# Ratio and Percentage

We can use the concepts of ratio and percentage to compare quantities effectively. Taking a ratio in mathematics is equivalent to weighing a mass. When we take a ratio and percentage of quantities, the values that we get can tell us about the magnitude of these quantities. Let us see how!

## Ratio and Percentage

### Ratio

A ratio is a comparison of two values expressed as a quotient. For example, if a class has 12 girls and 18 boys, the ratio of girls to boys is 12/18. This ratio can also be expressed as an equivalent fraction 2/3. Therefore, the ratio between girls and boys= 2:3

### Percent

The word percent is an abbreviation of the Latin phrase ‘per centum’ which means per hundred or for every hundred. When we say that a man gives 30 percent of his income as income tax. This means that he pays Rs 30 out of every hundred rupees of his income as income tax. The symbol % is used for the term percent.

### Percent as a Fraction

If we say x is 35% of y, then it means that:

$$x=35\%\quad of\quad y\\ x=\frac { 35 }{ 100 } y$$

Thus, a fraction with its denominator 100 is equal to that percent as is the numerator. To convert a fraction to a percent we multiply the fraction by 100 and put the percent sign %. For example,

$$\frac { x }{ y } =0.8\\ x=0.8\times 100\quad \%\quad of\quad y\\ x=80 \%\quad of\quad y$$

### Percent as a Ratio

A percent can be expressed as a ratio with its second term 100 and first term equal to the given percent. For example,

$$8 \%=\frac { 8 }{ 100 } =8:100\quad or\\ 8 \%=\frac { 8 }{ 100 } =\frac { 2 }{ 25 } =2:25\\$$

In order to convert a given ratio into a percent, we first convert a given ratio into the fraction and then multiply the fraction obtained by 100. For example,

$$\\ 3:25=\frac { 3 }{ 25 } \times 100\quad \%\quad =12 \%$$

### Percent in Decimal Form

To convert a given percent in decimal form, we express it as a fraction with the denominator as 100 and then the fraction is written in decimal form. For example, 65% =65/100 =0.65.

In order to convert a given decimal into a percent, we move the decimal point on the right side by two digits and put the percent sign %. For example, 0.122 =12.2%

## Finding a Percentage of a Number

To find a percent of a given number, we proceed as follows: Obtain the number, say x. Obtain the required percent, say P% multiply x by P and divided by 100 to obtain the required P% of x:

$$P\quad \%\quad of\quad x\quad =\quad \frac { P }{ 100 } \times x$$

For example: Find 12% of 1200Rs.

Solution:  $$12\quad \%\quad of\quad 1200Rs\quad =\quad \frac { 12 }{ 100 } \times 1200\quad =144Rs$$

## Solved Examples for You

### Type I

Question 1: If 23% of a is 46, then find a.

Solution: we know that – $$23\quad \%\quad of\quad a\quad =\quad \frac { 23 }{ 100 } \times a$$

But, 23% of a is given as 46. Therefore,

$$46=\frac { 23 }{ 100 } \times a\\ a=100\times \frac { 46 }{ 23 } \\ a=200$$

Question 2: A football team won 10 games from the total they played. This was 40% of the total. How many games were played in all?

Solution: Let x be the total number of games played. Then given that –

$$40\quad \%\quad of\quad x\quad =\quad 10$$

$$40\quad \%\quad of\quad x\quad =\quad \frac { 40 }{ 100 } \times x$$

$$10\quad =\frac { 40 }{ 100 } \times x\\ x=\frac { 100 }{ 40 } \times 10\\ x=25$$

Hence, in all 25 games were played.

### Type II

Question 3: A nursery has 5000 plants. 5% of the plants are roses and 1% are mangos plants. What is the total number of other plants?

Solution: We have, the total number of plants =5000. let’s assume there are x plats of rose, y plants of mangos and z plants of other types, then the number of rose plants

$$x=5\quad \%\quad of\quad 5000=\frac { 5 }{ 100 } \times 5000\\ x=250$$

Number of mango plants

$$y=1\quad \%\quad of\quad 5000=\frac { 1 }{ 100 } \times 5000\\ y=50$$

Therefore, the number of other plants

$$z=5000-x-y\\ z=5000-250-50\\ z=4700$$

Question 4: A certain company has 80 engineer employees. In this company, engineers constitute 40% of its workforce. How many people are employed in the company?

Solution:  Let x people be employed in the company. Since 40% of its workforce are engineers. This means that 40% of x is equal to the total number engineers.

$$40\quad \%\quad of\quad x\quad =\frac { 40 }{ 100 } \times x\\ 80=\frac { 40 }{ 100 } \times x\\ x=\frac { 100\times 80 }{ 40 } \\ x=200$$

Hence, 200 people are employed in the company.

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