Depth · one concept from first principles · step 2 of 4
A denominator that cancels to 1 leaves a whole number
Find the natural number for that makes the result the smallest possible whole number.
Number system
Your answer
How to solve
Strategy Identify Subproblems — Tidy the expression down to a single fraction first. Once everything that can cancel has cancelled, what is left is one denominator, and the question becomes the much easier one of what makes that denominator disappear.
1STEP 1
Turn the mixed number into a fraction
A mixed number cannot be cancelled while it is split into a whole and a fraction, so write it as one improper fraction first.
2 3/4 = 11/4
One fraction is easier to cancel than two pieces.
6.NS.A.1Identify Subproblems2STEP 2
Cancel the 11 and see what is left
Dividing by 11 is multiplying by 1/11, and the 11 on top cancels it. All that is left is a single quarter of the blank.
11/4 ÷ 11 × ■ = ■/4
The whole expression is just the blank over 4.
6.NS.A.1Identify SubproblemsThe whole expression is just the blank divided by .
Why?
is eleven quarters, and sharing eleven quarters equally among eleven leaves one quarter each, because dividing undoes the multiplying that made eleven of them in the first place.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
Multiplying that one quarter by the blank takes that many copies of it, so the result is the blank quarters.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
3STEP 3
Decide what makes a quarter whole
A fraction with denominator 4 is a whole number exactly when the numerator is a multiple of 4, so the blank has to be 4, 8, 12, and so on.
■ = 4, 8, 12, …
Multiples of 4, and nothing else.
6.NS.B.4Look For A Pattern4STEP 4
Take the smallest one
The results for those blanks are 1, 2, 3, and so on, so the smallest whole-number result comes from the smallest multiple of 4.
■ = 4 → 4/4 = 1
The blank is 4 and the result is 1.
6.NS.A.1Work BackwardsAnswer
4
Check it: 2 3/4 ÷ 11 × 4 = 11/4 × 1/11 × 4 = 1, a whole number.
Takeaway
Cancel everything you can first -- what is left over is usually a much smaller question.
- Turn the mixed number into a fraction
- Cancel the 11 and see what is left
- Decide what makes a quarter whole
- Take the smallest one
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested
A parent dashboard for the family lives at sensimlab.com.
The same shape shows up later on the AMC, as Small-Domain Constrained Search. Why the shapes repeat →