find a number between 1 5/8 and 1.62501

To find a number between two numbers, if you're allowed to pick any number you want between those two numbers, one natural choice is the midpoint... the point that is exactly halfway between the two numbers.

Finding a midpoint

To find the midpoint of two numbers, divide their sum by 2:

[tex]\text{midpoint}=\dfrac{\text{first number}+\text{second number}}{2}[/tex]

It is convenient to have both numbers in decimal form or both numbers in fraction form for adding and dividing by 2.

Fraction form:

To convert 1.62501 to a fraction, we note that the last decimal digit is in the hundred-thousandths place, so we'll need a fraction with 100,000 in the denominator.

[tex]1.62501=1\frac{62501}{100000}[/tex]

Using our midpoint formula:

[tex]\text{midpoint} = \dfrac{ 1 \frac{5}{8} + 1 \frac{62501}{100000} } {2}[/tex]

find a common denominator...

[tex]\text{midpoint} = \dfrac{ 1 +(\frac{5}{8})*\frac{12500}{12500} + 1 +\frac{62501}{100000} } {2}[/tex]

[tex]\text{midpoint} = \dfrac{ 1 + \frac{62500}{100000} + 1 +\frac{62501}{100000} } {2}[/tex]

combining like terms...

[tex]\text{midpoint} = \dfrac{ 2 +\frac{125001}{100000} } {2}[/tex]

rewriting the division as multiplication of a reciprocal....

[tex]\text{midpoint} = \left(2 +\frac{125001}{100000} \right) *\dfrac{1}{2}[/tex]

distributing...

[tex]\text{midpoint} = \left(2 *\dfrac{1}{2} +\frac{125001}{100000} *\dfrac{1}{2}\right)[/tex]

[tex]\text{midpoint} = 1 +\frac{125001}{200000}[/tex]

[tex]\text{midpoint} = 1 \frac{125001}{200000}[/tex]

So, [tex]1 \frac{125001}{200000}[/tex] is the midpoint of the two numbers, and is a number that is between them.

Decimal form:

To convert [tex]1 \frac{5}{8}[/tex] to a decimal, we divide 5 by 8. 5 divided by 8 is 0.625.

So, [tex]1 \frac{5}{8}=1.625[/tex]

To find a midpoint of 1.625 and 1.62501, we use our midpoint formula:

Using our midpoint formula:

[tex]\text{midpoint} = \dfrac{ 1.625 + 1.62501} {2}[/tex]

[tex]\text{midpoint} = \dfrac{ 3.25001} {2}[/tex]

[tex]\text{midpoint} =1.625005[/tex]

So, 1.625005 is midpoint of the two numbers, and is a number that is between them.

Extension with decimals:

Knowing the decimal form of the two numbers, 1.625 and 1.62501, we can include trailing zeros at the end of these numbers (since it is to the right of the decimal point), so that 1.625 is equivalent to 1.625000, and 1.62501 is equivalent to 1.625010.

Note that the last two digits are in the millionths place, and that changing the digit in the millionths place for the first number (while leaving all other digits as they are) will also be a number between the two numbers (all 9 of the numbers with a star written below):

1.625000

1.625001*

1.625002*

1.625003*

1.625004*

1.625005*

1.625006*

1.625007*

1.625008*

1.625009*

1.625010