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

1. ## 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.

2. ## 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.

3. ## 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)
$\displaystyle 5+5-\frac55=9$
$\displaystyle 5+5\times\frac55=10$

4. ## Re: Use four 5s and certain operations to make expressions equal to integers 1 throug

Originally Posted by Monoxdifly

$\displaystyle 5-\frac5{\frac55}=4$
.
.
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.

5. ## 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.

6. ## Re: Use four 5s and certain operations to make expressions equal to integers 1 throug

Originally Posted by Monoxdifly
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?

7. ## Re: Use four 5s and certain operations to make expressions equal to integers 1 throug

Originally Posted by greg1313
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)

8. ## Re: Use four 5s and certain operations to make expressions equal to integers 1 throug

Originally Posted by Monoxdifly
$\displaystyle \frac{{\frac{\frac55}}{.5}}{.5}=4$ (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.

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

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

$\displaystyle \dfrac{5*5 \ + \ 5}{5} \ = \ 6$

9. ## Re: Use four 5s and certain operations to make expressions equal to integers 1 throug

Originally Posted by greg1313
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...