← Estimate a product to solve an inequality · Pin Down a Number from Digit and Range Conditions

Estimate a product to solve an inequality · 10 practice problems

3.NBT.A.33.OA.D.8

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Generated variants — 10

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy

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

$19 \times \square > 95$

Your answer

Show solution

1 · Understandwhat's really being asked

Using a single digit from 0 to 9 in the $\square$ , I need to count how many of those digits make 19 times the $\square$ greater than 95.

Givens

The inequality is 19 times the $\square$ > 95.
The $\square$ 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 $\square$ .
The statement must be strictly greater than 95 (equal does not count).

2 · Planchoose the strategy

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

Estimate first: 19 is close to 20, so 19 times a digit is a bit under 20 times that digit. That points to a boundary, and because 19 times the $\square$ grows steadily as the $\square$ 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.

3 · Execute4 carry out the plan

1Estimate the boundary digit

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

Round 19 up to 20. Multiplying 20 by small digits gives a quick size estimate that locates roughly where the product first passes 95.

20 \times 6 = 120 \approx 19 \times 6

Rounding 19 to 20 gives a quick size estimate of the product without exact multiplying.

2Check the boundary exactly

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

Compute 19 times 5 to confirm that digit fails.

19 \times 5 = 95 \not> 95

95 is not greater than 95, so 5 does not work.

3Check the first working digit

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

Compute 19 times 6 to confirm the first digit that works.

19 \times 6 = 114 > 95

114 is greater than 95, so 6 works; any digit bigger than 6 gives an even larger product.

4Count the digits that work

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

Because the product grows as the

\square

grows, every digit from 6 up makes it true. The working digits are 6, 7, 8, 9, giving 4.

\{6, 7, 8, 9\} \Rightarrow 4 \text{ numbers}

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

Answer: 4

4 · Reviewdoes it hold up?

95 divided by 19 is about 5.00, so the $\square$ must be at least 6. That leaves 4 single digit(s) that reach it. Spot check: 19 times 6 = 114 (true).

Another way: Convert to division (tool 11): solve 95 divided by 19 is about 5.00, so the $\square$ must be that much or more, and counting the qualifying digits in 0 to 9 gives 4.

Standardsmin grade 3

3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Estimating 19 times the digit by rounding 19 to 20 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.

💡Takeaway. Round 19 to 20 to guess the cutoff, then check around it: Grade 3 estimation does the job!

Variant 2 easy

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

$26 \times \square > 130$

Your answer

Show solution

1 · Understandwhat's really being asked

Using a single digit from 0 to 9 in the $\square$ , I need to count how many of those digits make 26 times the $\square$ greater than 130.

Givens

The inequality is 26 times the $\square$ > 130.
The $\square$ 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 $\square$ .
The statement must be strictly greater than 130 (equal does not count).

2 · Planchoose the strategy

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

Estimate first: 26 is close to 30, so 26 times a digit is a bit under 30 times that digit. That points to a boundary, and because 26 times the $\square$ grows steadily as the $\square$ 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.

3 · Execute4 carry out the plan

1Estimate the boundary digit

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

Round 26 up to 30. Multiplying 30 by small digits gives a quick size estimate that locates roughly where the product first passes 130.

30 \times 6 = 180 \approx 26 \times 6

Rounding 26 to 30 gives a quick size estimate of the product without exact multiplying.

2Check the boundary exactly

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

Compute 26 times 5 to confirm that digit fails.

26 \times 5 = 130 \not> 130

130 is not greater than 130, so 5 does not work.

3Check the first working digit

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

Compute 26 times 6 to confirm the first digit that works.

26 \times 6 = 156 > 130

156 is greater than 130, so 6 works; any digit bigger than 6 gives an even larger product.

4Count the digits that work

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

Because the product grows as the

\square

grows, every digit from 6 up makes it true. The working digits are 6, 7, 8, 9, giving 4.

\{6, 7, 8, 9\} \Rightarrow 4 \text{ numbers}

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

Answer: 4

4 · Reviewdoes it hold up?

130 divided by 26 is about 5.00, so the $\square$ must be at least 6. That leaves 4 single digit(s) that reach it. Spot check: 26 times 6 = 156 (true).

Another way: Convert to division (tool 11): solve 130 divided by 26 is about 5.00, so the $\square$ must be that much or more, and counting the qualifying digits in 0 to 9 gives 4.

Standardsmin grade 3

3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Estimating 26 times the digit by rounding 26 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.

💡Takeaway. Round 26 to 30 to guess the cutoff, then check around it: Grade 3 estimation does the job!

Variant 3 easy

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

$34 \times \square > 170$

Your answer

Show solution

1 · Understandwhat's really being asked

Using a single digit from 0 to 9 in the $\square$ , I need to count how many of those digits make 34 times the $\square$ greater than 170.

Givens

The inequality is 34 times the $\square$ > 170.
The $\square$ 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 $\square$ .
The statement must be strictly greater than 170 (equal does not count).

2 · Planchoose the strategy

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

Estimate first: 34 is close to 40, so 34 times a digit is a bit under 40 times that digit. That points to a boundary, and because 34 times the $\square$ grows steadily as the $\square$ 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.

3 · Execute4 carry out the plan

1Estimate the boundary digit

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

Round 34 up to 40. Multiplying 40 by small digits gives a quick size estimate that locates roughly where the product first passes 170.

40 \times 6 = 240 \approx 34 \times 6

Rounding 34 to 40 gives a quick size estimate of the product without exact multiplying.

2Check the boundary exactly

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

Compute 34 times 5 to confirm that digit fails.

34 \times 5 = 170 \not> 170

170 is not greater than 170, so 5 does not work.

3Check the first working digit

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

Compute 34 times 6 to confirm the first digit that works.

34 \times 6 = 204 > 170

204 is greater than 170, so 6 works; any digit bigger than 6 gives an even larger product.

4Count the digits that work

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

Because the product grows as the

\square

grows, every digit from 6 up makes it true. The working digits are 6, 7, 8, 9, giving 4.

\{6, 7, 8, 9\} \Rightarrow 4 \text{ numbers}

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

Answer: 4

4 · Reviewdoes it hold up?

170 divided by 34 is about 5.00, so the $\square$ must be at least 6. That leaves 4 single digit(s) that reach it. Spot check: 34 times 6 = 204 (true).

Another way: Convert to division (tool 11): solve 170 divided by 34 is about 5.00, so the $\square$ must be that much or more, and counting the qualifying digits in 0 to 9 gives 4.

Standardsmin grade 3

3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Estimating 34 times the digit by rounding 34 to 40 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.

💡Takeaway. Round 34 to 40 to guess the cutoff, then check around it: Grade 3 estimation does the job!

Variant 4 medium

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

$38 \times \square > 190$

Your answer

Show solution

1 · Understandwhat's really being asked

Using a single digit from 0 to 9 in the $\square$ , I need to count how many of those digits make 38 times the $\square$ greater than 190.

Givens

The inequality is 38 times the $\square$ > 190.
The $\square$ 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 $\square$ .
The statement must be strictly greater than 190 (equal does not count).

2 · Planchoose the strategy

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

Estimate first: 38 is close to 40, so 38 times a digit is a bit under 40 times that digit. That points to a boundary, and because 38 times the $\square$ grows steadily as the $\square$ 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.

3 · Execute4 carry out the plan

1Estimate the boundary digit

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

Round 38 up to 40. Multiplying 40 by small digits gives a quick size estimate that locates roughly where the product first passes 190.

40 \times 6 = 240 \approx 38 \times 6

Rounding 38 to 40 gives a quick size estimate of the product without exact multiplying.

2Check the boundary exactly

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

Compute 38 times 5 to confirm that digit fails.

38 \times 5 = 190 \not> 190

190 is not greater than 190, so 5 does not work.

3Check the first working digit

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

Compute 38 times 6 to confirm the first digit that works.

38 \times 6 = 228 > 190

228 is greater than 190, so 6 works; any digit bigger than 6 gives an even larger product.

4Count the digits that work

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

Because the product grows as the

\square

grows, every digit from 6 up makes it true. The working digits are 6, 7, 8, 9, giving 4.

\{6, 7, 8, 9\} \Rightarrow 4 \text{ numbers}

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

Answer: 4

4 · Reviewdoes it hold up?

190 divided by 38 is about 5.00, so the $\square$ must be at least 6. That leaves 4 single digit(s) that reach it. Spot check: 38 times 6 = 228 (true).

Another way: Convert to division (tool 11): solve 190 divided by 38 is about 5.00, so the $\square$ must be that much or more, and counting the qualifying digits in 0 to 9 gives 4.

Standardsmin grade 3

3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Estimating 38 times the digit by rounding 38 to 40 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.

💡Takeaway. Round 38 to 40 to guess the cutoff, then check around it: Grade 3 estimation does the job!

Variant 5 medium

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

$29 \times \square > 208$

Your answer

Show solution

1 · Understandwhat's really being asked

Using a single digit from 0 to 9 in the $\square$ , I need to count how many of those digits make 29 times the $\square$ greater than 208.

Givens

The inequality is 29 times the $\square$ > 208.
The $\square$ 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 $\square$ .
The statement must be strictly greater than 208 (equal does not count).

2 · Planchoose the strategy

#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 a boundary, and because 29 times the $\square$ grows steadily as the $\square$ 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.

3 · Execute4 carry out the plan

1Estimate the boundary digit

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

Round 29 up to 30. Multiplying 30 by small digits gives a quick size estimate that locates roughly where the product first passes 208.

30 \times 8 = 240 \approx 29 \times 8

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

2Check the boundary exactly

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

Compute 29 times 7 to confirm that digit fails.

29 \times 7 = 203 \not> 208

203 is not greater than 208, so 7 does not work.

3Check the first working digit

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

Compute 29 times 8 to confirm the first digit that works.

29 \times 8 = 232 > 208

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

4Count the digits that work

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

Because the product grows as the

\square

grows, every digit from 8 up makes it true. The working digits are 8, 9, giving 2.

\{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

4 · Reviewdoes it hold up?

208 divided by 29 is about 7.17, so the $\square$ must be at least 8. That leaves 2 single digit(s) that reach it. Spot check: 29 times 8 = 232 (true).

Another way: Convert to division (tool 11): solve 208 divided by 29 is about 7.17, so the $\square$ must be that much or more, and counting the qualifying digits in 0 to 9 gives 2.

Standardsmin 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.

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

Variant 6 medium

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

$31 \times \square > 217$

Your answer

Show solution

1 · Understandwhat's really being asked

Using a single digit from 0 to 9 in the $\square$ , I need to count how many of those digits make 31 times the $\square$ greater than 217.

Givens

The inequality is 31 times the $\square$ > 217.
The $\square$ 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 $\square$ .
The statement must be strictly greater than 217 (equal does not count).

2 · Planchoose the strategy

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

Estimate first: 31 is close to 40, so 31 times a digit is a bit under 40 times that digit. That points to a boundary, and because 31 times the $\square$ grows steadily as the $\square$ 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.

3 · Execute4 carry out the plan

1Estimate the boundary digit

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

Round 31 up to 40. Multiplying 40 by small digits gives a quick size estimate that locates roughly where the product first passes 217.

40 \times 8 = 320 \approx 31 \times 8

Rounding 31 to 40 gives a quick size estimate of the product without exact multiplying.

2Check the boundary exactly

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

Compute 31 times 7 to confirm that digit fails.

31 \times 7 = 217 \not> 217

217 is not greater than 217, so 7 does not work.

3Check the first working digit

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

Compute 31 times 8 to confirm the first digit that works.

31 \times 8 = 248 > 217

248 is greater than 217, so 8 works; any digit bigger than 8 gives an even larger product.

4Count the digits that work

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

Because the product grows as the

\square

grows, every digit from 8 up makes it true. The working digits are 8, 9, giving 2.

\{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

4 · Reviewdoes it hold up?

217 divided by 31 is about 7.00, so the $\square$ must be at least 8. That leaves 2 single digit(s) that reach it. Spot check: 31 times 8 = 248 (true).

Another way: Convert to division (tool 11): solve 217 divided by 31 is about 7.00, so the $\square$ must be that much or more, and counting the qualifying digits in 0 to 9 gives 2.

Standardsmin grade 3

3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Estimating 31 times the digit by rounding 31 to 40 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.

💡Takeaway. Round 31 to 40 to guess the cutoff, then check around it: Grade 3 estimation does the job!

Variant 7 medium

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

$42 \times \square > 252$

Your answer

Show solution

1 · Understandwhat's really being asked

Using a single digit from 0 to 9 in the $\square$ , I need to count how many of those digits make 42 times the $\square$ greater than 252.

Givens

The inequality is 42 times the $\square$ > 252.
The $\square$ 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 $\square$ .
The statement must be strictly greater than 252 (equal does not count).

2 · Planchoose the strategy

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

Estimate first: 42 is close to 50, so 42 times a digit is a bit under 50 times that digit. That points to a boundary, and because 42 times the $\square$ grows steadily as the $\square$ 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.

3 · Execute4 carry out the plan

1Estimate the boundary digit

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

Round 42 up to 50. Multiplying 50 by small digits gives a quick size estimate that locates roughly where the product first passes 252.

50 \times 7 = 350 \approx 42 \times 7

Rounding 42 to 50 gives a quick size estimate of the product without exact multiplying.

2Check the boundary exactly

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

Compute 42 times 6 to confirm that digit fails.

42 \times 6 = 252 \not> 252

252 is not greater than 252, so 6 does not work.

3Check the first working digit

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

Compute 42 times 7 to confirm the first digit that works.

42 \times 7 = 294 > 252

294 is greater than 252, so 7 works; any digit bigger than 7 gives an even larger product.

4Count the digits that work

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

Because the product grows as the

\square

grows, every digit from 7 up makes it true. The working digits are 7, 8, 9, giving 3.

\{7, 8, 9\} \Rightarrow 3 \text{ numbers}

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

Answer: 3

4 · Reviewdoes it hold up?

252 divided by 42 is about 6.00, so the $\square$ must be at least 7. That leaves 3 single digit(s) that reach it. Spot check: 42 times 7 = 294 (true).

Another way: Convert to division (tool 11): solve 252 divided by 42 is about 6.00, so the $\square$ must be that much or more, and counting the qualifying digits in 0 to 9 gives 3.

Standardsmin grade 3

3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Estimating 42 times the digit by rounding 42 to 50 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.

💡Takeaway. Round 42 to 50 to guess the cutoff, then check around it: Grade 3 estimation does the job!

Variant 8 hard

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

$47 \times \square > 280$

Your answer

Show solution

1 · Understandwhat's really being asked

Using a single digit from 0 to 9 in the $\square$ , I need to count how many of those digits make 47 times the $\square$ greater than 280.

Givens

The inequality is 47 times the $\square$ > 280.
The $\square$ 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 $\square$ .
The statement must be strictly greater than 280 (equal does not count).

2 · Planchoose the strategy

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

Estimate first: 47 is close to 50, so 47 times a digit is a bit under 50 times that digit. That points to a boundary, and because 47 times the $\square$ grows steadily as the $\square$ 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.

3 · Execute4 carry out the plan

1Estimate the boundary digit

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

Round 47 up to 50. Multiplying 50 by small digits gives a quick size estimate that locates roughly where the product first passes 280.

50 \times 6 = 300 \approx 47 \times 6

Rounding 47 to 50 gives a quick size estimate of the product without exact multiplying.

2Check the boundary exactly

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

Compute 47 times 5 to confirm that digit fails.

47 \times 5 = 235 \not> 280

235 is not greater than 280, so 5 does not work.

3Check the first working digit

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

Compute 47 times 6 to confirm the first digit that works.

47 \times 6 = 282 > 280

282 is greater than 280, so 6 works; any digit bigger than 6 gives an even larger product.

4Count the digits that work

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

Because the product grows as the

\square

grows, every digit from 6 up makes it true. The working digits are 6, 7, 8, 9, giving 4.

\{6, 7, 8, 9\} \Rightarrow 4 \text{ numbers}

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

Answer: 4

4 · Reviewdoes it hold up?

280 divided by 47 is about 5.96, so the $\square$ must be at least 6. That leaves 4 single digit(s) that reach it. Spot check: 47 times 6 = 282 (true).

Another way: Convert to division (tool 11): solve 280 divided by 47 is about 5.96, so the $\square$ must be that much or more, and counting the qualifying digits in 0 to 9 gives 4.

Standardsmin grade 3

3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Estimating 47 times the digit by rounding 47 to 50 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.

💡Takeaway. Round 47 to 50 to guess the cutoff, then check around it: Grade 3 estimation does the job!

Variant 9 hard

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

$53 \times \square > 318$

Your answer

Show solution

1 · Understandwhat's really being asked

Using a single digit from 0 to 9 in the $\square$ , I need to count how many of those digits make 53 times the $\square$ greater than 318.

Givens

The inequality is 53 times the $\square$ > 318.
The $\square$ 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 $\square$ .
The statement must be strictly greater than 318 (equal does not count).

2 · Planchoose the strategy

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

Estimate first: 53 is close to 60, so 53 times a digit is a bit under 60 times that digit. That points to a boundary, and because 53 times the $\square$ grows steadily as the $\square$ 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.

3 · Execute4 carry out the plan

1Estimate the boundary digit

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

Round 53 up to 60. Multiplying 60 by small digits gives a quick size estimate that locates roughly where the product first passes 318.

60 \times 7 = 420 \approx 53 \times 7

Rounding 53 to 60 gives a quick size estimate of the product without exact multiplying.

2Check the boundary exactly

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

Compute 53 times 6 to confirm that digit fails.

53 \times 6 = 318 \not> 318

318 is not greater than 318, so 6 does not work.

3Check the first working digit

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

Compute 53 times 7 to confirm the first digit that works.

53 \times 7 = 371 > 318

371 is greater than 318, so 7 works; any digit bigger than 7 gives an even larger product.

4Count the digits that work

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

Because the product grows as the

\square

grows, every digit from 7 up makes it true. The working digits are 7, 8, 9, giving 3.

\{7, 8, 9\} \Rightarrow 3 \text{ numbers}

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

Answer: 3

4 · Reviewdoes it hold up?

318 divided by 53 is about 6.00, so the $\square$ must be at least 7. That leaves 3 single digit(s) that reach it. Spot check: 53 times 7 = 371 (true).

Another way: Convert to division (tool 11): solve 318 divided by 53 is about 6.00, so the $\square$ must be that much or more, and counting the qualifying digits in 0 to 9 gives 3.

Standardsmin grade 3

3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Estimating 53 times the digit by rounding 53 to 60 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.

💡Takeaway. Round 53 to 60 to guess the cutoff, then check around it: Grade 3 estimation does the job!

Variant 10 hard

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

$58 \times \square > 348$

Your answer

Show solution

1 · Understandwhat's really being asked

Using a single digit from 0 to 9 in the $\square$ , I need to count how many of those digits make 58 times the $\square$ greater than 348.

Givens

The inequality is 58 times the $\square$ > 348.
The $\square$ 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 $\square$ .
The statement must be strictly greater than 348 (equal does not count).

2 · Planchoose the strategy

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

Estimate first: 58 is close to 60, so 58 times a digit is a bit under 60 times that digit. That points to a boundary, and because 58 times the $\square$ grows steadily as the $\square$ 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.

3 · Execute4 carry out the plan

1Estimate the boundary digit

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

Round 58 up to 60. Multiplying 60 by small digits gives a quick size estimate that locates roughly where the product first passes 348.

60 \times 7 = 420 \approx 58 \times 7

Rounding 58 to 60 gives a quick size estimate of the product without exact multiplying.

2Check the boundary exactly

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

Compute 58 times 6 to confirm that digit fails.

58 \times 6 = 348 \not> 348

348 is not greater than 348, so 6 does not work.

3Check the first working digit

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

Compute 58 times 7 to confirm the first digit that works.

58 \times 7 = 406 > 348

406 is greater than 348, so 7 works; any digit bigger than 7 gives an even larger product.

4Count the digits that work

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

Because the product grows as the

\square

grows, every digit from 7 up makes it true. The working digits are 7, 8, 9, giving 3.

\{7, 8, 9\} \Rightarrow 3 \text{ numbers}

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

Answer: 3

4 · Reviewdoes it hold up?

348 divided by 58 is about 6.00, so the $\square$ must be at least 7. That leaves 3 single digit(s) that reach it. Spot check: 58 times 7 = 406 (true).

Another way: Convert to division (tool 11): solve 348 divided by 58 is about 6.00, so the $\square$ must be that much or more, and counting the qualifying digits in 0 to 9 gives 3.

Standardsmin grade 3

3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Estimating 58 times the digit by rounding 58 to 60 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.

💡Takeaway. Round 58 to 60 to guess the cutoff, then check around it: Grade 3 estimation does the job!

Where next?

← Back to the problem: Estimate a product to solve an inequality