Each letter represents a different digit. Solve this equation:

       M A D E
     + M E A D
    -----------
       E D A M

Would you believe that we can use Algebra to figure out all the numbers? We can also use trial and error.

Here is a summary of all the steps that you will use to solve this equation. Down below, I will take you through each step:

You can use Algebra to figure out all of the numbers. Here are the steps that I recommend that you follow:

• Can "D" equal zero? Answer that question by looking at the one's column.
  This statement is false: E+0=M because E<>M
• Will there be a carry from the one's column to the ten's column?
  Look at the ten's column. D+A=A
  If there is no carry, then D is equal to zero, but D is not equal to zero.
• Will there be a carry from the ten's column to the hundred's column?
  Look at the ten's column, with the carry from the one's column:
  1 + D + A = A
  There must be a carry, because 1+D <> 0
  This makes the equation for the ten's column
  1 + D + A = 10 + A
• Solve this equation using Algebra:
  1 + D + A = 10 + A
  This will tell you what D is equal to.
• Substitute that value (D=?) into your original equation.
• Look at the hundred's column. Will there be a carry to the thousand's column?
  No, there will not be, for a reason that I will let you figure out.
• This means that: M+M=E or E = 2M
• Earlier we said that there will be a carry from the one's to the ten's column:
  Look at the one's column: E+9 = 10+M
• Use substitution. Substitute E=2M into E+9 = 10+M
  Solve the resulting equation.
  This will tell you what E and M must both be.
• Plug those values into your original equation.
• Solve for A using Algebra. I'm not going to explain that one. You can finish from here.

Try to do all of the above without looking down below. You will learn more by doing it yourself.

Okay, here's the process step by step:

Let's start by looking at the value for D. Can D be zero?

Before you scroll down to see my answer, figure it out for yourself.

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The one's column reads:   E + D = M

Can D be zero?  E + 0 = M

No, D cannot be zero, because then E = M, but the problem states that E does not equal M (different digits, remember).

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When we add the one's column, there will either be a carry (of one), or there won't be a carry.

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Now, let's look at the ten's column:  D + A = A

D cannot be zero:   0 + A = A

This tells us that there will be a carry from the one's column to the ten's column.

If you think about it, there will also be a carry from the ten's column to the hundred's column.

Before you scroll down, use your power of deduction and figure out what D must be equal to.

1 + D + A = 10 + A

We can use Algebra to figure out what D must be equal to, or we can use trial and error.

Figure out what D is equal to, then scroll down.

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Here's the ten's column, showing both carries:

1 + D + A = 10 + A

.Subtract A from both sides of the equation:

1 + D + A - A = 10 + A - A

1 + D = 10

Subtract 1 from both sides of the equation:

1 + D - 1 = 10 - 1

D = 9

D is equal to nine. With everything we have done so far, we have proved that D is equal to 9. Let's look at the problem again with those numbers filled in:

       M A 9 E
     + M E A 9
    -----------
       E 9 A M

Look at the hundred's column?  1 + A + E = 9

Can there be a carry to the thousand's column?  1 + A + E = 19

The largest possible values for A and E are 8 and 7.  1+8+7=16.

No, there is no carry to the thousand's column.

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Look at the thousand's column:  M + M = E

What are the only possible values for M?  1, 2, 3 or 4

What are the only possible values for E?  2, 4, 6 or 8

E must be double the value of M.  E = 2M

if M=1 then E=2
if M=2 then E=4
if M=3 then E=6
if M=4 then E=8

Now, I want you to look at the one's column:  E+9=10+M
(remember, we decided earlier that there would be a carry)

I want you to figure out, for yourself, what M and E must be equal to. You can figure it out with Algebra, or you can figure it out by trial and error.

You figure it out first, then scroll down and I'll review what you might have done.

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We have figured out two equations:

E = 2M
E + 9 = 10 + M

Substitute 2M for the E in the second equation and what do you get?

2M + 9 = 10 + M

Subtract M from both sides of the equation:

2M + 9 - M = 10 + M - M

M + 9 = 10

Subtract 9 from both sides of the equation:

M + 9 - 9 = 10 - 9

M = 1

The first equation was

E = 2M

Substitute what you know:

E = 2 * 1

E = 2

Plug those numbers into your equation.

       M A 9 E
     + M E A 9
    -----------
       E 9 A M

I bet you can finish the work, now.

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Good luck.

This material was written by John (webmaster (at) mazes.com) Knoderer (©'99). If your non-profit educational site would like to archive this material for free student use, permission is hereby granted. Please notify the author. Thank You