Compare decimals from the highest place down

4.NF.C.7 · take · grade 4

Archetype: Compare Fractions and Decimals by Structure · step in a 7-type progression

Using the digits 1 through 9, fill in the blanks A and B so that both inequalities below are true. Find the smallest possible value of A and the largest possible value of B.

$4.6 < 4.\boxed{A}$

$4.6 > \boxed{B}.6$

Here A is the tenths digit of the decimal $4.\square$ , and B is the ones digit of the decimal $\square.6$ .

Show solution

Understand

Using single digits 1 through 9, I must fill blank A in 4.6 < 4.A and blank B in 4.6 > B.6 so both inequalities hold. A is the tenths digit of 4.A; B is the ones digit of B.6. I want the smallest A that works and the largest B that works.

Givens

4.6 < 4.A, where A is a tenths digit (a single digit 1-9).
4.6 > B.6, where B is a ones digit (a single digit 1-9).

Unknowns

The smallest possible value of A.
The largest possible value of B.

Constraints

A and B are each one of the digits 1, 2, ..., 9.
Compare decimals by their highest place first.

Plan

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

Each blank has only the 9 digits to test, so I can reason place-by-place. For A the ones places are equal (both 4), so the tenths digit decides; for B the tenths places are equal (both 6), so the ones digit decides. Testing the boundary digit confirms the smallest A and largest B.

Execute

#6 Guess and Check 4.NF.C.7

Both numbers have ones digit 4, so the ones place is tied. To decide, compare the tenths: we need A > 6. Among digits 1-9 the values greater than 6 are 7, 8, 9.

4.6 < 4.A \Rightarrow A > 6

When the whole-number parts match, the bigger decimal is the one with the bigger tenths digit.

#5 Look for a Pattern 4.NF.C.7

The valid digits for A are 7, 8, 9. The smallest of these is 7.

A_{\min} = 7

Just above 6 is 7, so 7 is the smallest tenths digit that still beats 4.6.

#6 Guess and Check 4.NF.C.7

Both numbers have the same tenths digit 6, so the tenths place is tied. To decide, compare the ones: we need 4 > B, that is B < 4. Among digits 1-9 the values less than 4 are 1, 2, 3.

4.6 > B.6 \Rightarrow B < 4

With equal tenths, the smaller number is the one with the smaller ones digit.

#5 Look for a Pattern 4.NF.C.7

The valid digits for B are 1, 2, 3. The largest of these is 3.

B_{\max} = 3

Just below 4 is 3, so 3 is the largest ones digit that keeps B.6 under 4.6.

Answer: A = 7, B = 3

Review

Check A = 7: 4.6 < 4.7 is true. Check B = 3: 3.6 > would fail order, but 4.6 > 3.6 is true. Both inequalities hold, and trying A = 6 (4.6 < 4.6 false) or B = 4 (4.6 > 4.6 false) shows 7 and 3 are the boundary digits.

Make a systematic list (tool 2): for A list 1.6...4.9 outcomes and for B list 1.6...9.6, marking which satisfy the inequality; the smallest passing A is 7 and the largest passing B is 3.

Standards · min grade 4

4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Comparing 4.6 with 4.A and with B.6 place by place to bound A and B.

💡 Compare decimals from the highest place down: same ones? look at tenths; same tenths? look at ones - pure Grade 4 place value!