Grade 4 · 3.NF.A.3 · 3.NF.A.3B

Making an equivalent fraction by multiplying

A quiche is cut into slices, and then every slice is cut again. The pieces change but the quiche does not, so what happens to the fraction?

You need

Seeing equivalent fractions in pictures and on the number lineGrade 3

Unlocks

Common denominators and the least common denominatorGrade 4 Simplifying a fraction to simplest formGrade 4

Where you are 4 in Equivalent fractions

Fractions, ratio and percent3

Fractions15

Equivalent fractions4

Seeing equivalent fractions in pictures and on the number line

Making an equivalent fraction by multiplying

Common denominators and the least common denominator Simplifying a fraction to simplest form

When you reach for it

Ece, Halyna and Zoran are cooking brunch. There is a quiche, cut into slices, and there are waffles, which come cut into little squares. All morning the three of them talk in fractions: two thirds of a waffle, three quarters of the quiche.

Then somebody cuts the food again. A slice of quiche is cut into thinner slices so more people can share. A strip of waffle is cut into squares. The food on the plate has not changed at all, but the pieces have, so the fraction gets a new name.

You reach for this move whenever you want the same amount written with smaller pieces. You will need it to compare two fractions, to add them, and to write a fraction in hundredths. You already know the picture from Seeing equivalent fractions in pictures and on the number line: cutting every piece of a shape in two doubles both numbers of the fraction and changes nothing. This lesson cuts into any number of pieces and turns the picture into a rule. It has three parts, one for each day:

Part 1: cut every piece of a fraction model into the same number of smaller pieces.
Part 2: the rule, multiply the top and the bottom by the same number, and why it is true.
Part 3: find a missing top or bottom, and list a whole family of equal fractions.

The procedure

Part 1: cut every piece the same way

Zoran has a waffle cut into 3 equal strips. He spreads jam on 2 of the strips, so jam covers \tfrac{2}{3} of the waffle. Now he cuts every strip into 4 squares.

Figure 1Cut every third into 4 squares. The shaded part is the same, and now it is 8 of 12 squares, so \tfrac{2}{3} = \tfrac{8}{12}.

Count the squares. The waffle has 3 strips and each strip is now 4 squares, so the whole waffle is

3 \times 4 = 12 \text{ squares}

The jam is on 2 strips, and each of those is 4 squares too, so the jam covers

2 \times 4 = 8 \text{ squares}

The jam is 8 of the 12 squares, which is \tfrac{8}{12} of the waffle. But Zoran did not add jam or scrape any off. The jam covers exactly the same part of the waffle as before. So the two fractions name the same amount:

\frac{2}{3} = \frac{8}{12}

Two fractions that name the same amount of the same whole are called equivalent fractions.

Look at what the cut did to each number. The bottom number, the denominator, counts how many pieces make the whole. Every one of the 3 strips turned into 4 squares, so the denominator was multiplied by 4. The top number, the numerator, counts the pieces with jam. Each of those 2 strips also turned into 4 squares, so the numerator was multiplied by 4 as well.

Both numbers change together, and they have to. The knife went through every piece, the jam strips and the plain strip alike. Whatever happens to the pieces you count also happens to the pieces that make the whole. And each new square is 4 times smaller than a strip, while there are 4 times as many of them. Smaller pieces, more of them, same amount.

Part 2: the rule, and why it is true

The waffle shows a pattern that works for every fraction. Start with \tfrac{a}{b}: the whole is cut into b equal pieces and you count a of them. Here a and b are whole numbers and b is not 0. Now cut every piece into n equal smaller pieces, where n is a counting number (1, 2, 3 and so on).

The whole had b pieces, and each became n pieces. That is b groups of n, so the whole is now n \times b pieces.
The part you count had a pieces, and each became n pieces. So the part is now n \times a pieces.
Nothing was added and nothing was taken away, so the amount is the same.

That gives the rule. For any fraction \tfrac{a}{b} and any counting number n,

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

In words: multiplying the top and the bottom by the same counting number gives an equivalent fraction. It does not matter whether you write n \times a or a \times n; the two products are equal by the commutative property of multiplication.

The same fact on a number line. A fraction is also a point on the number line. To find \tfrac{3}{5}, cut the space from 0 to 1 into 5 equal jumps and take 3 of them. Now cut every one of those jumps into 3 smaller jumps.

Figure 2Every fifth is cut into 3 smaller jumps. Three fifths and nine fifteenths are the same point.

The space from 0 to 1 now holds 5 \times 3 = 15 small jumps, so each small jump is \tfrac{1}{15}. The point \tfrac{3}{5} was 3 big jumps from 0, and every big jump is now 3 small ones, so the point is 3 \times 3 = 9 small jumps from 0. The point did not move. Only the ruler got finer:

\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}

The same fact as multiplying by one. The fraction \tfrac{n}{n} means n pieces, each \tfrac{1}{n} of a whole. That is the whole thing, so \tfrac{n}{n} = 1. For example \tfrac{4}{4} = 1 and \tfrac{3}{3} = 1. When you go on to multiply one fraction by another, you will multiply the tops together and the bottoms together. Then the rule above reads

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

and since \tfrac{n}{n} is 1, and any number times 1 is the same number, the value cannot change. Multiplying as scaling uses exactly this. For now, the cutting picture is the reason; the times-one view is the same fact said another way. It is also a handy way to say what you did: “I multiplied by \tfrac{4}{4}” means “I cut every piece into 4.”

Part 3: missing numbers and families

Often you know the size of the pieces you want and need the rest of the name. Suppose

\frac{3}{5} = \frac{\square}{15}

The bottom tells you how the pieces were cut. To turn 5 pieces into 15 pieces, every fifth must be cut into 3, because 5 \times 3 = 15. Every piece was cut into 3, so the top must be multiplied by 3 too: 3 \times 3 = 9. The missing number is 9, which matches the number line in Part 2.

A missing bottom works the same way, read from the top. If \tfrac{3}{5} = \tfrac{12}{\square}, the 3 counted pieces became 12, so every piece was cut into 4, because 3 \times 4 = 12. Then the whole became 5 \times 4 = 20 pieces, and the missing number is 20.

A family of equivalent fractions. Cut the pieces of \tfrac{3}{5} into 1, 2, 3, 4 or 5 parts and you get a list of names for the same amount:

Cut every piece into	1	2	3	4	5
New name	\tfrac{3}{5}	\tfrac{6}{10}	\tfrac{9}{15}	\tfrac{12}{20}	\tfrac{15}{25}

Read across the table. The tops go 3, 6, 9, 12, 15, counting by 3. The bottoms go 5, 10, 15, 20, 25, counting by 5. They are the multiples of 3 and of 5, matched up in order. The list never ends, because you can always cut the pieces into one more part. Every fraction has an endless family like this one.

Worked examples

Examples 1 and 2 belong to Part 1. Examples 3 and 4 belong to Part 2. Examples 5 to 7 belong to Part 3.

Example 1

Cutting the quiche thinner

Ece cuts a quiche into 4 equal slices. She puts spinach on 3 of the slices. More friends are coming, so she cuts every slice into 3 thinner slices. How many slices are there now, and how many have spinach? Write the spinach part as two equivalent fractions.

Solution

Figure 3Every quarter cut into 3 thinner slices. Three quarters of the quiche is 9 twelfths of it.

Before the cut, the spinach covers 3 of 4 slices, which is \tfrac{3}{4} of the quiche.

Every slice becomes 3 slices. The quiche had 4 slices, so it now has 4 \times 3 = 12 slices. The spinach was on 3 slices, and each of those also became 3, so the spinach is on 3 \times 3 = 9 slices.

Cutting moved no spinach. It covers the same part of the quiche, now counted in twelfths:

\frac{3}{4} = \frac{9}{12}

Check by looking at one quarter: it holds 3 thin slices. Three quarters hold 3 + 3 + 3 = 9 thin slices. That agrees.

Example 2

Two different cuts of one half

Halyna and Zoran each have a waffle of the same size, and each puts strawberries on the left half. Halyna cuts each half of her waffle into 6 squares. Zoran cuts each half of his into 5 strips. Write the strawberry part of each waffle as a fraction. Are the two fractions equal?

Solution

Halyna’s waffle: there are 2 halves and each became 6 squares, so the waffle has 2 \times 6 = 12 squares. The strawberry half is 1 \times 6 = 6 of them. So

\frac{1}{2} = \frac{6}{12}

Zoran’s waffle: each of the 2 halves became 5 strips, so there are 2 \times 5 = 10 strips, and the strawberry half is 1 \times 5 = 5 of them. So

\frac{1}{2} = \frac{5}{10}

Both fractions name one half of the same size of waffle, so they name the same amount as each other:

\frac{6}{12} = \frac{5}{10}

Notice that neither of these comes from the other by cutting: no whole number turns 10 pieces into 12. They are equal because both are names for \tfrac{1}{2}. Two fractions in the same family are always equivalent, even when you cannot step straight from one to the other.

Example 3

A fraction bigger than one

Mark \tfrac{5}{4} on a number line that runs from 0 to 2 and is cut into quarters. Then cut every quarter into 3 equal smaller jumps. What is the new name for the same point?

Solution

The point \tfrac{5}{4} is 5 quarter-jumps from 0: four of them reach 1, and one more goes past it.

Cutting every quarter into 3 makes each whole into 4 \times 3 = 12 small jumps, so each small jump is \tfrac{1}{12}. Each of the 5 quarter-jumps to the point is now 3 small jumps, so the point is 5 \times 3 = 15 small jumps from 0:

\frac{5}{4} = \frac{5 \times 3}{4 \times 3} = \frac{15}{12}

The rule does not care that the top is bigger than the bottom. It only uses the fact that every piece was cut the same way. Check: the whole number 1 was \tfrac{4}{4} and is now \tfrac{12}{12}, and \tfrac{15}{12} is 3 twelfths past it, just as \tfrac{5}{4} is one quarter past 1, and one quarter is 3 twelfths.

Example 4

From tenths to hundredths

Halyna’s waffle maker makes square waffles with 10 rows and 10 columns of little squares. She pours syrup down 3 of the 10 columns. What fraction of the waffle has syrup, written in tenths and in hundredths?

Solution

The columns cut the waffle into 10 equal strips and syrup covers 3, so syrup covers \tfrac{3}{10} of the waffle.

Each column is 10 little squares, so every tenth is cut into 10 pieces. That is multiplying the top and the bottom by 10, or multiplying by \tfrac{10}{10}, which is 1:

\frac{3}{10} = \frac{3 \times 10}{10 \times 10} = \frac{30}{100}

Check by counting squares: 3 columns of 10 squares is 30 squares, out of 10 \times 10 = 100. Writing tenths as hundredths like this is the step that lets you add them (Adding tenths and hundredths as fractions).

Example 5

A missing top and a missing bottom

Find each missing number.

\frac{4}{3} = \frac{\square}{12} \qquad \frac{3}{8} = \frac{9}{\square}

Solution

First fraction: the bottom tells us the cut. 3 pieces became 12, and 3 \times 4 = 12, so every third was cut into 4. The top must be multiplied by the same 4: 4 \times 4 = 16. So \tfrac{4}{3} = \tfrac{16}{12}.

Second fraction: here the top tells us the cut. 3 counted pieces became 9, and 3 \times 3 = 9, so every eighth was cut into 3. The bottom must be multiplied by the same 3: 8 \times 3 = 24. So \tfrac{3}{8} = \tfrac{9}{24}.

In both, you find the number n from the pair of numbers you know, then use the same n on the other one. It must be the same n, because one cut goes through every piece.

Example 6

Three quarters of a waffle in squares

Zoran has one of Halyna’s waffles, with 100 little squares. He covers \tfrac{3}{4} of it with blueberries. How many squares have blueberries?

Solution

We want \tfrac{3}{4} = \tfrac{\square}{100}. To turn 4 pieces into 100, each quarter must hold 25 squares, because 4 \times 25 = 100. So multiply the top by 25 too:

\frac{3}{4} = \frac{3 \times 25}{4 \times 25} = \frac{75}{100}

75 squares have blueberries. Check: one quarter is 25 squares, and three quarters is 25 + 25 + 25 = 75 squares.

Example 7

Finding one member of a family

Zoran lists the family of \tfrac{2}{5}. Halyna asks him for the member whose top and bottom add up to 35. Which fraction is it?

Solution

Every member of the family is \tfrac{2 \times n}{5 \times n} for some counting number n. Make a table and add the top and bottom each time:

n	1	2	3	4	5
Fraction	\tfrac{2}{5}	\tfrac{4}{10}	\tfrac{6}{15}	\tfrac{8}{20}	\tfrac{10}{25}
Top plus bottom	7	14	21	28	35

The sums go up by 7 each time. That is because the top grows by 2 and the bottom by 5 with every step, and 2 + 5 = 7. So the sums are the multiples of 7, and 35 = 7 \times 5 means n = 5. The fraction is

\frac{2 \times 5}{5 \times 5} = \frac{10}{25}

Check: 10 + 25 = 35, and \tfrac{10}{25} is \tfrac{2}{5} with every fifth cut into 5.

Boundaries

The same number on top and bottom. The rule needs one n for both, because one cut goes through every piece. A fraction made with two different numbers names a different amount (unless the top is 0, since 0 pieces is 0 however you cut).
A counting number, not zero. Cutting each piece into n parts only makes sense for n = 1, 2, 3, \ldots. With n = 0 the bottom would be 0, and a whole cut into 0 pieces names no fraction at all. With n = 1 nothing is cut, and you get the fraction you started with.
The same whole. \tfrac{2}{3} of a big waffle and \tfrac{8}{12} of a small one are not the same amount of waffle. Equivalent fractions name the same part of the same whole.
Only multiples of the bottom. Multiplying turns \tfrac{a}{b} into a fraction whose bottom is b, 2 \times b, 3 \times b and so on. So \tfrac{2}{3} has no name in tenths with a whole-number top, because no whole number times 3 makes 10. Some equal pairs need the opposite move first: \tfrac{4}{6} = \tfrac{6}{9} is true, but you reach it by making bigger pieces, \tfrac{4}{6} = \tfrac{2}{3}, before cutting again (Simplifying a fraction to simplest form).

Where this goes next

The opposite move, dividing the top and the bottom by the same number, makes bigger pieces and writes a fraction with the smallest numbers possible (Simplifying a fraction to simplest form). Together the two moves let you rename any fraction.

The most common reason to rename is to make two fractions use the same size of piece. Then comparing them, or adding them, is just counting pieces (Common denominators and the least common denominator). Another way to compare is to measure each fraction against a friendly one such as \tfrac{1}{2} (Benchmark fractions, Comparing against a benchmark), and equivalent fractions tell you what \tfrac{1}{2} looks like in any size of piece: \tfrac{3}{6}, \tfrac{4}{8}, \tfrac{5}{10} and so on.

Exercises

Part 1: cut every piece

A waffle is cut into 4 equal strips, and 1 strip has butter. Every strip is cut into 3 squares. What fraction of the waffle has butter, written in twelfths?
Zoran cuts a quiche into 3 slices and puts ham on 1 slice. Then he cuts every slice into 4 thinner slices. Write the ham part as two equivalent fractions.
A waffle has 5 strips, and 4 have honey. Every strip is cut into 2 squares. How many squares are there, and how many have honey?
Ece cuts each half of a quiche into 4 slices. One half has mushrooms. Write the mushroom part in halves and in eighths.
Halyna’s waffle has 6 strips, and 1 strip has syrup. She cuts every strip into 2 squares. Write the syrup part in twelfths.
Ece cuts her quiche into 4 slices and eats 1. Zoran cuts a quiche of the same size into 8 slices and eats 2. Who ate more quiche? Use cutting to explain.
A waffle has 5 strips and 1 has jam. Halyna cuts every strip into 2 squares, and then cuts every square into 10 tiny squares. Write the jam part three ways: in fifths, tenths and hundredths.

Part 2: the rule

Multiply the top and the bottom by the given number, and write the equivalent fraction.

\tfrac{3}{7}, by 2
\tfrac{5}{8}, by 3
\tfrac{7}{4}, by 5
\tfrac{9}{10}, by 10
Halyna wrote \tfrac{4}{5} = \tfrac{16}{20}. Which fraction equal to 1 did she multiply by?
Are \tfrac{4}{6} and \tfrac{10}{15} equivalent? Show that both come from \tfrac{2}{3}.
A number line from 0 to 2 is cut into thirds, and a dot sits at \tfrac{4}{3}. Every third is cut into 2 equal jumps. What is the new name for the dot?
Ece multiplies the top and the bottom of \tfrac{2}{9} by 1. What does she get, and why?

Part 3: missing numbers and families

Find the missing number.

\tfrac{3}{4} = \tfrac{\square}{8}
\tfrac{5}{6} = \tfrac{\square}{12}
\tfrac{7}{10} = \tfrac{\square}{100}
\tfrac{2}{3} = \tfrac{14}{\square}
\tfrac{3}{2} = \tfrac{12}{\square}
Write the next four members of the family of \tfrac{5}{3}, cutting every piece into 2, 3, 4 and 5.
A waffle has 100 little squares. Zoran puts bananas on \tfrac{2}{5} of it. How many squares have bananas?
Which of \tfrac{6}{9}, \tfrac{8}{12}, \tfrac{10}{16} and \tfrac{12}{18} are in the family of \tfrac{2}{3}?

Reasoning

Explain why \tfrac{2}{3} = \tfrac{\square}{10} has no whole number that fits the box.
Halyna says: “When I cut every square of a waffle into 4, each piece gets 4 times smaller, but there are 4 times as many pieces, so the jam covers the same amount.” Use her idea to explain why multiplying the top and the bottom of a fraction by the same number gives an equivalent fraction.

Answers

\tfrac{3}{12}. The whole is 4 \times 3 = 12 squares and the butter strip is 1 \times 3 = 3 of them, so \tfrac{1}{4} = \tfrac{3}{12}.
\tfrac{1}{3} = \tfrac{4}{12}. The quiche has 3 \times 4 = 12 thin slices and the ham is on 1 \times 4 = 4 of them.
10 squares, 8 with honey. 5 \times 2 = 10 and 4 \times 2 = 8, so \tfrac{4}{5} = \tfrac{8}{10}.
\tfrac{1}{2} = \tfrac{4}{8}. The quiche has 2 \times 4 = 8 slices and one half is 1 \times 4 = 4 of them.
\tfrac{2}{12}. 6 \times 2 = 12 squares, and the syrup strip is 1 \times 2 = 2 of them, so \tfrac{1}{6} = \tfrac{2}{12}.
They ate the same amount. Cut each of Ece’s 4 slices into 2 and her quiche is in 4 \times 2 = 8 slices, with her slice becoming 1 \times 2 = 2 of them. So \tfrac{1}{4} = \tfrac{2}{8}.
\tfrac{1}{5} = \tfrac{2}{10} = \tfrac{20}{100}. The first cut makes 5 \times 2 = 10 squares, 1 \times 2 = 2 with jam. The second makes 10 \times 10 = 100 tiny squares, 2 \times 10 = 20 with jam.
\tfrac{6}{14}, since \tfrac{3 \times 2}{7 \times 2} = \tfrac{6}{14}.
\tfrac{15}{24}, since \tfrac{5 \times 3}{8 \times 3} = \tfrac{15}{24}.
\tfrac{35}{20}, since \tfrac{7 \times 5}{4 \times 5} = \tfrac{35}{20}.
\tfrac{90}{100}, since \tfrac{9 \times 10}{10 \times 10} = \tfrac{90}{100}.
\tfrac{4}{4}. The bottom went from 5 to 20, and 5 \times 4 = 20; the top also went from 4 to 4 \times 4 = 16.
Yes. \tfrac{4}{6} = \tfrac{2 \times 2}{3 \times 2} and \tfrac{10}{15} = \tfrac{2 \times 5}{3 \times 5}. Both equal \tfrac{2}{3}, so they equal each other.
\tfrac{8}{6}. Each whole is now 3 \times 2 = 6 small jumps, and the dot’s 4 thirds are 4 \times 2 = 8 small jumps.
\tfrac{2}{9} again. \tfrac{2 \times 1}{9 \times 1} = \tfrac{2}{9}: cutting every piece into 1 piece cuts nothing.
6. 4 \times 2 = 8, so 3 \times 2 = 6, and \tfrac{3}{4} = \tfrac{6}{8}.
10. 6 \times 2 = 12, so 5 \times 2 = 10, and \tfrac{5}{6} = \tfrac{10}{12}.
70. 10 \times 10 = 100, so 7 \times 10 = 70, and \tfrac{7}{10} = \tfrac{70}{100}.
21. 2 \times 7 = 14, so 3 \times 7 = 21, and \tfrac{2}{3} = \tfrac{14}{21}.
8. 3 \times 4 = 12, so 2 \times 4 = 8, and \tfrac{3}{2} = \tfrac{12}{8}.
\tfrac{10}{6}, \tfrac{15}{9}, \tfrac{20}{12}, \tfrac{25}{15}. The tops count by 5 and the bottoms by 3.
40 squares. 5 \times 20 = 100, so \tfrac{2}{5} = \tfrac{2 \times 20}{5 \times 20} = \tfrac{40}{100}.
\tfrac{6}{9}, \tfrac{8}{12} and \tfrac{12}{18}, from n = 3, 4 and 6. Not \tfrac{10}{16}: its top is 2 \times 5, so it would need the bottom 3 \times 5 = 15.
To get 10 pieces from 3, each third would have to be cut into n pieces with 3 \times n = 10. But 3 \times 3 = 9 and 3 \times 4 = 12, and no whole number lies between 3 and 4. Thirds cannot be cut evenly into tenths, so no whole-number top fits. (The amount \tfrac{2}{3} lies between \tfrac{6}{10} and \tfrac{7}{10}.)
A fraction \tfrac{a}{b} counts a pieces out of the b that make the whole. Cut every piece into n. Every piece in the whole becomes n pieces, so the whole is n \times b pieces, and every counted piece becomes n pieces, so the part is n \times a pieces. That is the new fraction \tfrac{n \times a}{n \times b}. Each new piece is n times smaller, but the part holds n times as many, which is Halyna’s point: the part covers exactly what it covered before. So \tfrac{n \times a}{n \times b} = \tfrac{a}{b}.