Watch outRegrouping feels free, but wrapping a subtraction flips the signs inside.
How to solve
StrategyOrganize Information in More Ways — Nothing here needs a new idea — every expression is already solvable by plain computation. What makes each one easy is re-sorting the pieces: pair the numbers that add to 100, subtract the matching terms in step, slide factors together until they make 1000, move a division next to the factor it divides exactly, or pull out the number that appears in two products. So the work is to look at each expression, spot the hidden pairing or common factor (that is the pattern-spotting part), regroup, and only then compute the few small numbers that are left. I finish by evaluating all six the plain way as a check, because a regrouping that is off by one term still produces a clean-looking wrong answer.
1STEP 1
The rule that makes all this regrouping legal
Start from the rule that permits regrouping.
a-b-c=a-(b+c) a ÷ b ÷ c=a÷(b × c)
Inserting parentheses is a Grade 5 skill, and it is the whole trick here: parentheses do not change the answer, they only change which small computation you do first.
5.OA.A.1Organize Information In More Ways
In a long sum the terms may be reordered and regrouped at will, which is what makes every shortcut on this page legal.
Why?
Swapping two terms in a sum cannot change what they come to, so the list may be rearranged into whatever order is convenient.
🧱Commutativity of additionPutting a with b or b with a piles up the same total — order does not change a sum.
Why?
Which two terms you push together first makes no difference either, so brackets can be drawn wherever they help.
🧱Associativity of additionAdding three piles, it does not matter which two you push together first.
This is just the distributive law read backwards — five columns each worth 15 means 15 × 11111 — and it turns a five-row column addition with carries into one easy multiplication.
Rearranging a sum minus a sum into a sum of differences is allowed because addition and subtraction can be reordered — and it replaces two three-digit column additions with five one-digit facts.
4.NBT.B.4Organize Information In More Ways
5STEP 5
(4) Slide the factors together to make 100 and 1000
Dividing early keeps the numbers small: doing it the printed way would mean multiplying 54 × 105=5670 and then dividing a four-digit number by 27 and again by 35, all to reach the number 6.
5.NBT.B.6Organize Information In More Ways
7STEP 7
(6) Pull out the factor the first two products share
Spotting a repeated factor is the single most useful regrouping move: here it replaces two four-digit-by-four-digit multiplications with the subtraction 1991-1989.
Checking by brute force is worth the minute it costs: the difference of the two big products has to equal 1990 × 2=3980, and seeing that number appear is what confirms the distributive shortcut was applied correctly.
5.NBT.B.5Identify Subproblems
Answer
166665, 500, 50, 3610, 706, 3538
15 × 11111 = 166665
Every answer is a whole number, as it must be, since each expression only adds, subtracts, multiplies, and divides exactly. The magnitudes make sense too: in (1) five numbers each a bit over 20000 should total a bit over 100000, and 166665 fits; in (2) ten numbers averaging about 50 are taken from 1000, so about 500 should be left; in (3) the first bracket beats the second by 10 in each of five places, so 50 is forced; in (4) the 3000 term dominates and 700-90 adds 610 to it; in (5) the second chain is tiny next to 700, so an answer just above 700 is right; and in (6) the two big products differ by only about 2000 because their factors differ by 2, so a four-digit answer, not a seven-digit one, is expected. Each regrouped result was also confirmed by evaluating the original expression term by term.
Takeaway
Before you calculate, look for the pieces that fit together — pairs that make 100, factors that make 1000, or a number both products share — and the hard-looking line becomes one you can do in your head.
The rule that makes all this regrouping legal
(1) Every digit shows up once in every column
(2) Pair the subtractions into hundreds
(3) Subtract term by term instead of sum by sum
(4) Slide the factors together to make 100 and 1000
(5) Do each division first, where it comes out exactly
(6) Pull out the factor the first two products share