Math Detours · Grade 7 · Around the corner · 3

SEND + MORE = MONEY

Each letter is a digit. Carrying, column by column, forces the answer, and there is only one.

Name ____________________Date ____________

Tonight's idea

In a cryptarithm each letter stands for a digit. Same letter, same digit. Different letters, different digits. No number starts with 0. You do not guess. You reason, column by column, as in ordinary addition.

S E N D + M O R E M O N E Y ? ? ? carries thousands hundreds tens units a carry
Each circle holds a carry: 0 or 1. The M of MONEY is the carry out of the thousands column.

The tool is the carry. Two digits add to at most 18, or 19 with a carry coming in. So a carry is always 0 or 1.

Far left. Two four-digit numbers make a five-digit number. The extra digit is a carry and is not 0. So M = 1.

Thousands. S + 1, plus perhaps a carry, must reach 10. It makes 10 or 11, so O is 0 or 1. M has 1, so O = 0.

Hundreds. E + 0, plus a carry, gives N. No carry would make N = E, which is not allowed. So the carry is 1 and N = E + 1. N is not 0 (O is), so this column passes nothing on. S got no help: S = 9.

Watch one: finish it

Tens. N + R, plus a carry, ends in E and passes on the 1 the hundreds needed. As N is E + 1, R plus the carry is 9. S has 9, so R = 8 and the units carry 1.

Units. D + E = 10 + Y. The digits left are 2 to 7, so D + E is at least 12: one of them is 7, the other 5 or 6. E = 7 makes N = 8, taken. So D = 7. E = 6 makes N = 7, taken. So E = 5, N = 6, Y = 2.

\[9567 + 1085 = 10652\]

Your turn

  1. Each of these has one answer. (a) A + A + A = BA. (b) I + BB = ILL.
  2. TO + GO = OUT. Start with the letter that must be a carry.
  3. AB + BA = CAC. Find C first. Then ask what A + B must be in the units column.
  4. TWO + TWO = FOUR has seven answers. (a) F is the same digit in all of them. Which digit, and why? (b) Why can R never be odd? (c) Find the answer with T = 7 and W = 6.

Stretch: a harder one

BASE + BALL = GAMES. There is one answer. Start with G. Then write down what the units column says and what the tens column says, and put the two together: they force L = 5. After that, try the few cases that are left.

Answers

  1. (a) Three times A ends in A. Only 0 and 5 do that, and A = 0 gives no B. So \(5 + 5 + 5 = 15\). (b) A two-digit number plus one digit is at most \(99 + 9\), so I = 1. To reach 100 at all, BB must be 99: \(1 + 99 = 100\).
  2. O is a carry, so O = 1. Units: \(1 + 1 = 2\), so T = 2. Tens: 2 + G = 10 + U. G = 9 would make U = 1, taken, so G = 8 and U = 0. \(21 + 81 = 102\).
  3. C is a carry, so C = 1. Units: B + A ends in 1. A and B both start a number, so neither is 0 and B + A = 11, carrying 1. Tens: \(11 + 1 = 12\), so A = 2 and then B = 9. \(29 + 92 = 121\).
  4. (a) F = 1: it is the carry from two three-digit numbers. (b) R is the last digit of O + O, and a number plus itself is even. (c) \(7 + 7 = 14\), so O is 4, or 5 with a carry. W = 6 gives \(6 + 6 = 12\), which does carry, so O = 5. Units: \(5 + 5 = 10\), so R = 0, carry 1. Tens: \(6 + 6 + 1 = 13\), so U = 3. \(765 + 765 = 1530\).

Stretch. G = 1. Units: E + L is S, or S + 10. Tens: S + L plus the carry ends in E. Put together, L + L equals 9 times the units carry plus 10 times the tens carry. L + L is even and not 0, so the units carry is 0, the tens carry is 1 and L = 5. Then S = E + 5. Hundreds: A + A + 1 ends in M. Thousands: B + B plus a carry is 10 + A. Trying the cases leaves only B = 7, A = 4, M = 9, then E = 3, S = 8.

\[7483 + 7455 = 14938\]

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