FRACTIONS, DECIMALS, APPROXIMATION AND PERCENTAGES

Hi guys. Today we’ll be looking to completely exhaust fractions, decimals, approximation and percentages based on jamb syllabus.

If you feel you need a good textbook instead, see the recommended textbook jamb asked students to read for mathematics.

However, for those who feel they have completely exhausted this topic, click here to see tough questions from jamb past question with solution

Having said all that, let’s proceed.

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1.1 Numbers

A number is used as a concept of quantity. It is used to denote how many times a unit quantity is derivable from the number being considered. A numeral is a symbol denoting that number. The commonest numerals (symbols) we use are the Hindu-Arabic numerals 0,1,2,3,… etc.

 

Kinds of real numbers

There are two kinds of real numbers, namely;

1. Rational numbers

2. Irrational numbers

 

1.1.1 Rational numbers

Rational numbers are further divided into; integers and non-integers. As we progress in this book we will dwell more on integer but for now, we will be dwelling on non-integers

 

Non-integers

Non-integers are mainly fractions and the various forms it can be expressed; vulgar forms of fraction, decimal forms of fraction and percentage forms of fractions as shown respectively as \frac{6}{10}, 0\cdot 6, 60%

Fractions

A fraction is a number denoted by mainly two numerals separated by a division line. Where the numeral above the line is called the numerator or dividend and the numeral below the line is called the denominator or divisor. A fraction(common fraction) can be seen as a short form of writing a division problem.

Examples of fractions are \frac{2}{3}, \frac{14}{5}, \frac{8}{10}, \frac{111}{13} etc. where 2, 14, 8, 111 are the numerator or dividend and 3, 5, 10, 13 are the denominators or divisor and the lines between the numerals indicates division of the numbers i.e
\frac{2}{3}=2\div3,\\ \frac{14}{5}=14\div5,\\ \frac{8}{10}=8\div10, and\\ \frac{111}{13}=111\div13.

Decimal

Decimal is another form of expressing a fraction. Generally, a decimal can be expressed as a\cdot b, where a is the integer, \cdot is the decimal point and b is the fractional part of the decimal.

An example of a decimal is 0\cdot 4

In the above example, zero(0) is the integer and 4 is the fractional part which can be expressed as \frac{4}{10} in fraction.

Other examples of decimals are 0\cdot 8, 1\cdot2, 13\cdot5, 5\cdot6 etc.

 

1.1.2 Irrational numbers

These are numbers that cannot be expressed as an exact fraction or numbers that have no perfect roots. Examples are \sqrt{2}, \sqrt[3]{5}. More will be touched on this topic as we progress in this tutorial.

 

1.2 Approximation

Approximation is the correcting of numbers to the nearest whole number or unit of measure, especially with measurements. Approximation sometimes involves estimating a rough calculation that points to the correct answer. It is another form of rounding up or down.

 

1.2.1 Significant figures (sf)

Significant figures are gotten by counting from the first non-zero numeral encountered starting from the left of the number.

Example 1.1

What is the first significant figure in the following:

1. 19750

2. 0{\cdot}0409

3. 0{\cdot}000092

Solution

In 19750; 1 is the first non-zero numeral when counting from the left of 19750
\therefore 1 is the answer

In 0{\cdot}0409; 4 is the first non-zero numeral when counting from the left to the right
\therefore 4 is the answer

In 0{\cdot}000092; 9 is the first non-zero numeral when counting to the right of the decimal point in the decimal.
\therefore 9 is the answer

 

1.2.2 Decimal Places (dp)

Decimal places can be gotten by counting to the right of the decimal point in a decimal.

Example 1.2

How many decimal places are the following numbers expressed:

1. 0{\cdot}001

2. 1{\cdot}01

3. 2{\cdot}2002

Solution

0{\cdot}001 has 3 decimal places because there are three digits after the decimal point.

1{\cdot}01 has 2 decimal places because there are two digits after the decimal point.

2{\cdot}2002 has 4 decimal places because there are four digits after the decimal point.

 

Problems on approximation involving significant figures and decimal places

Example 1.3

Approximate the following to 2sf

1. 1{\cdot}000123

2. 0{\cdot}0000923

3. 42{\cdot}98

4. 7356

Solution

1{\cdot}000123 \approx 1{\cdot}0, where 1 is the first non-zero numeral and zero is the number after it, making it 2sf .

0{\cdot}0000923 \approx 0{\cdot}000092, where 9 is the first non-zero numeral and 2 is the number after it, making it 2sf. Here, 3 was rounded down to zero(0).

42{\cdot}98 \approx 43, where 4 is the first non-zero numeral and 3 is the number after it making it 2sf .
When approximating, we round up or round down the number immediately after the significant figure. Like in the example above, 9 is rounded up and added to 2 to make it 3. The decimal point is not necessary because placing zeros after it is irrelevant without a non-zero numeral.

7356 \approx 7400, where 7 is the first non-zero numeral and 4 is the number after it making it 2sf.
Here, 5 was rounded up and added to 3 to make it 4. Unlike the previous example, the zeros after the significant figures are necessary because they are part of a whole number and not after a decimal point. Nevertheless, the answer to the previous example can also be written as 43{\cdot}00.

Example 1.4

Approximate the following to 2dp

1. 0{\cdot}1009

2. 1{\cdot}0156

3. 2{\cdot}0009

Solution

0{\cdot}1009 \approx 0{\cdot}10, where 1 is the first number after the decimal point and zero is the second number after the decimal point making it 2dp. The third number after the decimal point was rounded down, so it has no effect on the answer.

1{\cdot}0156 \approx 1{\cdot}02, where zero is the first number after the decimal point and one is the second number after the decimal point making it 2dp. Since the third number after the decimal point(5) was rounded up, it was added to the second number after the decimal point. Hence the answer is 1{\cdot}02 and not 1{\cdot}01.

2{\cdot}0009 \approx 2{\cdot}00, unlike significant figures where the first significant figure must be a non-zero numeral, the first decimal place can be zero.
In the above example, the first and second number after the decimal place is zero and the third number was rounded down, which had no effect on the answer.

 

1.3 Percentage

A percentage is a fraction whose denominator is equal to 100. For example, if there are 11 red balls in a basket of 100 balls, the fraction of red balls will be \frac{11}{100}, and it can also be written as 11% (read as 11 percent). When the denominator of a fraction is not 100 and is to be expressed in percentage, the fraction will be multiplied by 100 as shown below.

\frac{2}{25} to percentage = \frac{2}{25}\times 100=2\times 4=8%

 

1.3.1 Error

Error is the magnitude of the difference between the actual measurement and calculated measurement.

Note: Error must be positive

error = |actual\quad measurement\quad -\quad calculated\quad measurement|

 

1.3.2 Percentage error

This is the ratio of the error to the actual measurement expressed in percentage.

%error=\frac{Error}{Actual\quad measurement}\times 100%

Example 1.5

Calculate the percentage error when a stick of 10cm was erroneously measured as 12cm.

Solution

actual\quad measurement=10cm \\ wrong\quad measurement=12cm \\ error=|10-12|cm \\ error=|-2|cm \\ error=2cm(error\quad must\quad be\quad positive)
%error=\frac{error}{actual\quad measurement}\times 100%
%error=\frac{2}{10}\times100%
%error=20%

 

1.4 Simple interest

Simple interest is the additional money a business man or woman makes after investing money(capital or principal) after a given period of time at a given rate.

Simple interest(I)=\frac{principal(P)\times Rate(R)\times Time(T)}{100}

The product of principal, rate and time is divided by 100 because the rate is always expressed in percentage.

Example 1.6

Find the simple interest a man would make for investing #10000 at a rate of 2% after 4 years.

Solution

I=\frac{PRT}{100}=\frac{10000 \times 2 \times 4}{100}=#800
The interest after 4 years will be #800

 

1.5 Profit and Loss

Profit is made when the selling price of a commodity is greater than the cost price of that commodity.

Loss is made when the cost price of a commodity is greater than the selling of that commodity.

Selling price(sp)- This is the price a business man uses in giving out his goods in the market.

Cost price(cp)- This is the price a business man uses in collecting goods from the market.

Example1.7

A man bought a bag for #25 and sold it for #30, did he make profit or loss?

solution

The man made a profit because the selling price #30 is greater than the cost price #25

profit=sp-cp=30-25=#5

The man made a profit of #5

Example 1.8

A man bought a chair for #39 and sold it for #38, did he make profit or loss?

Solution

The man made a loss because the selling price is less than the cost price.

\therefore Loss=cp-sp=39-38=#1

The man made a loss of #1

Note: When the Sp is equal to the Cp, neither a profit or a loss is made.

% profit=\frac{profit}{Cp}\times 100%

% loss=\frac{loss}{Cp}\times 100%

 

1.6 Ratio

A ratio is a relationship between two or more groups of people or thing that is represented by two or more numbers showing how much larger one group is than the others. If a whole class is separated into a number of fractional parts where each fraction has the same denominator, the numerators of the fractions form a ratio.

Example 1.9

If there are 7 red balls and 3 blue balls in a basket,

a. what is the ratio of red balls to blue balls?

b. what is the ratio of blue balls to red balls?

Solution

Ratio of red balls to blue balls =7:3

Ratio of blue balls to red balls =3:7

In terms of fraction, there are \frac{7}{10} red balls and \frac{3}{10} blue balls in the basket. Since both fractions have the same denominator, the numerator can be written as a ratio.

Example 1.1o

What is the ratio of red balls to green balls in a container if there are \frac{3}{4} red balls, \frac{1}{6} green balls and \frac{1}{12} blue balls?

Solution

Comparing both fractions (\frac{3}{4} and \frac{1}{6}) they have different denominators, therefore their numerators cannot be taken directly as ratio. Since 4 and 6 have a LCM of 12, multiply the numerator and denominator of the red ball fraction by 3 and the numerator and denominator of the green ball fraction by 2.

\frac{3}{4}\times \frac{3}{3}=\frac{9}{12} red balls

\frac{1}{6}\times \frac{2}{2}=\frac{2}{12} green balls

Since the denominators are the same, the numerators can now be taken as ratios i.e 9:2 (ratio of red balls to green balls)

 

1.7 Proportion

Proportion can be seen as sharing something into parts. With the aid of ratios, quantities can be shared properly into various parts.

If a quantity has a ratio A:B, it means it was divided into part A and part B. If A+B=C, then the whole quantity was divided into C parts.

Part A will be \frac{A}{C} of the whole quantity.

Part B will be \frac{B}{C} of the whole quantity.

Example 1.11

If #1000 was shared among James and John in a ratio of 13:12, how much will get to each of them?

Solution

13+12=25

where 13 is the portion that gets to James, 12 is the portion that gets to John and 25 is the total number of portions.

Since they shared 1000

The money that gets to James \rightarrow \frac{13}{25}\times 1000=#520

The money that gets to John \rightarrow \frac{12}{25}\times 1000=#480$

To check if your answer is correct, add both proportions. You’re supposed to get the amount shared.

 

1.8 Rate

Rate is the connecting of quantities of different kind mainly by division. Examples of rate are:

1. Rate of distance moved by time, also known as speed.
2. Rate of a given mass per unit length also known as linear density.
3. Rate of a given mass per unit volume also known as volume density.
4. Rate of pay per hour also known as wages etc.

Example 1.12

A town has an area of 1500hectares and a population of 90000. Calculate its population density (in persons/hectares)

Solution

Rate of persons per hectare=\frac{No\quad of\quad persons}{Hectares\quad of\quad land}

\frac{90000}{1500}=60persons/hectares

 

For solutions to more than 20 selected tough jamb past questions on this topic, click here

Click here to see the official jamb mathematics syllabus for yourself

Click here to see the recommended mathematics textbook from jamb

 

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