SOLUTION: If the digits of a two-digit positive interger are reversed, the result is 6 less than twice the original numder. Find all such intergers for which this is true.
How do you figu
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Question 175629This question is from textbook Algebra 1
: If the digits of a two-digit positive interger are reversed, the result is 6 less than twice the original numder. Find all such intergers for which this is true.
How do you figure this out?
This question is from textbook Algebra 1
Found 4 solutions by ankor@dixie-net.com, Mathtut, josmiceli, solver91311:
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
If the digits of a two-digit positive integer are reversed, the result is 6 less than twice the original number. Find all such integers for which this is true.
:
Let x = the 10's digit
Let y = the units digit
then
10x + y = original integer
and
10y + x = reversed digit number
:
Write an equation for what it says:
10y + x = 2(10x + y) - 6
:
10y + x = 20x + 2y - 6
:
10y - 2y = 20x - x - 6
:
8y = 19x - 6
y =
You can see there are not many values for x which will result in a single digit integer for y.
:
x=1 obviously not (13/8)
x=2
y =
y =
y = 32/8
y = 4
Our original number = 24, and you will find that this is the only one.
Answer by Mathtut(3670) (Show Source): You can put this solution on YOUR website!
lets call our digits a and b written as ab
:
now remember ab can also be written as 10a+b
and when it is reverse it is ba which can be written as 10b+a
:
so with that info lets write the equation: they only gave us enough info for one equation so we are going to have to do some surmising on this one:
:
10b+a=2(10a+b)-6
:
10b+a=20a+2b-6
:
8b-19a=-6
:
the only possibilities for this scenario is digits 0-9 for both a and b. the question is can we find an integer value for both a and b that satisfies this equation.
the value of 19a always has to be more that the value of 8b in order to get -6
:
so for a=1 b can only be 1 or 2 (8-19),(16-19)...neither of which works
a=2 so 19a=38.. so we need 8b=32...8(4)=32 so we have a pair a=2 b=4.
a=3 so 19a=57...so we need 8b=51....no integer value will work
a=4 so 19a=76...so we need 8b=70....no integer value will work
a=5 so 19a=95...so we need 8b=89---> we have exhaused our possibilities here. as you can see in order for 8b=89 we would have to multiply by two digit number and as the value of 19a gets higher the same will hold true
:
so we have found the only integers for which this is true and
:
they are
Answer by josmiceli(19441) (Show Source): You can put this solution on YOUR website!
Here's how to find the value of a number if
you are given the digits:
Let the 10s digit =
Let the 1s digit =
The value of the number is
Given:
The key to going on from here is to realize
that and are single digits
and can only be 0 - 9.
can't be , you can't start a 2-digit
number with . I'll find and
solve for
----------
m -- n
1 -- 13/8 not a digit
2 -- 4
3 -- 51/8 not a digit
4 -- 35/4 not a digit
5 -- 89/8 not a digit
6 -- 27/2 not a digit
7 -- 127/8 not a digit
8 -- 73/4 not a digit
9 -- 165/8 not a digit
The only number found is 24
check:
OK
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
Let the tens digit be represented by and the ones digit be represented by . That means you can represent any two digit positive number by if you restrict and to the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
If our original number is then the number with reversed digits is . Then the conditions of the problem give us:
Simplify and solve for
Now remember that both and are restricted to the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. That means that we have to find which of the given set of numbers, when substituted for , yield also an element of the given set. We know that since the denominator is 19 and there are only ten consecutive integer possibilities, we have at most 1 solution.
We can also see that must be an integer multiple of 19.
The first five integer multiples of 19 are 19, 38, 57, 76, and 95. So:
→ → not an integer.
→ → IS an integer. This is our solution, but let's continue just to make sure.
→ → not an integer.
→ → not an integer.
→ → not an integer AND so we can stop looking.
Now we know the ones digit is 4, so using we know the tens digit is 2.
The only positive two-digit integer that satisfies the given conditions is 24.
Check the answer:
2 times 24 is 48 which is 6 more than 42 which is 24 with the digits reversed. Checks.
By the way, the word is integer, not interger.
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