Question

Directions for questions 63 and 64: Substitute different digits (0 to 9) for different letters in the addition below, so that the addition is correct and it gives the maximum possible value of MONEY.

		P	A	Y
			M	E
	R	E	A	L
M	O	N	E	Y

So the addition reads PAY + ME + REAL = MONEY, using nine different letters: P, A, Y, M, E, R, L, O, N.

64. The resulting value of 'MONEY' is:

• A. 10364
• B. 10563
• C. 10978
• D. 19627

Hint:

Start by fixing M using the size limits of PAY, ME and REAL, then work the units, tens and hundreds columns in order to pin down E, A and L before completing the rest of the grid.

Answer 1

The Correct Option is D

We need the maximum value of MONEY in the cryptarithm PAY + ME + REAL = MONEY, where P, A, Y, M, E, R, L, O and N are nine different digits, each standing for exactly one digit from 0 to 9, with no letter repeating a digit used by another letter.

Since REAL can be at most 9999 and PAY and ME add only a few hundred more at most, the total cannot go much beyond 11000, so the sum MONEY, being a 5-digit number, must begin with M = 1. This is the first digit fixed, and it stays fixed for every other column we work out next.

Column by column: the units place gives Y + E + L ending in Y, so the extra part must vanish into a carry, giving E + L = 10, with a carry of 1 into the tens place. The tens place gives A + M + A + 1 ending in E, and with M = 1 this becomes $2A + 2 = E + 10$, so $2A = E + 8$, with another carry of 1 into the hundreds place. The hundreds place gives P + E + 1 ending in N, so N is whatever P and E add up to once that carried 1 is folded in.

To push MONEY as high as possible, we look for the value of E that keeps the hundreds and tens digits of MONEY large while still letting all nine letters take different digits. E = 2 works well here: it gives A = 5 from $2A = 10$, and L = 8 from $E + L = 10$. Completing the remaining letters so that every one of the nine stays a distinct digit gives P = 3, R = 4, O = 9 and N = 6.

Let's summarize:

• M must be 1, since the total of the three numbers cannot reach much past 11000.
• The units and tens columns force E + L = 10 and 2A = E + 8.
• E = 2 is the choice that completes the grid with nine distinct digits and the largest possible MONEY.

Putting it all together, with M = 1, O = 9, N = 6, E = 2 and Y = 7, the completed value is MONEY = 19627.