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# Chapter 2: Fractions

## 2.1 Fractions, decimals and percentages

Last year you learnt that a fraction is a part of a whole. A common fraction is written as one number on top of another number: $\dfrac{\text{numerator}}{\text{denominator}}$

• The denominator represents all the parts of a whole. For example, in the fraction $\frac{3}{5}$, each whole is divided into 5 parts.
• The numerator represents only the parts of a whole we are dealing with. For example, in the fraction $\frac{3}{5}$, we are dealing with only 3 out of 5 parts.

The fraction $\frac{3}{5}$ may be represented as follows:

When dealing with common fractions, remember:

• A proper fraction is smaller than 1. The numerator is smaller than the denominator, for example $\frac{5}{8}$.
• An improper fraction is larger than 1. The numerator is larger than the denominator, for example $\frac{8}{5}$.
• An improper fraction may be expressed as a mixed number. A mixed number consists of a whole number and a proper fraction, for example $1\frac{3}{5}$.

### Decimal fractions

From last year, you know that we may write a common fraction as a decimal fraction. A decimal fraction always has a power of ten as its denominator. We use decimal notation to write decimal fractions. In this notation, a decimal point separates whole numbers from proper fractions. The numerator is written after the decimal point and the denominator is not written down.

decimal fraction A decimal fraction has a power of ten as its denominator. The fraction is written in decimal notation.

decimal notation In decimal notation, a decimal point separates whole numbers from proper fractions. The fraction is written as a numerator after the decimal point and the denominator is not written down.

In decimal notation, the number of digits after the decimal point shows us which power of ten is used in the fraction's denominator.

\begin{array}{|l|l|l|l|} \hline \textbf{Name} & \textbf{Common fraction} & \textbf{Powers of ten} & \textbf{Decimal fraction} \newline \hline \text{One tenth} & \frac{1}{10} & \frac{1}{10}=10^{-1} & 0.1 \newline \hline \text{One hundredth} & \frac{1}{100} & \frac{1}{10\times10}=10^{-2} & 0.01 \newline \hline \text{One thousandth} & \frac{1}{1,000} & \frac{1}{10\times10\times10}=10^{-3} & 0.001 \newline \hline \text{One ten thousandth} & \frac{1}{10,000} & \frac{1}{10\times10\times10\times10}=10^{-4} & 0.0001 \newline \hline \text{One hundred thousandth} & \frac{1}{100,000} & \frac{1}{10\times10\times10\times10\times10}=10^{-5} & 0.00001 \newline \hline \end{array}

We normally do not write zeros at the end of decimal fractions. For example, we write 0.5 rather than 0.50 or 0.500. This is because $\frac{5}{10}=\frac{50}{100}=\frac{500}{1,000}$.

### Converting numbers in decimal notation to fractions

The fraction 0.5 and the fraction $\frac{1}{2}$ represent exactly the same portion of a whole, namely a half. Similarly, the fraction 2.5 is equivalent to the mixed number $2\frac{1}{2}$ and the improper fraction $\frac{5}{2}$. We may write fractions in decimal notation or as a fraction.

### Worked example 2.1: Converting numbers in decimal notation to common fractions

Write 4.08 as a common fraction.

1. Step 1: Use the number of digits after the decimal point to determine which power of ten is the denominator.

There are 2 digits after the decimal point. Therefore, the denominator is:

2. Step 2: Starting from the first non-zero digit, write the digits without the decimal point.

This becomes the numerator of the common fraction.

3. Step 3: Find the simplest form of the fraction obtained in Step 2 to get the correct improper fraction.

### Worked example 2.2: Converting numbers in decimal notation to mixed numbers

Write 205.05 as a mixed number.

1. Step 1: Write down the whole number and the fraction separately.

Whole number: 205
Fraction: 0.05

2. Step 2: Use the method in worked example 2.1 to convert the decimal fraction to a proper fraction.

There are 2 digits after the decimal point:

In its simplest form:

3. Step 3: Write down the whole number and the proper fraction.

### Exercise 2.1: Convert numbers in decimal notation to fractions and mixed numbers

1. Convert the following decimal fractions to common fractions. You may leave your answers as improper fractions.

1. 0.128

There are 3 digits after the decimal point:

In its simplest form:

1. 0.025

There are 3 digits after the decimal point:

In its simplest form:

1. 0.0008

There are 4 digits after the decimal point:

In its simplest form:

1. 3.25

There are 2 digits after the decimal point:

In its simplest form:

1. 2.005

There are 3 digits after the decimal point:

In its simplest form:

2. Convert the following decimal fractions to mixed numbers.

1. 1.24

Whole number: 1
Fraction: 0.24

There are 2 digits after the decimal point:

In its simplest form:

The mixed number is $1 \frac{6}{25}$.

1. 18.02

Whole number: 18
Fraction: 0.02

There are 2 digits after the decimal point:

In its simplest form:

The mixed number is $18 \frac{1}{50}$.

1. 103.3

Whole number: 103
Fraction: 0.355

There are 3 digits after the decimal point:

In its simplest form:

The mixed number is $103 \frac{71}{200}$.

### Converting fractions and mixed numbers to decimal notation

When a common fraction has a denominator that is a power of ten, it is easy to convert it to a decimal fraction. You have learnt how to use place value tables for decimal fractions.

Consider the fraction $\frac{35}{1,000}$. It means we have 35 thousandths. We may write this in a place value table as follows:

\begin{array}{|c|c|c|c|c|c|} \hline (1) & \text{.} & \left( \frac{1}{10} \right) & \left( \frac{1}{100} \right) & \left( \frac{1}{1,000} \right) & \left( \frac{1}{10,000} \right) \newline \text{u} & \text{.} & \text{t} & \text{h} & \text{th} & \text{tth} \newline \hline 0 & \text{.} & 0 & 3 & 5 & \newline \hline \end{array}

Note that we have 5 thousandths, 3 hundredths and zero tenths.

The number of digits after the decimal point is equal to the power of ten in the denominator: $1,000=10\times10\times10=10^3$.

### Worked example 2.3: Converting fractions with denominators that are powers of ten to decimal notation

Write these fractions in decimal notation.

1. Step 1: Remember the correct first steps you must follow.

• For a proper fraction, write down a zero and a decimal point: 0.
• For a mixed number, write down the whole number and a decimal point: 5.
• For an improper fraction, write down the full numerator: 503
2. Step 2: Determine the power of ten of the denominator.

The power of ten of the denominator is equal to the number of digits that there must be after the decimal point.

$100=10\times10=10^2$
There must be 2 digits after the decimal point.

3. Step 3: Write the digits after the decimal point.

• For a proper fraction, write the numerator after the decimal point. If there are fewer digits in the numerator than the number from Step 2, insert zeros after the decimal point: $\frac{3}{100}=0.03$
• For a mixed number, do the same as for a proper fraction. Then include the whole number in your answer: $5\frac{3}{100}=5.03$
• For an improper fraction, simply insert the decimal point in the correct place: 5.03

When the denominator of a fraction is not a power of ten, we use division to write a common fraction or mixed number in decimal notation.

### Worked example 2.4: Converting fractions with denominators that are not powers of ten to decimal notation

Write $44\frac{5}{6}$ in decimal notation.

1. Step 1: Divide the denominator into the numerator. If it cannot divide, write a zero.

2. Step 2: Put a decimal point after the zero and after the numerator.

3. Step 3: Add zeros after the decimal point in the numerator.

The numbers 5; 5.0; 5.00; 5.000; 5.0000, and so on, all represent the value 5.

4. Step 4: Carry on with the division.

5. Step 5: If the answer to the division is a recurring decimal, insert a dot above the digit that repeats.

A recurring decimal is a decimal fraction in which one or more digits repeat in the same pattern forever. We write a dot above the digit or digits that repeat.

6. Step 5: If your original number was a mixed number, replace the zero in the answer to the division with the whole number in the original mixed number.

### Exercise 2.2: Convert fractions and mixed numbers to decimal notation

Write the following fractions and mixed numbers in decimal notation.

1. There must be 1 digit after the decimal point: 710.3

2. There must be 4 digits after the decimal point: 0.0020

We write this as: 0.002

3. The whole number is written before the decimal point.

There must be 2 digits after the decimal point: 33.05

4. We keep on getting the same remainder, so this is a recurring decimal: $8.\dot{6}$

5. Replace the zero with the whole number: 15.8

6. This is not a recurring decimal: 0.4375

### Percentages

You have learnt that a percentage is a fraction in which the denominator is always 100. We write down only the numerator, followed by a percentage symbol. For example, 25% means $\frac{25}{100}$.

percentage A percentage is a fraction in which the denominator is 100, and where only the numerator is written down, followed by a percentage symbol.

### Converting percentages to common fractions

To convert percentages to common fractions, we use the form $\frac{\text{numerator}}{\text{denominator}}$. We know that the denominator is always 100, so we have $\frac{\text{numerator}}{100}$. The numerator is the number before the percentage symbol.

### Worked example 2.5: Converting mixed number percentages to common fractions

Write $24\frac{2}{3}\%$ as a fraction.

1. Step 1: Write the mixed number as an improper fraction.

2. Step 2: Use the improper fraction as the numerator and 100 as the denominator, because this is a percentage.

3. Step 3: Simplify the fraction from Step 2.

You have learnt that dividing by a fraction is the same as multiplying by the reciprocal of the fraction. We may write 100 as $\frac{100}{1}$. Therefore, dividing by 100 is the same as multiplying by $\frac{1}{100}$.

In its simplest form, the fraction is:

### Worked example 2.6: Converting percentages in decimal notation to common fractions

Write 5.75% as a fraction.

1. Step 1: Start by using the number in decimal notation as the numerator and 100 as the denominator.

2. Step 2: Multiply the numerator by a power of 10 so that it will no longer have a decimal point. Multiply the denominator by the same number.

There are two digits after the decimal point in the numerator, so we must multiply by $10^2=10\times10=100$.

3. Step 3: Simplify the fraction from Step 2.

### Exercise 2.3: Convert percentages to common fractions

Write the following percentages as common fractions.

1. 72%

2. 115%

3. 80.25%

There are 2 digits after the decimal point in the numerator, so we must multiply by 100.

Simplify the fraction.

4. 0.08%

There are 2 digits after the decimal point in the numerator, so we must multiply by 100.

Simplify the fraction.

5. Write the mixed number as an improper fraction.

Simplify the fraction.

### Converting common fractions and mixed numbers to percentages

Last year you learnt that you have to determine how many parts out of 100 you are dealing with when you want to convert a common fraction to a percentage. You used equivalent fractions to do this. For example:

You also learnt that "of" may be interpreted as multiplication. For example, if we want to know how many parts of 100 is represented by $\frac{2}{5}$, we may write this as $\frac{2}{5}\text{ of } 100$, which is the same as $\frac{2}{5}\times100$.

### Worked example 2.7: Converting mixed numbers to percentages

Write $1\frac{4}{5}$ as a percentage.

1. Step 1: Write the mixed number as an improper fraction.

2. Step 2: Multiply by 100.

As soon as you multiply by 100, you must add the percentage sign.

3. Step 3: Simplify the fraction from Step 2.

### Exercise 2.4: Convert common fractions and mixed numbers to percentages

Write the following as percentages.

1. Simplify the fraction.

2. Simplify the fraction.

3. Simplify the fraction.

4. Simplify the fraction.

## 2.2 Ratio and proportion

Suppose we cut a cake into 9 slices. After 2 slices are eaten, we may show the cake that is left over like this:

By now you know that the fraction of the cake that is eaten is $\frac{2}{9}$ and the fraction of the cake that is left over is $\frac{7}{9}$.

We may use a ratio to compare the part of the cake that is eaten to the part of the cake that is left over. We write it as $2:7$. A ratio compares the size of two amounts or quantities.

ratio A ratio gives us the relationship between the size of two quantities.

Fractions and ratios are related, but they are not exactly the same thing. A fraction gives us a specific number of parts in a whole, for example the part of the cake that was eaten. A ratio compares different parts of a whole. There are four different ratios we can look at for the cake:

• The ratio of cake eaten to cake left over is $2:7$.
• The ratio of cake left over to cake eaten is $7:2$.
• The ratio of cake eaten to original cake available is $2:9$.
• The ratio of cake left over to original cake available is $7:9$.

We often express ratios as common fractions to help us work with them in practice. The number on the left usually becomes the numerator and the number on the right the denominator. For example, the ratio $2:7$ may be written as $\frac{2}{7}$ and the ratio $7:9$ may be written as $\frac{7}{9}$. Just like fractions, ratios are simplified until the two numbers in the ratio do not have any common factors.

### Expressing two quantities as ratios and fractions

There are various ways in which two quantities can be expressed as a ratio or a common fraction.

### Worked example 2.8: Expressing two quantities as a ratio

Balarabe spends 7,500 kobo on chocolates and Talatu spends ₦90 on chocolates. Express the amount spent by Balarabe to the amount spent by Talatu as a ratio.

1. Step 1: Write both amounts in the same unit.

Balarabe: $$\frac{7,500}{100}= ₦\,75$$
Talatu: $$₦\,90$$

2. Step 2: Place the amount before the word "to" (Balarabe's amount) on the left in the ratio, and place the amount after the word "to" (Talatu's amount) on the right.

$$₦\,75: ₦\,90$$
3. Step 3: Because it is a relationship, you can remove the unit, "₦". Simplify the ratio until the two numbers do not have any common factors.

\begin{align} ₦\,75: ₦\,90&=75:90\newline &=15:18\newline &=5:6 \end{align}
4. Step 4: Interpret the ratio and check that it makes sense.

The ratio $5:6$ means that, for every ₦5 that Balarabe spent, Talatu spent ₦6. This makes sense, because the original information tells us that Talatu spent more money than Balarabe.

### Worked example 2.9: Expressing two quantities as common fractions of each other

Balarabe spends 7,500 kobo on chocolates and Talatu spends ₦90 on chocolates. Express the amount spent by each person as a fraction of the amount spent by the other person.

1. Step 1: Write both amounts in the same unit.

Balarabe: $$\frac{7,500}{100}= ₦\,75$$
Talatu: $$₦\,90$$

2. Step 2: Write appropriate ratios for the amounts given in the question.

\begin{align} \text{spent by Balarabe }:\text{spent by Talatu} &= ₦\,75: ₦\,90 \newline &=5:6 \end{align} \begin{align} \text{spent by Talatu }:\text{spent by Balarabe} &= ₦\,90: ₦\,75 \newline &=6:5 \end{align}
3. Step 3: Convert the ratios to fractions.

Usually, the number on the left becomes the numerator and the number on the right becomes the denominator.

4. Step 4: Interpret the fractions and check that they make sense.

\begin{align} \text{amount spent by Balarabe}&=\frac{5}{6} \times ₦\,90 \newline &= ₦\,\frac{5\times90}{6} \newline &= ₦\,75 \end{align} \begin{align} \text{amount spent by Talatu}&=\frac{6}{5} \times ₦\,75 \newline &= ₦\,\frac{6\times75}{5} \newline &= ₦\,90 \end{align}

Both answers agree with the original information.

### Exercise 2.5: Express quantities as ratios and fractions

1. In a class of 48 students, there are 30 girls.

1. Give the fraction of the class that is female.
1. Give the fraction of the class that is male.
\begin{align} &\,1-\frac{5}{8} \newline =&\frac{8}{8}-\frac{5}{8} \newline =&\frac{3}{8} \end{align}
1. Use your previous answer to calculate the number of boys in the class.
1. Express the number of girls to boys as a ratio.
2. A bag contains 20Â kg of rice. A measuring cup is used to remove 160Â g of rice from the bag.

1. Express the rice removed as a fraction of the original amount of rice in the bag.
1. Give the ratio of the rice removed to the rice left.
\begin{align} &\,1-\frac{1}{125} \newline =&\frac{125}{125}-\frac{1}{125} \newline =&\frac{124}{125} \end{align}
1. Express the rice removed as a fraction of the rice left.
1. Use your previous answer to calculate the mass of rice left.
3. During a practice session, an athlete sprints for 120 seconds and then jogs for 5 minutes.

1. Express the time sprinted as a fraction of the total practice time.
1. Express the time jogged as a fraction of the time sprinted.
1. Give the ratio of time jogged to the total practice time.
4. Every student in a Mathematics class should have a textbook. There is a shortage of textbooks in one class. There are only 35 textbooks available for the class. The teacher tells the principal that the ratio of books to students is $7:9$.

1. Tell the principal what fraction of the total number of books that the class should receive is currently available.

The ratio $7:9$ means there are only 7 books for every 9 students. Fraction available:

1. Express the students as a fraction of the books.
1. Calculate how many books should be available for the class.
5. The scale on a building plan is 1 : 200. This means 1Â cm on the plan represents 200Â cm in real life.

1. Express distances on the plan as a fraction of distances in real life.
1. Express distances in real life as a fraction of distances on the plan.
1. The length of a wall is 6Â cm on the plan. Calculate the length of the wall in real life.
1. The height of a door in real life is 3Â m. Calculate the distance that will be drawn for this on the plan.

### Sharing a quantity in a specific ratio

Suppose you want to divide a bag of apples in the ratio $2:1:3$. All the parts of the ratio add up to 6: $2+1+3=6$. We can express each part of the ratio as a fraction of all the parts: $\frac{2}{6};\space\frac{1}{6};\space\frac{3}{6}$. If we add these fractions, we get 1: $\frac{2}{6}+\frac{1}{6}+\frac{3}{6}=\frac{6}{6}=1$.

### Worked example 2.10: Sharing a quantity in a specific ratio

Share a bag of 56 cashew nuts in the ratio $2:8:6$.

1. Step 1: Simplify the ratio if possible.

2. Step 2: Determine the total parts in the simplified ratio.

3. Step 3: Write each part of the ratio as a fraction.

4. Step 4: Calculate how many cashew nuts each fraction represents.

The sweets must be shared $7:28:21$.

If you worked correctly, the numbers you calculated must add up to the number of cashew nuts in the bag.

### Exercise 2.6: Share quantities in specific ratios

1. Share 5 kg garri between two people in the ratio 9 : 11.

The total parts is greater than 5, so it will be easier to convert to grams: $5\text{ kg}=5,000\text{ g}$

2. Adedamola was late for school. He walked most of the 8 km to school, but ran in between to make up time. The ratio of the distance he ran to the distance he walked is $1:5$. Calculate the distance he ran and the distance he walked.

3. A mother shares ₦2,500 between her three children according to their ages. If they are aged 8 years, 12 years and 20 years, calculate how much money each child will get.

$\frac{2}{10}\times2,500=\frac{2\times2,500}{10}=\frac{2\times250}{1}= ₦\,500$ $\frac{3}{10}\times2,500=\frac{3\times2,500}{10}=\frac{3\times250}{1}= ₦\,750$ $\frac{5}{10}\times2,500=\frac{5\times2,500}{10}=\frac{5\times250}{1}= ₦\,1,250$

## 2.3 Practical applications

Fractions and ratios are very useful for solving problems in everyday life. We may work with fractions as common fractions, decimal fractions or percentages when we deal with practical problems.

### Exercise 2.7: Use fractions and ratios to solve problems

1. According to the map of a schoolground, the school buildings cover 0.48 of the schoolground. Express this as a common fraction.

2. A piece of land has a surface area of 800 hectares. The owner decides to plant yam on 512 hectares.

1. Determine what fraction of the land will be planted with yam.
1. Write the fraction of the land that will be planted with yam as a percentage.
3. The discount on a dress of ₦4,500 is advertised as $33\frac{1}{3}\%$. Express the discount as a fraction.

4. A Mathematics test counted a total of 75 marks.

1. Akeju scored 60 out of 75 for the test. Calculate her percentage.
1. Akinbode scored 60% for the test. Calculate his mark out of 75.
5. Express $\frac{9}{16}$ of one kilometre in kilometres.

6. A total of 90 Mathematics students are divided into three classes in the ratio $2:4:3$. Calculate the number of students in the largest class.

7. Share 3.6 litres of palm oil in the ratio $8:10$.

8. A map has a scale of $1:100,000$. The distance between two towns on the map is 15Â cm. Calculate the distance between the two towns in real life.

9. A Mathematics test counted a total of 40 marks. For recording purposes, each student's mark out of 40 must be expressed as a mark out of 60. This mark must then be recorded. Adanna scored 36 out of 40 for the test.

1. Write the total that must be recorded as a fraction of the real total.
1. Express Adanna's mark as a mark out of 60.
10. Two shops, A and B, sell yam in bags of 5Â kg. The ratio of the price at shop A to the price as shop B is $8:9$. If the bag costs ₦300 at shop A, calculate the price at shop B.

$\therefore \text{price B} = ₦\,337.5$

## 2.4 Summary

• A common fraction is written as one whole number on top of another whole number: $\frac{\text{numerator}}{\text{denominator}}$
• A decimal fraction has a power of ten as its denominator. The fraction is written in decimal notation.
• In decimal notation, a decimal point separates whole numbers from proper fractions. The fraction is written as a numerator after the decimal point, and the denominator is not written down.
• In decimal notation, the number of digits after the decimal point shows us which power of ten is used in the fraction's denominator.
• We may convert numbers in decimal notation to common fractions or mixed numbers. We may also convert common fractions and mixed numbers to decimal notation.
• A percentage is a fraction in which the denominator is 100 and where only the numerator is written down, followed by a percentage symbol.
• We convert percentages to common fractions by dividing by 100. We convert common fractions to percentages by multiplying by 100.
• A ratio gives us the relationship between the size of two quantities. It compares the different parts of a whole.
• We may express fractions as ratios. The numerator is usually written on the left in the ratio.
• We may express ratios as fractions. The number on the left usually becomes the numerator.
• A quantity can be shared in a specific ratio.