Question

1. determine a fractional number between each pair of decimal numbers.

a) 3.5, 3.6
b) -2.5, -2.55
c) ( 5.overline{3} ), ( 5.overline{4} )
d) -1.7, ( -1.overline{7} )
e) ( 4.overline{23} ), ( 4.2overline{3} )
f) 5.2, ( 5.overline{2} )
g) ( -3.overline{4} ), ( -3.overline{41} )
h) ( 1.overline{71} ), ( 1.overline{711} )
i) ( 5.overline{123} ), ( 5.overline{1234} )
j) ( -2.00overline{1} ), -2.001

Explanation:

Step1: Find midpoint of 3.5, 3.6

Midpoint formula: $\frac{3.5 + 3.6}{2} = \frac{7.1}{2} = 3.55 = \frac{71}{20}$

Step2: Find midpoint of -2.5, -2.55

$\frac{-2.5 + (-2.55)}{2} = \frac{-5.05}{2} = -2.525 = -\frac{101}{40}$

Step3: Convert repeating decimals to fractions

$5.\overline{3} = \frac{48}{9}$, $5.\overline{4} = \frac{49}{9}$. Midpoint: $\frac{\frac{48}{9} + \frac{49}{9}}{2} = \frac{97}{18}$

Step4: Convert repeating decimal to fraction

$-1.\overline{7} = -\frac{16}{9}$, $-1.7 = -\frac{17}{10}$. Midpoint: $\frac{-\frac{17}{10} + (-\frac{16}{9})}{2} = -\frac{313}{180}$

Step5: Convert repeating decimals to fractions

$4.\overline{23} = \frac{419}{99}$, $4.2\overline{3} = \frac{381}{90}$. Midpoint: $\frac{\frac{419}{99} + \frac{381}{90}}{2} = \frac{8001}{1980} = \frac{889}{220}$

Step6: Convert repeating decimal to fraction

$5.\overline{2} = \frac{47}{9}$, $5.2 = \frac{26}{5}$. Midpoint: $\frac{\frac{26}{5} + \frac{47}{9}}{2} = \frac{469}{90}$

Step7: Convert repeating decimals to fractions

$-3.\overline{4} = -\frac{31}{9}$, $-3.\overline{41} = -\frac{338}{99}$. Midpoint: $\frac{-\frac{31}{9} + (-\frac{338}{99})}{2} = -\frac{679}{198}$

Step8: Convert repeating decimal to fraction

$1.\overline{71} = \frac{170}{99}$, $1.\overline{711} = \frac{1710}{999}$. Midpoint: $\frac{\frac{170}{99} + \frac{1710}{999}}{2} = \frac{34010}{19782} = \frac{17005}{9891}$

Step9: Convert repeating decimals to fractions

$5.\overline{123} = \frac{5118}{999}$, $5.\overline{1234} = \frac{51229}{9999}$. Midpoint: $\frac{\frac{5118}{999} + \frac{51229}{9999}}{2} = \frac{102397}{19998}$

Step10: Convert repeating decimal to fraction

$-2.00\overline{1} = -\frac{1999}{999}$, $-2.001 = -\frac{2001}{1000}$. Midpoint: $\frac{-\frac{1999}{999} + (-\frac{2001}{1000})}{2} = -\frac{3998999}{1998000}$

Answer:

a) $\frac{71}{20}$
b) $-\frac{101}{40}$
c) $\frac{97}{18}$
d) $-\frac{313}{180}$
e) $\frac{889}{220}$
f) $\frac{469}{90}$
g) $-\frac{679}{198}$
h) $\frac{17005}{9891}$
i) $\frac{102397}{19998}$
j) $-\frac{3998999}{1998000}$

Note: Any fraction between the pairs is valid; midpoints are used here as a consistent method.