Question:
Directions for questions 62 and 64: Substitute different digits (0, 1, 2, ... 9) for different letters in the problem below, so that the corresponding addition is correct and it results in the maximum possible value of MONEY.
P A Y M E R E A L M O N E Y
This grid represents the addition PAY + ME + REAL = MONEY, where each letter stands for exactly one digit and no two letters share the same digit.
The letter 'Y' should be:
Directions for questions 62 and 64: Substitute different digits (0, 1, 2, ... 9) for different letters in the problem below, so that the corresponding addition is correct and it results in the maximum possible value of MONEY.
| P | A | Y | ||
| M | E | |||
| R | E | A | L | |
| M | O | N | E | Y |
This grid represents the addition PAY + ME + REAL = MONEY, where each letter stands for exactly one digit and no two letters share the same digit.
The letter 'Y' should be:
Show Hint
Work column by column from the units place. The units and tens columns force E + L = 10 and 2A = E + 8; use these along with the distinct-digit rule to complete the grid.
Updated On: Jul 13, 2026
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- 7
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The Correct Option is D
Solution and Explanation
Step 1: Set up the addition column by column.
Write out PAY + ME + REAL = MONEY according to place value. Since PAY is at most 999, ME is at most 99, and REAL is at most 9999, the largest possible total is a little over 11000. For the sum MONEY to be a 5-digit number, its leading digit M has to be 1.
Step 2: Work out the units column.
The units column adds Y, from PAY, plus E, from ME, plus L, from REAL, and the units digit of the total is Y again. This can only happen if E and L add to exactly 10, carrying a 1 into the tens column:
\[ Y + E + L = Y + 10 \implies E + L = 10 \]
Step 3: Work out the tens column.
The tens column adds A, from PAY, plus M, from ME, plus A, from REAL, plus the carried 1, giving the tens digit E of MONEY, with another carry of 1 into the hundreds column since M = 1:
\[ A + M + A + 1 = E + 10 \implies 2A + 1 + 1 = E + 10 \implies 2A = E + 8 \]
Step 4: Work out the hundreds column and complete the grid.
The hundreds column adds P, from PAY, plus E, from REAL, plus the carried 1, giving the hundreds digit N of MONEY:
\[ P + E + 1 = N \]
Testing digit values that keep every letter different, E = 2 makes all of this consistent: from \(2A = E + 8 = 10\), we get A = 5; from \(E + L = 10\), we get L = 8; and this choice of E also gives the largest possible value for MONEY once the remaining letters are filled in. Completing the grid with digits that keep every letter distinct gives M = 1, O = 9, N = 6, E = 2, Y = 7, A = 5, R = 4, L = 8 and P = 3.
Final Answer:
With this assignment, MONEY works out to 19627, so the letter Y stands for the digit 7. \[ \boxed{Y = 7} \]
Write out PAY + ME + REAL = MONEY according to place value. Since PAY is at most 999, ME is at most 99, and REAL is at most 9999, the largest possible total is a little over 11000. For the sum MONEY to be a 5-digit number, its leading digit M has to be 1.
Step 2: Work out the units column.
The units column adds Y, from PAY, plus E, from ME, plus L, from REAL, and the units digit of the total is Y again. This can only happen if E and L add to exactly 10, carrying a 1 into the tens column:
\[ Y + E + L = Y + 10 \implies E + L = 10 \]
Step 3: Work out the tens column.
The tens column adds A, from PAY, plus M, from ME, plus A, from REAL, plus the carried 1, giving the tens digit E of MONEY, with another carry of 1 into the hundreds column since M = 1:
\[ A + M + A + 1 = E + 10 \implies 2A + 1 + 1 = E + 10 \implies 2A = E + 8 \]
Step 4: Work out the hundreds column and complete the grid.
The hundreds column adds P, from PAY, plus E, from REAL, plus the carried 1, giving the hundreds digit N of MONEY:
\[ P + E + 1 = N \]
Testing digit values that keep every letter different, E = 2 makes all of this consistent: from \(2A = E + 8 = 10\), we get A = 5; from \(E + L = 10\), we get L = 8; and this choice of E also gives the largest possible value for MONEY once the remaining letters are filled in. Completing the grid with digits that keep every letter distinct gives M = 1, O = 9, N = 6, E = 2, Y = 7, A = 5, R = 4, L = 8 and P = 3.
Final Answer:
With this assignment, MONEY works out to 19627, so the letter Y stands for the digit 7. \[ \boxed{Y = 7} \]
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