Build largest and smallest mixed numbers from cards

4.NF.B.3 · take · grade 4

Archetype: Build the Largest or Smallest Value from Digit Cards · step in a 7-type progression

From $4$ number cards, choose $2$ of them and use each one exactly once to make a mixed number whose denominator is $7$ . Find the sum of the largest mixed number and the smallest mixed number you can make this way.

Number cards: $5$ , $8$ , $4$ , $6$

Show solution

Understand

From the four cards 5, 8, 4, 6 I pick two cards. One becomes the whole part and the other becomes the numerator over a denominator of 7, making a mixed number like W N/7. I want the biggest and smallest such mixed numbers, then their sum.

Givens

Number cards available: 5, 8, 4, 6.
Use exactly two cards, each once, to form a mixed number.
The denominator is fixed at 7.
A mixed number has the form (whole) (numerator/7).

Unknowns

The largest mixed number that can be made.
The smallest mixed number that can be made.
The sum of the largest and smallest.

Constraints

Numerator must be less than 7 for a proper fraction part (5, 4, and 6 qualify; 8 cannot be a numerator).
Each card is used at most once within one mixed number.

Plan

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

To get the biggest number, make the whole part as big as possible, then the numerator as big as possible; to get the smallest, make the whole part as small as possible. Test the card choices and check the constraint that a numerator must be under 7.

Execute

#6 Guess and Check 4.NF.A.2

The whole part matters most, so use the biggest card, 8, as the whole number. For the fraction part the numerator should be as large as possible but under 7; from the remaining cards 5, 4, 6 the largest valid numerator is 6. So the largest mixed number is 8 6/7.

8\tfrac{6}{7}

A bigger whole part beats any fraction part, so maximize the whole number first.

#2 Make a Systematic List 4.NF.A.2

Now make the whole part as small as possible: the smallest card is 4. For the smallest value the numerator should be as small as possible from the rest (5, 8, 6); but 8 is too big to be a numerator (it must be under 7), so the smallest valid numerator is 5. The smallest mixed number is 4 5/7.

4\tfrac{5}{7}

A smaller whole part makes a smaller number; the numerator must still be a legal fraction under 7.

#6 Guess and Check 4.NF.B.3

Add 8 6/7 + 4 5/7. Whole parts: 8 + 4 = 12. Fraction parts: 6/7 + 5/7 = 11/7 = 1 4/7. Combine: 12 + 1 4/7 = 13 4/7.

8\tfrac{6}{7}+4\tfrac{5}{7}=12+\dfrac{11}{7}=12+1\tfrac{4}{7}=13\tfrac{4}{7}

Add wholes and like-denominator fractions separately, then regroup 11/7 into 1 4/7.

Answer: 13 4/7

Review

Largest ≈ 8.86 and smallest ≈ 4.71 sum to about 13.57, which matches 13 4/7. The key catch — 8 cannot be a numerator because the fraction part must be under 7 — was respected, so the smallest correctly uses 5/7 not 8/7.

Systematically list all valid mixed numbers (tool 2): whole from {4,5,6,8}, numerator from the remaining cards under 7. The max is 8 6/7 and the min is 4 5/7, giving the same sum 13 4/7.

Standards · min grade 4

4.NF.A.2 Compare two fractions with different numerators and different denominators — Reasoning about which card choices make the mixed number largest or smallest.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the two mixed numbers and regrouping 11/7 into a whole plus a proper fraction.

💡 This only needs Grade 4 comparing and adding fractions — pick the biggest whole part first, but keep the numerator under 7!