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Ratios and Percents

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Ratio is a mathematical relationship between two quantities, amounts or numbers. It is a measure of number of times one of the value is greater compared to the other.

When something is divided into $100$ parts, the term percent would mean a part of it. Percent is a number of a hundredth.

Definition

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Ratio can be defined as a comparison between two values of the same parameter. It shows how many times bigger or smaller the numerator is with respect to the denominator. It is a quotient of two mathematical expressions. The symbol used to denote ratio is "$:$". There is another symbol to denote ratio which is the division symbol or '$/$'.

Percent can be defined as a ratio whose second term is always constant which is $100$. It can also be defined as a fraction whose denominator is always $100$. The name 'Percent' has been derived from the Latin word per centum which mean per hundred. The symbol used to denote percentage is '$ \% $'. The abbreviated version of the term 'percent' is 'pct'.

Formula

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Formula for Ratio: If we are trying to find out the ratio between two variables $a$ and $b$ of the same kind, then the ratio formula to be used is $a : b$ = $\frac{a}{b}$.

Formula for Percentage: If we are trying to find out the percentage say $x \%$, then the percent formula would be is $/$ of = $\frac{\%}{100}$ or $\frac{part}{whole}$ = $\frac{\%}{100}$ or $\frac{x}{100}$. The value of the percentage is kept in the numerator and the denominator of the percentage is always $100$.

How Ratio and Percentage are Related

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Ratio and percentage are inter related such that we can convert ratio to percentage and percentage to ratio.
Conversion of Ratio to Percentage: Let the ratio which we need to convert into percentage be $a : b$. The following steps need to be followed to convert the ratio into percentage.

Step 1: Write down the ratio $a : b$

Step 2: Convert the ratio into fraction form by changing the ratio symbol '$:$' into the fraction symbol '$/$'. So, $a : b$ becomes $\frac{a}{b}$ in the fraction form

Step 3: Multiply the fraction $\frac{a}{b}$ we got in step $2$ with $100$. So, it would be like $\frac{a}{b}$ $\times\ 100$

Step 4: The answer we get after cancellation and simplification is the answer in percentage form. The answer should always be either in whole number or in decimals

Step 5: We need to put the percentage symbol '$ \%$' beside the answer to denote that the result is in percentage
Conversion of Percentage to Ratio: Let the percent which need to be converted into ratio be $x \%$. The following steps need to be followed to convert percent into ratio.

Step 1: Write down the percentage $x \%$

Step 2: Convert the percentage $x \%$ into the form of fraction by dividing the value of the percent by $100$. So, it would be like $x \%$ into fraction form $\frac{x}{100}$

Step 3: Never forget to remove the percentage symbol '$ \%$' because it is now converted into fraction form and is no more in percentage

Step 4: Reduce the fraction obtained $\frac{x}{100}$ in the reduced form by dividing both the numerator and denominator by common factor if any. So, $\frac{x}{100}$ is changed into its simplest form if its reducible.

Step 5: Write the reduced form of fraction in ratio form which replaces the fraction symbol '$/$' by the ratio symbol '$:$'

Step 6: Thus, we get the answer in ratio form.

Examples

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Example 1:  

Convert the following ratio into percentage:

a) $3 : 5$

b) $7 : 20$

c) $8 : 25$

d) $9 : 10$

Solution: 

a) The ratio is converted to percentage by replacing the ratio symbol '$:$' to fraction symbol '$/$'. So, $3 : 5$ becomes $\frac{3}{5}$. Next, we multiply it with $100$.

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

We place a percentage symbol $\%$ after the value. Thus, the answer is $60 \%$

b) The ratio is converted to percentage by replacing the ratio symbol '$:$' to fraction symbol '$/$'. So, $7 : 20$ becomes $\frac{7}{20}$. Next, we multiply it with $100$.

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

We place a percentage symbol $\%$ after the value. Thus, the answer is $35 \%$

c) The ratio is converted to percentage by replacing the ratio symbol '$:$' to fraction symbol '$/$'. So, $8 : 25$ becomes $\frac{8}{25}$. Next, we multiply it with $100$.

$\frac{8}{25}$ $\times 100$ = $32$

We place a percentage symbol $\%$ after the value. Thus, the answer is $32 \%$

d) The ratio is converted to percentage by replacing the ratio symbol '$:$' to fraction symbol '$/$'. So, $9 : 10$ becomes $\frac{9}{10}$. Next, we multiply it with $100$.

$\frac{9}{10}$ $\times 100$ = $90$

We place a percentage symbol $ \%$ after the value. Thus, the answer is $90 \%$
Example 2: 

Express each of the following percentages into ratios in the simplest form:

a) $45 \%$

b) $25 \%$

Solution: 

a) To convert the percentage into ratio we first convert it into fraction. That is removing the $\%$ symbol and divides the value by $100$. So,

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

Next we reduce the fraction into the simplest form by dividing both numerator and denominator by common factors if any. So, as both $45$ and $100$ have a common factor $5$, it is reducible to

   $\frac{45}{100}$

= $\frac{9}{20}$

We replace the division symbol with ratio symbol : to get the result.

Thus, the answer in ratio and that too in its simplest form is $9 : 20$

b) To convert the percentage into ratio we first convert it into fraction. That is removing the $\%$ symbol and divides the value by $100$. So,

$25 \%$ = $\frac{25}{100}$

Next we reduce the fraction into the simplest form by dividing both numerator and denominator by common factors if any. So, as both $25$ and $100$ have a common factor $25$, it is reducible to

   $\frac{25}{100}$

= $\frac{1}{4}$

We replace the division symbol with ratio symbol : to get the result.

Thus, the answer in ratio and that too in its simplest form is $1 : 4$
Example 3: 

A bag contains $50$ paise, $25$ paise and $10$ paise coins in the ratio $5 : 9 : 4$. The total amount in the bag is $206$. Find out the number of coins in each denomination.

Solution: 

As the ratio of content of each type of coin is $5 : 9 : 4$, so number of $50$ paise coins is $5x$, number of $25$ paise coins is $9x$ and number of $10$ paise coins is $10x$. Total amount of $50$ paise coins  = $0.50 \times 5x$ = $2.5x$, total number of $25$ paise coins = $0.25 \times 9x$ = $2.25x$ and amount of $10$ paise coins = $0.10 \times 4x$ = $0.4x$.

So, the total amount in the bag is $2.5x + 2.25x + 0.4x$ = $206$

$5.15x$ = $206$

$X$ = $40$

Thus, the number of $50$ paise coins is $5x$ = $5 \times 40$ = $200$, number of $25$ paise coins is $9x$ = $9 \times 40$ = $360$ and number of $10$ paise coins is $4x$ = $4 \times 40$ = $160$
Example 4: 

There are two vessels $A$ and $B$ containing solution of milk and water. Vessel $A$ contains solution in the ratio $4 : 3$ and vessel $B$ contains solution in the ratio $9 : 1$. Both the solutions are mixed in certain ratio such that the ratio of milk to water in the final solution is $3 : 2$. Find out the ratio in which the two solutions are mixed

Solution: 

Let the amount of first solution from vessel $A$ taken is $x$ lts. Amount of milk in $x$ lts of solution would be $\frac{4 }{(4 + 3)}$ $\times x$ = $\frac{4x}{7}$. Amount of water in $x$ lts of solution would be $\frac{3}{(4 + 3)}$ $\times x$ = $\frac{3x}{7}$. Let the amount of first solution from vessel $B$ taken is $y$ lts. Similarly, amount of milk in the solution kept in vessel $B$ is $\frac{9}{(9 + 1)}$ $\times y$ = $\frac{9y}{10}$ and amount of water in $y$ lts of solution is $\frac{1}{(1 + 9)}$ $\times y$ = $\frac{y}{10}$. Total amount of milk from both the vessels $A$ and $B$ = $\frac{4x}{7}$ + $\frac{9y}{10}$ and total amount of water from both the vessels $A$ and $B$ = $\frac{3x}{7}$ + $\frac{y}{10}$.

Amount of milk content in the final solution = $\frac{3}{5}$ $\times (x + y)$

$\frac{4x}{7}$ + $\frac{9y}{10}$ = $\frac{3}{5}$ $\times (x + y)$

On simplifying,

$\frac{4x}{7}$ + $\frac{9y}{10}$ = $\frac{3}{5}$ $\times (x + y)$

$\frac{4x}{7}$ + $\frac{9y}{10}$ = $\frac{3x}{5}$ + $\frac{3y}{5}$

Multiplying throughout with the least common denominator 70,

$40x + 63y$ = $42x + 42y$

$40x + 63y$ = $42 \times 700$

$40x + 63y$ = $29400$ ..... equation 1

Amount of water content in the final solution = $\frac{2}{5}$ $\times (x + y)$

$\frac{3x}{7}$ + $\frac{y}{10}$ = $\frac{2}{5}$ $\times (x + y)$

$\frac{3x}{7}$ + $\frac{y}{10}$ = $\frac{2x}{5}$ + $\frac{2y}{5}$

Multiplying throughout with the least common denominator $70$,

$30x + 7y$ = $28x + 28y$

$30x + 7y$ = $28 \times 700$

$30x + 7y$ = $19600$ ....  $\times 9$

$270x + 63y$ = $176400$ .... equation 2

Using elimination method to solve the two equations,

$270x- 40x + 63y - 63y$ = $176400 - 29400$

$230x$ = $14700$

$X$ = $639$

$Y$ = $\frac{(19600 - 30 \times 639)}{7}$

$Y$ = $61$

Thus, $639$ lts of solution from vessel $A$ and $61$ lts of solution from vessel $B$ are mixed.