What the numerator and the denominator each tell you
A fraction is written with two numbers, one above the bar and one below. What does each of them tell you, and why does it take both?
Where you are 15 in Fractions
Why this exists
In Area and length models of a fraction a fraction was a picture: a whole cut into equal parts, some of them shaded. The fraction \displaystyle \frac{3}{8} was written with a 3 on top and an 8 below, and the two numbers were given names, numerator and denominator. This lesson is about what each of those numbers says, and why a fraction needs both.
The two numbers do different jobs. One of them names a kind of piece, the way “inch” or “dollar” names a unit. The other counts how many of those pieces there are. “Three eighths” works like “three inches”: the 3 is a count, and “eighths” says what is being counted. Once that is clear, a lot follows. A fraction is so many copies of one small piece, the unit fraction. Cutting the same whole into more pieces makes each piece smaller. And the short reading “three out of eight” turns out to be true only some of the time.
Definition
In the fraction
the number b below the bar is the denominator and the number a above it is the numerator. The line between them is the fraction bar.
- The denominator names the piece. It says how many equal pieces make one whole. A whole made of b equal pieces has pieces called bths: halves for 2, thirds for 3, fourths for 4, sixths for 6, eighths for 8. One such piece is the unit fraction \displaystyle \frac{1}{b}.
- The numerator counts the pieces. It says how many of those pieces the fraction is made of.
So \displaystyle \frac{a}{b} is a pieces, each of size \displaystyle \frac{1}{b} of the whole. It is said with the count first and the piece name second: \displaystyle \frac{3}{8} is “three eighths”, \displaystyle \frac{2}{3} is “two thirds”.
The words come from Latin. A numerator is a counter, and a denominator is a namer: the denominator gives the pieces their name.
In this lesson the numerator is never more than the denominator, so the fraction is never more than one whole. The denominators used are 2, 3, 4, 6 and 8.
Intuition
The lesson runs over three days. Part 1 gives each number its job. Part 2 builds a fraction from copies of one piece and asks what the denominator does to the size of that piece. Part 3 compares fractions with the same numerator and looks hard at the reading “out of”.
Part 1: one number names, the other counts
Aatu cuts a loaf of banana bread into 8 equal slices and gives 3 away (Figure 1). To say how much he gave away, two questions need answers, and they are different questions.
What size is each slice? Eight equal slices make the loaf, so each slice is one eighth of it. That is the denominator, 8. Notice that the 8 is decided by the cutting, before anyone gives a slice away.
How many slices went? Three. That is the numerator, 3.
So Aatu gave away \displaystyle \frac{3}{8} of the loaf, three eighths. The 8 is the same whether he gives away one slice or all of them; only the numerator changes as slices go. And if he had cut the loaf into 6 slices instead, the denominator would change, because the slices would be a different size.
Part 2: copies of one piece, and the size of the piece
A single slice of Aatu’s loaf is \displaystyle \frac{1}{8} of it, a unit fraction. Three slices are three copies of that piece, so
Every fraction is built this way: \displaystyle \frac{a}{b} is a copies of \displaystyle \frac{1}{b}. The denominator picks the piece and the numerator says how many times to take it.
The denominator also says how big the piece is. Take two paper chains of the same length. Elina makes hers from 6 equal links, Raili from 3. Each of Elina’s links is \displaystyle \frac{1}{6} of a chain and each of Raili’s is \displaystyle \frac{1}{3}. Raili’s links are longer: the same length shared among fewer links gives each link more. A bigger denominator means smaller pieces, as long as the whole is the same.
That last condition is not a small print detail. One third of a short chain can be the same length as one sixth of a long one.
Part 3: same count, different pieces, and “out of”
Two pieces of lasagna can be a lot or a little. If the pan is cut into 4 equal pieces, two of them, \displaystyle \frac{2}{4}, is half the pan. If the same pan is cut into 8 equal pieces, two of them, \displaystyle \frac{2}{8}, is much less. The numerators are the same; the pieces are not. Comparing two fractions needs both numbers.
A fraction like \displaystyle \frac{2}{8} is often read “two out of eight”. That reading is right when the whole really is cut into 8 equal pieces and 2 of them are taken. But “out of” leaves out the word equal, and the pieces must be equal: a pan cut into 6 pieces of different sizes does not turn 2 of them into \displaystyle \frac{2}{6}. And “out of” cannot stretch past one whole: there is no way to take 5 out of 4, yet five fourths is a perfectly good amount (see Fractions greater than one (improper fractions)). The better reading is the one from Part 2: \displaystyle \frac{2}{8} is 2 copies of \displaystyle \frac{1}{8}.
Results
The denominator names the piece. In \displaystyle \frac{a}{b}, the denominator b says that b equal pieces make one whole, so each piece is \displaystyle \frac{1}{b} of the whole.
The numerator counts the pieces. The numerator a says how many of those pieces the fraction holds.
A fraction is copies of a unit fraction. \displaystyle \frac{a}{b} is a copies of \displaystyle \frac{1}{b}. In particular b copies of \displaystyle \frac{1}{b}, which is \displaystyle \frac{b}{b}, make the whole.
A bigger denominator means a smaller piece. For the same whole, \displaystyle \frac{1}{6} is smaller than \displaystyle \frac{1}{3}, and \displaystyle \frac{1}{8} is smaller than \displaystyle \frac{1}{4}. With the same numerator the same is true of the whole fraction: \displaystyle \frac{2}{8} is less than \displaystyle \frac{2}{4} of the same whole.
“Out of” needs equal pieces. \displaystyle \frac{a}{b} is “a out of b” exactly when the whole is cut into b equal pieces and a of them are taken, so a is at most b.
Why it is true
Why a bigger denominator means a smaller piece. Picture one whole and cut it into 3 equal pieces. Now cut each of those pieces in half. The whole is now 6 equal pieces, and each new piece is half of an old one. So \displaystyle \frac{1}{6} is smaller than \displaystyle \frac{1}{3}: it takes two sixths to fill one third. The same cutting in half turns fourths into eighths, so each eighth is half of a fourth. In general, more equal pieces from the same whole leaves less for each piece, because the pieces together must make up exactly the whole.
Why the same numerator follows suit. \displaystyle \frac{2}{8} is 2 eighths and \displaystyle \frac{2}{4} is 2 fourths. Each eighth is smaller than each fourth, so two eighths are less than two fourths. Two of a smaller thing are less than two of a bigger thing.
Why the whole must be the same. The size of \displaystyle \frac{1}{b} depends on the whole it is cut from. A whole twice as long has pieces twice as long. So the denominator alone does not give a size; it gives a size of a named whole. Two pieces from different wholes cannot be compared by their denominators (Example 5).
Why “out of” needs equal pieces. The denominator means b equal pieces make the whole. Count pieces that are not equal and the count says nothing about size, since a big piece and a sliver count the same (Example 7).
Worked examples
Part 1: banana bread given away
Aatu’s loaf of banana bread is cut into 8 equal slices, and he gives 3 of them to a neighbor (Figure 1). What fraction of the loaf did he give away? Say what the numerator and the denominator each tell you. What fraction of the loaf is left?
SolutionFirst the size of the pieces. The loaf is cut into 8 equal slices, so 8 slices make the whole loaf and each slice is one eighth of it. The denominator is 8.
Then the count. He gave away 3 slices. The numerator is 3.
So he gave away \displaystyle \frac{3}{8} of the loaf, three eighths. The denominator 8 tells you each slice is \displaystyle \frac{1}{8} of the loaf; the numerator 3 tells you how many of those slices went.
The slices left are the same size, still eighths, so the denominator stays 8. There are 5 of them, so \displaystyle \frac{5}{8} of the loaf is left. Check: 3 + 5 = 8, all eight eighths.
Part 1: from a fraction to a picture
Raili says that \displaystyle \frac{5}{6} of her lasagna pan has spinach on top. Draw a picture of the pan that shows this. What part of the pan has no spinach?
Read the denominator first, because it tells you how to cut. The 6 says that 6 equal pieces make the whole pan. A rectangle cut into 2 rows of 3 gives 6 pieces, and since every piece has the same width and the same height, they are equal.
Then read the numerator. The 5 says to take 5 of those pieces, so shade any 5 of them (Figure 2).
The one piece left over is \displaystyle \frac{1}{6} of the pan, and it has no spinach. Check: 5 + 1 = 6.
Any cutting into 6 equal pieces would do: 3 rows of 2, or 6 long strips. The denominator fixes how many equal pieces there are, not how they are arranged.
Part 2: three eighths as copies of one eighth
Raili has one slice from a loaf like Aatu’s, cut into 8 equal slices (Figure 3). What fraction of the loaf is her slice? How many slices like hers make \displaystyle \frac{3}{8} of the loaf? How many make \displaystyle \frac{5}{8}? How many make the whole loaf?
The loaf is 8 equal slices, so one slice is \displaystyle \frac{1}{8} of it, a unit fraction.
\displaystyle \frac{3}{8} is three eighths: the denominator names the piece, one eighth, and the numerator counts 3 of them. So 3 slices like Raili’s make \displaystyle \frac{3}{8} of the loaf. In the same way \displaystyle \frac{5}{8} is 5 copies of \displaystyle \frac{1}{8}, so 5 slices.
The whole loaf is the 8 slices it was cut into, 8 copies of \displaystyle \frac{1}{8}. Written as a fraction that is \displaystyle \frac{8}{8}, eight eighths, which is one whole loaf.
In every case the slice stays the same and only the count changes. That is what it means for \displaystyle \frac{a}{b} to be a copies of \displaystyle \frac{1}{b}. Counting the copies one at a time, “one eighth, two eighths, three eighths”, is the next lesson, Counting by unit fractions.
Part 2: one sixth and one third of a chain
Elina and Raili each make a paper chain 24 inches long out of equal links. Elina’s chain has 6 links and Raili’s has 3 (Figure 4). What fraction of a chain is one of Elina’s links? One of Raili’s? Whose links are longer, and by how much?
Elina’s chain is 6 equal links, so each link is \displaystyle \frac{1}{6} of her chain. Raili’s chain is 3 equal links, so each link is \displaystyle \frac{1}{3} of hers.
To see which is longer, find the lengths. Six equal links make 24 inches, and 6 \times 4 = 24, so each of Elina’s links is 24 \div 6 = 4 inches. Three equal links make 24 inches, and 3 \times 8 = 24, so each of Raili’s is 24 \div 3 = 8 inches.
Raili’s links are longer: 8 inches against 4, twice as long. The chains are the same length, and Raili shared that length among fewer links, so each link got more.
So \displaystyle \frac{1}{3} of the chain is longer than \displaystyle \frac{1}{6} of it, though 6 is the bigger number. The denominator counts pieces, and more pieces of the same whole are smaller pieces. Two of Elina’s links, 4 + 4 = 8 inches, fill exactly one of Raili’s.
Part 2: when the wholes are different
Aatu makes a paper chain 24 inches long from 3 equal links. Elina makes one 48 inches long from 6 equal links (Figure 5). Each takes off one link. Aatu’s link is \displaystyle \frac{1}{3} of his chain and Elina’s is \displaystyle \frac{1}{6} of hers. Is Aatu’s link longer?
Aatu’s links: 3 equal links make 24 inches, so each is 24 \div 3 = 8 inches.
Elina’s links: 6 equal links make 48 inches, and 6 \times 8 = 48, so each is 48 \div 6 = 8 inches.
The two links are the same length, 8 inches each. One third is not longer here, because it is one third of a shorter chain. Elina’s chain is twice as long as Aatu’s, so although it has twice as many links, each link comes out the same.
“A bigger denominator means a smaller piece” is a statement about pieces of the same whole. In Example 4 the chains were the same length and one third was longer. Here the wholes differ, and the denominators alone cannot tell you which piece is bigger.
Part 3: two fourths and two eighths
Raili has two lasagna pans of the same size. She cuts one into 4 equal pieces and the other into 8 equal pieces, and takes 2 pieces from each (Figure 6). Write each amount as a fraction of a pan. Which is more lasagna?
From the first pan she takes 2 pieces, each \displaystyle \frac{1}{4} of the pan: \displaystyle \frac{2}{4}. From the second she takes 2 pieces, each \displaystyle \frac{1}{8} of the pan: \displaystyle \frac{2}{8}.
Both numerators are 2, so she took the same number of pieces. The denominators say the pieces are different sizes. The pans are the same, and cutting each fourth in half gives the eighths (look at Figure 6: 2 rows of 2 become 2 rows of 4). So each eighth is half a fourth, and \displaystyle \frac{2}{8} is less than \displaystyle \frac{2}{4}.
The same count of pieces is not the same amount. The numerator says how many; only the denominator says how big. (One amount with two names, such as \displaystyle \frac{2}{4} and \displaystyle \frac{4}{8}, is Seeing equivalent fractions in pictures and on the number line, and comparing fractions in general comes later, starting with Comparing fractions with the same denominator or numerator.)
Part 3: when “out of” does not work
Raili cuts a lasagna pan into 6 pieces as in Figure 7. The dashed lines mark the pan into 8 small squares, 2 rows of 4. The top row is cut into 2 large pieces and the bottom row into 4 small ones. She takes the 2 large pieces. She has 2 pieces out of 6. Is that \displaystyle \frac{2}{6} of the pan? If not, what fraction is it?
A denominator of 6 would mean that 6 equal pieces make the pan. These 6 pieces are not equal: each large piece covers 2 small squares and each small piece covers 1. So the pan is not cut into sixths, and “2 out of 6” does not give a fraction of the pan.
Use the equal parts that are there, the 8 small squares. Each is \displaystyle \frac{1}{8} of the pan. The two large pieces cover 2 + 2 = 4 small squares, so Raili has \displaystyle \frac{4}{8} of the pan. That is the whole top row, which is half the pan.
Compare with \displaystyle \frac{2}{6}: two sixths would be less than half the pan, since 3 sixths make a half. Raili has more than that.
“Out of” is a fair reading of a fraction only when the pieces being counted are equal and make up the whole. When they are, the denominator counts them and the numerator counts the ones taken.
Boundaries
The pieces must be equal. The denominator counts equal pieces of the whole. A count of unequal pieces is not a denominator (Example 7).
The whole must be named, and the same. The denominator gives the size of a piece only as a part of its whole. “A bigger denominator means a smaller piece” holds for pieces of one whole, or of wholes the same size, and fails otherwise (Example 5).
The denominator is never zero. A whole cannot be cut into 0 pieces, so \displaystyle \frac{1}{0} names no piece at all.
A numerator of zero is allowed. \displaystyle \frac{0}{4} is no fourths, none of the whole.
At most one whole, in this lesson. Here the numerator is never more than the denominator. Five copies of \displaystyle \frac{1}{4} is a real amount, more than one whole, but “five out of four” makes no sense; that is where “copies of a unit fraction” keeps working and “out of” stops. Counting on past one is Counting by unit fractions, and those amounts are Fractions greater than one (improper fractions).
Same numerator only. With the same numerator, the bigger denominator is the smaller fraction. With different numerators and different denominators, neither number alone decides: \displaystyle \frac{6}{6} has the bigger denominator and is still more than \displaystyle \frac{1}{2}, because it is the whole.
Equal parts of a group. When the whole is a set of things, like a bag of marbles, the denominator counts equal shares of the set; that is A fraction of a set.
Where this goes next
The next lesson, Counting by unit fractions, counts copies of a unit fraction one at a time and keeps going past the whole, which leads to Fractions greater than one (improper fractions) and to Whole numbers written as fractions, where three thirds is exactly 1. Placing \displaystyle \frac{3}{8} on a line uses the same two jobs: the denominator says how to cut the stretch from 0 to 1, the numerator says how many steps to take, in Locating a fraction on the number line.
Seeing that 2 eighths fill 1 fourth opens Seeing equivalent fractions in pictures and on the number line. Comparing fractions uses both numbers on purpose, in Comparing fractions with the same denominator or numerator. And writing a fraction as its copies of a unit fraction, joined by plus signs, is Decomposing a fraction into unit fractions.
Exercises
Part 1: one number names, the other counts
Use Figure 8 for Exercises 1 to 3. For each picture, write the fraction that is shaded and say what its numerator and its denominator each tell you.
- Picture A, a strip.
- Picture B, a lasagna pan.
- Picture C, a paper chain.
- A loaf of banana bread is cut into 6 equal slices and Elina eats 1. What fraction of the loaf did she eat, and what fraction is left? Which number is the same in both fractions, and why?
- A lasagna pan is cut into 3 equal pieces and 2 of the pieces have mushrooms. What fraction of the pan has mushrooms? Say it in words.
- Aatu wants to draw \displaystyle \frac{3}{4} of a loaf of banana bread. How many equal slices should his loaf have, and how many should he shade?
- A paper chain is made of 8 equal links, and \displaystyle \frac{5}{8} of the chain is blue. How many links are blue? How many are not, and what fraction of the chain is that?
Part 2: copies of one piece
- How many copies of \displaystyle \frac{1}{6} make \displaystyle \frac{4}{6}? How many make the whole?
- Raili has 3 slices of banana bread, and each slice is \displaystyle \frac{1}{4} of the same loaf. What fraction of the loaf does she have?
- A paper chain 30 inches long has 6 equal links. How long is each link? How long is \displaystyle \frac{4}{6} of the chain?
- Two paper chains are each 16 inches long. One has 8 equal links and the other has 4. How long is a link of each? Which is longer, \displaystyle \frac{1}{8} of a chain or \displaystyle \frac{1}{4}?
- Three loaves of banana bread are the same size. One is cut into halves, one into fourths and one into eighths. Put one slice from each in order, smallest first, and write each as a fraction of a loaf.
- Aatu’s chain is 12 inches long with 3 equal links. Elina’s chain is 36 inches long with 6 equal links. Whose single link is longer, and by how many inches?
- A paper chain has 4 equal links, and each link is 6 inches long. How long is the chain? What fraction of the chain is 3 links, and how long is that?
Part 3: same count, different pieces
- Two lasagna pans are the same size. Which is more lasagna, \displaystyle \frac{3}{4} of a pan or \displaystyle \frac{3}{8} of a pan? Explain using what each number tells you.
- A paper chain is 18 inches long. How long is \displaystyle \frac{2}{3} of it? How long is \displaystyle \frac{2}{6} of it?
- A lasagna pan is cut into 3 pieces: one piece is half the pan, and the other two are each one fourth of the pan. Elina takes the biggest piece. What fraction of the pan does she have? Why is it not \displaystyle \frac{1}{3}?
- For the same pan, which is more: \displaystyle \frac{1}{6} or \displaystyle \frac{1}{2}? Which is more: \displaystyle \frac{5}{6} or \displaystyle \frac{5}{8}?
Reasoning
- A fraction has denominator 1, such as \displaystyle \frac{1}{1}. What does the denominator say about the pieces, and how much of the whole is \displaystyle \frac{1}{1}?
- Is a fraction with a bigger denominator always less than one with a smaller denominator? Give an example to back your answer.
- Explain, using copies of \displaystyle \frac{1}{4}, why \displaystyle \frac{4}{4} of a pan is the whole pan.
- Draw a shape cut into 3 pieces with 2 shaded, where the shaded part is not \displaystyle \frac{2}{3} of the shape. Explain what makes it fail.
- Raili’s lasagna pan is much bigger than Aatu’s. Can \displaystyle \frac{1}{8} of Raili’s pan be more lasagna than \displaystyle \frac{1}{4} of Aatu’s? Explain.
Answers
- \displaystyle \frac{3}{4}. The denominator 4 says 4 equal lengths make the strip, so each is a fourth; the numerator 3 says 3 of those fourths are shaded.
- \displaystyle \frac{7}{8}. The pan is 2 rows of 4 equal pieces, 8 in all, so each is an eighth (the denominator); 4 + 3 = 7 of them are shaded (the numerator).
- \displaystyle \frac{5}{6}. The chain is 6 equal links, so each link is a sixth (the denominator); 5 links are shaded (the numerator).
- She ate \displaystyle \frac{1}{6} and \displaystyle \frac{5}{6} is left, since 6 - 1 = 5. The denominator 6 is the same in both, because both are counted in slices of the same size, sixths of the same loaf.
- \displaystyle \frac{2}{3}, two thirds. Three equal pieces make the pan, so each is a third, and 2 of them have mushrooms.
- 4 equal slices, with 3 shaded. The denominator says how many equal slices make the loaf; the numerator says how many to shade.
- 5 links are blue, and 8 - 5 = 3 are not, which is \displaystyle \frac{3}{8} of the chain.
- 4 copies make \displaystyle \frac{4}{6}, and 6 copies, \displaystyle \frac{6}{6}, make the whole.
- \displaystyle \frac{3}{4} of the loaf: 3 copies of \displaystyle \frac{1}{4}.
- Each link is 5 inches, since 6 \times 5 = 30. Then \displaystyle \frac{4}{6} is 4 links, 4 \times 5 = 20 inches.
- A link of the 8 link chain is 16 \div 8 = 2 inches, and a link of the 4 link chain is 16 \div 4 = 4 inches. \displaystyle \frac{1}{4} of the chain is longer.
- \displaystyle \frac{1}{8}, then \displaystyle \frac{1}{4}, then \displaystyle \frac{1}{2}. The loaves are the same, so more slices means smaller slices.
- Elina’s, by 2 inches. Aatu’s links are 12 \div 3 = 4 inches and Elina’s are 36 \div 6 = 6 inches. Elina’s link is only \displaystyle \frac{1}{6} of her chain, but her chain is longer.
- The chain is 4 \times 6 = 24 inches. Three links are \displaystyle \frac{3}{4} of the chain, and they are 3 \times 6 = 18 inches.
- \displaystyle \frac{3}{4} is more. Both numerators are 3, so both are 3 pieces. The pans are the same size, so fourths are bigger pieces than eighths, and 3 bigger pieces are more lasagna.
- \displaystyle \frac{2}{3} is 12 inches: each third is 18 \div 3 = 6 inches and 2 \times 6 = 12. \displaystyle \frac{2}{6} is 6 inches: each sixth is 18 \div 6 = 3 inches and 2 \times 3 = 6.
- \displaystyle \frac{1}{2} of the pan. The 3 pieces are not equal, so they are not thirds. The two small pieces are 2 fourths, half the pan, so the big piece is the other half.
- \displaystyle \frac{1}{2} is more: the same pan cut into 2 equal pieces has bigger pieces than when it is cut into 6. \displaystyle \frac{5}{6} is more than \displaystyle \frac{5}{8}: both are 5 pieces, and sixths of the same pan are bigger than eighths.
- The whole is 1 piece, so that piece is the whole itself, and \displaystyle \frac{1}{1} is one whole.
- No. For the same loaf, \displaystyle \frac{6}{6} has a bigger denominator than \displaystyle \frac{1}{2} but is more: \displaystyle \frac{6}{6} is all 6 sixths, the whole loaf, and \displaystyle \frac{1}{2} is only part of it. The rule holds only for the same numerator and the same whole.
- Four equal fourths make the pan, because that is what the denominator 4 means. \displaystyle \frac{4}{4} is 4 copies of \displaystyle \frac{1}{4}, all four pieces, so it is the whole pan.
- Any drawing with pieces of different sizes works, for example a rectangle cut into one wide piece and two thin ones with a thin one and the wide one shaded. The denominator 3 needs 3 equal pieces; these are not equal, so counting them gives no fraction.
- Yes. The size of \displaystyle \frac{1}{8} depends on its whole. If Raili’s pan is more than twice as big as Aatu’s, one eighth of hers is more than one fourth of his. “Bigger denominator, smaller piece” holds only for the same whole.