Question

Question: Each of the letters below represents a different digit. Solve for each letter.

(Please give a full explanation and not just the answer. I would like to know how to do more of these problems in the future.)
FORTY
+ TEN
+ TEN
-----------
SIXTY

Answer 1

To solve this problem, we need to assign unique digits to each letter in the equation FORTY + TEN + TEN = SIXTY.

Let's start by looking at the units place. From the equation, we can see that the units digit of SIXTY is Y. Since Y is the units digit, it must be equal to 0, because the sum of the units digits in FORTY, TEN, and TEN should be equal to the units digit in SIXTY.

Now, let's move on to the tens place. The tens digit in SIXTY is X. We know that Y is 0, so the sum of the tens digits in FORTY, TEN, and TEN should be equal to X. The only way to achieve this is if X is 1. Therefore, X = 1.

Now, let's look at the hundreds place. The hundreds digit in SIXTY is O. Since X is 1, we need to carry over 1 to the hundreds place. Therefore, O + T + T + 1 = X. Since X is 1, O + T + T + 1 = 1. The only way to achieve this is if O = 9 and T = 0.

Finally, let's look at the thousands place. The thousands digit in FORTY is F. Since T is 0, we need to carry over 1 to the thousands place. Therefore, F + 1 = T. Since T is 0, F + 1 = 0. The only way to achieve this is if F = 9.

So, the solution to the equation FORTY + TEN + TEN = SIXTY is:
9 0 1 9 0 + 1 0 + 1 0 = 6 0.

Answer 2

To solve the given equation, let's follow these steps:

1. Assign variables to each letter in the equation:

F = a

O = b

R = c

T = d

Y = e

E = f

N = g

I = h

X = i

U = j

2. Rewrite the equation using the assigned variables:

abcdy + defg + defg = fghijy

3. Simplify the equation:

abcdy + 2(defg) = fghijy

4. Start by examining the rightmost column (the column with the highest place value). Since 'y' appears in three places (tens, ones, and thousands), it must be equal to zero. This is because adding any two numbers and getting a sum ending in zero requires the summands to be multiples of ten. So, 'y' = 0, and the equation becomes:

abcd0 + 2(defg) = fghij0

5. Moving to the next column, let's determine 'g'. Notice that 'g' appears in three places (ones, hundreds, and thousands), but it only appears once on the right side of the equation. Therefore, 'g' must be equal to 5, as 1 + 4 = 5.

The equation becomes:

abcd0 + 2(def5) = f5hij0

6. Now, let's find 'd'. Observe that 'd' is in the thousands place. The only possible values for 'd' are 6, 7, 8, or 9. However, if 'd' is 6, then 'e' would also be 6, which is not possible since each letter represents a different digit. Similarly, if 'd' is 7, 'e' would be 7, which is also not possible. Continuing this logic, we find that 'd' must be 8 and 'e' must be 9.

The equation becomes:

abc809 + 2(f95) = f5hij0

7. Next, let's determine 'c'. Looking at the hundreds place, 'c' must be equal to 2. This is because the sum of two digits in the hundreds place can only be a multiple of 10 if the two digits sum to 10 or 20.

The equation becomes:

ab2809 + 2(f95) = f5hij0

8. Now, let's solve for 'b'. The sum of 'b' and 'f' (which is 1) must be equal to 10 since it is the only way to get a sum ending in 9 (from '9' in the ones place of the sum on the right side of the equation). Therefore, 'b' = 8.

The equation becomes:

a82809 + 2(195) = f5hij0

9. Finally, let's solve for 'a'. Since we know the sum of the thousands place digits on the left side of the equation must be 1 (from '1' in the thousands place of the sum on the right side), 'a' must be equal to 1.

The equation becomes:

182809 + 2(195) = f5hij0

10. To find the values for 'f', 'h', 'i', and 'j', we can calculate the sum on the right side of the equation:

182809 + 2(195) = 183199

Therefore, the solution to the equation is:

F = 1

O = 8

R = 2

T = 8

Y = 0

E = 9

N = 3

I = 1

X = not enough information

U = not enough information

So, FORTY + TEN + TEN = SIXTY is true when each letter is replaced with the corresponding digit:

182809 + 195 + 195 = 183199

Note: The values for 'X' and 'U' cannot be determined based on the given equation. Additional information or constraints would be required to find their values.

real