Fractions

Introduction

Class VII.A of a junior high school held an election for class president. The candidates were Danang, Fatimah, and Zalfa. After the votes were counted, Danang received $\dfrac{1}{3}$ of Fatimah’s votes, while Fatimah received $\dfrac{3}{4}$ of Zalfa’s votes. If there are 32 students in the class, how many votes did each candidate get?

Definition of Fractions

Quoted from the Mathematics Encyclopedia by ST. Negoro and B. Harahap:

A fraction is a number that describes part of a whole, part of an area, part of an object, or part of a set.

Look at the following image.

In a fraction written in the form $\dfrac{a}{b}$, where $a$ and $b$ are integers and $b \ne 0$, $a$ is called the numerator and $b$ is called the denominator.

Equivalent fractions are fractions that have the same value. To obtain an equivalent fraction, the numerator and denominator can be multiplied or divided by the same nonzero number. Examples are $\dfrac{2}{2}, \dfrac{2}{4}, \dfrac{3}{6}, \dfrac{4}{8}$, and $\dfrac{8}{16}$.

To compare two fractions with different denominators, first convert them into equivalent fractions with the same denominator.

Example

Compare the following pairs of fractions:

a. $\dfrac{2}{3}$ and $\dfrac{3}{4}$
b. $\dfrac{3}{5}$ and $\dfrac{5}{7}$

Answer

a. The denominators of $\dfrac{2}{3}$ and $\dfrac{3}{4}$ are 3 and 4. The LCM of 3 and 4 is 12.

So:

Therefore:

b. The denominators of $\dfrac{3}{5}$ and $\dfrac{5}{7}$ are 5 and 7. The LCM of 5 and 7 is 35.

So:

Therefore:

Also read:

• Integers
• Exponents
• Ratios

Example

Mrs. Intan has IDR 15,000.00. If she spends IDR 10,000.00, what fraction of her money has been spent?

Answer

The ratio of the money spent to the total amount is:

Decimal Fractions

Decimal fractions are fractions with denominators 10, 100, 1,000, and so on.

Example

1. Convert the following fractions to decimal form: a. $\dfrac{1}{4}$
   b. $\dfrac{6}{25}$

2. Convert the following decimals to ordinary fractions: a. $0.4$
   b. $0.016$
   c. $5.25$

Answer

1. a. $\dfrac{1}{4}=\dfrac{1\times 25}{4\times 25}=\dfrac{25}{100}=0.25$
   b. $\dfrac{6}{25}=\dfrac{6\times 4}{25\times 4}=\dfrac{24}{100}=0.24$

2. a. $0.4=\dfrac{4}{10}=\dfrac{4 : 2}{10 : 2}=\dfrac{2}{5}$
   b. $0.016=\dfrac{16}{1,000}=\dfrac{16 : 8}{1,000 : 8}=\dfrac{2}{125}$
   c. $5.25=\dfrac{525}{100}=\dfrac{525 : 25}{100 : 25}=\dfrac{21}{4}=5\dfrac{1}{4}$

Percent

Percent is a fraction with a denominator of 100 and is denoted by %. To convert an ordinary fraction to percent form, you can:

a. change the original fraction into an equivalent fraction with denominator 100, or
b. multiply the fraction by $100\%$.

Example

1. Convert the following fractions into percent form: a. $\dfrac{1}{4}$
   b. $\dfrac{6}{25}$

2. Convert the following percent forms to ordinary fractions: a. $20\%$
   b. $75\%$
   c. $150\%$

Answer

1. a. $\dfrac{1}{4}\times 100\%=25\%$
   b. $\dfrac{6}{25}\times 100\%=24\%$

2. a. $20\%=\dfrac{20}{100}=\dfrac{1}{5}$
   b. $75\%=\dfrac{75}{100}=\dfrac{3}{4}$
   c. $150\%=\dfrac{150}{100}=\dfrac{3}{2}$

Permille

Permille is a fraction with denominator 1,000. It is usually used to express salinity. For example, the salinity of the Red Sea reaches 40‰. That means every 1 kg of seawater contains 40 g of salt.

To convert an ordinary fraction to permille form, you can:

a. change the fraction into an equivalent fraction with denominator 1,000, or
b. multiply the fraction by 1,000‰.

Example

1. Convert the following fractions into permille form: a. $\dfrac{3}{5}$
   b. $\dfrac{5}{8}$

2. Convert the following permille forms to ordinary fractions: a. \(25‰\)
   b. \(1{,}500‰\)

Answer

1. a. \(\dfrac{3}{5}=\dfrac{3\times 200}{5\times 200}=\dfrac{600}{1,000}=600‰\)
   b. \(\dfrac{5}{8}\times 1{,}000‰=625‰\)

2. a. \(25‰=\dfrac{25}{1,000}=\dfrac{1}{40}\)
   b. \(1{,}500‰=\dfrac{1,500}{1,000}=\dfrac{3}{2}\)

Arithmetic Operations with Fractions

Addition

Fractions with the same denominator can be added by adding only the numerators, while the denominator stays the same. In general:

If the fractions have different denominators, first make the denominators equal by finding the LCM.

Example

Calculate the sum of the following fractions:

a. $\dfrac{3}{7}+\dfrac{1}{7}$
b. $\dfrac{1}{4}+\dfrac{2}{3}$

Answer

a. $\dfrac{3}{7}+\dfrac{1}{7}=\dfrac{3+1}{7}=\dfrac{4}{7}$

b. The LCM of 4 and 3 is 12, so:

Subtraction

Fractions with the same denominator can be subtracted by subtracting the numerators, while the denominator stays the same. In general:

If the denominators are different, first make them equal.

Example

Determine the results of the following subtractions:

a. $\dfrac{5}{7}-\dfrac{2}{7}$
b. $\dfrac{3}{4}-\dfrac{1}{5}$
c. $4\dfrac{2}{3}-2\dfrac{1}{5}$

Answer

a. $\dfrac{5}{7}-\dfrac{2}{7}=\dfrac{5-2}{7}=\dfrac{3}{7}$

b. The LCM of 4 and 5 is 20, so:

c. The LCM of 3 and 5 is 15, so:

Multiplication

Multiplying fractions means multiplying a number by a fraction. It may involve a whole number and a fraction, or two fractions. In general:

Example

Determine the products of the following fractions:

a. $\dfrac{3}{4}\times \dfrac{7}{10}$
b. $\dfrac{5}{6}\times \dfrac{6}{7}$

Answer

a. $\dfrac{3}{4}\times \dfrac{7}{10}=\dfrac{3\times 7}{4\times 10}=\dfrac{21}{40}$

b.

The numerator and denominator are divided by the GCF of 30 and 42, namely 6.

Division

Dividing by a fraction is the same as multiplying by the reciprocal of the divisor. In general:

Example

Determine the results of the following divisions:

a. $\dfrac{1}{2}\div \dfrac{3}{4}$
b. $2 \div \dfrac{1}{4}$
c. $\dfrac{4}{5}\div 4$

Answer

a. $\dfrac{1}{2}\div \dfrac{3}{4}=\dfrac{1}{2}\times \dfrac{4}{3}=\dfrac{4}{6}=\dfrac{2}{3}$

b. $2\div \dfrac{1}{4}=2\times \dfrac{4}{1}=8$

c. $\dfrac{4}{5}\div 4=\dfrac{4}{5}\times \dfrac{1}{4}=\dfrac{4\times 1}{5\times 4}=\dfrac{1}{5}$

Sequence of Calculation Operations

If you find a mathematical sentence that mixes add, subtract, multiply, divide and exponent operations, then we use the following rules for the order of arithmetic operations.

1. Complete the power operation.

2. Complete the multiply and divide operations.

3. Complete the add and subtract operations. Operations in parentheses are performed first.

Evaluation

To evaluate your understanding of fractions, try solving the examples in each section again without looking at the solutions first. Then compare your work with the discussion in the article.

Pay special attention to these skills:

1. comparing fractions,
2. converting fractions to decimals, percentages, and permille,
3. performing arithmetic operations with fractions,
4. applying the correct order of operations.

Closing

That concludes this discussion of fractions. If you find any part that needs correction, please leave a note in the comments.