Problem
Find a number satisfying product bounds
Choose a number from 1 through 9 where the number times 6 is greater than 40, 4 times the number is less than 35, and 3 times the number is greater than 22 — find that number.
Operations
Your answer
Watch out Don't stop once a number passes just one condition — only a number that satisfies all three is the answer.
How to solve
Strategy Guess and Check — Turn each condition into a range of allowed numbers using the times tables, then find the single value common to all three ranges.
1STEP 1
Apply condition 1
6×7=42 clears 40, so condition 1 needs the number to be 7 or more.
6×6=36, 6×7=42
The product with 6 is six added that many times, so each extra number adds another 6. 6×6=36 falls short of 40 but 6×7=42 clears it, so every number below 7 fails condition 1.
3.OA.A.4Guess And Check2STEP 2
Apply condition 2
4×8=32 stays under 35 but 4×9=36 doesn't, so condition 2 needs the number to be 8 or less.
4×8=32, 4×9=36
The product with 4 also grows by 4 with each extra number. 4×8=32 stays under 35 but 4×9=36 goes over, so every number 9 or above fails condition 2.
3.OA.A.4Guess And Check3STEP 3
Apply condition 3 and combine
3×8=24 clears 22, so condition 3 also needs 8 or more — the only number inside all three ranges is 8.
{7,8,9} ∩ {1,...,8} ∩ {8,9} = {8}
The product with 3 grows by 3 each time the number goes up. 3×8=24 clears 22, and any number above 8 makes even more groups of 3 so it stays past 22; but 3×7=21 falls short, and any number below 7 makes fewer groups of 3 so it stays under 22. So condition 3 needs 8 or more, and the only value that also fits 7-or-more and 8-or-less is 8.
3.OA.C.7Systematic ListCondition 3 says 3 times the number is more than 22, and that rules out every number below 8, so the number must be 8 or more.
Why?
3 times 8 makes 24, which clears 22, and any number bigger than 8 makes even more threes, so 8 and 9 both pass condition 3.
Why?
3 times 8 means eight equal groups of 3, and eight threes stack up to 24.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
24 is 22 with 2 more, so it sits above 22.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Going from 8 to 9 adds one more group of 3, so the product only grows and stays past 22.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
3 times 7 is only 21, which does not reach 22, and any number below 7 makes fewer threes, so 7 and every smaller number fail.
Why?
3 times 7 means seven equal groups of 3, and seven threes stack up to only 21.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
21 and 1 more make 22, so 21 falls just short of 22.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Each step down from 7 takes away a group of 3, so the product only shrinks and stays under 22.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Answer
8
{7,8,9} ∩ {1,...,8} ∩ {8,9} = {8}
Check — with 8: 6×8=48 is greater than 40, 4×8=32 is less than 35, and 3×8=24 is greater than 22, so all three conditions hold.
Takeaway
Turn each condition into a range with the times table, then keep only the number where all the ranges overlap.
- 6 times the number is greater than 40 → number is 7 or more
- 4 times the number is less than 35 → number is 8 or less
- 3 times the number is greater than 22 → number is 8 or more
- Where all three overlap → 8
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