Find a number satisfying product bounds

3.OA.A.43.OA.C.7 · take · grade 3

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

Find a number, chosen from $1$ through $9$ , that satisfies all of the following conditions.

The number times $6$ is greater than $40$ .
$4$ times the number is less than $35$ .
$3$ times the number is greater than $22$ .

Show solution

Understand

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.

Givens

The number is a whole number from 1 to 9.
Condition 1: the number x 6 > 40.
Condition 2: 4 x the number < 35.
Condition 3: 3 x the number > 22.

Unknowns

The single number satisfying all three conditions.

Constraints

All three conditions must hold at the same time.

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Turn each condition into a range of allowed numbers using the times tables, then find the single value common to all three ranges.

Execute

#6 Guess and Check 3.OA.A.4

Find numbers whose product with 6 is more than 40: 6 x 6 = 36 (too small), 6 x 7 = 42 (more than 40). So the number is 7 or more.

6 \times 7 = 42 > 40

Checking the six-times facts shows 7 is the first value pushing the product past 40.

#6 Guess and Check 3.OA.A.4

Find numbers whose product with 4 is less than 35: 4 x 8 = 32 (under 35) but 4 x 9 = 36 (over 35). So the number is 8 or less.

4 \times 8 = 32 < 35,\ 4 \times 9 = 36

The four-times facts show 8 is the largest value still under 35.

#2 Make a Systematic List 3.OA.C.7

Condition 3 needs 3 x number > 22: 3 x 7 = 21 (too small) but 3 x 8 = 24 (more than 22), so the number is 8 or more. Combining all three (7 or more, 8 or less, 8 or more) leaves only 8.

3 \times 8 = 24 > 22

Only 8 sits inside every range at once, so it is the common number.

Answer: 8

Review

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.

You could test each number 1-9 against all three conditions in a table and keep only the row that passes all three, which is 8.

Standards · min grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding which numbers satisfy each product condition.
3.OA.C.7 Fluently multiply and divide within 100 — Computing the six-, four-, and three-times facts to combine the ranges.

💡 Turn each clue into a range with the times tables, then keep the number that fits them all -- Grade 3 facts do it!