Digit Problems: Find the Number

Date: 01/29/98 at 19:45:52
From: Kara
Subject: Algebra 1 (Digit Problems)

My teacher explained digit problems in school. I thought I had gotten 
it, and so far I have found the right answers to the first three 
homework problems. I thought I was getting the hang of it until I 
stumbled upon one I couldn't figure out - and to my dismay the next 
four were the same. Here they are:

1. A two-digit number is 6 more than 4 times the sum of its digits.  
   The digits from left to right are consecutive even integers.  
   Find the number.

2. A two-digit number is five times the sum of its digits. The digits 
   from left to right name consecutive integers. Find the number.

3. A two digit number is 5 times the sum of its digits. When 9 is 
   added to the number, the result is the original number with its 
   digits reversed. Find the number. Hint:  The number with its digits 
   reversed is 10u +t.

4. The sum of the digits of a two digit number is 9. The number is 27 
   more than the original number with its digits reversed. Find the 
   number.

5. The units digit of a two-digit number is 2 less than the tens 
   digit. The number is two more than 6 times the sum of the digits.  
   Find the number.

I am pretty sure I am setting them up right, for example No. 5, the 
one I understood best:

       t-u = 2
  6(t+u)+2 = 10t+u

Numbers 1-4 are the hardest. I would appreciate it if you did one of 
those and showed me it step by step. Thanks Doc!

Date: 02/01/98 at 11:13:54
From: Doctor Wolf
Subject: Re: Algebra 1 (Digit Problems)

Okay, Kara - Let's get started!

You've done a nice job of setting up no. 5, and I'm assuming you
can solve it from there. I will do problem no. 1 completely, and set 
up no. 2 for you.

Problem 1:
A two-digit number is 6 more than 4 times the sum of its digits. The 
digits from left to right are consecutive even integers. Find the 
number.

The fact that the digits from left to right are consecutive even 
integers means that the number must look like 24, or 46, or possibly
68. Moreover, if we let t = the tens digit (as you did in no. 5), then

   t+2 = the units digit

Okay so far?

A two-digit number is 6 more than 4 times the sum of its digits can 
now be written as:

     10t + (t+2) = 6 + 4(t + t+2), or
     10t +  t+2  = 6 + 4(2t + 2), now use the distributive property
     10t +  t+2  = 6 + 8t + 8, now combine like terms    
     11t + 2     = 8t + 14, now subtract 8t from both sides of "="
      3t + 2     =      14, now subtract 2 from both sides of "="
      3t         =      12, or
               t = 4    So the tens digit, or t, is 4!

This means the units digit must be 6 and the next consecutive even 
integer, and so our number is 46. Let's check this out.

    46 is 6 more than 4 times (4+6), or 46 is 6 more than 40   
                                 
Yes!

I'll get you started on no. 2.

A two-digit number is five times the sum of its digits. The digits 
from left to right name consecutive integers. Find the number.

If the digits from left to right name consecutive integers, then the
number must look like 12, or 34, or 89, etc. This means if you let
t = the tens digit, then t+1 will represent the units digit.
                
   Therefore  10t + (t+1) = 5(t + t+1) 

Take it from there Kara!

I hope this helped, and don't hesitate to stop in again.

-Doctor Wolf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/