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(Please tell me exactly how you found the answer and what you did because I'm a Year 7 and this is a Year 9 question.)

Mr. Rich decides to pay a bill using internet banking. The bill is a two-digit whole number $ab$. He accidentally inserts an extra digit after the b. Later he finds out he overpaid his bill by $647. What was his original bill?

I started working it out like this

$(100a+10b+c)-(10a+b)=9(10a+b)+c$

But then I got stuck. What I don't understand is how you're meant to proceed. Are you meant to do anything with the $c$? Plus, are the digits different by placing two different pro numerals or are they the same? Can anyone help me?

gymbvghjkgkjkhgfkl
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bio
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  • You should include your own work in your question. – Zelos Malum Mar 06 '17 at 11:08
  • Between this and your other questions, it really looks like you are just posting your homework here for us to do for you. Try the problem yourself first, and show us some of your efforts. Indicate where you are getting stuck. – lulu Mar 06 '17 at 11:09
  • I'm just trying Year 9 and 10 questions for extension. This isn't homework – bio Mar 06 '17 at 11:13
  • We cannot know if you are or not and appearence is all we can go by so if you make it look like it, we will assume you are. It is often better to state purpose, show a genuine attempt and the more you can say what you know and hwat you do NOT understand, the more legitimate it'll appear to be and we will adapt our response accordingly. – Zelos Malum Mar 06 '17 at 11:17
  • What is 7 year? – marshal craft Mar 06 '17 at 11:18
  • @ marshal craft It says year 7 – bio Mar 06 '17 at 11:19
  • That it an invalid answer to the question I asked. You knew what I was referring and can bother to correct me, however wont explain what I had asked? – marshal craft Mar 06 '17 at 11:29
  • @Zelos Malum It's hard to follow your periphrases. – Jean Marie Mar 06 '17 at 11:30
  • Guys don't bother I've worked out that Mr. Rich paid $718 and he should have paid 71 – bio Mar 06 '17 at 11:38
  • You found the overpayment is $9(10a+b)+c$. When you divide it by $9$ you'll get a remainder of $c$. But you know the overpayment is $647$. Can you obtain $c$ from this? (EDIT: I can see I'm late, too long betwen starting a comment and actually posting it.) – CiaPan Mar 06 '17 at 11:55

2 Answers2

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You are already on the right way:

Actual bill: $10a+b$.

Actually Paid: $100a + 10b+c$

Overpaid: $647=(100a+10b+c)-(10a+b)=90a+9b+c=647$

With the last equation and with the knowledge that $a,b,c\in\{0,1,\ldots,9\}$ (i.e. they can be only integers from 0 to 9) you can solve this.

Hint: Try e.g. $a=2$ and see that in this case either $b$ or $c$ need to be greater than 9.

Maximilian Matthé
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$(100a+10b+c)-(10a+b)=647$.

$90a+9b+c=647$.

There are three unknowns and one equation. We also require integer solutions from the field $\Bbb Z_{10}$.

marshal craft
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