Common fractions: meaning, simplifying and comparing

In this lesson you will learn

Explain a fraction as part of a whole and as a division
Convert improper fractions to mixed numbers and back
Use the basic property to simplify fractions and rewrite them with a new denominator
Bring fractions to a common denominator, compare them and mark them on the number line

Leyla cut her pizza into 8 equal slices and ate 3 of them. Murad cut a pizza of the same size into 4 equal slices and ate 2. Who ate more? Leyla ate more slices, but her slices are half as big. Leyla ate 3/8 of her pizza and Murad ate 2/4 of his, which is exactly half. Since 2/4 = 4/8 and 4/8 > 3/8, Murad ate more. In this lesson you will learn to answer questions like this quickly and precisely.

Two helpers from the lesson “Greatest common divisor and least common multiple” will do most of the work: the GCD lets us simplify a fraction, and the LCM gives us a common denominator. Adding, subtracting, multiplying and dividing fractions is the topic of the next lesson, “Operations with fractions”.

What is a fraction: part of a whole and a division

Divide a whole — a pizza, a strip, 1 hour, 1 km of road — into b equal parts and take a of them. The amount you took is written as the fraction a/b. The number below the fraction bar gives the parts their “name” (eighths, quarters), and the number above counts them. One condition matters: the parts must be equal — if a pizza is cut into 8 slices of different sizes, one slice is not 1/8.

Definition

Common fraction

A number written as a/b, where a and b are natural numbers. a is the numerator, b is the denominator, and the line between them is the fraction bar. 3/8 is read “three eighths”.

a/b = a ÷ ba/b = a ÷ b

where:

anumerator — how many equal parts are taken (the dividend)
bdenominator — how many equal parts the whole is divided into (the divisor, b ≠ 0)

The fraction bar is a division sign: the quotient of two natural numbers can always be written as a fraction.

Why is a/b = a ÷ b? Share 3 cakes equally among 4 friends. Cut every cake into 4 equal parts: that makes 12 quarters, and each friend gets 3 quarters, that is 3/4 of a cake. So 3 ÷ 4 = 3/4. Two consequences follow: every natural number is a fraction with denominator 1 (5 = 5/1), and a fraction whose numerator equals its denominator is 1 (7/7 = 1).

The meaning of a fraction

1) A year has 12 months. What part of the year do the summer months make up?
2) 5 m of ribbon is shared equally among 8 children. How many metres does each child get?
3) What numbers are 18/6 and 7/7 equal to?

Show solution

1) The whole is 12 months and the part is the 3 summer months: 3/12. A little later we will see that this is 1/4.
2) A fraction is a division: 5 ÷ 8 = 5/8 m.
3) 18/6 = 18 ÷ 6 = 3; 7/7 = 7 ÷ 7 = 1.

Interactive

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Change the numerator and the denominator and see the fraction as a pie and a bar, with equivalent fractions, the decimal and the percentage.

Proper and improper fractions, mixed numbers

Definition

Proper and improper fraction

A fraction whose numerator is less than its denominator is a proper fraction (3/8, 5/6); it is always less than 1. A fraction whose numerator is equal to or greater than its denominator is an improper fraction (7/4, 9/9); it is at least 1.

Definition

Mixed number

The sum of a natural number and a proper fraction written without the “+” sign: 1 3/4 = 1 + 3/4, read “one and three quarters”. Here 1 is the whole part and 3/4 is the fractional part.

An improper fraction hides whole parts inside it. 7/4 of a pizza means 7 quarters: 4 of them make one whole pizza and 3 are left over. So 7/4 = 1 3/4. In other words, we divide 7 by 4 with a remainder: the quotient gives the whole part and the remainder gives the new numerator.

a/b = q r/b, a = b · q + ra/b = q r/b, a = b · q + r

where:

qthe quotient of a ÷ b — the whole part
rthe remainder (r < b) — the numerator of the fractional part
bthe denominator stays the same

To turn an improper fraction into a mixed number, divide the numerator by the denominator with a remainder. If the remainder is 0, the fraction equals a natural number.

From an improper fraction to a mixed number

Write as a mixed number or a natural number: 1) 17/5; 2) 23/4; 3) 24/6.

Show solution

1) 17 ÷ 5 = 3 remainder 2, so 17/5 = 3 2/5.
2) 23 ÷ 4 = 5 remainder 3, so 23/4 = 5 3/4.
3) 24 ÷ 6 = 4 remainder 0, so 24/6 = 4.
Check: 3 · 5 + 2 = 17 and 5 · 4 + 3 = 23.

The reverse conversion uses the same idea: in 2 3/7, each whole is 7 sevenths, so two wholes are 14 sevenths; add 3 more sevenths and you have 17/7.

q r/b = (q · b + r)/bq r/b = (q · b + r)/b

where:

qthe whole part
r/br/bthe fractional part

Multiply the whole part by the denominator, add the numerator, keep the denominator.

From a mixed number to an improper fraction

1) Write 2 3/7 and 4 1/6 as improper fractions.
2) Write 5 as a fraction with denominator 3.

Show solution

1) 2 3/7 = (2 · 7 + 3)/7 = 17/7; 4 1/6 = (4 · 6 + 1)/6 = 25/6.
2) There is no fractional part (r = 0): 5 = (5 · 3)/3 = 15/3. Check: 15 ÷ 3 = 5.

The basic property of a fraction. Simplifying

Take half a pizza: that is 1/2. Cut every slice in two: there are twice as many slices (4) and twice as many taken (2), but the amount of pizza is the same: 1/2 = 2/4. Cut again and you get 4/8. The numerator and the denominator grew by the same factor, while the value of the fraction stayed the same.

a/b = (a · k)/(b · k) = (a ÷ m)/(b ÷ m)a/b = (a · k)/(b · k) = (a ÷ m)/(b ÷ m)

where:

kany natural number: the fraction gets a new, larger denominator
ma common divisor of a and b: the fraction is simplified

The basic property of a fraction: multiplying or dividing the numerator and the denominator by the same natural number does not change its value. Such fractions are called equivalent fractions.

Rewriting with a new denominator

1) Write 3/5 as a fraction with denominator 20.
2) Fill in the gap: 5/8 = 35/?
3) Can 3/4 be written as a fraction with denominator 10?

Show solution

1) Extra factor: 20 ÷ 5 = 4. 3/5 = (3 · 4)/(5 · 4) = 12/20.
2) The numerator went from 5 to 35, that is, it became 7 times larger. The denominator must also become 7 times larger: 8 · 7 = 56, so 5/8 = 35/56.
3) 10 ÷ 4 is not a natural number, so 3/4 cannot be written with denominator 10 and a natural numerator. Denominator 100 works, though: 3/4 = 75/100.

Definition

Simplifying a fraction

Dividing the numerator and the denominator by a common divisor. A fraction whose numerator and denominator are coprime (GCD = 1) is in lowest terms; dividing by the GCD gets there in one step.

1
Find the GCD
Find the greatest common divisor of the numerator and the denominator (by listing divisors or by prime factors).
2
Divide both
Divide the numerator and the denominator by the GCD.
3
Check
The new numerator and denominator must have no common divisor other than 1. If the GCD is hard to find, divide by small common divisors (2, 3, 5 …) one after another — the result is the same.

Simplifying fractions

Simplify: 1) 18/24; 2) 42/56; 3) 35/63; 4) 120/180.

Show solution

1) GCD(18, 24) = 6: 18/24 = 3/4.
2) GCD(42, 56) = 14: 42/56 = 3/4.
3) GCD(35, 63) = 7: 35/63 = 5/9.
4) Step by step: 120/180 = 12/18 (divided by 10) = 2/3 (divided by 6). In one step: GCD(120, 180) = 60.
Now the first example is settled too: 3/12 = 1/4, so summer is a quarter of the year.

Common denominator and comparing fractions

Which is larger, 5/6 or 7/8? Sixths and eighths cannot be compared directly — the pieces have different sizes. The basic property helps: we replace both fractions with equivalent fractions that have the same denominator. This common denominator must be divisible by both denominators, and the smallest one is their LCM, which keeps the numbers small.

K = LCM(b, d), a/b = (a · m)/K, m = K ÷ bK = LCM(b, d), a/b = (a · m)/K, m = K ÷ b

where:

b, dthe denominators of the fractions
Kthe least common denominator
mthe extra factor — K divided by the denominator

Multiply the numerator and the denominator of each fraction by its own extra factor; the same for the second fraction: c/d = (c · (K ÷ d))/K.

1
Find the LCM of the denominators
Try the multiples of the larger denominator: for 6 and 8 check 8, 16, 24 — 24 is divisible by 6, so K = 24.
2
Find the extra factors
Divide K by each denominator: 24 ÷ 6 = 4, 24 ÷ 8 = 3.
3
Multiply and compare
Multiply the numerator and the denominator of each fraction by its factor: 5/6 = 20/24, 7/8 = 21/24. Since 21 > 20, 7/8 > 5/6.

Two special cases save time. If one denominator is divisible by the other, the larger one is the common denominator (12 for 3/4 and 5/12). If the denominators are coprime, the common denominator is their product (35 for 2/5 and 3/7).

Finding a common denominator

Bring to the least common denominator: 1) 3/4 and 5/12; 2) 2/5 and 3/7; 3) 1/4, 5/6 and 7/9.

Show solution

1) 12 is divisible by 4, so K = 12: 3/4 = 9/12, and 5/12 stays as it is.
2) 5 and 7 are coprime, so K = 35: 2/5 = 14/35, 3/7 = 15/35.
3) LCM(4, 6, 9) = 36; the extra factors are 9, 6 and 4: 1/4 = 9/36, 5/6 = 30/36, 7/9 = 28/36.

Case	Rule	Example
Same denominators	The larger numerator gives the larger fraction	5/9 > 4/9
Same numerators	The smaller denominator gives the larger fraction: the pieces are bigger	5/7 > 5/9
Different numerators and denominators	Bring to a common denominator, then compare the numerators	2/3 = 10/15 > 9/15 = 3/5
Proper and improper fraction	proper fraction < 1 ≤ improper fraction	11/12 < 1 < 12/11

Comparing and ordering

1) Compare 2/3 and 3/5.
2) Put 2/3, 3/5 and 7/10 in increasing order.
3) Which is larger: 11/12 or 12/11?

Show solution

1) K = 15: 2/3 = 10/15, 3/5 = 9/15. 10 > 9, so 2/3 > 3/5.
2) K = 30: 2/3 = 20/30, 3/5 = 18/30, 7/10 = 21/30. The numerators are 18 < 20 < 21, so 3/5 < 2/3 < 7/10.
3) 11/12 is proper (< 1) and 12/11 is improper (> 1), so 12/11 > 11/12 — no calculation needed.

Fractions on the number line

On the number line, divide the unit segment from 0 to 1 into b equal parts and count a parts to the right of 0 — that is the point a/b. Proper fractions lie between 0 and 1, improper fractions at 1 or to the right of it, and the mixed number q r/b lies between q and q + 1. Equivalent fractions are the same point (1/2, 2/4 and 4/8 coincide), and of two fractions the one further right is larger.

5/4 = 1 1/4 and 3/2 = 6/4 = 1 1/2: improper fractions lie to the right of 1.

Using the number line

1) The unit segment is 6 squares long. How many squares to the right of 0 are the points 5/6, 7/6 and 1 1/2?
2) Find a fraction that lies between 1/3 and 1/2.
3) Which natural numbers lie between 17/5 and 31/4?

Show solution

1) One square is 1/6 of the unit. 5/6 is 5 squares, 7/6 is 7 squares, and 1 1/2 = 3/2 = 9/6 is 9 squares.
2) With denominator 6 we get 2/6 and 3/6 — no fraction with denominator 6 fits between them. Take denominator 12: 4/12 and 6/12, and 5/12 lies between them.
3) 17/5 = 3 2/5 and 31/4 = 7 3/4. Between them lie 4, 5, 6 and 7.

Check yourself

1.15/25 = 3/
2.2 1/3 = /3
3.29/6 = 4 /6
4.3/8 = /40
5.The least common denominator of 4/9 and 5/12:
6.2/3 5/8 (write >, < or =)

If you can simplify fractions, bring them to a common denominator and compare them, you are ready to calculate with them: in the lesson “Operations with fractions” you will need all these skills for adding and subtracting.

Key points

In a/b, the denominator b tells how many equal parts the whole is divided into and the numerator a how many are taken; the fraction bar means division: a/b = a ÷ b.
A proper fraction is less than 1; an improper fraction becomes a mixed number by division with remainder (23/4 = 5 3/4), and the way back is (q · b + r)/b.
Multiplying or dividing the numerator and the denominator by the same natural number keeps the value; dividing by the GCD gives lowest terms at once.
The least common denominator is the LCM of the denominators, and each fraction’s extra factor is LCM ÷ denominator.
With equal denominators the larger numerator wins, with equal numerators the smaller denominator wins; on the number line the larger fraction lies further right.

Check yourself

12 questions. Every correct answer earns XP.

1 / 12

What is the denominator of 5/8?