Problem
Estimate a product to bound an unknown
Find every one-digit number that can replace the square so that (square) times 95 is greater than 20 times 30.
Base-ten numbersOperations
Your answer
Watch out It's tempting to report only the smallest digit that works, but stopping at one answer misses that every larger digit up to 9 also satisfies the inequality.
How to solve
Strategy Guess and Check — First compute the fixed right side (20 x 30 = 600). Then estimate: each step up in the square adds about 95. Test the boundary digit to find where the product first passes 600, then list every digit at or above it.
1STEP 1
Compute the right side
Multiply 20 by 30 to get the number the left side must beat.
20 × 30 = 600
Multiplying two multiples of 10 is 2 x 3 with two zeros tacked on.
3.NBT.A.3Analyze The Units2STEP 2
Estimate where (square) x 95 passes 600
Since 95 is close to 100, test near 6 -- 7 x 95 = 665 is the first to pass 600.
6 × 95 = 570 less than 600, 7 × 95 = 665 greater than 600
Estimating 95 as about 100 quickly shows which digit is the turning point.
3.OA.C.7Guess And CheckThe product (single digit) x 95 first rises above 600 when the digit reaches 7, since 6 x 95 = 570 stays under 600 but 7 x 95 = 665 clears it.
Why?
Both products can be found by splitting 95 into 90 and 5, multiplying the digit by each part, then adding the two results.
Why?
A digit times 95 equals that digit times 90 plus that digit times 5, because 95 is just 90 and 5 joined together.
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
Why?
A part like 7 x 90 is 7 x 9 tens, which is 63 tens, and 63 tens bundle up into 630.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
570 falls short of 600 while 665 goes past it, because 570 holds only 5 hundreds but 600 and 665 each hold 6 hundreds.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
Once one digit pushes the product over 600, every larger digit does too, so the turning point is the smallest digit that works.
Why?
A larger digit means more equal groups of 95, and more equal groups always pile up to a bigger total.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
3STEP 3
List all digits that work
Since the product keeps growing, every digit from 7 up works -- so 7, 8, and 9 are the answer.
7 × 95 = 665, 8 × 95 = 760, 9 × 95 = 855
Once a digit works, every larger digit works too, since the product keeps increasing.
3.OA.C.7Guess And CheckAnswer
7, 8, and 9
The boundary 6 x 95 = 570 is just under 600 and 7 x 95 = 665 is just over, so 7 being the smallest that works makes sense; all one-digit values are 7, 8, 9.
Takeaway
Treat 95 as almost 100 and you can guess the boundary -- Grade 3 estimating points right to 7, 8, 9!
- Compute the right side
- Estimate where (square) x 95 passes 600
- List all digits that work
Where next?
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