Question

1. a researcher found that the sea level rose \\(\frac{1}{5}\\) of a centimeter in \\(\frac{2}{3}\\) of a year. at what rate did the sea level rise?

a \\(\frac{2}{5}\\) centimeter per year
b \\(\frac{3}{5}\\) centimeter per year
c \\(\frac{2}{15}\\) centimeter per year
d \\(\frac{3}{10}\\) centimeter per year

Explanation:

Step1: Recall rate formula

Rate is defined as the amount of change (rise in sea level) divided by the time taken. So, rate \( r=\frac{\text{Change in sea level}}{\text{Time taken}} \).

Step2: Substitute values

The change in sea level is \( \frac{1}{5} \) centimeter and the time taken is \( \frac{2}{3} \) year. So, \( r = \frac{\frac{1}{5}}{\frac{2}{3}} \).

Step3: Divide fractions

Dividing by a fraction is the same as multiplying by its reciprocal. So, \( \frac{\frac{1}{5}}{\frac{2}{3}}=\frac{1}{5}\times\frac{3}{2}=\frac{3}{10} \)? Wait, no, wait. Wait, \( \frac{1}{5}\div\frac{2}{3}=\frac{1}{5}\times\frac{3}{2}=\frac{3}{10} \)? Wait, no, wait the options: Wait, maybe I made a mistake. Wait the problem says "the sea level rose \( \frac{1}{5} \) of a centimeter in \( \frac{2}{3} \) of a year". So rate is (1/5) cm per (2/3) year. So to find per year, we do (1/5) ÷ (2/3) = (1/5)*(3/2) = 3/10? But wait option B is 3/5? Wait no, maybe I misread the problem. Wait the problem: "the sea level rose \( \frac{1}{5} \) of a centimeter in \( \frac{2}{3} \) of a year". Wait no, maybe the first fraction is \( \frac{1}{5} \) or \( \frac{2}{5} \)? Wait the image: Let me check again. The problem: "the sea level rose \( \frac{1}{5} \) of a centimeter in \( \frac{2}{3} \) of a year". Wait no, maybe the first fraction is \( \frac{2}{5} \)? Wait no, the user's image: "the sea level rose \( \frac{1}{5} \) of a centimeter in \( \frac{2}{3} \) of a year". Wait but let's recalculate. Wait (1/5) divided by (2/3) is (1/5)*(3/2)=3/10. But option D is 3/10? Wait no, the options: A. 2/5, B. 3/5, C. 2/15, D. 3/10. Wait so 3/10 is option D? Wait no, wait maybe I messed up the numerator and denominator. Wait, maybe the sea level rose \( \frac{2}{5} \) cm? Wait the user's image: "the sea level rose \( \frac{1}{5} \) of a centimeter" – no, maybe the first fraction is \( \frac{2}{5} \)? Wait no, the user's problem: "the sea level rose \( \frac{1}{5} \) of a centimeter in \( \frac{2}{3} \) of a year". Wait let's check again. Rate = distance/time = (1/5) cm / (2/3) year = (1/5)*(3/2) = 3/10 cm per year. So option D is 3/10. But wait maybe the problem was \( \frac{2}{5} \) cm? Wait no, the user's image: "the sea level rose \( \frac{1}{5} \) of a centimeter". Wait maybe I made a mistake. Wait let's check the options again. Option B is 3/5, D is 3/10. Wait let's recalculate: (1/5) ÷ (2/3) = 3/10. So the answer should be D? Wait no, wait maybe the problem is "rose \( \frac{2}{5} \) of a centimeter"? Let me check the original problem again. The user's image: "the sea level rose \( \frac{1}{5} \) of a centimeter in \( \frac{2}{3} \) of a year". So that's correct. So (1/5) divided by (2/3) is 3/10. So option D. But wait maybe I misread the first fraction. Wait the first fraction: is it \( \frac{1}{5} \) or \( \frac{2}{5} \)? The user's image: "the sea level rose \( \frac{1}{5} \) of a centimeter". So 1/5. So the calculation is correct. So the rate is 3/10 centimeter per year, which is option D. Wait no, wait (1/5) ÷ (2/3) = (1×3)/(5×2)=3/10. Yes. So the answer is D. Wait but let's check again.

Wait, maybe the problem was "the sea level rose \( \frac{2}{5} \) of a centimeter in \( \frac{2}{3} \) of a year". Then (2/5) ÷ (2/3) = (2/5)*(3/2)=3/5, which is option B. Maybe there's a typo in the problem, or maybe I misread the first fraction. Let me check the user's image again. The problem: "the sea level rose \( \frac{1}{5} \) of a centimeter in \( \frac{2}{3} \) of a year". So 1/5. So (1/5) ÷ (2/3) = 3/10. So option D. But let's confirm the division of fractions. \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \). So \( \frac{1}{5} \div \frac{2}{3} = \frac{1}{5} \times \frac{3}{2} = \frac{3}{10} \). Yes. So the rate is \( \frac{3}{10} \) centimeter per year, which is option D. Wait but the options: D is \( \frac{3}{10} \) centimeter per year. So that's correct.

Answer:

D. \( \frac{3}{10} \) centimeter per year