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The image contains two math problems involving percentage approval changes over time. Let's solve these questions one by one.

**Problem 23:**
In 1983, 43% of U.S. adults approved of interracial marriage between blacks and whites. From 1983 to 2007, the approval rate increased by 1.5% each year. The question asks by which year all American adults will approve of interracial marriage, assuming the trend continues.

Starting with 43% approval in 1983, we need to find the year when the approval reaches 100%. The formula to calculate the approval percentage in year \( n \) is:
\[ \text{Approval percentage in year } n = 43\% + 1.5\% \times (n - 1983) \]
We set this equal to 100% and solve for \( n \).

Let's do this calculation.For problem 23, if the trend of increasing approval for interracial marriage continued by 1.5% each year starting from 1983, all American adults would have approved of interracial marriage by the year 2021.

**Problem 24:**
In 1986, 43% of U.S. adults approved of men and women living together without being married. From 1986 to 2007, the approval percentage increased by 0.6% each year. The question asks by which year 61% of all American adults will approve of cohabitation if this trend continues.

Starting with 43% approval in 1986, we need to find the year when the approval reaches 61%. The formula to calculate the approval percentage in year \( n \) is:
\[ \text{Approval percentage in year } n = 43\% + 0.6\% \times (n - 1986) \]
We set this equal to 61% and solve for \( n \).

Let's calculate this.For problem 24, if the trend of increasing approval for cohabitation continued by 0.6% each year starting from 1986, 61% of all American adults would have approved of cohabitation by the year 1998.

Would you like to explore further details or have questions about these calculations? Here are some follow-up topics you might find interesting:

1. How would a decrease in the yearly approval change rate impact the years predicted in both scenarios?
2. What other sociopolitical factors could influence the rates of approval over time?
3. How do these mathematical models apply to other areas of sociological research?
4. Can we predict the approval rate for the current year based on these trends?
5. What are the limitations of linear projection in social sciences?

Tip: When making predictions based on historical data, always consider potential changes in societal norms and external influences that could alter trends.

In 1983, 43% of U.S. adults approved of marriage between blacks and whites. For the period from 1983 through 2007, the percentage approving of interracial marriage increased on average by 1.5 each year. If this trend continues, by which year will all American adults approve of interracial marriage? In 1986, 43% of U.S. adults approved of men and women living together without being married. For the period from 1986 through 2007, the percentage approving of cohabitation increased on average by approximately 0.6 each year. If this trend continues, by which year will 61% of all American adults approve of cohabitation?

This problem involves predicting future approval rates of interracial marriage and cohabitation based on historical trends. Learn how to use linear projection and solve for unknown variables using algebra and percentage changes. Suitable for Grades 8-10.

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