To convert \( 5 \frac{3}{16} \) to a decimal using long division, we first need to convert the mixed number into an improper fraction.
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Convert \( 5 \frac{3}{16} \) to an improper fraction: \[ 5 \frac{3}{16} = \frac{(5 \times 16) + 3}{16} = \frac{80 + 3}{16} = \frac{83}{16} \]
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Now, we perform the long division of \( 83 \) by \( 16 \).
- \( 16 \) goes into \( 83 \) a total of \( 5 \) times (since \( 16 \times 5 = 80 \)).
- Subtract \( 80 \) from \( 83 \) to get \( 3 \).
- Now, we need to continue dividing \( 3 \) with decimal points.
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We can express \( 3 \) as \( 3.000 \) for the long division.
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\( 16 \) goes into \( 30 \) a total of \( 1 \) time (since \( 16 \times 1 = 16 \)).
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Subtract \( 16 \) from \( 30 \) to get \( 14 \).
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Bring down the next \( 0 \) to make it \( 140 \).
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\( 16 \) goes into \( 140 \) a total of \( 8 \) times (since \( 16 \times 8 = 128 \)).
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Subtract \( 128 \) from \( 140 \) to get \( 12 \).
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Bring down the next \( 0 \) to make it \( 120 \).
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\( 16 \) goes into \( 120 \) a total of \( 7 \) times (since \( 16 \times 7 = 112 \)).
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Subtract \( 112 \) from \( 120 \) to get \( 8 \).
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Bring down the next \( 0 \) to make it \( 80 \).
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\( 16 \) goes into \( 80 \) exactly \( 5 \) times (since \( 16 \times 5 = 80 \)).
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Subtract \( 80 \) from \( 80 \) to get \( 0 \).
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Putting it all together:
\[ 5 \quad (1) \quad (8) \quad (7) \quad (5) \ +0 \quad 1 \quad 7 \quad 5 \]
Thus, \( 83 \div 16 = 5.1875\).
Therefore, the decimal conversion of \( 5 \frac{3}{16} \) is:
\[ \boxed{5.1875} \]