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The problem asks for how many years a glacier has been decreasing in size, given the following information:

- The glacier decreases by \( \frac{2}{7} \) feet per year.
- The total decrease in size is \( -\frac{6}{11} \) feet.

We can find the number of years (\( t \)) by using the equation:

\[
\text{Total decrease} = \text{Rate of decrease} \times \text{Time}
\]

Thus:

\[
-\frac{6}{11} = \frac{2}{7} \times t
\]

To solve for \( t \), divide both sides by \( \frac{2}{7} \):

\[
t = \frac{-\frac{6}{11}}{\frac{2}{7}}
\]

Now, dividing two fractions means multiplying by the reciprocal of the second fraction:

\[
t = -\frac{6}{11} \times \frac{7}{2}
\]

Simplify the multiplication:

\[
t = -\frac{6 \times 7}{11 \times 2} = -\frac{42}{22}
\]

Now simplify \( \frac{42}{22} \):

\[
t = -\frac{21}{11}
\]

Therefore, the glacier has been decreasing for \( \frac{21}{11} \) years, or approximately 1.91 years.

Would you like further clarification or details?

Here are 5 related questions for further exploration:

1. How would the equation change if the glacier was shrinking at a faster rate, say \( \frac{3}{7} \) feet per year?
2. How can you convert \( \frac{21}{11} \) years into months and days?
3. What would the time be if the total decrease were \( -1 \) foot instead of \( -\frac{6}{11} \)?
4. How can you verify that multiplying by the reciprocal is valid when dividing fractions?
5. What other real-world applications can be modeled using similar rate equations?

**Tip:** Always check if your fractions can be simplified early on to make calculations easier!

A glacier decreases in size by 2/7 feet per year. The total change in the size of the glacier is -6/11 feet. How many years has the glacier been decreasing?

Learn how to calculate the number of years a glacier has been shrinking based on a rate of 2/7 feet per year and a total size reduction of -6/11 feet. This algebraic problem involves working with fractions and rates of change, suitable for Grades 7-8.

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