Compare the next digit to fill a blank in decimals

4.NF.C.7 · take · grade 4

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

Each $\square$ can be any digit from 0 to 9. List the labels in order from least to greatest.

$\text{(A)}\ 79.\square 98 \qquad \text{(B)}\ 7\square.096 \qquad \text{(C)}\ 70.0\square 2$

Show solution

Understand

Each box can hold any digit 0-9. Three labeled numbers are (A) 79.□98, (B) 7□.096, (C) 70.0□2. No matter what digits fill the boxes, order the three labels from least to greatest.

Givens

(A) = 79.□98 (the box is a tenths digit).
(B) = 7□.096 (the box is a ones digit).
(C) = 70.0□2 (the box is a hundredths digit).
Each box can be any digit from 0 to 9.

Unknowns

The order of labels A, B, C from least to greatest.

Constraints

The ordering must hold for every allowed choice of the box digits.

Plan

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

Each number can only range over a small interval as its box runs 0-9. If those intervals do not overlap, the order is fixed for all choices. So we find the smallest-and-largest possible value of each label and compare the ranges.

Execute

#2 Make a Systematic List 5.NBT.A.3

Let each box run from 0 to 9. (C) 70.0□2 ranges from 70.002 to 70.092. (B) 7□.096 ranges from 70.096 to 79.096. (A) 79.□98 ranges from 79.098 to 79.998.

C\in[70.002,70.092],\ B\in[70.096,79.096],\ A\in[79.098,79.998]

Plugging in the smallest and largest digit shows the whole span each number can cover.

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

C's largest value is 70.092, and B's smallest value is 70.096. Since 70.092 < 70.096, every C is less than every B - the ranges do not overlap.

70.092 < 70.096 \Rightarrow C < B

If C can never reach as high as B's lowest, C is always smaller.

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

B's largest value is 79.096, and A's smallest value is 79.098. Since 79.096 < 79.098, every B is less than every A.

79.096 < 79.098 \Rightarrow B < A

B tops out just below A's floor, so B is always smaller than A.

Answer: C, B, A

Review

The three ranges [70.002, 70.092], [70.096, 79.096], [79.098, 79.998] are disjoint and line up in the order C, then B, then A, so the ordering holds for every digit choice. Spot check (boxes all 9): C=70.092 < B=79.096 < A=79.998. Correct.

Guess and Check the extreme cases: put 9 in C and 0 in B (C=70.092, B=70.096) and 9 in B, 0 in A (B=79.096, A=79.098); even these worst cases keep C < B < A.

Standards · min grade 5

4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Comparing C with B and B with A using place value.
5.NBT.A.3 Read, write, and compare decimals to thousandths — Reading the thousandths-place numbers and finding each label's range.

💡 Find the lowest and highest each number can be - if the ranges don't overlap, the order is locked in!