Question

MEDAL!!!


The sum of the digits of a certain two-digit number is 8. When you reverse its digits you decrease the number by 18. Find the number. (Solve this using elimination.)

Answer 1

@SolomonZelman

Answer 2

\(\color{#000000 }{ \displaystyle \underline{b}\underline{a} =\underline{a}\underline{b} +18 }\)
\(\color{#000000 }{ \displaystyle a+b=8 }\)

Answer 3

\(\color{#000000 }{ \displaystyle \underline{b}\underline{a} =\underline{a}\underline{b} +18 }\)
can be also written as

\(\color{#000000 }{ \displaystyle 10b+a=10a+b+18 }\)
And with the \(a+b=8\) you get,

\(\color{#000000 }{ \displaystyle 10b+a=10a+b+18 }\)
\(\color{#000000 }{ \displaystyle a+b=8 }\)

Answer 4

So you just have a system of equations to solve
(hope it's clear that when I underline I mean digits)

Answer 5

I came up with the correct answer for the two digit number and the inversed-digit number, but I have to give you a chance

Answer 6

Solve \[x+y=8\] and \[10x + y - 18 = x + 10y\]\[9x-9y = 18\]\[x-y=2\]Add the two equations gives\[2x=10\]\[x=5\]Therefore, y=3, and the number is 53.

Answer 7

you are not supposed to give direct answers, this is against the site's policy...

Answer 8

I'm well aware. My apologies, I assumed this was a challenge.

Answer 9

Well, I didn't see the question say "challenge", but I suppose this time its fine; I'm not a moderator anyway:)

Answer 10

Also the number is not 53 (look at the question)

Answer 11

@SolomonZelman I still don't understand how you got your equations.

Answer 12

I warn others about providing direct answers ALL the time.

Answer 13

Ok, do you agree that 38 can be recorded as 30•10+8?

Answer 14

Yes

Answer 15

Ok, and 83 can be recorded as 8•10+3, right?

Answer 16

I beg to differ on the correctness of my answer. The sume of the digits of 53 is eight and when you reverse its digits, i.e. 35, you decrease the number by 18.

Answer 17

you are looking for the original number, not when the digits are revered.
So 35, not 53.

Answer 18

Try again.

Answer 19

calculusxy, don't want to bug.... u there?

Answer 20

Yes I agree that 83 can be recorded as 8•10+3.

Answer 21

Ok, now back to the problem,

The sum of the digits of a certain two-digit number is 8.
When you reverse its digits you decrease the number by 18.
Find the number. (Solve this using elimination.)

So let's call our digits a and b.

Answer 22

ok

Answer 23

Our original number is \(\underline{a}\underline{b}\)
(I am underlining to denote that they are digits, not a•b)

Answer 24

\(\underline{a}\underline{b}=a\cdot 10+b\)
alternatively, like that we can record our original number.

Answer 25

We are given that when the digits are reversed, the number increases by 18, or in equation world that is,

\(\underline{b}\underline{a}=18+\underline{a}\underline{b}\)
(You need to add 18 to original number to make it = to the number with reversed digits -- that is the number with revered digits is +18 greater)

Answer 26

And we also know that the sum of digits is 8,
\(a+b=8\)

Answer 27

the number is decreased by 18

Answer 28

Question states the number is decreased by 18 when the digits are reversed, not increased.

Answer 29

I apologize @ospreytriple :)

Answer 30

So when the original number is decreased by 18, we have
\(\underline{b}\underline{a}=\underline{a}\underline{b}-18\)

Answer 31

You've got it well under control. Sorry for my earlier indiscretion.

Answer 32

calculusxy, are you lost?

Answer 33

no you can proceed

Answer 34

but i am confused as to how can i solve it when there is not a certain result to the equation (numerically)?

Answer 35

\(\underline{a}\underline{b}\) - original number
\(\underline{b}\underline{a}\) - number with reversed digits
sum of digits is 8, - \(a+b=8\)

Alternatively,
\(10a+b\) - original number
\(10b+a \) - number with reversed digits

When you reverse digits number decreases by 18,
\(\underline{a}\underline{b}-18=\underline{b}\underline{a}\)

or alternatively,
\(10a+b-18=10b+a \)

Answer 36

Therefore, you have your system of equations;

\(10a+b-18=10b+a \)
\(a+b=8 \)

Answer 37

questions about this so far?

Answer 38

wait. i am just looking over what you just posted.

Answer 39

:) take your time

Answer 40

where did you get 10b + a?

Answer 41

or how did you get 10b + a ?

Answer 42

82 can be recorded as 8•10+2, right?

Answer 43

yes

Answer 44

and 28 can also be recorded as 2•10+8. Correct?

Answer 45

oh okay

Answer 46

This is how I am getting
10a+b
10b+a

Answer 47

10a+b, is the original number \(\underline{a}\underline{b}\)
10b+a, is the reversed-digits number \(\underline{b}\underline{a}\)

Answer 48

i tried to simplify the first equation \(10a + b - 18 = 10b + a\) into \(-18 = 9b - 9a\).

Answer 49

\(\color{#000000 }{ \displaystyle 10a+b-18=10b+a }\)

\(\color{#000000 }{ \displaystyle -18=9b-9a }\)
And you know that sum of digits is 8 -- i.e. \(a+b=8\)

Answer 50

so i can use the system of equations:
\(-18 = 9b - 9a\)
\(a + b = 8\)

Answer 51

yes, you can solve this system of equations

Answer 52

\(\color{#000000 }{ \displaystyle -18=9b-9a }\)
\(\color{#000000 }{ \displaystyle 8=b+a }\)

\(\color{#000000 }{ \displaystyle -18=9b-9a }\)
\(\color{#000000 }{ \displaystyle 72=9b+9a }\)

and then add the equations (one way to do this)

Answer 53

a = 5
b = 3

number is 53?

Answer 54

\(\color{#000000 }{ \displaystyle 54=18b }\)
\(\color{#000000 }{ \displaystyle b=3}\)

a+b=8 ---> a+3=8 ----> a=5

Answer 55

Yes, you are correct

Answer 56

thank you!

Answer 57

So the original number (the one requested) is 53.

Answer 58

yw