Estimate a product to solve an inequality

3.NBT.A.33.OA.D.8 · take · grade 3

Archetype: Pin Down a Number from Digit and Range Conditions · step in a 9-type progression

Among the numbers from $0$ to $9$ , how many different numbers can go in the $\square$ to make the statement true?

$29 \times \square > 208$

Show solution

Understand

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.

Givens

The inequality is 29 times the box > 208.
The box can hold any whole number from 0 to 9.

Unknowns

How many of the digits 0 through 9 make the inequality true.

Constraints

Only single digits 0,1,2,...,9 are allowed in the box.
The statement must be strictly greater than 208 (equal does not count).

Plan

#6 Guess and Check · also uses: #5 Look for a Pattern

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.

Execute

#6 Guess and Check 3.NBT.A.3

Round 29 up to 30. Since 30 times 7 = 210 is just over 208, the answer is near 7, so I check 7 and the digits just above it.

30 \times 7 = 210 \approx 29 \times 7

Rounding 29 to 30 gives a quick size estimate of the product without exact multiplying.

#6 Guess and Check 3.OA.D.8

Compute 29 times 7 to test the boundary precisely.

29 \times 7 = 203 \not> 208

203 is less than 208, so 7 does not make the statement true.

#6 Guess and Check 3.OA.D.8

Compute 29 times 8 to see whether the next digit works.

29 \times 8 = 232 > 208

232 is greater than 208, so 8 works; any digit bigger than 8 gives an even larger product.

#5 Look for a Pattern 3.OA.D.8

Because the product grows as the box grows, every digit from 8 up makes it true while every digit 7 or below fails. The working digits are 8 and 9.

\{8, 9\} \Rightarrow 2 \text{ numbers}

Once the inequality first holds, it keeps holding for all larger digits, so I only count from the boundary up.

Answer: 2

Review

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).

Convert to division (tool 8/11): solve 208 divided by 29 is about 7.17, so the box must be 8 or more, and within 0 to 9 that is the two values 8 and 9.

Standards · min grade 3

3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Estimating 29 times the digit by rounding 29 to 30 to locate the boundary.
3.OA.D.8 Solve two-step word problems using four operations within 100 — Checking exact products and counting which digits satisfy the inequality.

💡 Round 29 to 30 to guess the cutoff, then check around it: Grade 3 estimation does the job!