Reasoning · Grade 4-2 Adding and Subtracting Decimals

Problem

Complete a decimal subtraction from digit cards

There is one card for each digit from 1 to 8. The 6, the 1 and the 7 are already placed in the subtraction. The leftover cards 2, 3, 4, 5 and 8 fill the five boxes, one each. Place them so the result is as large as possible.
Your answer
How to solve
Strategy Make a Systematic List — Five cards can be dropped into five boxes in 120 different ways, which is far too many to try one at a time. So instead of guessing whole arrangements, I rewrite the horizontal sentence as a lined-up column subtraction. Then each column becomes its own tiny subproblem: the hundredths column alone pins down two of the cards, the tenths column pins down two more, and the ones column just falls out. That turns 120 guesses into three short lists, and the only real choice left is which of two survivors gives the bigger answer.
1STEP 1

Line the sentence up in columns

Rewrite it as a column subtraction.

6.A0 - 1.7B = C.DE
2STEP 2

The hundredths column forces a borrow

The last column has to borrow.

10 - B = E, B + E = 10
3STEP 3

List every pair of leftover cards that adds to 10

The only leftover pair adding to 10 is 2 and 8.

2+8=10, 3+7 (7 used), 4+6 (6 used), 5+5 (only one 5)
4STEP 4

Choose B = 2 so the answer is as big as possible

For a bigger answer the subtrahend ends in 2.

1.72 ＜ 1.78 → 6.A - 1.72 ＞ 6.A - 1.78
5STEP 5

The tenths column pins down A and D

The next column fixes 3 and 5.

(A-1+10)-7=D → D=A+2
6STEP 6

The ones column hands over the last card

So the biggest is 6.3 − 1.72 = 4.58.

6-1-1=4, 6.3-1.72=4.58
Answer
4.58
6.3 − 1.72 = 4.58
Do the subtraction straight out to be sure: 6.30 - 1.72 = 4.58. Check the cards: 6, 3, 1, 7, 2, 4, 5, 8 is each of the digits 1 through 8 exactly once, none repeated and none missing. The size is sensible too - the minuend can never exceed 6.5 and the subtrahend can never be smaller than 1.72, so no arrangement could push the result past about 4.8, and 4.58 sits just under that ceiling. The only other arrangement that works at all is 6.3 - 1.78 = 4.52, which is smaller, so 4.58 really is the maximum.
Takeaway

Stack the decimals in columns and work from the right - each column tells you which cards could possibly fit, so you never have to try all 120 arrangements.

  • Line the sentence up in columns
  • The hundredths column forces a borrow
  • List every pair of leftover cards that adds to 10
  • Choose B = 2 so the answer is as big as possible
  • The tenths column pins down A and D
  • The ones column hands over the last card