Equivalent Fractions, Ordering, Decimals and Percentages for JSS 1

Fractions

Performance Objectives

By the end of the lesson, students should be able to:

1. find equivalent fractions of any given fractions;
2. arrange given fractions either in ascending or descending order of magnitude;
3. convert fractions to decimals, decimals to fractions, fractions to percentages, and percentages to fractions;
4. apply equivalent fractions in sharing of items such as money;
5. solve problems on quantitative aptitude involving fractions.

Meaning of Fractions

A fraction is a part of a whole object, group or quantity.

When an object is divided into equal parts, each part represents a fraction of the entire object. A fraction is written in the form $\frac{a}{b}$, where $a$ is the numerator and $b$ is the denominator. The denominator represents the total number of equal parts into which the whole is divided, and the numerator represents the number of parts being considered.

Equivalent Fractions

Equivalent fractions are fractions that have different numerators and denominators but represent the same value or portion of a whole.

For example, dividing a circular cake into two equal parts and taking one part represents $\frac{1}{2}$ of the cake. Dividing the same cake into four equal parts and taking two parts represents $\frac{2}{4}$ of the cake. The physical amount of cake taken in both cases is identical. Therefore, $\frac{1}{2}$ and $\frac{2}{4}$ are equivalent fractions.

Observe the diagram below to see how different fractions represent the same physical quantity.

Methods of Finding Equivalent Fractions

Equivalent fractions can be formed using two main mathematical operations:

1. Multiplication: Multiplying both the numerator and the denominator of a fraction by the same non-zero whole number.
2. Division: Dividing both the numerator and the denominator of a fraction by the same common factor. This process is also known as simplifying or reducing the fraction to its lowest terms.

Worked Examples

Example 1: Find three equivalent fractions for $\frac{3}{5}$.

To find equivalent fractions, multiply both the numerator and the denominator by the same whole numbers, such as $2$, $3$, and $4$:

First equivalent fraction:

$$ \frac{3 \times 2}{5 \times 2} = \frac{6}{10} $$

Second equivalent fraction:

$$ \frac{3 \times 3}{5 \times 3} = \frac{9}{15} $$

Third equivalent fraction:

$$ \frac{3 \times 4}{5 \times 4} = \frac{12}{20} $$

The three equivalent fractions are $\frac{6}{10}$, $\frac{9}{15}$, and $\frac{12}{20}$.

Example 2: Find an equivalent fraction of $\frac{24}{36}$ that has a denominator of $3$.

To change the denominator from $36$ to $3$, determine the number that divides $36$ to give $3$:

$$ 36 \div 3 = 12 $$

Now, divide both the numerator and the denominator of $\frac{24}{36}$ by $12$:

$$ \frac{24 \div 12}{36 \div 12} = \frac{2}{3} $$

The equivalent fraction with a denominator of $3$ is $\frac{2}{3}$.

Applying Equivalent Fractions in Sharing Items

Equivalent fractions are useful when sharing items, such as money, among different groups to ensure that the proportions remain fair and equal.

Worked Examples

Example 1: A father wants to share $\text{₦}2,400$ among his three children. He gives the eldest child $\frac{4}{8}$ of the money, the second child $\frac{2}{4}$ of the money, and the youngest child $\frac{1}{2}$ of the money. Calculate the amount of money each child receives and show that their shares are equivalent.

Calculate the share for each child:

Share for the eldest child:

$$ \frac{4}{8} \times \text{₦}2,400 = \text{₦}1,200 $$

Share for the second child:

$$ \frac{2}{4} \times \text{₦}2,400 = \text{₦}1,200 $$

Share for the youngest child:

$$ \frac{1}{2} \times \text{₦}2,400 = \text{₦}1,200 $$

Each child receives $\text{₦}1,200$. This shows that $\frac{4}{8}$, $\frac{2}{4}$, and $\frac{1}{2}$ are equivalent fractions, resulting in equal shares of the money.

Example 2: Two business partners, Ada and Bola, agree to share profits in the ratio of their contributions. Ada is entitled to $\frac{15}{25}$ of the profits, while Bola is entitled to $\frac{3}{5}$ of the profits. If the total profit is $\text{₦}50,000$, determine if the shares are equal and calculate the amount each partner receives.

First, simplify Ada’s fraction by dividing the numerator and denominator by their highest common factor, which is $5$:

$$ \frac{15 \div 5}{25 \div 5} = \frac{3}{5} $$

Since Ada’s simplified fraction is $\frac{3}{5}$, her share is equivalent to Bola’s share of $\frac{3}{5}$.

Calculate the monetary value of each share:

$$ \frac{3}{5} \times \text{₦}50,000 = \text{₦}30,000 $$

Both partners receive an equal share of $\text{₦}30,000$ each.

Ordering of Fractions

Ordering fractions means arranging them either in ascending order (from the smallest to the largest) or in descending order (from the largest to the smallest).

To compare and order fractions with different denominators, they must first be converted to equivalent fractions with a common denominator. The common denominator is found by calculating the Least Common Multiple (LCM) of the original denominators.

Observe the number line below to see how fractions of different values are positioned relative to one another.

Worked Examples

Example 1: Arrange the fractions $\frac{2}{3}$, $\frac{3}{4}$, and $\frac{5}{6}$ in ascending order of magnitude.

The denominators are $3$, $4$, and $6$. Find the Least Common Multiple (LCM) of $3$, $4$, and $6$:

Multiples of $3$: $3, 6, 9, 12, 15, \dots$

Multiples of $4$: $4, 8, 12, 16, \dots$

Multiples of $6$: $6, 12, 18, \dots$

The LCM of $3$, $4$, and $6$ is $12$.

Convert each fraction to an equivalent fraction with a denominator of $12$:

For $\frac{2}{3}$:

$$ \frac{2 \times 4}{3 \times 4} = \frac{8}{12} $$

For $\frac{3}{4}$:

$$ \frac{3 \times 3}{4 \times 3} = \frac{9}{12} $$

For $\frac{5}{6}$:

$$ \frac{5 \times 2}{6 \times 2} = \frac{10}{12} $$

Compare the numerators of the new fractions: $8 < 9 < 10$.

Therefore, the fractions in ascending order are:

$$ \frac{2}{3}, \frac{3}{4}, \frac{5}{6} $$

Example 2: Arrange the fractions $\frac{1}{2}$, $\frac{3}{5}$, and $\frac{3}{10}$ in descending order of magnitude.

Find the LCM of the denominators $2$, $5$, and $10$. The LCM is $10$.

Convert each fraction to an equivalent fraction with a denominator of $10$:

For $\frac{1}{2}$:

$$ \frac{1 \times 5}{2 \times 5} = \frac{5}{10} $$

For $\frac{3}{5}$:

$$ \frac{3 \times 2}{5 \times 2} = \frac{6}{10} $$

For $\frac{3}{10}$:

$$ \frac{3}{10} $$

Compare the numerators of the new fractions: $6 > 5 > 3$.

Therefore, the fractions in descending order are:

$$ \frac{3}{5}, \frac{1}{2}, \frac{3}{10} $$

Conversion of Fractions, Decimals and Percentages

Fractions, decimals, and percentages are different ways of expressing the same value. It is important to be able to convert from one form to another.

Converting Fractions to Decimals

To convert a fraction to a decimal, divide the numerator by the denominator using long division.

Example 1: Convert $\frac{3}{4}$ to a decimal.

Divide $3$ by $4$:

$$ 3 \div 4 = 0.75 $$

Therefore, $\frac{3}{4} = 0.75$.

Example 2: Convert $\frac{5}{8}$ to a decimal.

Divide $5$ by $8$:

$$ 5 \div 8 = 0.625 $$

Therefore, $\frac{5}{8} = 0.625$.

Converting Decimals to Fractions

To convert a decimal to a fraction, write the decimal as a fraction with a denominator that is a power of $10$ (such as $10$, $100$, or $1000$), depending on the number of decimal places. Then, simplify the fraction to its lowest terms.

Example 1: Convert $0.35$ to a fraction.

Since there are two decimal places, write the number over $100$:

$$ 0.35 = \frac{35}{100} $$

Simplify by dividing both the numerator and denominator by their highest common factor, which is $5$:

$$ \frac{35 \div 5}{100 \div 5} = \frac{7}{20} $$

Therefore, $0.35 = \frac{7}{20}$.

Example 2: Convert $0.125$ to a fraction.

Since there are three decimal places, write the number over $1000$:

$$ 0.125 = \frac{125}{1000} $$

Simplify by dividing both the numerator and denominator by their highest common factor, which is $125$:

$$ \frac{125 \div 125}{1000 \div 125} = \frac{1}{8} $$

Therefore, $0.125 = \frac{1}{8}$.

Converting Fractions to Percentages

To convert a fraction to a percentage, multiply the fraction by $100\%$.

Example 1: Convert $\frac{3}{5}$ to a percentage.

Multiply the fraction by $100\%$:

$$ \frac{3}{5} \times 100\% = 3 \times 20\% = 60\% $$

Therefore, $\frac{3}{5} = 60\%$.

Example 2: Convert $\frac{7}{20}$ to a percentage.

Multiply the fraction by $100\%$:

$$ \frac{7}{20} \times 100\% = 7 \times 5\% = 35\% $$

Therefore, $\frac{7}{20} = 35\%$.

Converting Percentages to Fractions

To convert a percentage to a fraction, write the percentage value over $100$ and simplify the resulting fraction to its lowest terms.

Example 1: Convert $45\%$ to a fraction.

Write the percentage over $100$:

$$ 45\% = \frac{45}{100} $$

Simplify by dividing both the numerator and denominator by $5$:

$$ \frac{45 \div 5}{100 \div 5} = \frac{9}{20} $$

Therefore, $45\% = \frac{9}{20}$.

Example 2: Convert $12\frac{1}{2}\%$ to a fraction.

First, convert the mixed number percentage to an improper fraction:

$$ 12\frac{1}{2}\% = \frac{25}{2}\% $$

To convert to a fraction, divide by $100$:

$$ \frac{25}{2} \div 100 = \frac{25}{2} \times \frac{1}{100} $$

Multiply the fractions:

$$ \frac{25 \times 1}{2 \times 100} = \frac{25}{200} $$

Simplify by dividing both the numerator and denominator by $25$:

$$ \frac{25 \div 25}{200 \div 25} = \frac{1}{8} $$

Therefore, $12\frac{1}{2}\% = \frac{1}{8}$.

Quantitative Aptitude Involving Fractions

Quantitative aptitude problems test logical reasoning and problem-solving skills using mathematical patterns, shapes, and relationships involving fractions.

Observe the pattern in the diagram below to understand the relationship between the numbers in the circles.

Worked Examples

Example 1: In the quantitative aptitude pattern shown above, the number in the top circle is obtained by multiplying the numbers in the two bottom circles. Find the missing value represented by the question mark ($?$) in the second diagram.

Analyze the first diagram to confirm the rule:

$$ \text{Bottom-left} \times \text{Bottom-right} = \text{Top} $$
$$ \frac{1}{4} \times \frac{2}{3} = \frac{2}{12} = \frac{1}{6} $$

The rule is correct. Now apply this rule to the second diagram to find the missing value ($?$):

$$ \frac{3}{5} \times \frac{1}{2} = \frac{3 \times 1}{5 \times 2} = \frac{3}{10} $$

The missing value is $\frac{3}{10}$.

Example 2: Find the missing term in the following sequence of fractions:

$$ \frac{1}{2}, \frac{3}{4}, \frac{9}{8}, \text{ ?} $$

Analyze the relationship between consecutive terms in the sequence:

To get from the first term to the second term:

$$ \frac{1}{2} \times \frac{3}{2} = \frac{3}{4} $$

To get from the second term to the third term:

$$ \frac{3}{4} \times \frac{3}{2} = \frac{9}{8} $$

The pattern is to multiply each term by $\frac{3}{2}$ to get the next term. Apply this rule to find the missing term:

$$ \frac{9}{8} \times \frac{3}{2} = \frac{27}{16} $$

Converting the improper fraction to a mixed number:

$$ \frac{27}{16} = 1\frac{11}{16} $$

The missing term is $\frac{27}{16}$ (or $1\frac{11}{16}$).

Evaluation Exercises

Forming Equivalent Fractions

1. Write three equivalent fractions for each of the following:
   1. $\frac{2}{7}$
   2. $\frac{5}{6}$
2. Find the missing value in each of the following equivalent fractions:
   1. $\frac{3}{8} = \frac{x}{24}$
   2. $\frac{45}{60} = \frac{3}{y}$

Ordering Fractions

1. Arrange the following fractions in ascending order of magnitude:
   1. $\frac{1}{3}, \frac{2}{5}, \frac{3}{10}$
   2. $\frac{5}{8}, \frac{3}{4}, \frac{1}{2}$
2. Arrange the following fractions in descending order of magnitude:
   1. $\frac{2}{3}, \frac{5}{6}, \frac{7}{12}$
   2. $\frac{3}{5}, \frac{7}{10}, \frac{1}{2}$

Conversions

1. Convert the following fractions to decimals:
   1. $\frac{4}{5}$
   2. $\frac{7}{8}$
2. Convert the following decimals to fractions in their simplest form:
   1. $0.64$
   2. $0.075$
3. Convert the following fractions to percentages:
   1. $\frac{9}{20}$
   2. $1\frac{2}{5}$
4. Convert the following percentages to fractions in their simplest form:
   1. $80\%$
   2. $37\frac{1}{2}\%$

Quantitative Aptitude Problems

1. Find the missing terms in the following fraction sequences:
   1. $\frac{1}{3}, \frac{2}{6}, \frac{4}{12}, \text{ ?}$
   2. $\frac{1}{16}, \frac{1}{8}, \frac{1}{4}, \text{ ?}$
2. A mathematical box pattern defines the top number as the sum of the two bottom numbers. If the bottom numbers are $\frac{1}{3}$ and $\frac{2}{5}$, calculate the top number.