Grade 4 · 4.NF.B.4 · 4.NF.B.4A

A fraction times a whole number

Five cats at a cat rescue each get three fourths of a cup of food at feeding time. How much food is that in all, and why does it come to fifteen fourths of a cup?

You need

Decomposing a fraction into unit fractionsGrade 4 Fractions greater than one (improper fractions)Grade 3 What multiplication meansGrade 2

Unlocks

A fraction of a whole numberGrade 4 A fraction times a fractionGrade 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

Volkan, Bohdan and Zanele help out at a cat rescue. At feeding time every cat gets the same amount of dry food, measured out with a scoop. Five cats each get \displaystyle \frac{3}{4} of a cup. How much food has to come out of the bag?

Equal groups call for multiplication. Recall that 5 \times 3 means five groups of three (see What multiplication means). Here each group is a fraction of a cup, so the question is \displaystyle 5 \times \frac{3}{4}: five copies of \displaystyle \frac{3}{4}.

Reach for this move whenever the same fraction is taken again and again: the same food for each cat, the same part of a can for each kitten, the same amount every day for a week. The whole number counts the copies. The fraction says how big each copy is.

The procedure

This lesson takes three days, one part a day.

Part 1: a whole number times a unit fraction

Bohdan’s smallest scoop holds \displaystyle \frac{1}{4} cup. A small kitten gets one scoop. Five kittens get five scoops, five copies of \displaystyle \frac{1}{4}. Five copies of a number is five times that number, so the food is \displaystyle 5 \times \frac{1}{4} cups. A whole number times a fraction means the same thing 5 \times 3 means: that many copies of the number, added up.

\begin{aligned} 5 \times \frac{1}{4} &= \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} \\ &= \frac{5}{4} \end{aligned}

The sum is easy to read by counting pieces. Each copy is one fourth of a cup, and there are five copies, so there are five fourths. The count of pieces is the numerator, 5. The size of a piece, fourths, is the denominator, 4.

Five fourths is more than a cup, because four fourths already fill one cup (see Fractions greater than one (improper fractions)). The first four scoops fill a cup, and the fifth starts a second one: \displaystyle \frac{5}{4} = 1\frac{1}{4} cups.

Figure 1Five scoops of \displaystyle \frac{1}{4} cup fill one cup and one fourth of another, so \displaystyle 5 \times \frac{1}{4} = \frac{5}{4}.

A number line shows the same product as lengths. Cut each whole into 4 equal steps. Start at 0 and make five hops, each \displaystyle \frac{1}{4} long. Four hops reach 1, and the fifth lands on \displaystyle \frac{5}{4}.

Figure 2Five hops of \displaystyle \frac{1}{4} from 0 land on \displaystyle \frac{5}{4}.

Now read the same idea the other way. Every fraction is a whole number times its own unit fraction. Take \displaystyle \frac{3}{4}. It means three pieces, each of size \displaystyle \frac{1}{4}, which is the sum \displaystyle \frac{1}{4} + \frac{1}{4} + \frac{1}{4} (see writing a fraction as a sum of unit fractions). Three copies of \displaystyle \frac{1}{4} added up is \displaystyle 3 \times \frac{1}{4}, so

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

In the same way \displaystyle \frac{2}{5} = 2 \times \frac{1}{5} and \displaystyle \frac{7}{6} = 7 \times \frac{1}{6}. In words: a fraction is its numerator times its unit fraction. We say \displaystyle \frac{3}{4} is a multiple of \displaystyle \frac{1}{4}, just as 15 is a multiple of 5 because 15 = 3 \times 5.

Part 2: a whole number times any fraction

Back to the five grown cats, each with \displaystyle \frac{3}{4} cup. Zanele fills five bowls:

\begin{aligned} 5 \times \frac{3}{4} &= \frac{3}{4} + \frac{3}{4} + \frac{3}{4} + \frac{3}{4} + \frac{3}{4} \end{aligned}

Count the pieces again. Each bowl holds 3 fourths. Five bowls of 3 pieces is 5 \times 3 = 15 pieces, and every piece is still a fourth of a cup. So the cats eat \displaystyle \frac{15}{4} cups.

Figure 3Five bowls of \displaystyle \frac{3}{4} cup hold 5 \times 3 = 15 pieces, and every piece is a fourth of a cup.

The same count can be written in symbols, and it shows why the rule works for every product like this. From Part 1, \displaystyle \frac{3}{4} = 3 \times \frac{1}{4}. Put that in:

\begin{aligned} 5 \times \frac{3}{4} &= 5 \times \left(3 \times \frac{1}{4}\right) \\ &= (5 \times 3) \times \frac{1}{4} \\ &= 15 \times \frac{1}{4} \\ &= \frac{15}{4} \end{aligned}

The second line moves the parentheses. That is allowed by the associative property of multiplication: when three numbers are multiplied, how you group them does not change the product. The last line is Part 1 again: fifteen copies of \displaystyle \frac{1}{4} is \displaystyle \frac{15}{4}.

The same steps work for any whole number n and any fraction \displaystyle \frac{a}{b}. Multiply the whole number by the numerator, and keep the denominator:

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

The denominator does not change because copying a fraction never changes the size of a piece. Every piece in every bowl is a fourth of a cup, however many bowls there are. Only the count of pieces grows.

How much food is \displaystyle \frac{15}{4} cups? Four fourths make one cup. Since 15 = 12 + 3 and 12 fourths is 3 cups,

\begin{aligned} \frac{15}{4} &= \frac{12}{4} + \frac{3}{4} \\ &= 3 + \frac{3}{4} \end{aligned}

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

The cats eat \displaystyle 3\frac{3}{4} cups. Turning a fraction like \displaystyle \frac{15}{4} into a mixed number is its own skill, Mixed numbers and improper fractions: rewriting and comparing.

On a number line, five copies of \displaystyle \frac{3}{4} are five hops, each three fourths long. They land on \displaystyle \frac{15}{4}, which sits between 3 and 4.

Figure 4Five hops of \displaystyle \frac{3}{4} land on \displaystyle \frac{15}{4}, three fourths of the way from 3 to 4.

Part 3: word problems, and which whole numbers the answer sits between

In a word problem, look for two things: the size of one share (words like “each”, “every” and “per” point at it), and the count of shares (the number of cats, kittens or days). The count is the whole number. The share is the fraction.

A comparison can call for the same product. A kitten at the rescue eats \displaystyle \frac{1}{3} cup at dinner, and Mittens, a big old cat, eats 3 times as much as the kitten. “3 times as much” means three copies of the kitten’s amount (see Times as many):

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

Mittens eats exactly one cup.

An answer like \displaystyle \frac{15}{4} cups is exact, but a helper at the rescue also wants to know: is 3 cups of food enough? Is 4? That asks between which two whole numbers the answer lies. Find the multiples of the denominator just below and just above the numerator. For \displaystyle \frac{15}{4}, those are 12 and 16, and

\frac{12}{4} = 3 \qquad \frac{16}{4} = 4

Since 15 is between 12 and 16, \displaystyle \frac{15}{4} is between 3 and 4. Three cups is not enough; four cups is plenty.

A quick look before any arithmetic also helps. Each cat’s \displaystyle \frac{3}{4} cup is less than one cup, so five cats need less than 5 \times 1 = 5 cups. Each share is more than half a cup, so five cats need more than five halves, \displaystyle 2\frac{1}{2} cups. The answer \displaystyle 3\frac{3}{4} fits between those, which is a good sign it is right.

Worked examples

Example 1

Seven scoops of a third

Zanele feeds 7 kittens, and each kitten gets one scoop of \displaystyle \frac{1}{3} cup. Write a multiplication for the food, and find how much it is.

Solution

There are 7 equal shares of \displaystyle \frac{1}{3} cup, so the food is \displaystyle 7 \times \frac{1}{3} cups. Seven copies of one third are seven thirds:

7 \times \frac{1}{3} = \frac{7}{3}

Three thirds make one cup. Since 7 = 3 + 3 + 1,

\begin{aligned} \frac{7}{3} &= \frac{3}{3} + \frac{3}{3} + \frac{1}{3} \\ &= 1 + 1 + \frac{1}{3} \end{aligned}

That is \displaystyle 2\frac{1}{3} cups. On a number line in thirds, seven hops of \displaystyle \frac{1}{3} pass 1 after three hops, pass 2 after six, and land one third past 2.

Example 2

A fraction as a count of scoops

A cat with a small appetite gets \displaystyle \frac{5}{8} cup of food. Volkan can only find a scoop that holds \displaystyle \frac{1}{8} cup. How many scoops does he need? Write \displaystyle \frac{5}{8} as a whole number times a unit fraction.

Solution

The fraction \displaystyle \frac{5}{8} means 5 pieces, each of size \displaystyle \frac{1}{8}. Each scoop is one of those pieces, so Volkan needs 5 scoops, and

\frac{5}{8} = 5 \times \frac{1}{8}

Check by multiplying: \displaystyle 5 \times \frac{1}{8} = \frac{5 \times 1}{8} = \frac{5}{8}.

Example 3

Wet food for three kittens

Three kittens each get \displaystyle \frac{2}{5} of a can of wet food. How many cans does Bohdan use?

Solution

Three equal shares of \displaystyle \frac{2}{5} can is \displaystyle 3 \times \frac{2}{5}. Write each share as a multiple of the unit fraction, \displaystyle \frac{2}{5} = 2 \times \frac{1}{5}, and regroup with the associative property:

\begin{aligned} 3 \times \frac{2}{5} &= 3 \times \left(2 \times \frac{1}{5}\right) \\ &= (3 \times 2) \times \frac{1}{5} \\ &= \frac{6}{5} \end{aligned}

So \displaystyle 3 \times \frac{2}{5} = \frac{6}{5}. Five fifths is one can, and 6 = 5 + 1, so \displaystyle \frac{6}{5} = 1\frac{1}{5} cans. Bohdan opens two cans: he uses all of the first and \displaystyle \frac{1}{5} of the second.

Example 4

One product, written three ways

The cats in one room get \displaystyle 4 \times \frac{3}{8} cups of food. Zanele writes the same amount as \displaystyle 12 \times \frac{1}{8}, Bohdan as \displaystyle 6 \times \frac{2}{8}, and Volkan as \displaystyle 3 \times \frac{4}{8}. Are all four the same amount?

Solution

Count eighths in each one, since every piece in every product is an eighth of a cup.

Product	Copies	Eighths in each copy	Eighths in all
\displaystyle 4 \times \frac{3}{8}	4	3	12
\displaystyle 12 \times \frac{1}{8}	12	1	12
\displaystyle 6 \times \frac{2}{8}	6	2	12
\displaystyle 3 \times \frac{4}{8}	3	4	12

Each one is \displaystyle \frac{12}{8} cups, because 4 \times 3, 12 \times 1, 6 \times 2 and 3 \times 4 all equal 12. They are the same twelve eighths of a cup, shared into bowls in different ways. As a mixed number, \displaystyle \frac{12}{8} = \frac{8}{8} + \frac{4}{8} = 1\frac{4}{8} cups.

Example 5

A product that lands on a whole number

Six cats each get \displaystyle \frac{2}{3} cup of food at breakfast. How many cups is that?

Solution

Six shares of \displaystyle \frac{2}{3} cup:

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

Three thirds make one cup, and 12 = 3 + 3 + 3 + 3, so twelve thirds is exactly 4 cups. That matches 4 \times 3 = 12: the numerator 12 is a multiple of the denominator 3, so the answer is a whole number and no cup is left part full.

Example 6

Between which two whole numbers?

Nine cats each get \displaystyle \frac{3}{8} cup at dinner. Between which two whole numbers of cups is the total? Is 3 cups enough?

Solution

Nine shares of \displaystyle \frac{3}{8} cup:

9 \times \frac{3}{8} = \frac{9 \times 3}{8} = \frac{27}{8}

The multiples of 8 just below and just above 27 are 24 and 32. Since \displaystyle \frac{24}{8} = 3 and \displaystyle \frac{32}{8} = 4, the total is between 3 and 4 cups. Exactly, 27 = 24 + 3, so \displaystyle \frac{27}{8} = 3\frac{3}{8} cups.

Three cups is not enough: it is \displaystyle \frac{3}{8} cup short. Four cups is enough.

A quick look agrees. Each share, \displaystyle \frac{3}{8}, is less than half a cup, so nine cats need less than nine halves, \displaystyle 4\frac{1}{2} cups.

Example 7

Is there enough food?

Zanele has 5 cups of food left in the bag. Seven cats each need \displaystyle \frac{3}{4} cup. Is there enough? If not, how many cats can get a full share?

Solution

Seven shares of \displaystyle \frac{3}{4} cup:

7 \times \frac{3}{4} = \frac{21}{4}

Five cups is \displaystyle \frac{20}{4}, and 21 is more than 20, so there is not enough. Since 21 = 20 + 1, the cats need \displaystyle 5\frac{1}{4} cups, and Zanele is \displaystyle \frac{1}{4} cup short.

Try one cat fewer:

6 \times \frac{3}{4} = \frac{18}{4} = 4\frac{2}{4}

That is less than 5 cups, so six cats can get a full share. What is left is \displaystyle \frac{20}{4} - \frac{18}{4} = \frac{2}{4} cup, less than the \displaystyle \frac{3}{4} cup the seventh cat needs. Six cats get a full share, and the seventh gets \displaystyle \frac{2}{4} cup until a new bag is opened.

Boundaries

The number of copies is a whole number. It can be 0, 1, 2, 3 and so on. Zero copies is nothing: \displaystyle 0 \times \frac{3}{4} = 0. One copy is the fraction itself: \displaystyle 1 \times \frac{3}{4} = \frac{3}{4}. Half a copy or three fourths of a copy is not a count of copies; that kind of product needs a new meaning, a part of an amount, taught in A fraction of a whole number and later A fraction times a fraction.
The denominator is not multiplied. Copying a fraction makes more pieces of the same size, so only the numerator changes: \displaystyle n \times \frac{a}{b} = \frac{n \times a}{b}.
Every copy is of the same whole. Five shares of \displaystyle \frac{3}{4} cup add up only because each is three fourths of the same size of cup.
The denominator cannot be 0. A whole cut into 0 parts has no pieces to count.
In this lesson the whole number comes first. \displaystyle \frac{3}{4} \times 5 has the same value as \displaystyle 5 \times \frac{3}{4}, by the commutative property of multiplication (switching the order of the factors does not change the product), but reading it as “three fourths of 5” is the next lesson’s idea.

Where this goes next

Copies of a fraction lead straight to taking a fraction of a number. Three fourths of 12 cans splits the cans into 4 equal groups and takes 3 of them, which is A fraction of a whole number. Later, when both factors are fractions, the same picture of pieces grows into A fraction times a fraction.

The idea of copies of a small piece also works for decimals: 4 \times 0.3 is four copies of three tenths, twelve tenths in all. That is Multiplying decimals.

Exercises

Part 1: a whole number times a unit fraction

Write \displaystyle \frac{4}{5} as a whole number times a unit fraction.
Write \displaystyle \frac{7}{10} as a whole number times a unit fraction.
Find \displaystyle 6 \times \frac{1}{8}.
Find \displaystyle 9 \times \frac{1}{4}, and write it as a mixed number.
Find \displaystyle 11 \times \frac{1}{6}, and write it as a mixed number.
Volkan feeds 8 kittens, and each gets one scoop of \displaystyle \frac{1}{3} cup. Write a multiplication for the food and find the total.
A number line is marked in fifths. Start at 0 and make 7 hops of \displaystyle \frac{1}{5}. Write the hops as a multiplication, and say where you land.
How many copies of \displaystyle \frac{1}{12} make \displaystyle \frac{5}{12}? Write it as a multiplication.

Part 2: a whole number times any fraction

Find \displaystyle 4 \times \frac{2}{7}, and write it as a mixed number.
Find \displaystyle 3 \times \frac{5}{6}, and write it as a mixed number.
Find \displaystyle 6 \times \frac{3}{5}, and write it as a mixed number.
Find \displaystyle 2 \times \frac{7}{8}, and write it as a mixed number.
Write \displaystyle 5 \times \frac{2}{3} as a whole number times \displaystyle \frac{1}{3}, then as one fraction.
Fill in the missing number: \displaystyle 4 \times \frac{5}{9} = ? \times \frac{1}{9}.
Bohdan writes \displaystyle 3 \times \frac{4}{10} = \frac{12}{30}. Is he right? Explain using pieces.
Find three different ways to write \displaystyle \frac{12}{6} as a whole number times a fraction with denominator 6.

Part 3: word problems

Zanele feeds 4 cats, and each gets \displaystyle \frac{2}{3} cup. How many cups is that? Between which two whole numbers is it?
Eight kittens each get \displaystyle \frac{3}{10} of a can of wet food. How many cans of food is that? How many cans must be opened?
Six cats each get \displaystyle \frac{5}{8} cup of food. How many cups is that? Between which two whole numbers is it?
Bohdan feeds 10 cats, and each gets \displaystyle \frac{1}{2} cup. How many cups is that?
An old cat eats \displaystyle \frac{2}{5} cup of soft food every day. How much does it eat in 7 days?
Volkan has 4 cups of food. Five cats each need \displaystyle \frac{5}{6} cup. Is there enough? If not, how much more food does he need?

Reasoning

Explain why \displaystyle 5 \times \frac{2}{9} = \frac{10}{9}, and why the denominator stays 9.
Without working out the product, explain why \displaystyle 7 \times \frac{3}{8} is less than 7.
Explain why \displaystyle 3 \times \frac{5}{6} and \displaystyle 5 \times \frac{3}{6} are the same amount.

Answers

\displaystyle \frac{4}{5} = 4 \times \frac{1}{5}. Four fifths is four pieces of size \displaystyle \frac{1}{5}.
\displaystyle \frac{7}{10} = 7 \times \frac{1}{10}.
\displaystyle \frac{6}{8}. Six copies of one eighth are six eighths.
\displaystyle \frac{9}{4} = 2\frac{1}{4}. Nine fourths; 8 fourths make 2 wholes, and one fourth is left.
\displaystyle \frac{11}{6} = 1\frac{5}{6}. Six sixths make 1, and 11 = 6 + 5.
\displaystyle 8 \times \frac{1}{3} = \frac{8}{3} = 2\frac{2}{3} cups, since 6 thirds make 2 cups and 8 = 6 + 2.
\displaystyle 7 \times \frac{1}{5} = \frac{7}{5}. You land on \displaystyle \frac{7}{5}, which is \displaystyle 1\frac{2}{5}: five hops reach 1, and two more go past it.
5 copies: \displaystyle \frac{5}{12} = 5 \times \frac{1}{12}.
\displaystyle 4 \times \frac{2}{7} = \frac{8}{7} = 1\frac{1}{7}, since 4 \times 2 = 8 and 8 = 7 + 1.
\displaystyle 3 \times \frac{5}{6} = \frac{15}{6} = 2\frac{3}{6}, since 3 \times 5 = 15 and 15 = 12 + 3.
\displaystyle 6 \times \frac{3}{5} = \frac{18}{5} = 3\frac{3}{5}, since 6 \times 3 = 18 and 18 = 15 + 3.
\displaystyle 2 \times \frac{7}{8} = \frac{14}{8} = 1\frac{6}{8}, since 2 \times 7 = 14 and 14 = 8 + 6.
\displaystyle 5 \times \frac{2}{3} = 5 \times \left(2 \times \frac{1}{3}\right) = 10 \times \frac{1}{3} = \frac{10}{3}, regrouping by the associative property.
20. Four copies of 5 ninths are 4 \times 5 = 20 ninths, so \displaystyle 4 \times \frac{5}{9} = 20 \times \frac{1}{9} = \frac{20}{9}.
No. Each copy is 4 tenths, and three copies are 12 tenths: \displaystyle 3 \times \frac{4}{10} = \frac{12}{10}. The pieces stay tenths; \displaystyle \frac{12}{30} would be pieces of size \displaystyle \frac{1}{30}, which are smaller.
Any whole number n and numerator a with n \times a = 12 work, for example \displaystyle 12 \times \frac{1}{6}, \displaystyle 6 \times \frac{2}{6} and \displaystyle 4 \times \frac{3}{6} (also \displaystyle 3 \times \frac{4}{6} and \displaystyle 2 \times \frac{6}{6}). Each is 12 sixths.
\displaystyle 4 \times \frac{2}{3} = \frac{8}{3} = 2\frac{2}{3} cups, between 2 and 3, since \displaystyle \frac{6}{3} = 2 and \displaystyle \frac{9}{3} = 3.
\displaystyle 8 \times \frac{3}{10} = \frac{24}{10} = 2\frac{4}{10} cans. That is between 2 and 3, so 3 cans must be opened.
\displaystyle 6 \times \frac{5}{8} = \frac{30}{8} = 3\frac{6}{8} cups, between 3 and 4, since \displaystyle \frac{24}{8} = 3 and \displaystyle \frac{32}{8} = 4.
\displaystyle 10 \times \frac{1}{2} = \frac{10}{2} = 5 cups, since two halves make each cup.
\displaystyle 7 \times \frac{2}{5} = \frac{14}{5} = 2\frac{4}{5} cups, since 14 = 10 + 4.
No. \displaystyle 5 \times \frac{5}{6} = \frac{25}{6} = 4\frac{1}{6} cups, and 4 cups is \displaystyle \frac{24}{6}. He needs \displaystyle \frac{1}{6} cup more.
\displaystyle \frac{2}{9} is two pieces, each a ninth. Five copies are 5 \times 2 = 10 pieces. Copying does not cut or join any piece, so each piece is still a ninth, and the total is \displaystyle \frac{10}{9}.
\displaystyle \frac{3}{8} is less than 1, so each of the 7 copies is less than 1. Seven copies each less than 1 add to less than 7 \times 1 = 7.
\displaystyle 3 \times \frac{5}{6} is 3 copies of 5 sixths, 15 sixths. \displaystyle 5 \times \frac{3}{6} is 5 copies of 3 sixths, also 15 sixths, because 3 \times 5 = 5 \times 3 (the commutative property). Both are \displaystyle \frac{15}{6}.