Problem
Estimate a product to solve an inequality
Using a single digit from 0 to 9 in the box, I need to count how many of those digits make 29 times the box greater than 208.
Base-ten numbersOperations
Your answer
numbers
Watch out Rounding 29 up to 30 suggests 7 works, since 30 × 7 = 210 is more than 208 — but you must check the exact value: 29 × 7 = 203, which is not greater than 208.
How to solve
Strategy Guess and Check — Estimate first: 29 is close to 30, so 29 times a digit is a bit under 30 times that digit. That points to the boundary near 7, and because 29 times the box grows steadily as the box grows, once a digit works every larger digit works too. So I just check the digits around the boundary and count the ones that pass.
1STEP 1
Estimate the boundary digit
Round 29 up to 30. Since 30 times 7 = 210 is just over 208, the answer is near 7, so check 7 and the digits above it.
30 × 7 = 210 ≈ 29 × 7
Rounding 29 to 30 gives a quick size estimate of the product without exact multiplying.
3.NBT.A.3Guess And Check2STEP 2
Check digit 7 exactly
Compute 29 times 7 to test the boundary precisely: 29 × 7 = 203, which is not greater than 208.
29 × 7 = 203 is not greater than 208
203 is less than 208, so 7 does not make the statement true.
3.OA.D.8Guess And Check3STEP 3
Check digit 8 exactly
Compute 29 times 8 to see whether the next digit works: 29 × 8 = 232, which is greater than 208.
29 × 8 = 232 greater than 208
232 is greater than 208, so 8 works; any digit bigger than 8 gives an even larger product.
3.OA.D.8Guess And Check4STEP 4
Count the digits that work
The product grows as the box grows, so every digit from 8 up works and 7 or below fails. Digits 8 and 9 work — 2 in all.
{8, 9} → 2 numbers
Once the inequality first holds, it keeps holding for all larger digits, so I only count from the boundary up.
3.OA.D.8Look For A PatternAmong the digits 0 through 9, exactly two of them, 8 and 9, make 29 times the box greater than 208, so the count is 2.
Why?
Each time the box goes up by one, 29 times the box climbs by another 29, so once a digit clears 208 every larger digit clears it too, and every smaller digit stays short.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
The jump happens right between 7 and 8: seven copies of 29 reach only 203, still under 208, but eight copies reach 232, above 208, so 8 is the smallest digit that works.
Why?
Treating 29 as 30 minus 1 makes the products easy: eight of them is 240 minus 8, which is 232, and seven of them is 210 minus 7, which is 203.
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
Why?
232 counts as more than 208 while 203 counts as less, which you settle by lining the numbers up and comparing them one place at a time from the highest place down.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Answer
2 numbers
{8, 9} → 2 numbers
208 divided by 29 is about 7.2, so the box must be at least 8. Digits 8 and 9 are the only single digits that reach that, giving 2 numbers. Spot checks confirm: 29 times 8 = 232 (true) and 29 times 7 = 203 (false).
Takeaway
Round 29 to 30 to guess the cutoff, then check around it: Grade 3 estimation does the job!
- Estimate the boundary digit
- Check digit 7 exactly
- Check digit 8 exactly
- Count the digits that work
Where next?
Another one like thissuggested
Step back and solidifysuggested