Problem
Multiplying both sides keeps the inequality direction
A quantity divided by 14 has to sit strictly between two given divisions. We need every natural number it could be.
ExpressionsNumber system
Your answer
How to solve
Strategy Convert to Algebra — Turn the two outside divisions into plain decimals so the condition becomes a range. Then multiply through by 14 -- which does not flip the signs, because 14 is positive -- and the blank is trapped between two numbers.
1STEP 1
Work out the two boundaries
Divide each of the outside expressions to see what range the middle one has to fall in.
26 ÷ 8 = 3.25, 42 ÷ 12 = 3.5
The middle value lies between 3.25 and 3.5.
6.NS.B.3Identify Subproblems2STEP 2
Multiply all three parts by 14
Multiplying every part of an inequality by the same positive number keeps the signs pointing the same way, and it clears the division under the blank.
3.25 × 14 = 45.5, 3.5 × 14 = 49
The blank sits between 45.5 and 49.
6.EE.B.5Convert To AlgebraMultiplying all three parts by keeps the inequality true.
Why?
Multiplying by means taking fourteen equal copies of each part, and fourteen copies of a smaller amount stay smaller than fourteen copies of a bigger one.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
3STEP 3
List the natural numbers inside
Both signs are strict, so 49 itself is out; the natural numbers strictly above 45.5 and strictly below 49 are the answer.
■ = 46, 47, 48
Three numbers work.
6.EE.B.8Make A Systematic ListAnswer
46, 47, 48
Test the ends: 46 ÷ 14 ≈ 3.29 and 48 ÷ 14 ≈ 3.43, both inside; 45 ÷ 14 ≈ 3.21 is too small and 49 ÷ 14 = 3.5 is not less than 3.5.
Takeaway
Multiplying every part of an inequality by the same positive number leaves the signs alone.
- Work out the two boundaries
- Multiply all three parts by 14
- List the natural numbers inside