Problem

Compare decimals from the highest place down

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 $\square.6$ .

Fractions

Show worked solution try it yourself first ✦

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Solve 4.6 < 4.A by the tenths place

#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.

🧠 Why? · Feynman

In 4.6 < 4.A the ones digits are both 4, so the comparison comes down to the tenths place, and A must be greater than 6.

Why?

The two numbers are equal in the ones place, so the shared 4 is the same in each and cannot make one larger than the other; whatever settles the comparison must come from the tenths place.

Why?

Each number splits into its ones part and its tenths part with no gap and no overlap, so when one shared piece is identical in both, only the remaining tenths pieces can make them differ.

🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

Why?

With the ones parts equal, the number holding more tenths is the larger one, and the tenths digit tells how many tenths there are, so a bigger tenths digit means a bigger number.

Why?

The tenths place always counts pieces of one fixed size, one tenth, so more of that same-size piece is genuinely a larger amount.

🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.

2Pick the smallest A

#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.

3Solve 4.6 > B.6 by the ones place

#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.

4Pick the largest B

#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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

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

▶ Practice — 10 problems