Question #0d788

2 Answers
Dec 13, 2017

Technically, #1.bar 6 =1 2/3#.
#1.6=1 6/10 or 1 3/5#.

Explanation:

Use your calculator or long division to divide 2 by 3. You get #2/3=.6666... or .bar 6 #
I think maybe your confusion comes from the notation using the bar over the repeating number.
#.6# is not the same as #.bar6#; when you see that bar over a number following a decimal, it means the number keeps repeating.

Also, you might know that #1/3=.3333... = .bar3#
And #1/3+1/3=2/3#, and

certainly #.333...+.333...=.666...#, or, more succinctly,
#.bar3+.bar3=.bar6#

Does that answer your question?

PS. As a math teacher, I would like to sincerely thank you for working on building conceptual understanding and facility with fractions!! Good work!

Dec 13, 2017

#1.6 = 1 6/10 = 1 3/5#

#1.66666666...# (recurring decimal) # = 1.bar6 = 1 2/3#

Explanation:

There are two types of rational decimal fractions.

  • terminating decimals - (those which stop)
    #" "0.8," " 3.875, " " 4.5, " "0.1378#

  • recurring decimals - those which repeat to infinity.
    #" "0.44444....," "3.151515...." " 5.653653653....#

These are indicated with a bar or dot on the recurring digits:
#0.bar4," "3.bar(15)," "5.bar(653)" "or 0.dot4," "3.dot1dot5," "5.dot6 5dot3" " #

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#1.6# (terminating decimal) is NOT actually equal to #1 2/3#

#1.6 = 1 6/10 = 1 3/5#

But, #1.66666666...# (recurring decimal) # = 1.bar6 = 1 2/3#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This arises because #1/3# means #1div3# and when you divide #1# by #3#, the answer keeps recurring,

Try it yourself with a calculator or by division:

(There is always a remainder of #1#)

#1 div 3 = 3|ul(1.0000000...)#
#color(white)(xxxxxxxx)0.3333333...#

The same happens with #2/3# which means #2div3#

When you divide #2# by #3#, the answer keeps recurring.

(There is always a remainder of #2#)

#2 div 3 = 3|ul(2.0000000...)#
#color(white)(xxxxxxxx)0.6666666...#

If you have a whole number and a fraction, this does not affect the decimal at all:

#1 2/3 = 1.bar6#

#2 2/3 = 2.bar6#

#5 2/3 = 5.bar6#