Grade 5 · 5.NF.B.4A · 5.NF.B.4B

A fraction times a fraction

Junko paints part of a banner blue, and Eshan paints part of the blue part gold. What fraction of the whole banner is gold?

You need

Simplifying a fraction to simplest formGrade 4 Mixed numbers and improper fractions: rewriting and comparingGrade 4 A fraction times a whole numberGrade 4

Unlocks

A fraction of a fraction of an amountGrade 5 Multiplying as scalingGrade 5 Multiplying mixed numbersGrade 5

Where you are 4 in Multiplying fractions

Fractions, ratio and percent3

Fractions15

Multiplying fractions4

A fraction times a whole number

A fraction times a fraction

Multiplying as scaling Multiplying mixed numbers

When you reach for it

The art club is painting paper banners for the school fair. Junko paints part of a banner blue. Then Eshan paints gold letters over part of the blue part. How much of the whole banner is gold? That is a part of a part, and finding it is multiplying a fraction by a fraction.

You already know that a fraction times a whole number takes a fraction of that number: \displaystyle \frac{1}{4} \times 8 is one fourth of 8 (A fraction times a whole number). The same word “of” carries over. \displaystyle \frac{1}{2} \times \frac{1}{3} means one half of one third. The answer is a fraction of the whole banner, and this lesson finds it with a picture first and then with a rule the picture explains:

\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}

Multiply the tops (the numerators) and multiply the bottoms (the denominators). The lesson takes four days. Day 1 is a part of a part when both parts are unit fractions. Day 2 is any two fractions and why the rule is true. Day 3 is the area of a rectangle whose sides are fractions. Day 4 is simplifying before you multiply, and fractions greater than 1.

The procedure

Day 1: a part of a part

Take \displaystyle \frac{1}{2} \times \frac{1}{3}. Draw the banner as a rectangle. Junko paints \displaystyle \frac{1}{3} of it blue: cut the banner into 3 equal columns and shade one.

Now Eshan paints gold on \displaystyle \frac{1}{2} of the blue part. Half of one column is found by cutting that column into 2 equal pieces. Cut the other columns the same way, with one line straight across the whole banner. Now the banner is 3 columns by 2 rows, so it has 3 \times 2 = 6 equal pieces. The gold part is 1 of them.

Figure 1Half of one third of the banner is one of six equal pieces, so \displaystyle \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}.

So one half of one third is one sixth:

\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}

The same thing happens for any two unit fractions. A unit fraction has 1 on top, like \displaystyle \frac{1}{5}. To take \displaystyle \frac{1}{a} of \displaystyle \frac{1}{b}, cut the banner into b columns and every column into a rows. That makes a \times b equal pieces, and the part of a part is one of them:

\frac{1}{a} \times \frac{1}{b} = \frac{1}{a \times b}

The pieces get smaller, not bigger. Cutting a third into halves makes sixths, and a sixth is less than a third. A bigger bottom number means more pieces, so each piece is smaller.

Day 2: any two fractions

Take \displaystyle \frac{2}{3} \times \frac{3}{5}: two thirds of three fifths. Musa paints \displaystyle \frac{3}{5} of a banner red, so cut it into 5 columns and shade 3. Then Junko paints \displaystyle \frac{2}{3} of the red part with white dots. Cut across into 3 rows and take 2 of them, but only inside the red columns.

Figure 2The darker part is 2 rows by 3 columns, 6 pieces, out of 3 rows by 5 columns, 15 pieces.

Count the whole banner: 3 rows by 5 columns is 3 \times 5 = 15 equal pieces. Count the dotted part: 2 rows by 3 columns is 2 \times 3 = 6 pieces. So

\frac{2}{3} \times \frac{3}{5} = \frac{2 \times 3}{3 \times 5} = \frac{6}{15}

This is why the rule works, for any two fractions \displaystyle \frac{a}{b} and \displaystyle \frac{c}{d}:

The bottoms multiply because the whole banner is cut b ways in one direction and d ways in the other. That makes b \times d pieces, all the same size, since every column is cut by the same lines.
The tops multiply because the part of a part is a rectangle of its own: a rows by c columns, which is a \times c pieces.

The answer \displaystyle \frac{6}{15} is correct, but it can be named in bigger pieces. Both 6 and 15 can be divided by 3, and dividing the top and the bottom by the same number gives an equal fraction (Simplifying a fraction to simplest form):

\frac{6}{15} = \frac{6 \div 3}{15 \div 3} = \frac{2}{5}

So \displaystyle \frac{2}{3} \times \frac{3}{5} = \frac{2}{5}. When 2 and 5 have no common factor but 1, the fraction is in simplest form.

Order does not matter. \displaystyle \frac{3}{5} \times \frac{2}{3} is the same picture turned on its side, so it is also \displaystyle \frac{6}{15}. In symbols, 3 \times 2 = 2 \times 3 on top and 5 \times 3 = 3 \times 5 on the bottom. This is the commutative property: you can swap the two factors and the product stays the same.

Day 3: rectangles with fraction sides

You know that a rectangle’s area is its length times its width, because the area is a count of unit squares in rows (Area of rectangles and squares). Does that still work when the sides are fractions?

Take a banner patch \displaystyle \frac{3}{4} meter wide and \displaystyle \frac{2}{3} meter tall. Draw a square 1 meter on each side. Its area is 1 square meter. Cut its width into fourths and its height into thirds. That makes 4 \times 3 = 12 equal tiles, each \displaystyle \frac{1}{4} meter wide and \displaystyle \frac{1}{3} meter tall. Twelve of them fill one square meter, so each tile has area \displaystyle \frac{1}{12} square meter.

Figure 3Twelve tiles fill the square meter, so each is \displaystyle \frac{1}{12} of a square meter. The rectangle covers 2 \times 3 = 6 of them.

The patch is 3 tiles wide and 2 tiles tall, so it covers 3 \times 2 = 6 tiles. Its area is \displaystyle \frac{6}{12} square meter, which is \displaystyle \frac{1}{2} square meter.

Multiply the side lengths and you get the same thing:

\frac{3}{4} \times \frac{2}{3} = \frac{3 \times 2}{4 \times 3} = \frac{6}{12} = \frac{1}{2}

The match is not luck. The bottoms 4 \times 3 count the tiles in the square meter, which names the tile size. The tops 3 \times 2 count the tiles in the patch. So length times width gives the area for fraction sides too.

When the two sides have the same bottom number, the tiles are squares. With fourths on both sides, each tile is a square \displaystyle \frac{1}{4} meter on a side, and 4 \times 4 = 16 of them fill a square meter, so each has area \displaystyle \frac{1}{16} square meter.

Day 4: simplify first, and fractions greater than 1

Take \displaystyle \frac{3}{4} \times \frac{5}{6}. Multiplying straight through gives

\frac{3}{4} \times \frac{5}{6} = \frac{3 \times 5}{4 \times 6} = \frac{15}{24}

Both 15 and 24 can be divided by 3, so \displaystyle \frac{15}{24} = \frac{5}{8}. That works, but there is a shorter road. Look at the step before the multiplying, \displaystyle \frac{3 \times 5}{4 \times 6}. The 3 on top and the 6 on the bottom share the factor 3. Write 6 as 2 \times 3:

\frac{3 \times 5}{4 \times 6} = \frac{3 \times 5}{4 \times 2 \times 3}

Now divide the top and the bottom by 3. That gives an equal fraction, the same move as on Day 2. The top becomes 5 and the bottom becomes 4 \times 2 = 8, so the product is \displaystyle \frac{5}{8}, the same answer with smaller numbers along the way.

The 3 came from the first fraction and the 6 from the second. Pairing them is allowed. Once the tops are all on one line and the bottoms on another, it does not matter which fraction each number came from. The commutative property lets the factors on top be put in any order, and the same for the bottom. So you may divide any top and any bottom by a common factor, before you multiply. In short, write the new numbers in place: 3 becomes 1, 6 becomes 2, and

\frac{3}{4} \times \frac{5}{6} = \frac{1}{4} \times \frac{5}{2} = \frac{5}{8}

If you divide by every common factor first, the product comes out in simplest form.

Fractions greater than 1. A fraction like \displaystyle \frac{9}{4}, with a top bigger than its bottom, is more than one whole. The rule is the same: tops times tops, bottoms times bottoms, and simplify first if you can. The answer may also be more than 1. It can be left as a fraction or written as a mixed number, a whole number and a fraction, such as \displaystyle \frac{15}{2} = 7\frac{1}{2} (Mixed numbers and improper fractions: rewriting and comparing turns one form into the other).

Worked examples

Examples 1 and 2 go with Day 1, Examples 3 and 4 with Day 2, Examples 5 and 6 with Day 3, and Examples 7 and 8 with Day 4.

Example 1

A fourth of a third

Musa paints \displaystyle \frac{1}{3} of a banner green. Junko paints yellow stars on \displaystyle \frac{1}{4} of the green part. What fraction of the whole banner has yellow stars?

Solution

The stars cover \displaystyle \frac{1}{4} of \displaystyle \frac{1}{3} of the banner, which is \displaystyle \frac{1}{4} \times \frac{1}{3}.

Draw the banner and cut it into 3 columns; shade one green. To take a fourth of that column, cut it into 4 equal rows, and carry the lines across the whole banner so every piece is the same size. The banner now has 3 \times 4 = 12 pieces, and the starred part is one of them:

\frac{1}{4} \times \frac{1}{3} = \frac{1}{12}

So \displaystyle \frac{1}{12} of the banner has yellow stars.

If Junko had painted first and Musa second, \displaystyle \frac{1}{3} of \displaystyle \frac{1}{4}, the picture would be 4 columns cut into 3 rows: still 12 pieces, still one of them. So \displaystyle \frac{1}{3} \times \frac{1}{4} = \frac{1}{12} too, as the commutative property says.

Example 2

A third of four fifths

Eshan paints \displaystyle \frac{4}{5} of a banner orange. Then he paints a white border over \displaystyle \frac{1}{3} of the orange part. What fraction of the banner has the border?

Solution

The border covers \displaystyle \frac{1}{3} \times \frac{4}{5} of the banner.

Cut the banner into 5 columns and shade 4 orange. Cut across into 3 rows. One row of the orange part is a third of it, because each orange column is split into 3 equal pieces and one piece of each is taken.

The whole banner has 3 \times 5 = 15 pieces. The border is 1 row across 4 columns, so it is 1 \times 4 = 4 pieces:

\frac{1}{3} \times \frac{4}{5} = \frac{1 \times 4}{3 \times 5} = \frac{4}{15}

The border covers \displaystyle \frac{4}{15} of the banner. Check: \displaystyle \frac{4}{5} is \displaystyle \frac{12}{15}, and a third of 12 fifteenths is 4 fifteenths.

Example 3

Three fifths of three fourths

Musa paints \displaystyle \frac{3}{4} of a banner red. Junko paints white dots on \displaystyle \frac{3}{5} of the red part. What fraction of the banner is dotted?

Solution

The dotted part is \displaystyle \frac{3}{5} \times \frac{3}{4} of the banner.

In the area model the banner is cut into 4 columns, with 3 red, and into 5 rows, with 3 taken inside the red part. The whole has 5 \times 4 = 20 pieces, and the dotted part is 3 rows by 3 columns, 3 \times 3 = 9 pieces:

\frac{3}{5} \times \frac{3}{4} = \frac{3 \times 3}{5 \times 4} = \frac{9}{20}

The only common factor of 9 and 20 is 1, so \displaystyle \frac{9}{20} is in simplest form. The dotted part is \displaystyle \frac{9}{20} of the banner, a little less than half.

Example 4

Simplifying a product

Eshan paints \displaystyle \frac{3}{8} of a banner purple. Musa writes the club’s name over \displaystyle \frac{4}{9} of the purple part. What fraction of the banner has the name on it, in simplest form?

Solution

The name covers \displaystyle \frac{4}{9} \times \frac{3}{8} of the banner. Multiply the tops and the bottoms:

\frac{4}{9} \times \frac{3}{8} = \frac{4 \times 3}{9 \times 8} = \frac{12}{72}

The area model would have 72 tiny pieces, with 12 of them covered. To name that in bigger pieces, look for a number that divides both 12 and 72. The number 12 itself does, because 72 = 12 \times 6:

\frac{12}{72} = \frac{12 \div 12}{72 \div 12} = \frac{1}{6}

The name covers \displaystyle \frac{1}{6} of the banner. Dividing by 2 and then by 6, or by 4 and then by 3, reaches the same \displaystyle \frac{1}{6}; dividing by the biggest common factor just does it in one step.

Example 5

Tiles with the same bottom number

A banner panel is \displaystyle \frac{3}{4} meter wide and \displaystyle \frac{5}{4} meters tall. Find its area by tiling, then check by multiplying.

Solution

Both sides are counted in fourths of a meter, so use square tiles \displaystyle \frac{1}{4} meter on a side. A square meter is 4 tiles by 4 tiles, 16 tiles, so each tile has area \displaystyle \frac{1}{16} square meter.

The panel is 3 tiles wide and 5 tiles tall. It is taller than 1 meter, so it sticks out one row past the square meter, which is fine: the tiles are the same size everywhere.

Figure 4Sixteen quarter-meter tiles fill a square meter. The rectangle covers 3 \times 5 = 15 of them.

The panel covers 3 \times 5 = 15 tiles, so its area is \displaystyle \frac{15}{16} square meter.

Multiplying the sides gives the same answer:

\frac{3}{4} \times \frac{5}{4} = \frac{3 \times 5}{4 \times 4} = \frac{15}{16}

The panel is more than 1 meter tall but has less than 1 square meter of area, because it is less than 1 meter wide.

Example 6

Which banner is bigger?

Eshan’s banner is \displaystyle \frac{2}{3} meter wide and \displaystyle \frac{9}{10} meter tall. Musa’s is \displaystyle \frac{4}{5} meter wide and \displaystyle \frac{3}{4} meter tall. Which uses more paper?

Solution

Find each area as length times width.

Eshan’s banner:

\frac{2}{3} \times \frac{9}{10} = \frac{2 \times 9}{3 \times 10} = \frac{18}{30} = \frac{3}{5}

dividing 18 and 30 by 6. In tiles, a square meter cut into thirds one way and tenths the other has 30 tiles, and the banner covers 2 \times 9 = 18 of them.

Musa’s banner:

\frac{4}{5} \times \frac{3}{4} = \frac{4 \times 3}{5 \times 4} = \frac{12}{20} = \frac{3}{5}

dividing 12 and 20 by 4.

Both banners have area \displaystyle \frac{3}{5} square meter, so they use the same amount of paper, even though they have different shapes.

Example 7

Simplify, then multiply

Find \displaystyle \frac{8}{9} \times \frac{15}{16} in simplest form.

Solution

Multiplying straight through gives \displaystyle \frac{120}{144}, large numbers to simplify. Look for common factors first, pairing any top with any bottom.

The top 8 and the bottom 16 share 8: 8 \div 8 = 1 and 16 \div 8 = 2. The top 15 and the bottom 9 share 3: 15 \div 3 = 5 and 9 \div 3 = 3. So

\frac{8}{9} \times \frac{15}{16} = \frac{1}{3} \times \frac{5}{2} = \frac{5}{6}

The tops left are 1 and 5, which share no factor with the bottoms 3 and 2, so \displaystyle \frac{5}{6} is in simplest form. Check the long way: \displaystyle \frac{120}{144} divided top and bottom by 24 is \displaystyle \frac{5}{6}.

Example 8

A banner bigger than a square meter

The banner for the gym wall is \displaystyle \frac{10}{3} meters long and \displaystyle \frac{9}{4} meters tall. What is its area?

Solution

Area is length times width, \displaystyle \frac{10}{3} \times \frac{9}{4}. Both fractions are greater than 1, and the rule does not change.

Simplify first. The top 10 and the bottom 4 share 2: they become 5 and 2. The top 9 and the bottom 3 share 3: they become 3 and 1. So

\frac{10}{3} \times \frac{9}{4} = \frac{5}{1} \times \frac{3}{2} = \frac{15}{2}

As a mixed number, \displaystyle \frac{15}{2} is 7 wholes and 1 half, because 7 \times 2 = 14 and one half is left: \displaystyle \frac{15}{2} = 7\frac{1}{2}. The banner has area \displaystyle 7\frac{1}{2} square meters.

A rough check: the banner is a bit more than 3 meters by a bit more than 2 meters, so its area should be a bit more than 3 \times 2 = 6 square meters. It is.

Boundaries

The bottoms can never be 0. A fraction with 0 on the bottom names no amount, so \displaystyle \frac{a}{b} \times \frac{c}{d} needs b and d to be counting numbers.

“Of” means times when the second amount is the whole you take the part from. Eshan’s gold is \displaystyle \frac{1}{2} of the blue part, and the blue part is \displaystyle \frac{1}{3} of the banner, so the gold is \displaystyle \frac{1}{2} \times \frac{1}{3} of the banner. The answer is always a fraction of the original whole. When that whole is an amount such as a number of people, there is one more step; that is A fraction of a fraction of an amount.

The rule is for multiplying only. Adding fractions does not add the tops and add the bottoms; it needs a common bottom first (Adding and subtracting fractions with unlike denominators). In the same way, simplifying before multiplying pairs a top with a bottom. It never divides two tops, or two bottoms, by a number, because that would change the value.

A single square can only show a product of fractions that are each at most 1. For a fraction greater than 1 the picture needs more than one whole, as in Example 5, but the rule is the same. A mixed number such as \displaystyle 2\frac{1}{4} is not ready for the rule as written; it is turned into a fraction first, which is Multiplying mixed numbers.

Where this goes next

Every product in this lesson can be judged before it is worked out. Taking a part of a part always gave something smaller than the part you started with, and a factor greater than 1 gave something bigger. Multiplying as scaling makes that idea exact. Next in this unit, Multiplying mixed numbers uses this rule on numbers like \displaystyle 2\frac{1}{4}, and A fraction of a fraction of an amount uses it in stories about amounts.

The rule also opens two later doors. Dividing by a fraction turns into multiplying by one (Dividing by multiplying by the reciprocal), and the tiling of Day 3 grows into boxes whose edges are fractions (Volume of a box with fraction edge lengths).

Exercises

Exercises 1 to 6 go with Day 1, 7 to 13 with Day 2, 14 to 18 with Day 3, 19 to 23 with Day 4, and the reasoning group draws on all four days.

Day 1: a part of a part

\displaystyle \frac{1}{2} \times \frac{1}{4}
\displaystyle \frac{1}{3} \times \frac{1}{5}
\displaystyle \frac{1}{6} \times \frac{1}{3}
Eshan paints \displaystyle \frac{1}{4} of a banner green, then paints \displaystyle \frac{1}{5} of the green part yellow. What fraction of the banner is yellow?
Describe an area model for \displaystyle \frac{1}{4} \times \frac{3}{5}: how many columns and rows, and how many pieces are shaded twice? Use it to find the product.
Junko says \displaystyle \frac{1}{5} of \displaystyle \frac{1}{2} of a banner is the same amount as \displaystyle \frac{1}{2} of \displaystyle \frac{1}{5} of a banner. Find both, and name the property that says she is right.

Day 2: any two fractions

Give each answer in simplest form.

\displaystyle \frac{2}{5} \times \frac{2}{3}
\displaystyle \frac{3}{4} \times \frac{3}{7}
\displaystyle \frac{5}{6} \times \frac{1}{4}
\displaystyle \frac{2}{3} \times \frac{3}{8}
\displaystyle \frac{4}{5} \times \frac{5}{6}
Junko paints \displaystyle \frac{5}{8} of a banner purple, then paints white stripes on \displaystyle \frac{2}{3} of the purple part. What fraction of the banner has stripes?
Musa’s area model shows a banner cut into 4 columns and 5 rows. He shades 3 columns, then shades 2 rows of those columns again. Which product does his model show, and what is it?

Day 3: rectangles with fraction sides

A patch is \displaystyle \frac{1}{2} meter wide and \displaystyle \frac{3}{4} meter tall. How many of its tiles fill a square meter, how many cover the patch, and what is its area?
Find the area of a banner panel \displaystyle \frac{4}{5} meter wide and \displaystyle \frac{2}{3} meter tall.
A square patch has sides \displaystyle \frac{3}{5} meter long. Find its area by tiling with squares \displaystyle \frac{1}{5} meter on a side.
A banner is \displaystyle \frac{7}{4} meters long and \displaystyle \frac{3}{4} meter tall. Find its area by tiling with quarter-meter squares, and write it as a mixed number.
Eshan’s banner is \displaystyle \frac{5}{6} meter by \displaystyle \frac{3}{5} meter. Junko’s is \displaystyle \frac{2}{3} meter by \displaystyle \frac{7}{8} meter. Whose uses more paper, and by how much?

Day 4: simplify first

Simplify before multiplying. Write answers greater than 1 as mixed numbers too.

\displaystyle \frac{4}{9} \times \frac{3}{10}
\displaystyle \frac{5}{12} \times \frac{8}{15}
\displaystyle \frac{7}{8} \times \frac{6}{35}
\displaystyle \frac{5}{3} \times \frac{9}{10}
\displaystyle \frac{14}{5} \times \frac{15}{4}

Reasoning

Eshan finds \displaystyle \frac{6}{25} \times \frac{10}{9} by dividing 6 and 9 by 3 and dividing 10 and 25 by 5 first. Junko multiplies first and then simplifies. Show that they get the same answer, and explain why Eshan may divide the 6 and the 9 even though they are in different fractions.
Use an area model to explain why the bottom of \displaystyle \frac{a}{b} \times \frac{c}{d} is b \times d, and why it would make no sense for it to be b + d.

Answers

\displaystyle \frac{1}{8}. Halving a fourth makes 4 \times 2 = 8 equal pieces; one is taken.
\displaystyle \frac{1}{15}. 3 \times 5 = 15 pieces, one taken.
\displaystyle \frac{1}{18}. 6 \times 3 = 18 pieces, one taken.
\displaystyle \frac{1}{20} of the banner. It is \displaystyle \frac{1}{5} \times \frac{1}{4} = \frac{1}{5 \times 4}.
\displaystyle \frac{3}{20}. Cut the banner into 5 columns and shade 3; cut it into 4 rows and shade 1 row of the shaded columns again. There are 4 \times 5 = 20 pieces, and 1 \times 3 = 3 are shaded twice.
Both are \displaystyle \frac{1}{10}: \displaystyle \frac{1}{5} \times \frac{1}{2} = \frac{1}{10} and \displaystyle \frac{1}{2} \times \frac{1}{5} = \frac{1}{10}. She is right by the commutative property of multiplication: swapping the factors does not change the product.
\displaystyle \frac{4}{15}. \displaystyle \frac{2 \times 2}{5 \times 3}; 4 and 15 share no factor but 1.
\displaystyle \frac{9}{28}. \displaystyle \frac{3 \times 3}{4 \times 7}.
\displaystyle \frac{5}{24}. \displaystyle \frac{5 \times 1}{6 \times 4}.
\displaystyle \frac{1}{4}. \displaystyle \frac{2 \times 3}{3 \times 8} = \frac{6}{24}, and dividing by 6 gives \displaystyle \frac{1}{4}.
\displaystyle \frac{2}{3}. \displaystyle \frac{4 \times 5}{5 \times 6} = \frac{20}{30}, and dividing by 10 gives \displaystyle \frac{2}{3}.
\displaystyle \frac{5}{12} of the banner. \displaystyle \frac{2}{3} \times \frac{5}{8} = \frac{10}{24}, and dividing by 2 gives \displaystyle \frac{5}{12}.
\displaystyle \frac{2}{5} \times \frac{3}{4} = \frac{3}{10}. The 4 columns with 3 shaded show \displaystyle \frac{3}{4}; the 5 rows with 2 taken show \displaystyle \frac{2}{5} of that. That is 6 of 20 pieces, \displaystyle \frac{6}{20} = \frac{3}{10}.
8 tiles fill a square meter (2 across by 4 down), each \displaystyle \frac{1}{2} meter by \displaystyle \frac{1}{4} meter. The patch covers 1 \times 3 = 3 of them, so its area is \displaystyle \frac{3}{8} square meter, and \displaystyle \frac{1}{2} \times \frac{3}{4} = \frac{3}{8}.
\displaystyle \frac{8}{15} square meter. \displaystyle \frac{4 \times 2}{5 \times 3} = \frac{8}{15}.
\displaystyle \frac{9}{25} square meter. 25 tiles fill a square meter; the patch is 3 by 3, which is 9 tiles.
\displaystyle \frac{21}{16} = 1\frac{5}{16} square meters. 16 tiles fill a square meter; the banner is 7 tiles by 3 tiles, 21 tiles, and 21 = 16 + 5.
Junko’s, by \displaystyle \frac{1}{12} square meter. Eshan’s is \displaystyle \frac{5}{6} \times \frac{3}{5} = \frac{15}{30} = \frac{1}{2}, which is \displaystyle \frac{6}{12}. Junko’s is \displaystyle \frac{2}{3} \times \frac{7}{8} = \frac{14}{24} = \frac{7}{12}.
\displaystyle \frac{2}{15}. 4 and 10 become 2 and 5; 3 and 9 become 1 and 3; so \displaystyle \frac{2}{3} \times \frac{1}{5} = \frac{2}{15}.
\displaystyle \frac{2}{9}. 5 and 15 become 1 and 3; 8 and 12 become 2 and 3; so \displaystyle \frac{1}{3} \times \frac{2}{3} = \frac{2}{9}.
\displaystyle \frac{3}{20}. 7 and 35 become 1 and 5; 6 and 8 become 3 and 4; so \displaystyle \frac{1}{4} \times \frac{3}{5} = \frac{3}{20}.
\displaystyle \frac{3}{2} = 1\frac{1}{2}. 5 and 10 become 1 and 2; 9 and 3 become 3 and 1; so \displaystyle \frac{1}{1} \times \frac{3}{2} = \frac{3}{2}.
\displaystyle \frac{21}{2} = 10\frac{1}{2}. 14 and 4 become 7 and 2; 15 and 5 become 3 and 1; so \displaystyle \frac{7}{1} \times \frac{3}{2} = \frac{21}{2}.
Both get \displaystyle \frac{4}{15}. Eshan: \displaystyle \frac{2}{5} \times \frac{2}{3} = \frac{4}{15}. Junko: \displaystyle \frac{6 \times 10}{25 \times 9} = \frac{60}{225}, and dividing by 15 gives \displaystyle \frac{4}{15}. Eshan’s step is allowed because the product is one fraction, \displaystyle \frac{6 \times 10}{25 \times 9}, with all the tops multiplied on one line and all the bottoms on the other. The commutative property lets the factors on each line be put in any order, so it does not matter which fraction the 6 and the 9 came from. Dividing that one top and that one bottom by 3 gives an equal fraction.
Draw a whole cut into d columns for \displaystyle \frac{c}{d}, then cut every column into b rows for \displaystyle \frac{a}{b}. The lines run across the whole, so the pieces are all the same size and there are b rows of d, which is b \times d pieces. That number of equal pieces is what the bottom counts. Adding cannot be right: cutting a third into halves makes sixths, not fifths, as \displaystyle \frac{1}{2} \times \frac{1}{3} on Day 1 shows; 6 pieces, not 3 + 2 = 5.