Steps: Multiply to remove the decimal: $$0.79\times100=79$$ Write as a fraction: $$0.79=\frac{79}{100}$$ Simplify if possible: $$\gcd(79,100)=1\text{, so the fraction is already in lowest terms}$$ Therefore, $0.79=\frac{79}{100}$.

Question

6. After the world noticed the sweet calculator watch Mr. Passwater was wearing one day, everyone began wanting to emulate him and get their own calculator watch. As a result, one factory had to build an industrial machine to help manufacture and package the watches so they could be shipped to customers around the world. In a certain simulation, the total number of watches the machine could produce after n hours can be modeled using an arithmetic sequence. The machine had produced 4306 watches after 2 hours, and 15,071 total watches after 7 hours. Based on the simulation, how many total watches will the machine produce by the end of hour 12?

Accepted Answer

The correct answer is: $25836$

Explanation

The total watches form an arithmetic sequence $a_n=a_1+(n-1)d$. We are given $a_2=4306$ and $a_7=15071$, so we can find the common difference $d$ and then compute $a_{12}$.

Steps:

Difference between terms: $$a_7-a_2=(7-2)d \Rightarrow 15071-4306=5d$$
Solve for $d$: $$10765=5d \Rightarrow d=2153$$
Find $a_1$ and then $a_{12}$: $$a_2=a_1+d \Rightarrow 4306=a_1+2153 \Rightarrow a_1=2153$$

$$a_{12}=a_1+11d=2153+11(2153)=2153\cdot12=25836$$

Therefore, by the end of hour 12 the machine will have produced 25,836 watches.