8th grade word problem

Question

In the puzzle below, each letter stands for a different digit from 0 to 9. the same letter stands for the same digit throughout the problem. what digit does each letter represent?

 

Y + W = D

C + M = C

D - G = R

Y + Y = W

W/Y = G

Y - G = A 

H x A = H

H x G = L

 

 

 

Answer 1

The best approach is to note that there is an identity of Addition (C + 0 = C). Therefore, from equation 2:

M = 0;

Then there is another identity: (H x 1 = H). Therefore from equation 7:

A = 1;

By substitution, W  = Y + Y and W/Y = G, So (Y+Y)/Y = G > 2Y/Y = G >

G = 2;

From this, the substitution of values gives the equations:

Y + W = D

C + 0 = C

D - 2 = R

2Y = W

W/Y = 2

Y - 2 = 1

H = H

H x 2 = L

Note that some equations eliminate themselves (Like H = H and C = C)

1) Y + W = D

2) D - 2 = R

3) 2Y = W

4) W/Y = 2

5) Y - 2 = 1

6) H x 2 = L

Also, note that equations 3 and 4 above are the same:

1) Y + W = D

2) D - 2 = R

3) 2Y = W

4) Y - 2 = 1

5) H x 2 = L

From equation 4 above:

Y = 3;

From substitution in equation 3

W = 6;

Substituting into equation 1

D = 9;

Substituting into equation 2

R = 7;

Therefore, there are three variables left and three values remaining from the set 0 to 9:

H x 2 = L

C = C

Values: 4,5,8;

Note 4 x 2 = 8, so:

H = 4;

L = 8;

And finally,

C = 5.

 

In Summary:

M = 0;

A = 1;

G = 2;

Y = 3;

H = 4;

C = 5;

W = 6;

R = 7;

L = 8;

D = 9;

Answer 2

This puzzle is like a game!  Look at the equations one at a time to see if you can figure out the value of any of the letters.  Do them in the order in which you can find clues.  Here is the order which worked for me, but others may work, too.

 

1.  H x A = H          A has to equal 1                         A = 1

2.  C + M = C         M has to equal 0                         M = 0

  

3.  Y + Y = W         so, W = 2Y

4.  Y + W = D         so, Y + 2Y = D, and D = 3Y

 

5.  Y - G = A     and we already know A = 1, so

                  Y - G = 1.  Therefore G = Y - 1

6.  W/Y = G      and we already know W = 2Y, so

                  2Y/Y = G.  Therefore G = 2                    G = 2

     We know from #5 that G = Y - 1, so 2 = Y - 1      Y = 3

     Therefore, Y = 3, 2Y (which is W) = 6, and          W = 6

     3Y (which is D) = 9                                            D = 9

     

7.  D - G = R     so 9 - 2 = R = 7                              R = 7

 

8.  H x G = L     so H x 2 = L, and L =2H

 

We have found the values of all letters except C, H, and L.

We have letters for all digits except 4, 5, and 8.

For L and H we need one digit that is twice another digit.

  8 is twice as large as 4, so                                     H = 4

                                                                             L = 8

 

That leaves only C and 5.                                        C = 5