Depth · one concept from first principles · step 2 of 4

Build largest and smallest mixed numbers from cards

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: $5$ , $8$ , $4$ , $6$

Fractions

Your answer

How to solve

Strategy Guess and Check — 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.

1STEP 1

Build the largest mixed number

Whole part counts most: use 8 (biggest card) as the whole, then the largest card under 7 as numerator — 6. Largest is 8 $\frac{6}{7}$ .

8 \frac{6}{7}

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

4.NF.A.2Guess And Check

The largest mixed number is made by putting the biggest card, 8, in the whole place and the biggest still-legal card, 6, in the numerator over 7, giving 8 $\frac{6}{7}$ .

Why?

A mixed number is a whole-number part plus a fraction part that is less than one whole, so the whole part settles which mixed number is bigger before the fraction even matters, and 8 is the biggest whole part available.

Why?

The fraction part here is at most $\frac{6}{7}$ , and any number of sevenths below seven is less than one whole.

Why?

One whole splits into exactly seven equal sevenths, so having fewer than seven of them is having less than the whole.

🧱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?

When you compare two numbers you weigh the biggest place first, and one whole unit outranks any part of a unit the same way a ten outranks the ones.

Why?

Each higher place is a bundle of the place below it, so a single unit up in the whole place is worth more than anything sitting in the fraction below it.

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

Why?

With the whole part fixed as large as possible, the mixed number grows when the fraction grows, and among the leftover cards 5, 4, 6 the largest one still below seven is 6.

Why?

Over the same denominator 7, a bigger numerator names more of the same-size seventh-pieces, which is a larger fraction.

Why?

A fraction like $\frac{6}{7}$ is just six copies of the piece $\frac{1}{7}$ added together, and adding more equal pieces makes a larger total.

🧱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?

The numerator has to stay below 7 or the fraction part would be a whole or more, so 8 is not allowed as a numerator and 6 is the biggest card that is.

Why?

Seven sevenths already rebuild one full whole, so eight sevenths would run past a whole and stop being a proper fraction part.

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

2STEP 2

Build the smallest mixed number

Make the whole small: smallest card is 4. Numerator should be small too, but 8 can't be one (under 7 only), so use 5. Smallest is 4 $\frac{5}{7}$ .

4 \frac{5}{7}

3STEP 3

Add the largest and smallest

Add 8 $\frac{6}{7}$ + 4 $\frac{5}{7}$ . Whole parts: 8 + 4 = 12. Fraction parts: $\frac{6}{7}$ + $\frac{5}{7}$ = $\frac{11}{7}$ = 1 $\frac{4}{7}$ . Combine: 12 + 1 $\frac{4}{7}$ = 13 $\frac{4}{7}$ .

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

Answer

13\frac{4}{7}

\frac{6}{7}

+ 4

\frac{5}{7}

= 13

\frac{4}{7}

Largest ≈ 8.86 and smallest ≈ 4.71 sum to about 13.57, which matches 13

\frac{4}{7}

. Since 8 is not under 7, it can never be a numerator — so it can only serve as a whole-number part, which is exactly how it's used in the largest number (8

\frac{6}{7}

Takeaway

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

Build the largest mixed number
Build the smallest mixed number
Add the largest and smallest

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