Problem
Simplify products by decomposing factors
Three sets of worked multiplications are shown. Each set follows its own hidden shortcut. The shortcut must be read off the examples. Use each one on 78 × 99, 62 × 68 and 89 × 29.
Reasoning · Grade 4-1 Multiplication
Your answer
Watch out One shortcut seems to cover all, but each set differs — read them apart.
How to solve
Strategy Look for a Pattern — Each set is a small code to crack, so I look for a pattern that links the digits of the two factors to the digits of the answer. Because the three sets are independent, I treat them as three separate subproblems and finish one before starting the next. And I never trust a guessed rule straight away: I test it on all the examples in its set first, and only then apply it to the new product.
1STEP 1
Set (1): read the shortcut off the examples
Set (1) is times 100, then subtract once.
n × 99 = n × (100 - 1) = n × 100 - n
Ninety-nine copies of something is a hundred copies with one copy removed — the same take-one-away trick used to work out 9 × 7 as 70 - 7.
3.OA.B.5Look For A PatternNinety-nine copies of a number is a hundred copies with one copy taken away.
Why?
A hundred groups can be split into ninety-nine groups and one more, so multiplying by 99 and by 100 differ by exactly one copy.
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
Why?
Taking that one copy away undoes adding it, so the easy hundred-times product can be corrected in a single subtraction.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
2STEP 2
Set (1): apply it to 78 × 99
So 78 × 99 = 7722.
78 × 99 = 7800 - 78 = 7722
Multiplying by 100 only shifts the digits two places, so the whole job comes down to one easy subtraction instead of a two-digit-by-two-digit multiplication.
4.NBT.B.5Identify Subproblems3STEP 3
Set (2): split each answer into two halves and hunt for the pattern
In set (2) the front half is the tens digit times the next number.
43 × 47: 4 × 5 = 20, 3 × 7 = 21 → 2021
Chopping a four-digit answer at the hundreds boundary is just reading it as so many hundreds plus so many ones, which is the place-value reading of any number.
3.OA.B.5Look For A Pattern4STEP 4
Set (2): check the rule on all three examples before using it
The rule fits all three examples.
36 × 34: 3 × 4 = 12, 6 × 4 = 24 → 1224
A rule that matches one example might be luck; a rule that matches three different ones is worth using.
4.NBT.B.5Guess And Check5STEP 5
Set (2): apply it to 62 × 68
So 62 × 68 = 4216.
62 × 68: 6 × 7 = 42, 2 × 8 = 16 → 4216
Checking that the numbers really fit the set's description first is what makes it safe to use the shortcut at all.
4.NBT.B.5Identify Subproblems6STEP 6
Why the set (2) rule works
It works because the ones digits add to 10.
60 × 60 + 60×(8+2) + 2 × 8 = 3600 + 600 + 16 = 4216
Splitting each factor into tens and ones and multiplying the pieces is the area-rectangle picture of multiplication; the ones digits adding to 10 is what makes the middle two pieces join into one tidy round number.
3.OA.B.5Identify Subproblems7STEP 7
Set (3): find the pattern in the halves again
In set (3) the ones match, so add that digit to the front.
45 × 65: 4 × 6 + 5 = 29, 5 × 5 = 25 → 2925
The front half needed a small extra addition this time, which is why testing the guess on all three examples matters — one example alone would not have revealed the added ones digit.
3.OA.B.5Look For A Pattern8STEP 8
Set (3): apply it to 89 × 29
So 89 × 29 = 2581.
89 × 29: 8 × 2 + 9 = 25, 9 × 9 = 81 → 2581
Two one-digit multiplications and one small addition replace a full two-digit multiplication, so the whole product can be done in your head.
4.NBT.B.5Identify SubproblemsAnswer
7722, 4216, 2581
7800 − 78 = 7722
Each answer sits in a sensible range. 78 × 99 must be a little under 78 × 100 = 7800, and 7722 is 78 below it. 62 × 68 must be near 65 × 65, roughly 4225, and 4216 is just under. 89 × 29 must be near 89 × 30 = 2670 but less by one 89, and 2670 - 89 = 2581 — an exact confirmation. Ordinary long multiplication also gives 7722, 4216 and 2581, so all three shortcuts landed correctly.
Takeaway
Test a shortcut on every example before you trust it — one match can be luck, three matches is a rule!
- Set (1): read the shortcut off the examples
- Set (1): apply it to 78 × 99
- Set (2): split each answer into two halves and hunt for the pattern
- Set (2): check the rule on all three examples before using it
- Set (2): apply it to 62 × 68
- Why the set (2) rule works
- Set (3): find the pattern in the halves again
- Set (3): apply it to 89 × 29