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Grade 4 · 4.NF.B.4 · 4.NF.B.4C

A fraction of a whole number

A train trip is 120 miles long, and the train has gone two thirds of the way. How many miles is that, and what does it mean to take a fraction of a number?

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When you reach for it

Zubin, Zlata and Anahera are taking a long train trip. All day, questions come up that ask for a part of an amount. Three fourths of the train’s cars carry passengers: how many cars is that? The trip is 120 miles, and the train has gone two thirds of the way: how far is that? Zlata reads for three fifths of an hour: how many minutes?

Each question names a fraction and a whole number, joined by the word “of”. The whole number is the amount you start with: a set of things you can count, or a measurement such as miles or minutes. The fraction says what part of it you want.

In the last lesson, A fraction times a whole number, a whole number counted copies of a fraction. Here the roles turn around. The fraction acts on the whole number: it cuts the amount into equal groups and takes some of them. By the end of Part 2 you will see that both readings give the same product.

The procedure

This lesson takes three days, one part a day.

Part 1: a fraction of a set

The train has 16 cars. Three fourths of them are passenger cars, and the rest are baggage cars. How many passenger cars are there?

Start with the denominator. Fourths means 4 equal parts, so sort the 16 cars into 4 equal groups. Sorting a set into equal groups and asking how many land in each group is division (see What division means):

16 \div 4 = 4

Each group has 4 cars, and each group is one fourth of the train. So one fourth of 16 is 4.

Now the numerator. Three fourths means 3 of those groups. Three groups of 4 cars is

3 \times 4 = 12

So \displaystyle \frac{3}{4} of 16 is 12. The train has 12 passenger cars. The group that is left is the other fourth, 4 baggage cars, and 12 + 4 = 16 accounts for every car.

\tfrac{1}{4}
\tfrac{1}{4}
\tfrac{1}{4}
\tfrac{1}{4}
 passenger cars baggage
Figure 1Sixteen cars in 4 equal groups make 4 cars in each group. Three of the groups, \displaystyle \frac{3}{4} of the cars, hold 12 cars.

The two numbers of the fraction do two different jobs. The denominator says how many equal groups to make. The numerator says how many of those groups to take. Making the groups is a division, so for a bigger set you use the division you already know (see Dividing by a single digit). For example, one fourth of 96 is 96 \div 4 = 24, and three fourths of 96 is 3 \times 24 = 72.

The groups must be equal. If the 16 cars were split into 4 groups of different sizes, “one group” would not be a fourth of the train, and the count would mean nothing.

Part 2: why “of” means times

The last lesson’s way of counting pieces finds \displaystyle 16 \times \frac{3}{4}. Sixteen copies of \displaystyle \frac{3}{4} are 16 \times 3 = 48 fourths:

16 \times \frac{3}{4} = \frac{16 \times 3}{4} = \frac{48}{4}

Four fourths make one whole, and 48 = 12 \times 4, so \displaystyle \frac{48}{4} = 12. That is the same number as \displaystyle \frac{3}{4} of 16. It is not luck. Here are two ways to see why.

Take three fourths of each one. Think of 16 as sixteen separate ones, like sixteen full bottles of water. Pour out three fourths of every bottle. You have taken three fourths of all 16, and what you poured is sixteen amounts of \displaystyle \frac{3}{4} each, which is \displaystyle 16 \times \frac{3}{4}.

Follow the groups with symbols. One fourth of 16 is 16 \div 4, and a division can be written as a fraction, \displaystyle 16 \div 4 = \frac{16}{4} (see A fraction as division). Three fourths is three of those groups:

\begin{aligned} \frac{3}{4} \text{ of } 16 &= 3 \times \frac{16}{4} \\ &= \frac{3 \times 16}{4} \\ &= \frac{48}{4} = 12 \end{aligned}

The second line is the rule from the last lesson, a whole number times a fraction: multiply the numerator and keep the denominator.

Both ways reach \displaystyle \frac{48}{4}. So taking a fraction of a number is multiplying by that fraction, and we write “of” as \times:

\frac{3}{4} \text{ of } 16 = \frac{3}{4} \times 16 = \frac{3 \times 16}{4} = 12

The order of the factors does not matter: \displaystyle \frac{3}{4} \times 16 = 16 \times \frac{3}{4}, by the commutative property of multiplication (switching the order of the factors does not change the product; see Properties of multiplication). So “three fourths of 16” and “sixteen copies of three fourths” are two stories for one product.

For any fraction \displaystyle \frac{a}{b} and any whole number n,

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

That gives two orders for the arithmetic, and they always agree.

  • Divide first: 16 \div 4 = 4, then 3 \times 4 = 12. This is Part 1’s picture: find one group, then take enough groups. It keeps the numbers small.
  • Multiply first: 3 \times 16 = 48, then 48 \div 4 = 12. This works even when the whole number does not split evenly into the groups, which is where Part 3 goes.

Part 3: a fraction of a measurement

A measurement can be cut into equal parts just as a set can, and the same move finds a part of it. The trip is 120 miles. When the train has gone \displaystyle \frac{2}{3} of the way, cut the trip into 3 equal parts and take 2 of them:

\begin{aligned} 120 \div 3 &= 40 \\ 2 \times 40 &= 80 \end{aligned}

So \displaystyle \frac{2}{3} of 120 miles is 80 miles. As one product, \displaystyle \frac{2}{3} \times 120 = \frac{240}{3} = 80. The last third, 40 miles, is still ahead.

40 miles 40 miles 40 miles 0 40 80 120 80 miles so far
Figure 2Three equal parts of 120 miles are 40 miles each. Two of them, \displaystyle \frac{2}{3} of the trip, are 80 miles.

The answer keeps the unit of the whole. A fraction of a number of miles is a number of miles.

A fraction of a larger unit. Zlata reads for \displaystyle \frac{3}{5} of an hour. A fraction of an hour is easier to picture in minutes, so first write the hour as 60 minutes. Then take three fifths of 60:

\frac{3}{5} \times 60 = \frac{3 \times 60}{5} = \frac{180}{5} = 36

Dividing first agrees: 60 \div 5 = 12 minutes in each fifth, and 3 \times 12 = 36. Zlata reads for 36 minutes.

12 12 12 12 12 0 12 24 36 48 60 36 minutes
Figure 3An hour is 60 minutes. Each fifth of it is 12 minutes, so \displaystyle \frac{3}{5} of an hour is 36 minutes.

The same idea turns a fraction of a foot into inches, a fraction of a pound into ounces, or a fraction of a day into hours: write the whole in the smaller unit, then take the fraction of that number (see Customary conversions).

When the answer is not a whole number. The train station is 2 miles from Zubin’s home, and he has ridden his bike \displaystyle \frac{3}{4} of the way. Two miles do not split into 4 equal parts of a whole number of miles. That is no problem for a length, because a mile can be cut. Multiply first:

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

Four fourths make one mile and 6 = 4 + 2, so \displaystyle \frac{6}{4} = 1\frac{2}{4} miles, which is the same as \displaystyle 1\frac{1}{2} miles (see Equivalent fractions). Dividing first still works too: each fourth of the way is \displaystyle \frac{2}{4} of a mile, and three of them make \displaystyle \frac{6}{4}.

A set of things that cannot be cut is different. Three fourths of 10 passengers is \displaystyle \frac{30}{4} = 7\frac{2}{4}, and no one is half a passenger. A question like that needs a second look at what the leftover means (see Remainders and what they mean).

Worked examples

Example 1

Pretzels on the ride

Zlata’s snack bag holds 15 pretzels. She eats \displaystyle \frac{2}{5} of them in the first hour of the ride. How many does she eat, and how many are left?

Solution

Fifths means 5 equal groups. Sharing 15 pretzels into 5 equal groups puts 15 \div 5 = 3 pretzels in each group, so one fifth of the bag is 3 pretzels. She eats 2 of the groups:

2 \times 3 = 6

Zlata eats 6 pretzels. The other 3 groups are left, 3 \times 3 = 9 pretzels. Check: 6 + 9 = 15.

Example 2

Seats in a train car

A train car has 24 seats, and \displaystyle \frac{5}{6} of them are taken. How many seats are taken? How many are empty?

Solution

Sixths means 6 equal groups, and 24 \div 6 = 4, so each sixth of the seats is 4 seats. Five sixths is 5 of those groups:

5 \times 4 = 20

So 20 seats are taken. The empty seats are the sixth group, 4 seats, and 20 + 4 = 24.

Example 3

Two ways to find two thirds of 18

The train line has 18 stops. Anahera rides for \displaystyle \frac{2}{3} of them before she gets off. How many stops does she ride? Find it by dividing first, by multiplying first, and as copies of \displaystyle \frac{2}{3}.

Solution

Divide first. Thirds means 3 equal groups: 18 \div 3 = 6 stops in each group. Two groups are 2 \times 6 = 12 stops.

Multiply first. Of means times, so

\frac{2}{3} \times 18 = \frac{2 \times 18}{3} = \frac{36}{3} = 12

since 36 = 12 \times 3.

Copies. By the commutative property, \displaystyle \frac{2}{3} \times 18 = 18 \times \frac{2}{3}, which is 18 copies of \displaystyle \frac{2}{3}: 18 \times 2 = 36 thirds, again \displaystyle \frac{36}{3} = 12.

All three agree. Anahera rides 12 stops.

Example 4

A child’s ticket

A full train ticket costs 35 dollars. A child’s ticket costs \displaystyle \frac{3}{5} of the full price. Zubin’s family buys one full ticket and one child’s ticket. How much do they pay?

Solution

The child’s ticket is \displaystyle \frac{3}{5} of 35 dollars. Divide first: 35 \div 5 = 7 dollars in each fifth, and three fifths is 3 \times 7 = 21 dollars. As one product,

\frac{3}{5} \times 35 = \frac{105}{5} = 21

The two tickets cost 35 + 21 = 56 dollars. A check that the size makes sense: \displaystyle \frac{3}{5} is more than half and less than one, so the child’s price should be more than half of 35 and less than 35, and 21 is.

Example 5

Asleep for part of the ride

The ride to the coast takes 3 hours. Zubin sleeps for \displaystyle \frac{3}{4} of it. How long does he sleep, in hours and minutes?

Solution

Write the whole in minutes first. Each hour is 60 minutes, so 3 hours are 3 \times 60 = 180 minutes. Now take three fourths of 180:

\begin{aligned} 180 \div 4 &= 45 \\ 3 \times 45 &= 135 \end{aligned}

Zubin sleeps 135 minutes. Since 135 = 120 + 15 and 120 minutes are 2 hours, that is 2 hours and 15 minutes.

The same answer comes from staying in hours: \displaystyle \frac{3}{4} \times 3 = \frac{9}{4} = 2\frac{1}{4} hours, and a fourth of an hour is 60 \div 4 = 15 minutes.

Example 6

A share that is not whole

The tracks run beside a lake for 10 miles. Zlata watches the water for \displaystyle \frac{2}{3} of that stretch. How many miles is that? Between which two whole numbers is it?

Solution

Ten does not split into 3 equal whole numbers, so multiply first:

\frac{2}{3} \times 10 = \frac{2 \times 10}{3} = \frac{20}{3}

Three thirds make one mile. The multiples of 3 just below and just above 20 are 18 and 21, and \displaystyle \frac{18}{3} = 6, \displaystyle \frac{21}{3} = 7, so the answer is between 6 and 7 miles. Exactly, 20 = 18 + 2, so

\frac{20}{3} = 6\frac{2}{3}

Zlata watches the lake for \displaystyle 6\frac{2}{3} miles. Dividing first gives the same thing: each third of the stretch is \displaystyle \frac{10}{3} = 3\frac{1}{3} miles, and two of them make \displaystyle 6\frac{2}{3} miles.

Example 7

Between two stops

The whole trip is 150 miles. The first stop comes after \displaystyle \frac{2}{5} of the trip. The second stop comes after \displaystyle \frac{5}{6} of the trip. How many miles apart are the two stops, and how far is it from the second stop to the end?

Solution

Both fractions are of the same whole, the 150-mile trip, so find each distance from the start.

First stop: 150 \div 5 = 30, and 2 \times 30 = 60 miles from the start.

Second stop: 150 \div 6 = 25, and 5 \times 25 = 125 miles from the start.

The stops are 125 - 60 = 65 miles apart. From the second stop to the end is 150 - 125 = 25 miles.

A check: after the second stop, \displaystyle \frac{1}{6} of the trip is left, and \displaystyle \frac{1}{6} of 150 is 150 \div 6 = 25 miles, which matches.

Boundaries

  • The parts must be equal. A fourth of a set is one of 4 equal groups. Groups of different sizes are not fourths, however many there are.
  • The fraction is of a named whole. \displaystyle \frac{2}{3} of a 120-mile trip and \displaystyle \frac{2}{3} of an hour are different amounts. Two fractions can be compared or combined this way only when they are of the same whole, as in the last example.
  • A set of things that cannot be cut needs an even split. For miles, minutes or pounds, any fraction of any whole number has an answer, perhaps a mixed number. For passengers or seats, an answer with a fraction left over has no plain meaning, and the leftover has to be read in the story.
  • A fraction greater than one gives more than the whole. \displaystyle \frac{5}{4} of 8 is \displaystyle \frac{40}{4} = 10: five groups when only four make the whole. That is a fair product, but it is no longer a part of the amount.
  • The denominator cannot be 0. An amount cannot be cut into 0 equal groups.
  • The amount here is a whole number. A fraction of a fraction, such as half of three fourths of a cup, needs the next idea, A fraction times a fraction.

Where this goes next

Two cuts in a row, a fraction of what an earlier fraction left, is A fraction of a fraction of an amount. Working backward, from a part and its fraction to the whole it came from, is Finding the whole when only a fraction of it is known. A percent is a fraction out of a hundred, so finding a percent of a number uses this same move, in Finding a percent of a quantity. The bars cut into equal parts in this lesson’s figures grow into the tape diagrams of Tape diagrams and sharing in a ratio.

Exercises

Part 1: a fraction of a set

  1. Find \displaystyle \frac{1}{3} of 21.
  2. Find \displaystyle \frac{1}{6} of 30.
  3. Find \displaystyle \frac{2}{3} of 27.
  4. Find \displaystyle \frac{3}{8} of 16.
  5. Find \displaystyle \frac{4}{5} of 20.
  6. A freight train has 14 cars, and \displaystyle \frac{3}{7} of them carry grain. How many cars carry grain?
  7. Anahera takes 36 photos on the trip, and \displaystyle \frac{5}{9} of them show mountains. How many show mountains? How many do not?
  8. Zlata says \displaystyle \frac{2}{4} of 8 and \displaystyle \frac{1}{2} of 8 are the same. Find both and say whether she is right.

Part 2: of means times

  1. Write \displaystyle \frac{3}{5} of 25 as a multiplication and find it.
  2. Find \displaystyle \frac{5}{6} \times 12 by multiplying first and by dividing first.
  3. Find \displaystyle 28 \times \frac{3}{4}.
  4. Find \displaystyle \frac{7}{10} of 50.
  5. Show that \displaystyle \frac{4}{7} of 21 and \displaystyle 21 \times \frac{4}{7} give the same number.
  6. A full ticket costs 40 dollars, and a senior’s ticket costs \displaystyle \frac{5}{8} of it. How much is a senior’s ticket?
  7. Find \displaystyle \frac{5}{3} of 9. Why is the answer more than 9?

Part 3: a fraction of a measurement

  1. How many minutes are in \displaystyle \frac{3}{4} of an hour?
  2. A trip is 200 miles. The train has gone \displaystyle \frac{3}{10} of the way. How many miles has it gone, and how many are left?
  3. Zubin is on trains for \displaystyle \frac{1}{3} of a day. How many hours is that?
  4. Anahera walks \displaystyle \frac{3}{4} of a 5-mile trail near the station. How far does she walk? Write the answer as a mixed number.
  5. A train window is 8 feet wide, and a shade covers \displaystyle \frac{2}{3} of it. How wide is the shade?
  6. A yard is 36 inches. How many inches are in \displaystyle \frac{2}{3} of a yard?

Reasoning

  1. Explain why \displaystyle \frac{1}{6} of 42 is the same as 42 \div 6, and find it.
  2. Without finding it exactly, explain why \displaystyle \frac{5}{8} of 48 is more than 24 and less than 48.
  3. Zubin says \displaystyle \frac{2}{3} of 7 miles cannot be found, because 7 does not split into 3 equal groups. Explain why he is wrong, and find the distance.
  4. Explain why finding \displaystyle \frac{4}{9} of 27 by dividing first and by multiplying first must give the same answer, and find it both ways.
Answers
  1. 7. Three equal groups of 21: 21 \div 3 = 7.
  2. 5, since 30 \div 6 = 5.
  3. 18. One third of 27 is 27 \div 3 = 9, and two thirds is 2 \times 9 = 18.
  4. 6. One eighth of 16 is 16 \div 8 = 2, and three eighths is 3 \times 2 = 6.
  5. 16. One fifth of 20 is 4, and four fifths is 4 \times 4 = 16.
  6. 6 cars. One seventh of 14 is 14 \div 7 = 2, and 3 \times 2 = 6.
  7. 20 show mountains and 16 do not. One ninth of 36 is 4, so five ninths is 5 \times 4 = 20, and 36 - 20 = 16.
  8. She is right; both are 4. One fourth of 8 is 2, so \displaystyle \frac{2}{4} of 8 is 2 \times 2 = 4, and \displaystyle \frac{1}{2} of 8 is 8 \div 2 = 4.
  9. \displaystyle \frac{3}{5} \times 25 = \frac{75}{5} = 15. Dividing first: 25 \div 5 = 5, and 3 \times 5 = 15.
  10. 10. Multiply first: \displaystyle \frac{5 \times 12}{6} = \frac{60}{6} = 10. Divide first: 12 \div 6 = 2, and 5 \times 2 = 10.
  11. 21. \displaystyle 28 \times \frac{3}{4} = \frac{84}{4} = 21, or 28 \div 4 = 7 and 3 \times 7 = 21.
  12. 35. One tenth of 50 is 5, and 7 \times 5 = 35.
  13. Both are 12. Groups: 21 \div 7 = 3 and 4 \times 3 = 12. Copies: \displaystyle 21 \times \frac{4}{7} = \frac{84}{7} = 12. They agree by the commutative property.
  14. 25 dollars. One eighth of 40 is 5, and 5 \times 5 = 25.
  15. 15. \displaystyle \frac{5}{3} \times 9 = \frac{45}{3} = 15. Three thirds make the whole 9, and five thirds is two thirds more than that, so the answer is more than 9.
  16. 45 minutes. An hour is 60 minutes, 60 \div 4 = 15, and 3 \times 15 = 45.
  17. 60 miles gone and 140 miles left. One tenth of 200 is 20, 3 \times 20 = 60, and 200 - 60 = 140.
  18. 8 hours. A day is 24 hours, and 24 \div 3 = 8.
  19. \displaystyle 3\frac{3}{4} miles. \displaystyle \frac{3}{4} \times 5 = \frac{15}{4}, and 15 = 12 + 3, so \displaystyle \frac{15}{4} = 3\frac{3}{4}.
  20. \displaystyle 5\frac{1}{3} feet. \displaystyle \frac{2}{3} \times 8 = \frac{16}{3}, and 16 = 15 + 1, so \displaystyle \frac{16}{3} = 5\frac{1}{3}.
  21. 24 inches. 36 \div 3 = 12, and 2 \times 12 = 24.
  22. 7. One sixth of 42 is one of 6 equal groups made from 42, and the size of each group when 42 is shared into 6 equal groups is exactly what 42 \div 6 means. So both are 42 \div 6 = 7.
  23. Half of 48 is 24, and \displaystyle \frac{5}{8} is more than \displaystyle \frac{4}{8}, which is one half, so \displaystyle \frac{5}{8} of 48 is more than 24. Also \displaystyle \frac{5}{8} is less than \displaystyle \frac{8}{8}, the whole, so it is less than 48. (Exactly, it is 30.)
  24. A length can be cut into parts that are not whole miles. Multiply first: \displaystyle \frac{2}{3} \times 7 = \frac{14}{3}, and 14 = 12 + 2, so the distance is \displaystyle 4\frac{2}{3} miles.
  25. Both are 12. Dividing first: 27 \div 9 = 3, and 4 \times 3 = 12. Multiplying first: \displaystyle \frac{4 \times 27}{9} = \frac{108}{9} = 12. Both orders find \displaystyle 4 \times \frac{27}{9}: one way works out \displaystyle \frac{27}{9} first, the other writes it as \displaystyle \frac{4 \times 27}{9}, and the rule for a whole number times a fraction says these are equal.
kg1128 · updated 2026-10-20 · CC-BY-SA-4.0