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Comparing fractions

Comparing positive fractions with the same denominator

Example 1

Which is greater? a) \latex{\frac{2}{6}} or \latex{\frac{5}{6}} b) \latex{\frac{2}{5}} or \latex{\frac{3}{5}}

Solution

Represent the fractions on a circle. Show the fractions on a number line.

a)

\latex{\frac{2}{6}}

\latex{\frac{5}{6}}

\latex{0}

\latex{\frac{2}{6}}

\latex{\frac{5}{6}}

\latex{1=\frac{6}{6}}

b)

\latex{\frac{2}{5}}

\latex{\frac{3}{5}}

\latex{0}

\latex{\frac{2}{5}}

\latex{\frac{3}{5}}

\latex{1=\frac{5}{5}}

In the case of positive fractions with the same denominator, the one with the larger numerator is greater.

Comparing positive fractions with the same numerator

Example 2

Which is greater? a) \latex{\frac{3}{8}} or \latex{\frac{3}{5}} b) \latex{\frac{2}{3}} or \latex{\frac{2}{7}}

Solution

Represent the fractions in a rectangle. Show the fractions on a number line.

a)

\latex{\frac{3}{8}}

\latex{\frac{3}{5}}

\latex{ 0 }

\latex{\frac{3}{5}}

\latex{\frac{3}{8}}

\latex{ 1 }

\latex{\frac{3}{8}}

\latex{ 1 }

\latex{\frac{3}{5}}

\latex{\frac{3}{8}}

\latex{ 0 }

b)

\latex{\frac{2}{3}}

\latex{\frac{2}{7}}

\latex{ 0 }

\latex{\frac{2}{7}}

\latex{ 1 }

\latex{\frac{2}{3}}

\latex{ 0 }

\latex{\frac{2}{3}}

\latex{ 1 }

In the case of positive fractions with the same numerator, the one with the smaller denominator is greater.

Comparing positive fractions with different numerators and denominators

When comparing fractions with different numerators and denominators, you should expand or simplify them, so that either their numerators or denominators become equal.

Example 3

Compare the fractions \latex{\frac{2}{3}} and \latex{\frac{3}{4}} .

Solution

\latex{ 12 } is a multiple of both \latex{ 3 } and \latex{ 4 }; therefore, expand both fractions so that their denominators become \latex{ 12 }.

Find a

common denominator

\latex{ 2 }

\latex{ 3 }

\latex{ 8 }

\latex{ 12 }

\latex{ \times4 }

\latex{ 3 }

\latex{ 4 }

\latex{ 9 }

\latex{ 12 }

\latex{ \times3 }

Since \latex{ \frac{8}{12}\lt \frac{9}{12}}, \latex{\frac{2}{3}\lt\frac{3}{4}}.

Representing it on a number line:

\latex{ 0 }

\latex{\frac{1}{12}}

\latex{\frac{2}{12}}

\latex{\frac{3}{12}}

\latex{\frac{4}{12}}

\latex{\frac{5}{12}}

\latex{\frac{6}{12}}

\latex{\frac{7}{12}}

\latex{\frac{8}{12}}

\latex{\frac{9}{12}}

\latex{\frac{10}{12}}

\latex{\frac{11}{12}}

\latex{ 1 }

\latex{\frac{1}{4}}

\latex{\frac{1}{3}}

\latex{\frac{2}{4}}

\latex{\frac{2}{3}}

\latex{\frac{3}{4}}

\latex{\frac{3}{3}=\frac{4}{4}}

Since fractions with the same numerator can also be compared, fractions can be expanded to a common numerator.

Example 4

Compare the fractions \latex{\frac{3}{13}} and \latex{\frac{5}{21}} .

Solution

Find a

common numerator

\latex{ 3 }

\latex{ 13 }

\latex{ 15 }

\latex{ 65 }

\latex{ \times5 }

\latex{ 5 }

\latex{ 21 }

\latex{ 15 }

\latex{ 63 }

\latex{ \times3 }

Since \latex{\frac{15}{65}\lt \frac{15}{63}}, \latex{\frac{3}{13}\lt\frac{5}{21}}.

Fractions can be compared by finding a common numerator or denominator.

In several cases, it is possible to determine which fraction is larger simply by interpreting the fractions.

Example 5

Which is greater: \latex{\frac{2000}{2001}} or \latex{\frac{2001}{2000}?}

Solution

By comparing the numerator and the denominator, it is possible to decide whether a fraction is greater or smaller than \latex{ 1 }.

Since \latex{\frac{2000}{2001}\lt1} and \latex{1\lt \frac{2001}{2000}}, thus \latex{\frac{2000}{2001}\lt\frac{2001}{2000}}.

Example 6

Which is less: \latex{\frac{5}{4}} or \latex{\frac{4}{3}?}

Solution

These fractions can be written as mixed numbers as well:

Since \latex{\frac{1}{4}\lt \frac{1}{3}}, thus \latex{\frac{5}{4}\lt\frac{4}{3}}.

\latex{\frac{5}{4}}

\latex{\frac{4}{3}}

\latex{\frac{5}{4}} is \latex{\frac{1}{4}} larger than \latex{ 1 }
whole.

\latex{\frac{4}{3}} is \latex{\frac{1}{3}} larger than \latex{ 1 }
whole.

\latex{1}

\latex{\frac{1}{4}}

\latex{\frac{1}{3}}

\latex{1}

Example 7

Compare the fractions \latex{\frac{7}{8}} and \latex{\frac{8}{9}}.

Solution

How much smaller are the fractions than \latex{ 1 } whole?

Since \latex{\frac{1}{9}\lt \frac{1}{8}}, less has to be added to \latex{\frac{8}{9}} to get \latex{ 1 } whole.

\latex{\frac{7}{8}} is \latex{\frac{1}{8}} less

\latex{\frac{8}{9}} is \latex{\frac{1}{9}} less

than \latex{ 1 } whole

So \latex{\frac{7}{8}\lt\frac{8}{9}}.

\latex{\frac{1}{8}}

\latex{\frac{1}{9}}

Fractions can be compared by comparing them to the same number.

Exercises

{{exercise_number}}. Which fraction is smaller? Justify your answer.

a) \latex{\frac{5}{7}} or \latex{\frac{5}{9}}

b) \latex{\frac{5}{6}} or \latex{\frac{6}{7}}

c) \latex{\frac{7}{13}} or \latex{\frac{5}{13}}

d) \latex{\frac{8}{7}} or \latex{\frac{10}{9}}

e) \latex{\frac{5}{8}} or \latex{\frac{4}{7}}

f) \latex{\frac{3}{5}} or \latex{\frac{4}{9}}

g) \latex{\frac{47}{45}} or \latex{\frac{34}{35}}

h) \latex{\frac{1999}{2000}} or \latex{\frac{2000}{2001}}

{{exercise_number}}. Replace the \latex{\square} with natural numbers to make the relations true. How many solutions are there? If possible, simplify or rewrite the fraction as a mixed number.

a) \latex{\frac{\square}{8}\lt1}

b) \latex{\frac{8}{\square}\lt1}

c) 1 \latex{\leq\frac{\square}{8}\leq2}

d) \latex{\frac{8}{\square}} is a whole number

{{exercise_number}}. Arrange the fractions in ascending order.

a) one-sixth, two-fifths, two-thirds, nineteen-tenths, fifteen-fifteenths, one-half
b) two-sevenths, eleven-eighths, three-fifths, five-fifths, twenty-one-sixths, eight-ninths

{{exercise_number}}. Which fraction is greater? Write down how you reached your conclusion.

a) \latex{\frac{30}{31}} or \latex{\frac{40}{41}}

b) \latex{\frac{10}{9}} or \latex{\frac{15}{14}}

c) \latex{\frac{9999}{10000}} or \latex{\frac{10000}{9999}}

d) \latex{\frac{2}{7}} or \latex{\frac{7}{9}}

e) \latex{\frac{17}{18}} or \latex{\frac{7}{8}}

f) \latex{\frac{23}{22}} or \latex{\frac{18}{17}}

{{exercise_number}}. What positive whole number can replace the △ to make the relations correct?

a) \latex{\frac{\triangle}{12}\gt \frac{5}{12}}

b) \latex{\frac{\triangle}{7}\lt \frac{8}{7}}

c) \latex{\frac{13}{\triangle}\lt\frac{13}{8}}

d) \latex{\frac{25}{18}\lt\frac{25}{\triangle}}

{{exercise_number}}. What was the final result of the greyhound race? Arrange the times in descending order.

A: \latex{1\frac{2}{3}} \latex{ minutes }

B: \latex{1\frac{1}{10}} \latex{ minutes }

C: \latex{1\frac{5}{6}} \latex{ minutes }

D: \latex{1\frac{3}{5}} \latex{ minutes }

E: \latex{1\frac{11}{15}} \latex{ minutes }

{{exercise_number}}. What positive whole numbers can replace the ◯ to make the relations correct?

a) \latex{\frac{7}{17}\lt \frac{\bigcirc}{17}\lt\frac{9}{17}}

b) \latex{\frac{13}{18}\lt \frac{\bigcirc}{18}\lt\frac{19}{18}}

c) \latex{\frac{3}{14}\lt\frac{3}{\bigcirc}\lt\frac{3}{11}}

d) \latex{\frac{10}{13}\lt\frac{10}{\bigcirc}\lt\frac{10}{9}}

{{exercise_number}}. Which of the following statements are true? Which are false?

a) The larger of two positive fractions with the same numerator is the one with the smaller denominator.
b) The larger of two positive fractions with the same denominators has the larger numerator.
c) The smaller of two positive fractions is the one whose numerator and denominator are smaller.
d) The smaller of two positive fractions with the same denominators has the larger numerator.

Quiz

Find fractions between \latex{\frac {3}{20}} and \latex{\frac{4}{25}} that have two-digit numerators and denominators.

Mathematics 5.

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