Answer to Question #246734 in Algebra for korina

Question

A college student sent a postcard to her parents with the message SEND + MORE = MONEY if each letter represents a digit (0, 1, 2, 3, 4, 5, 6, 7, 8, or 9), with different letters representing different digits and the same letter representing the same digits each time it occurs, how much money is being requested?

Answer 1

$\frac{\begin{matrix} &S&E&N&D\\ + \\ &M&O&R&E\\ \end{matrix}}{\begin{matrix} M&O&N&E&Y \end{matrix}}$

1) SEND+MORE<9999+9999=19998<20000. So, we must have M=1.

2) Now we have two cases: S+M+1=10+O or S+M=10+O.

S=8+O or S=9+O

O can’t be equal to 1 (we have M=1) and S$\leq$ 9. Therefore, O=0 and S=8 or S=9.

If S=8, then E+O=10+N or E+O+1=10+N.

E=10+N or E=9+N

Since N$\geq$ 2, we have that S$\neq$ 8.

So, S=9.

3) E+O=N or E+O+1=N

E=N or E+1=N

Since E and N are different letters, we have E+1=N.

We can get it, if N+R=10+E or N+R+1=10+E.

So, R=9 or R=8. Since S=9, it follows that R=8.

4) We got R=8 from N+R+1=10+E. It means, that D+E=10+Y.

D+E$\leq$ 6+7=13. So, 2$\leq$ Y$\leq3$ .

If Y=3, then (E=6 and D=7) or (E=7 and D=6).

If E=6, then N=E+1=7=D. If E=7, then N=E+1=8=R.

We have that Y=2.

5) If Y=2, then D+E=12.

3$\leq$ D$\leq7$

D=3, E=9=S

D=4, E=8=R

D=5, E=7, N=E+1=8=R

D=6, E=6=D

D=7, E=5, N=E+1=6

M=1, O=0, S=9, R=8, Y=2, D=7, E=5, N=6

ANSWER: 9567+1085=10652, MONEY=10652.