Reasoning · Grade 6-2 Mixed Fraction and Decimal Calculations

Problem

Simplify messy decimal computations

Five decimal expressions: 8.88×1.25; 9.1+9.2+…+10.9; 1994+199.4+19.94+1.994; 1990×198.9-1989×198.8; 5.5×6.6+6.6×7.7+7.7×8.8+8.8×9.9. Working left to right is slow and error-prone. Find the value of each of the five.
Your answer
How to solve
Strategy Organize Information in More Ways — Every one of these five is a re-sorting puzzle dressed up as arithmetic. Nothing is missing and nothing is hidden — the numbers just need to be split, grouped or lined up differently before any calculating starts, which is exactly what Organizing Information in More Ways means. Looking for a Pattern is what tells me which regrouping to choose: repeated digits in (3), a repeated factor of 1.1 in (5), a near-repeat of the same product in (4). And each expression is its own small job, so I treat the five as five separate subproblems and finish one before starting the next.
1STEP 1

Decide what to look for before calculating

Look for the hidden structure before calculating.

split into places or whole part + decimal part or pull out a common factor
2STEP 2

(1) Split 8.88 into place-value pieces, because 8 × 1.25 = 10

Split by place value, (1) is 11.1.

8.88 × 1.25 = (8 + 0.8 + 0.08) × 1.25 = 10 + 1 + 0.1 = 11.1
3STEP 3

(2) Count the terms, then add whole parts and decimal parts in two separate piles

Adding the two piles, (2) is 190.

(9 × 9) + (10 × 10) + (0.1 + 0.2 + … + 0.9) × 2 = 81 + 100 + 9 = 190
4STEP 4

(3) Notice the repeated digits and pull 1994 out front

In (3) the repeat comes out for a single multiplication.

1994 + 199.4 + 19.94 + 1.994 = 1994 × (1 + 0.1 + 0.01 + 0.001) = 1994 × 1.111
5STEP 5

(3) Finish with one whole-number multiplication and one decimal-point move

So (3) is 2215.334.

1994 × 1111 = 1994000 + 199400 + 19940 + 1994 = 2215334, so 1994 × 1.111 = 2215.334
6STEP 6

(4) Make the two products share a factor by rewriting 198.9

Making the products share a factor, (4) is 397.8.

1990 × 198.9 - 1989 × 198.8 = (1990-1989) × 198.8 + 1990 × 0.1 = 198.8 + 199 = 397.8
7STEP 7

(5) Spot the common factor 1.1 in every single number

Pulling out the common factor, (5) is 242.

1.21 × (5 × 6 + 6 × 7 + 7 × 8 + 8 × 9) = 1.21 × 200 = 242
Answer
11.1, 190, 2215.334, 397.8, 242
1.21 × 200 = 242
Every answer sits where an estimate says it should. (1) 8.88 is just under 9, and 9 × 1.25 = 11.25, so 11.1 is right below it. (2) The nineteen terms are spread evenly around 10.0, so the sum should be about 19 × 10 = 190 — and because the list is exactly symmetric about 10.0, that estimate is the exact answer, which double-checks 190. (3) The sum is a bit more than 1994 + 199 + 20 + 2 = 2215, and 2215.334 is a bit more than 2215. (4) Both products are near 396000, so their difference should be small; 397.8 is small, and it is also close to 1990 × 0.1 = 199 plus one copy of 198.8, as the working showed. (5) Four products each near 36, 51, 68 and 87 add to roughly 242, matching exactly.
Takeaway

Read a messy decimal expression before you compute it — the numbers are usually arranged so that one clever regrouping does almost all the work.

  • Decide what to look for before calculating
  • (1) Split 8.88 into place-value pieces, because 8 × 1.25 = 10
  • (2) Count the terms, then add whole parts and decimal parts in two separate piles
  • (3) Notice the repeated digits and pull 1994 out front
  • (3) Finish with one whole-number multiplication and one decimal-point move
  • (4) Make the two products share a factor by rewriting 198.9
  • (5) Spot the common factor 1.1 in every single number