Problem
Simplify the expression before comparing
A quantity built from a division has to sit strictly between two values. We need every natural number the blank could be.
Number systemExpressions
Your answer
How to solve
Strategy Convert to Algebra — Nothing can be compared while the parts are still tangled up. Simplify the left side to a single number and the middle to a single multiple of the blank; only then does the condition read as a range.
1STEP 1
Simplify the left side
Write both mixed numbers as improper fractions and turn the division into a multiplication by the reciprocal.
55/6 × 4/11 = 10/3
The left end is 10/3, a bit more than 3.
6.NS.A.1Work Backwards2STEP 2
Simplify the middle
Dividing by 7/■ means multiplying by ■/7, and 42 ÷ 7 = 6.
42 ÷ 7 = 6
The middle is just 6 times the blank.
6.NS.A.1Convert To Algebrais simply six times the blank.
Why?
Dividing by asks how many of that fraction fit into , which is the undoing of multiplying by it.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
Sevenths of the blank fit six times as often as whole blanks do, because is six sevens.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
3STEP 3
Turn the condition into a range for the blank
Divide all three parts by 6; dividing by a positive number keeps the signs pointing the same way.
10/3 ÷ 6 = 5/9, 28 ÷ 6 = 14/3
The blank is between 5/9 and 14/3.
6.EE.B.8Work Backwards4STEP 4
List the natural numbers inside
5/9 is under 1 and 14/3 is between 4 and 5, so the natural numbers from 1 to 4 fit.
■ = 1, 2, 3, 4
Four numbers work.
6.EE.B.8Make A Systematic ListAnswer
1, 2, 3, 4
Test the ends: with ■ = 4 the middle is 24, which is under 28; with ■ = 5 it is 30, which is not.
Takeaway
Simplify each side to a single number before comparing -- the range only appears once the clutter is gone.
- Simplify the left side
- Simplify the middle
- Turn the condition into a range for the blank
- List the natural numbers inside