Problem
Find a number satisfying product bounds
We need one number from 1 to 9 that meets all three conditions: 6 times it is more than 40, 4 times it is less than 35, and 3 times it is more than 22.
Operations
Your answer
Watch out The number must satisfy all three conditions at once, not just one or two -- 7 clears the first two but fails the third.
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 x 6 = 36 is too small, but 6 x 7 = 42 clears 40, so the number is 7 or more.
6 × 7 = 42 ＞ 40
Checking the six-times facts shows 7 is the first value pushing the product past 40.
3.OA.A.4Guess And Check2STEP 2
Apply condition 2
4 x 8 = 32 stays under 35 but 4 x 9 = 36 goes over, so the number is 8 or less.
4 × 8 = 32 ＜ 35, 4 × 9 = 36
The four-times facts show 8 is the largest value still under 35.
3.OA.A.4Guess And Check3STEP 3
Apply condition 3 and combine
3 x 7 = 21 misses 22 but 3 x 8 = 24 clears it, so combining all three ranges leaves only 8.
3 × 8 = 24 ＞ 22
Only 8 sits inside every range at once, so it is the common number.
3.OA.C.7Make A Systematic 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
Test 8 in all three: 6 x 8 = 48 ＞ 40, 4 x 8 = 32 ＜ 35, and 3 x 8 = 24 ＞ 22, so 8 satisfies every condition.
Takeaway
Turn each clue into a range with the times tables, then keep the number that fits them all -- Grade 3 facts do it!
- Apply condition 1
- Apply condition 2
- Apply condition 3 and combine
Where next?
Level upsuggested
Another one like thissuggested
Same look, different logicsuggested
Step back and solidifysuggested