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Thread: Use four 5s and certain operations to make expressions equal to integers 1 through 10

  1. #1
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    Use four 5s and certain operations to make expressions equal to integers 1 through 10

    .


    * Use exactly four 5s in every expression and no other digits/numbers

    * Choose from among addition, subtraction, division, and/or multiplication operations

    * You may use parentheses for grouping and/or multiplication, as needed

    * You may use decimal points, such as with .5, but not for a repeating decimal.

    * No concatenation is allowed. This includes: 55, 555, 5555, .55, .555, .5555,
    5.5, 55.5, 5.55, 555.5, 5.555, 55.55

    * No other characters or operations may be used


    Try to come up with one expression for each number.
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    Re: Use four 5s and certain operations to make expressions equal to integers 1 throug

    Here is an example of one of them:

    (5 + 5 + 5)/5 = 3

    Or, written a different way, it is

    $\displaystyle \dfrac{5 + 5 + 5}{5} \ = \ 3$


    Although (.5 + .5 + .5)/.5 also equals 3, all of the decimal points
    can be removed. It is not a primitive form. It is a redundant form.
    It is preferred to give solutions that are primitive forms.
    Last edited by greg1313; Aug 7th 2019 at 07:43 AM.
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    Re: Use four 5s and certain operations to make expressions equal to integers 1 throug

    $\displaystyle \frac55\times\frac55=1$
    $\displaystyle \frac55+\frac55=2$
    $\displaystyle 5-\frac{5+5}5=3$
    $\displaystyle 5-\frac5{\frac55}=4$
    $\displaystyle 5\times\frac5{\frac55}=5$
    $\displaystyle 5+\frac5{\frac55}=6$
    $\displaystyle 5+\frac{5+5}5=7$
    $\displaystyle \frac{5-\frac55}{.5}=8$ (I hate that I need to resort using .5)
    $\displaystyle 5+5-\frac55=9$
    $\displaystyle 5+5\times\frac55=10$
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    Re: Use four 5s and certain operations to make expressions equal to integers 1 throug

    Quote Originally Posted by Monoxdifly View Post

    $\displaystyle 5-\frac5{\frac55}=4$
    .
    .
    I cannot tell what those expressions are supposed
    to be for 4, 5, and 6. Where would grouping symbols
    go for them for instance? I tried working them by the
    Order of Operations, but I'm not getting 4, 5, or 6,
    respectively.
    Last edited by greg1313; Aug 7th 2019 at 08:47 PM.
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    Re: Use four 5s and certain operations to make expressions equal to integers 1 throug

    Oops, sorry. Miscalculations on my part. It should be $\displaystyle \frac55$ instead of $\displaystyle \frac5{\frac55}$ for those three.
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    Re: Use four 5s and certain operations to make expressions equal to integers 1 throug

    Quote Originally Posted by Monoxdifly View Post
    Oops, sorry. Miscalculations on my part. It should be $\displaystyle \frac55$ instead of $\displaystyle \frac5{\frac55}$ for those three.
    Each of those three updated expressions use only three 5s, for example

    5 - 5/5 = 4, but you must use exactly four 5s in each expression.

    Can you redo expressions to equal 4, 5, and 6 that use the required
    four 5s?
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    Re: Use four 5s and certain operations to make expressions equal to integers 1 throug

    Quote Originally Posted by greg1313 View Post
    Each of those three updated expressions use only three 5s, for example

    5 - 5/5 = 4, but you must use exactly four 5s in each expression.

    Can you redo expressions to equal 4, 5, and 6 that use the required
    four 5s?
    $\displaystyle \frac{{\frac{\frac55}}{.5}}{.5}=4$ (Again, I resent that I had to resort using .5)
    5 + (5 - 5) 5 = 5 (Didn't even realize that it was THAT simple)
    $\displaystyle \frac{5(.5)+0.5}{0.5}=6$ (Need about 1 hour to finally came up with this one)
    Last edited by topsquark; Aug 9th 2019 at 06:31 AM.
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    Re: Use four 5s and certain operations to make expressions equal to integers 1 throug

    Quote Originally Posted by Monoxdifly View Post
    $\displaystyle \frac{{\frac{\frac55}}{.5}}{.5}=4$ (Again, I resent that I had to resort using .5)
    5 + (5 - 5) 5 = 5 (Didn't even realize that it was THAT simple)
    $\displaystyle \frac{5(.5)+0.5}{0.5}=6$ (Need about 1 hour to finally came up with this one)
    The first one doesn't require any decimal points.
    The second one has at least one alternate variation.
    Your third one is not a primitive solution. All of those
    decimal points can be removed.

    $\displaystyle \dfrac{5*5 \ - \ 5}{5} \ = \ 4$

    $\displaystyle 5 \ + \ (5 - 5)/5 \ = \ 5 $

    $\displaystyle \dfrac{5*5 \ + \ 5}{5} \ = \ 6 $
    Last edited by greg1313; Aug 9th 2019 at 03:57 PM.
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    Re: Use four 5s and certain operations to make expressions equal to integers 1 throug

    Quote Originally Posted by greg1313 View Post
    The first one doesn't require any decimal points.
    The second one has at least one alternate variation.
    Your third one is not a primitive solution. All of those
    decimal points can be removed.

    $\displaystyle \dfrac{5*5 \ - \ 5}{5} \ = \ 4$

    $\displaystyle 5 \ + \ (5 - 5)/5 \ = \ 5 $

    $\displaystyle \dfrac{5*5 \ + \ 5}{5} \ = \ 6 $
    Gee... Silly me... Proof that I need to train more...
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