To explain that your interest has linear growth, we can focus on how simple interest works. In this case, the formula for simple interest is given by \( I = P \cdot r \cdot t \), where \( P \) is the principal amount, \( r \) is the interest rate, and \( t \) is the time in years.
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For a principal of $50 and an interest rate of 12%, the interest earned after 1 year (\( t = 1 \)) is:
\[ I = 50 \cdot 0.12 \cdot 1 = 6. \]
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After 5 years (\( t = 5 \)):
\[ I = 50 \cdot 0.12 \cdot 5 = 30. \]
These values show that the interest earned increases linearly with time. Now, let's consider the slope formed by the points (1, 6) and (5, 30):
- The change in interest from year 1 to year 5 is \( 30 - 6 = 24 \).
- The change in time from year 1 to year 5 is \( 5 - 1 = 4 \).
Thus, the slope \( m \) of the line between these two points is:
\[ m = \frac{\Delta I}{\Delta t} = \frac{24}{4} = 6. \]
In this case, every year you earn $6 from the original deposit of $50. This linearity is due to the interest being calculated as a constant percentage of the principal for each year.
Now, let's analyze the options you provided:
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"When you multiply 50 by 0.12, you get the same value as the slope between the points (1,6) and (5,30)." - This statement is not accurate because the slope is not equal to \( 50 \cdot 0.12 \); rather, the slope reflects the change in interest per unit of time.
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"You can find the product of the principal and the interest rate to see that it is the reciprocal of the slope." - This statement is misleading and incorrect in the context of linear growth analysis.
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"When you divide 50 by 12, you get the same value as the slope between the points (1, 6) and (5, 30)." - This is incorrect, as dividing $50 by 12 does not yield the slope.
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"You can look at the interest rate of 12% to see that the slope of the line between the points (1, 6) and (5,30) will be 12." - This is the most relevant and accurate explanation; while the actual change in interest per year is $6 (which is calculated by multiplying the $50 principal by 12%), the linear relationship can be observed through the constant increase in interest over time leading to uniform additions to the total.
Thus, the most fitting reasoning toward proving linear growth in interest is seen through the consistent yearly earning of $6, driven by the interest rate application to the principal amount. This clearly establishes that the growth of interest is linear over time.
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