Grade 5 · 5.NF.B.3

A fraction as division

Three lasagnas, four hungry people, and nobody wants to be left out. How much does each person get, and what does a fraction have to do with sharing?

You need

A fraction of a setGrade 3

Where you are 15 in Fractions

Fractions, ratio and percent3

Fractions15

▴ 10 above What the numerator and the denominator each tell you Whole numbers written as fractions

A fraction as division

Dividing fractions6 Fraction of a quantity3

Why this exists

It is potluck night. Charu has brought 3 lasagnas, and 4 people will share them equally: Charu, Goro, Ijeoma and Goro’s brother. How much lasagna does each person get?

Whole-number division gives a strange answer. 3 \div 4 is 0 with 3 left over: nobody gets a whole lasagna, and all three are left on the table. That is true, but nobody is going to leave the lasagnas uneaten. They will cut them. The question is how much each person ends up with, and the answer is less than one lasagna, so it cannot be a whole number.

Fractions answer it. Each person gets \displaystyle \frac{3}{4} of a lasagna. The same fraction that names “three of four equal pieces” also names “three wholes shared four ways”. That is the whole idea of this lesson: the bar in a fraction means divide.

Before this, you met fractions as parts of one whole (A fraction as part of a whole) and as parts of a group of objects (A fraction of a set: \displaystyle \frac{1}{4} of a set of 12 cookies is 3 cookies). Sharing objects that can be counted out is the second picture. This lesson shares wholes that have to be cut.

The lesson takes three days. Part 1 shares a few wholes among more people, so each share is less than one. Part 2 shares more wholes than people, and turns a division with a remainder into a mixed number. Part 3 reads word problems: which number is divided by which, and between which two whole numbers the answer lies.

Definition

In words. When a equal wholes are shared equally among b people (or b groups, or b boxes), each share is \displaystyle \frac{a}{b} of one whole. The number of wholes goes on top. The number of shares goes on the bottom.

In symbols. For any whole numbers a and b, with b not 0,

\frac{a}{b} = a \div b

Here a is the numerator (top number) and b is the denominator (bottom number), as in What the numerator and the denominator each tell you. The fraction \displaystyle \frac{a}{b} already had a meaning: a pieces, each of size \displaystyle \frac{1}{b} (a unit fraction). The definition says that the same number is also what you get by dividing a by b.

Intuition

Part 1: sharing a few wholes among more people

Go back to Charu’s 3 lasagnas and 4 people. There are two natural ways to cut them, and the figure shows both.

Figure 1Both ways give each person the same amount: 3 pieces of size \displaystyle \frac{1}{4}, or one fourth of the whole row of 3. Either way the share is \displaystyle \frac{3}{4} of a lasagna.

Cut every lasagna into 4. Each lasagna becomes 4 pieces, and each piece is \displaystyle \frac{1}{4} of a lasagna. That makes 3 \times 4 = 12 pieces in all. Shared among 4 people, each person gets 12 \div 4 = 3 pieces, one from each lasagna. Three pieces of size \displaystyle \frac{1}{4} is \displaystyle \frac{3}{4} of a lasagna.

Line them up and cut once. Put the 3 lasagnas end to end in one long row, and cut the row into 4 equal parts. Each part is \displaystyle \frac{1}{4} of all three lasagnas. How long is that part? The first cut lands three fourths of the way along the first lasagna, since 4 parts of that size cover the 12 fourths in the row. So one part is \displaystyle \frac{3}{4} of a lasagna, the same as before.

The two ways show something worth saying out loud:

\frac{1}{4} \text{ of } 3 \text{ lasagnas} = \frac{3}{4} \text{ of } 1 \text{ lasagna}

Both are the share when 3 is divided by 4. Writing it as a division, \displaystyle 3 \div 4 = \frac{3}{4}.

When there are fewer wholes than people, each share is less than one whole, so the fraction has a smaller top than bottom. When there are exactly as many wholes as people, each person gets one whole: \displaystyle 4 \div 4 = \frac{4}{4} = 1 (see Whole numbers written as fractions).

Part 2: more wholes than people

Goro brings 7 brownie pans, and 3 people share them: Charu, Goro and Ijeoma.

Hand out whole pans first. Each person can take 2 whole pans, which uses 2 \times 3 = 6 pans. That leaves 1 pan. Cut it into 3 equal strips, and each person takes one strip, \displaystyle \frac{1}{3} of a pan. Each person’s share is

2 + \frac{1}{3} = 2\frac{1}{3} \text{ pans}

Figure 2Each person gets 2 whole pans and one third of the last pan, so \displaystyle 7 \div 3 = 2\frac{1}{3}. Charu’s share is shaded.

Cut every pan into 3. Each of the 7 pans becomes 3 strips of size \displaystyle \frac{1}{3}, so there are 7 \times 3 = 21 strips. Each person gets 21 \div 3 = 7 strips, and 7 strips of size \displaystyle \frac{1}{3} is \displaystyle \frac{7}{3} of a pan.

Both ways measure the same share, so

7 \div 3 = \frac{7}{3} = 2\frac{1}{3}

A fraction with a bigger top than bottom, such as \displaystyle \frac{7}{3}, is more than one whole (see Fractions greater than one (improper fractions)), and \displaystyle 2\frac{1}{3} is the same number written as a mixed number: a whole number and a fraction added together (see Mixed numbers and improper fractions).

Look at the long division 7 \div 3. It gives 2 with a remainder (the amount left over) of 1. The 2 is the whole pans each person gets. The remainder 1 is the pan that is left, and sharing that last pan among the 3 people gives each \displaystyle \frac{1}{3}. So the remainder is not thrown away. It becomes the top of the fraction part, and the divisor becomes the bottom:

7 \div 3 = 2 \text{ remainder } 1 \quad \text{becomes} \quad 2\frac{1}{3}

This works for any whole-number division. Bigger numbers just need the long division you already know, or the estimating in Dividing by a two-digit number when the divisor has two digits.

Part 3: which number is divided by which

In a word problem, the numbers do not arrive labeled “top” and “bottom”. Ask two questions. What is being shared? That amount is divided, so it goes on top. Into how many equal shares? That number goes on the bottom. The order matters: 3 lasagnas among 4 people gives \displaystyle \frac{3}{4} each, but 4 lasagnas among 3 people gives \displaystyle \frac{4}{3} = 1\frac{1}{3} each.

You can also say, before doing any exact work, between which two whole numbers a share lies. Count how many whole wholes each share gets. For 7 pans among 3 people, 3 \times 2 = 6 is not more than 7 but 3 \times 3 = 9 is, so each share is more than 2 pans and less than 3. On the number line, \displaystyle \frac{7}{3} sits between 2 and 3 (see Fractions on the number line).

Figure 3\displaystyle \frac{7}{3} sits one third of the way from 2 to 3. A share of 7 pans among 3 people is between 2 and 3 pans.

Finally, since \displaystyle \frac{a}{b} is a \div b, a share can also be written as a decimal by carrying the division past the ones place. Three lasagnas among four people is 3 \div 4 = 0.75 of a lasagna. That move is the subject of Converting between fractions and decimals; here it only gives a second name for the same share.

Results

A fraction is a division. For whole numbers a and b with b not 0, \displaystyle \frac{a}{b} = a \div b.
Sharing. Sharing a equal wholes equally among b people gives each person \displaystyle \frac{a}{b} of one whole.
Two ways to see one share. \displaystyle \frac{1}{b} of a wholes is the same amount as \displaystyle \frac{a}{b} of one whole.
Remainders become fractions. If a \div b = q remainder r, then \displaystyle \frac{a}{b} = q\frac{r}{b}, and the share lies between q and q + 1 (it equals q when r = 0).

Why it is true

Say a wholes are shared among b people. Cut every whole into b equal pieces, each \displaystyle \frac{1}{b} of a whole. There are now a \times b pieces. Since there are b people, they can each take a pieces: b people times a pieces each is b \times a pieces, which is all of them, because b \times a = a \times b (the commutative property of multiplication). Every person has the same, and nothing is left over, so this is a fair share of all a wholes. And a pieces of size \displaystyle \frac{1}{b} is exactly what \displaystyle \frac{a}{b} means. So the share, a \div b, is \displaystyle \frac{a}{b}.

There is a second way to say it. Division undoes multiplication: a \div b is the number that gives a when you multiply it by b. Check that \displaystyle \frac{a}{b} does this. b copies of \displaystyle \frac{a}{b} is b \times a pieces of size \displaystyle \frac{1}{b}, and every b of those pieces make one whole, so together they make a wholes. So \displaystyle b \times \frac{a}{b} = a, which is what a \div b has to do.

For the remainder rule, suppose a \div b = q remainder r, which means a = q \times b + r with r less than b. Hand out q whole things to each person, using q \times b of them. The r that are left are shared too, and by the rule just proved each person gets \displaystyle \frac{r}{b} more. So each share is \displaystyle q + \frac{r}{b} = q\frac{r}{b}. Since r is less than b, the extra \displaystyle \frac{r}{b} is less than one whole, which is why the share is less than q + 1.

Worked examples

Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, and Examples 5 to 7 with Part 3.

Example 1

Two lasagnas among five people

Ijeoma’s family of 5 shares 2 lasagnas equally. How much lasagna does each person get? Show it both ways.

Solution

The lasagnas are being shared, so 2 is divided by 5: each person gets \displaystyle 2 \div 5 = \frac{2}{5} of a lasagna.

Cut each lasagna into 5. Each piece is \displaystyle \frac{1}{5} of a lasagna, and there are 2 \times 5 = 10 pieces. Shared among 5 people, that is 10 \div 5 = 2 pieces each, one from each lasagna. Two pieces of size \displaystyle \frac{1}{5} is \displaystyle \frac{2}{5}.

Line them up. Put the 2 lasagnas end to end and cut the row into 5 equal parts. The row is 10 fifths long, so each part is 2 fifths: \displaystyle \frac{2}{5} of a lasagna.

Check: 5 people with \displaystyle \frac{2}{5} each have \displaystyle \frac{10}{5} between them, and \displaystyle \frac{10}{5} = 2, the two lasagnas.

Example 2

Same share, bigger group

At the next table, 8 people share 6 brownie pans. Goro says each person there gets as much of a pan as each of Charu’s 4 people got of a lasagna when they shared 3 lasagnas. Is he right?

Solution

Each of the 8 people gets \displaystyle 6 \div 8 = \frac{6}{8} of a pan.

Now split the table in two: 4 people with 3 pans, and 4 people with the other 3 pans. Each half is exactly Charu’s situation, so each person gets \displaystyle \frac{3}{4} of a pan. Splitting the group did not change anyone’s share, so

\frac{6}{8} = \frac{3}{4}

The same thing shows up with Equivalent fractions: dividing the top and the bottom of \displaystyle \frac{6}{8} by 2 gives \displaystyle \frac{3}{4}. Goro is right: both shares are \displaystyle \frac{3}{4} of a whole.

Example 3

Eleven lasagnas among four tables

The potluck has 11 lasagnas to share equally among 4 tables. How much does each table get? Give the answer as a fraction and as a mixed number.

Solution

The lasagnas are shared, so 11 is divided by 4: each table gets \displaystyle \frac{11}{4} of a lasagna.

To write it as a mixed number, hand out whole lasagnas first. 4 \times 2 = 8 is not more than 11, but 4 \times 3 = 12 is, so each table gets 2 whole lasagnas, using 8. That leaves 11 - 8 = 3 lasagnas, and 3 shared among 4 is \displaystyle \frac{3}{4} each, by Part 1. So

\frac{11}{4} = 11 \div 4 = 2\frac{3}{4}

In long-division words: 11 \div 4 = 2 remainder 3, and the remainder over the divisor gives the fraction part.

Check: 4 tables with 2 whole lasagnas each use 8, and 4 tables with \displaystyle \frac{3}{4} each use \displaystyle \frac{12}{4} = 3 more. That is 8 + 3 = 11.

Example 4

A school potluck with a two-digit divisor

The school potluck has 75 brownie pans for 12 classrooms, shared equally. How many pans does each classroom get? Write the answer as a mixed number in lowest terms.

Solution

Each classroom gets \displaystyle 75 \div 12 = \frac{75}{12} pans.

Divide. 12 does not go into 7, so look at all 75 ones. Estimate: 12 \times 6 = 72 and 12 \times 7 = 84, so the quotient is 6, using 72 pans, with 75 - 72 = 3 left over. So 75 \div 12 = 6 remainder 3, and the 3 left-over pans shared among 12 classrooms give each \displaystyle \frac{3}{12} of a pan:

\frac{75}{12} = 6\frac{3}{12} = 6\frac{1}{4}

The last step divides the top and the bottom of \displaystyle \frac{3}{12} by 3, which keeps the value. A fraction is in lowest terms when its top and bottom share no factor but 1, and 1 and 4 share none.

Check: 12 \times 6 = 72 pans, and 12 classrooms with \displaystyle \frac{1}{4} of a pan each use 3 more pans: 72 + 3 = 75.

Example 5

Six and four, two ways round

Goro’s table has 4 people and 6 lasagnas. Ijeoma’s table has 6 people and 4 lasagnas. Write each person’s share at each table, and say which table gets more per person.

Solution

At Goro’s table the 6 lasagnas are shared among 4 people, so the division is 6 \div 4:

\frac{6}{4} = 1\frac{2}{4} = 1\frac{1}{2} \text{ lasagnas each}

since 4 goes into 6 once with 2 left, and \displaystyle \frac{2}{4} = \frac{1}{2}.

At Ijeoma’s table the 4 lasagnas are shared among 6 people, so the division goes the other way round, 4 \div 6:

\frac{4}{6} = \frac{2}{3} \text{ of a lasagna each}

Goro’s table gets more: more than one lasagna each, against less than one. The same two numbers in the opposite order give a very different share. The amount being shared always goes on top.

Example 6

Between which two whole numbers

There are 47 brownie pans for 9 groups. Without finishing the division, say between which two whole numbers each group’s share lies. Then find the exact share, and how many more pans would be needed for each group to get the larger whole number of pans.

Solution

Each group gets \displaystyle \frac{47}{9} pans. Find the whole pans first: 9 \times 5 = 45 is not more than 47, and 9 \times 6 = 54 is. So each group gets at least 5 whole pans but cannot get 6. The share is between 5 and 6.

Exactly: 47 \div 9 = 5 remainder 47 - 45 = 2, so

\frac{47}{9} = 5\frac{2}{9} \text{ pans}

For each group to get 6 pans, the potluck needs 9 \times 6 = 54 pans, which is 54 - 47 = 7 more.

Example 7

Working backward, then sharing again

At the end of the night, the 6 people at Charu’s table had each eaten \displaystyle 1\frac{2}{3} lasagnas, and all the lasagna at that table was gone. How many lasagnas were on the table? If the same lasagnas had been shared among 8 people instead, how much would each get? Give that share as a fraction, a mixed number and a decimal.

Solution

Each share is the number of lasagnas divided by 6, so the number of lasagnas is 6 times one share, because division undoes multiplication. Six shares of \displaystyle 1\frac{2}{3} are 6 whole lasagnas and 6 pieces of \displaystyle \frac{2}{3}:

6 + 6 \times \frac{2}{3} = 6 + \frac{12}{3} = 6 + 4 = 10

There were 10 lasagnas.

Shared among 8, each person gets \displaystyle 10 \div 8 = \frac{10}{8}. Since 10 \div 8 = 1 remainder 2,

\frac{10}{8} = 1\frac{2}{8} = 1\frac{1}{4}

and \displaystyle \frac{1}{4} = 0.25, so the share is 1.25 lasagnas. Carrying on the long division gives the same: 2 ones left is 20 tenths, 20 \div 8 is 2 tenths with 4 left, and 40 hundredths \div 8 is 5 hundredths, so 10 \div 8 = 1.25.

Check: \displaystyle 8 \times 1\frac{1}{4} = 8 + 2 = 10 lasagnas.

Boundaries

The bottom number cannot be 0. Sharing among 0 people has no answer, and \displaystyle \frac{a}{0} is not a number: no number times 0 gives a when a is not 0, and when a is 0 every number does, so there is no single answer.
The wholes must be the same size, and the shares equal. \displaystyle \frac{3}{4} of a lasagna means \displaystyle \frac{3}{4} of one of those lasagnas. If one lasagna were twice as big as the others, the fraction would not say how much each person got.
The wholes must be things that can be cut. If 7 people share 3 buses, nobody gets \displaystyle \frac{3}{7} of a bus. Then the whole-number answer with a remainder is the one that fits, and Remainders and what they mean decides what to do with what is left over.
The fraction is the share of one whole, not of the whole pile. Each of 4 people sharing 3 lasagnas gets \displaystyle \frac{3}{4} of one lasagna, which is \displaystyle \frac{1}{4} of all three. Both sentences are true; they measure against different wholes.
This lesson divides whole numbers. Dividing a fraction by a whole number, or by another fraction, comes later.

Where this goes next

The rest of this unit multiplies fractions. A fraction times a fraction finds a fraction of a fraction, and A fraction of a fraction of an amount takes a part of a part of an amount, which grows straight out of the “one fourth of three lasagnas” picture in Part 1. In the next unit, Dividing fractions: sharing into equal parts shares a fraction of a whole among several people, the same sharing story with a smaller pile.

Since \displaystyle \frac{a}{b} is a \div b, every fraction can be turned into a decimal by dividing, which is how Converting between fractions and decimals works. Later, a fraction is also read as a comparison of two amounts: What a ratio is: language and notation writes “3 lasagnas for every 4 people”, and Unit rates when the amounts are fractions asks how much goes to one person. In algebra, Translating between words and expressions writes “x shared among 5” as \displaystyle \frac{x}{5}, using the bar as the division sign, exactly as here.

Exercises

Exercises 1 to 8 go with Part 1, 9 to 16 with Part 2, and 17 to 22 with Part 3. The reasoning group draws on all three.

Part 1: sharing a few wholes

Write each fraction as a division: (a) \displaystyle \frac{2}{9}, (b) \displaystyle \frac{9}{2}, (c) \displaystyle \frac{1}{7}. Then (d) write 3 \div 10 as a fraction.
Charu and two friends, 3 people in all, share 2 lasagnas equally. How much lasagna does each get?
Six people share 5 brownie pans equally. How much of a pan does each person get?
Five people share 4 lasagnas. Describe how cutting each lasagna into 5 pieces gives the share, and say what the share is.
How much does each person get? (a) 1 lasagna shared among 6 people. (b) 6 lasagnas shared among 6 people.
Which gives each person more: 3 brownie pans shared among 7 people, or 3 brownie pans shared among 8 people? Explain using the shares.
Ijeoma says that 10 people sharing 4 lasagnas get the same share as 5 people sharing 2 lasagnas. Show that she is right.
At one table, each of 7 people got \displaystyle \frac{3}{7} of a lasagna, and nothing was left. How many lasagnas were shared?

Part 2: more wholes than people

Three people share 10 brownie pans. Write each share as a fraction and as a mixed number.
Six people share 11 lasagnas. Write each share as a fraction and as a mixed number.
Write each as a mixed number: (a) \displaystyle \frac{13}{5}, (b) \displaystyle \frac{22}{7}, (c) \displaystyle \frac{31}{6}.
There are 26 brownie pans for 8 classrooms. How many pans does each classroom get? Give a mixed number in lowest terms.
The town potluck has 86 brownie pans for 12 tables. How many pans does each table get? Give a mixed number in lowest terms.
Eight people share 20 lasagnas. Write each share as a fraction, then as a mixed number in lowest terms.
Write each division as a mixed number or a whole number: (a) 50 \div 9, (b) 61 \div 10, (c) 45 \div 15.
At Goro’s table, 3 people share 8 brownie pans. At Charu’s table, 5 people share 13 brownie pans. At which table does each person get more?

Part 3: reading the problem

Write each share as a fraction, and say which number is divided by which. (a) Nine people share four lasagnas. (b) Four people share nine lasagnas; give this share as a mixed number too.
Between which two whole numbers does each lie? Then give it as a mixed number. (a) \displaystyle \frac{23}{4}, (b) \displaystyle \frac{40}{9}, (c) \displaystyle \frac{59}{8}.
Four people share 33 brownie pans. Between which two whole numbers is each share? How many more pans would be needed for each person to get the larger whole number of pans?
Write each share as a fraction and as a decimal. (a) 3 lasagnas shared among 2 people. (b) 7 brownie pans shared among 4 people. (c) 13 lasagnas shared among 10 people.
Eight people each got \displaystyle 1\frac{3}{8} lasagnas, and all the lasagna was eaten. How many lasagnas were there?
Ijeoma writes 6 \div 10 for “six people share ten brownie pans”. Is she right? Find each person’s share.

Reasoning

Seven brownie pans are split between 2 people. Charu says the answer is “3 remainder 1”. Goro says it is \displaystyle 3\frac{1}{2}. Explain what each answer means, and describe a situation where each one is the answer you want.
Explain, without working out any exact answer, why sharing fewer wholes than people always gives each person less than one whole, and sharing more wholes than people always gives more than one.
Explain why the fraction part of a mixed number made from a division, such as the \displaystyle \frac{2}{5} in \displaystyle \frac{12}{5} = 2\frac{2}{5}, is always less than one whole. Use the remainder.

Answers

(a) 2 \div 9. (b) 9 \div 2. (c) 1 \div 7. (d) \displaystyle \frac{3}{10}. The top number is the one being divided.
\displaystyle \frac{2}{3} of a lasagna. Cut each lasagna into thirds: 6 pieces, 2 each.
\displaystyle \frac{5}{6} of a pan. Cut each pan into 6: 30 pieces, 30 \div 6 = 5 pieces each.
\displaystyle \frac{4}{5} of a lasagna. Cutting each into 5 makes 4 \times 5 = 20 pieces of size \displaystyle \frac{1}{5}; 20 \div 5 = 4 pieces each, one from each lasagna.
(a) \displaystyle \frac{1}{6} of a lasagna. (b) \displaystyle \frac{6}{6} = 1 whole lasagna each.
3 pans among 7 people. The shares are \displaystyle \frac{3}{7} and \displaystyle \frac{3}{8}: the same 3 pans cut into fewer shares makes each share bigger, so \displaystyle \frac{3}{7} is more than \displaystyle \frac{3}{8}.
Split the 10 people into two groups of 5, each with 2 lasagnas. Each group is the second situation, so the shares match: \displaystyle \frac{4}{10} = \frac{2}{5}.
3 lasagnas. The total is 7 shares of \displaystyle \frac{3}{7}, which is \displaystyle \frac{21}{7} = 3.
\displaystyle \frac{10}{3} = 3\frac{1}{3} pans. 10 \div 3 = 3 remainder 1.
\displaystyle \frac{11}{6} = 1\frac{5}{6} lasagnas. 11 \div 6 = 1 remainder 5.
(a) \displaystyle 2\frac{3}{5}, since 13 = 2 \times 5 + 3. (b) \displaystyle 3\frac{1}{7}, since 22 = 3 \times 7 + 1. (c) \displaystyle 5\frac{1}{6}, since 31 = 5 \times 6 + 1.
\displaystyle 3\frac{1}{4} pans. 26 \div 8 = 3 remainder 2, so \displaystyle 3\frac{2}{8} = 3\frac{1}{4}.
\displaystyle 7\frac{1}{6} pans. 12 \times 7 = 84, so 86 \div 12 = 7 remainder 2, and \displaystyle 7\frac{2}{12} = 7\frac{1}{6}.
\displaystyle \frac{20}{8} = 2\frac{4}{8} = 2\frac{1}{2} lasagnas. 20 \div 8 = 2 remainder 4.
(a) \displaystyle 5\frac{5}{9}, since 9 \times 5 = 45 leaves 5. (b) \displaystyle 6\frac{1}{10}. (c) 3, with no remainder, since 15 \times 3 = 45.
Goro’s table. The shares are \displaystyle \frac{8}{3} = 2\frac{2}{3} and \displaystyle \frac{13}{5} = 2\frac{3}{5}. Both have 2 whole pans, and \displaystyle \frac{2}{3} = \frac{10}{15} is more than \displaystyle \frac{3}{5} = \frac{9}{15} (see Comparing fractions).
(a) \displaystyle \frac{4}{9} of a lasagna: the 4 lasagnas are divided by 9, 4 \div 9. (b) \displaystyle \frac{9}{4} = 2\frac{1}{4} lasagnas: the 9 lasagnas are divided by 4, 9 \div 4.
(a) Between 5 and 6: \displaystyle 5\frac{3}{4}, since 4 \times 5 = 20 and 4 \times 6 = 24. (b) Between 4 and 5: \displaystyle 4\frac{4}{9}, since 9 \times 4 = 36 and 9 \times 5 = 45. (c) Between 7 and 8: \displaystyle 7\frac{3}{8}, since 8 \times 7 = 56 and 8 \times 8 = 64.
Between 8 and 9 pans: \displaystyle 33 \div 4 = 8\frac{1}{4}. For 9 each, 4 \times 9 = 36 pans are needed, so 3 more.
(a) \displaystyle \frac{3}{2} = 1.5 lasagnas. (b) \displaystyle \frac{7}{4} = 1\frac{3}{4} = 1.75 pans, since \displaystyle \frac{3}{4} = 0.75. (c) \displaystyle \frac{13}{10} = 1.3 lasagnas, 13 tenths.
11 lasagnas. Eight shares of \displaystyle 1\frac{3}{8} are 8 wholes and \displaystyle \frac{24}{8} = 3 more: 8 + 3 = 11.
No. The pans are being shared, so they are divided: \displaystyle 10 \div 6 = \frac{10}{6}, which is \displaystyle 1\frac{4}{6} = 1\frac{2}{3} pans each. Her \displaystyle 6 \div 10 = \frac{3}{5} would be six pans among ten people.
Both are correct answers to different questions. “3 remainder 1” says each person takes 3 whole pans and 1 pan is left uncut. That is the answer you want if the pans are carried home whole, for instance to return them. \displaystyle 3\frac{1}{2} says the last pan is cut in half too, so every bit of brownie is shared; that is the answer when everything is eaten. The two agree: the remainder 1 shared between 2 is the \displaystyle \frac{1}{2}.
If there are fewer wholes than people, giving everyone one whole would need more wholes than there are, so each share is less than one. If there are more wholes than people, everyone can take one whole and some are still left to share, so each share is more than one. (With equal numbers, each share is exactly one.)
In a \div b = q remainder r, the remainder is always less than b: if b or more were left, everyone could take one more whole. The fraction part is \displaystyle \frac{r}{b}, fewer than b pieces of size \displaystyle \frac{1}{b}, and it takes b of them to make one whole. For 12 \div 5 = 2 remainder 2, the \displaystyle \frac{2}{5} is 2 fifths, short of 5 fifths.