Ratio

The ratio is used to compare two quantities. These quantities must have the same units.

The ratio is represented by “:”, which is read as “ is to”. We can write it in the form of “fraction”.

 

Equivalent Ratio

The equivalent ratio is like the equivalent fractions so to find the equivalent ratio we need to write it in the form of a fraction. To find the equivalent ratio we need to multiply or divide the numerator and denominator with the same number.

Example

Find the two equivalent ratios of 3 : 14.

Solution

First multiply both numerator and denominator by 2.

     

Now, we can multiply by 3

     

Now, we can multiply by 5

     

So, the three equivalent ratios of  are ,  and .

 

To compare that the two ratios are equivalent or not we need to convert them in the form of a like fraction. Like fractions are the fractions with the same denominator.

Example

State whether the ratios 4 : 5 and 7 : 9 are equivalent are not?

Solution:

To check, first, we need to make their denominator same.

Converting ratios to fractions, we have  and .

LCM of denominators 5 and 9 is 45.

Hence we need to make denominators of both fractions equal to 45.

   

   

As , hence the ratio 4 : 5 and 7 : 9 are not equivalent ratios.

 

Proportion

Proportion shows the equality between two ratios. If two ratios are in proportion then these must be equal.

Example

If the cost of 12 mangoes is Rs. 180 then what will be the cost of 30 strawberries.

Solution

Using Unitary Method

Cost of 12 mangoes = ₹ 180

Cost of 1 mango = ₹ 180/12

Cost of 30 mangoes = ₹ (180/12) × 30

                 = ₹ 450

 

Using proportion method

Assuming the cost of 30 mangoes = ₹ x

Then, 12 : 30 = 180 : x

    i.e., 12 / 30 = 180 / x

Cross-multiplying, we get

    12x = 180 × 30

        x = (180 × 30)/12 = 450

Hence, cost of 30 mangoes will be ₹ 450.

 

Percentage

The percentage is another way of comparisons. In ratios we have to make the denominator same then only we can compare them but in percentage, we can compare by calculating the percentage of the given quantity.

The percentage is the numerator of the fraction with the 100 denominators.

Example

What is the percentage of boys and girls in the class of 100 students if the number of boys is 55 and the number of girls is 45?

Solution

Percentage if the total is not a hundred

If the total number of quantities is not hundred i.e., the denominator is not hundred then to find the percentage we need to make the denominator 100.

Example

Out of 15 students, 3 failed in mathematics. What percentage of students failed?

Solution

Unitary Method

Out of 15, the number of students who failed = 3

Hence, out of 100, the number of students failed =

By making denominator 100

Out of 15, the number of students who failed = 3

Hence,

   

Converting fractional numbers to percentage

Fractional numbers have different denominator and to convert them into percentage we have to multiply the fraction with the 100%.

Example

Out of 32 students, 12 are girls. What is the percentage of girls?

Solution

Percentage of girls =

 

Converting decimals to percentage

To convert the decimal into a percentage, first, we need to convert the decimal into fraction then multiply it by 100%.

Example

Convert 0.65 into a percentage.

Solution

Multiply the decimal with the 100%.

 

Converting Percentage to fractions or Decimals

We can reverse the above process to convert the percentage into fraction or decimal.

 

Parts always add to give a whole

If we know the one part of a whole then we can find the other part because all the parts together form a whole or 100%.

e.g., if there are 25% red balls, 45% blue balls and rest of the balls are green, then the green balls will be 30%.

Green balls    = 100% − ( red balls + blue balls)

         = 100% − ( 25% + 45% )

         = 30% green balls.

 

Example

What percent of the given figure is shaded?

Solution:

Out of total 6 parts of the figure, 5 are shaded.

Hence, fraction will be

Hence,

    Percentage of shaded portion = fraction × 100

                            =  %

Remember:

If we say that Deepak spends 40% of his salary, then it means that he spends Rs. 40 out of every Rs. 100 of his salary.

Conversion of percentages to numbers ( i.e., “how many” )

Example

If 25% out of 60 labourers get daily wage more than ₹ 500, then how many labourers get daily wage more than ₹ 500?

Solution

The number of labourers getting more than ₹ 500 = .

Hence, 15 labourers out of 60 get daily wage more than ₹ 500.

Conversion of Ratios to percent

Example

A prize money of ₹ 500000 is divided among 4 players, A, B, C and D, in such a way that A, B, C and D got the two parts, three parts, 1 part and four parts of profit respectively. How much money did each player get? What percent of the prize, did each player get?

Solution

The 4 players got prize in the ratio of 2 : 3 : 1 : 4, so the total of the parts is

2 + 3 + 1 + 4 = 10

For player A:

Amount of prize = ₹ 100000

Percentage of prize =

For player B:

Amount of prize = ₹ 150000

Percentage of prize =

For player C:

Amount of prize = ₹ 50000

Percentage of prize =

For player D:

Amount of prize = ₹ 200000

Percentage of prize =

 

Increase or decrease as Percent

 

Percentage increase or Percentage decrease =

Example

The cost of rice increased from ₹ 100 to ₹ 120. Find the increase in percentage.

Solution

Original amount = Initial cost of rice = ₹ 100

Amount of change = increase in the cost of rice = 120 – 100 = 30.

Hence,

Percentage increase or Percentage decrease =

                     =  = 20%

 

Cost Price

Cost price is the price at which you buy some product. It is written as CP.

 

Selling Price

Selling price is the price at which you sell something. It is written as SP.

These are the factors which tell us that the sale of some product is profitable or not.

Example

If the buying price (or CP) of a table is Rs 700 and the selling price (or SP) is Rs 820, then find the profit or loss.

Solution

As the SP is more than CP, so the seller earns the profit in the table.

 Profit made = SP – CP 

= Rs 820 – Rs 700

= Rs 80

 

Profit or loss percentage

The profit and loss can be converted into a percentage. It is always calculated on the cost price.

 

     

 

     

 

Example

If the cost price of a laptop is Rs.45000 and the selling price is Rs. 50000, then what is the profit or loss percentage?

Solution

 

Finding SP if CP and profit or loss % is given

 

     

     

 

Example

If the cost of a TV is Rs.25000 and shopkeeper sells it at a loss % of 5% then what is the selling price of the TV?

Solution

Hence, the shopkeeper sells it at the price of Rs. 23750

 

Finding CP if SP and profit or loss % is given?

 

     

 

     

 

Example

If the Selling price of a bookshelf is Rs 750 and the profit made by the seller is 10% then what is the cost price of the bookshelf?

Solution

Hence the seller bought the bookshelf at the cost of Rs. 682.

 

Simple Interest

When we borrow some money from the bank then we have to pay some interest to the bank.

The money which we borrow is called the Principal.

The amount which we have to pay to the bank to use that money is called interest.

At the end of the year, we return the money to the bank with interest, that money is called Amount.

Amount = Principal + interest

 

Where,

SI = Simple interest

P = Principal

R = Rate of Interest

T = time period

Example

Sunita borrows a loan of Rs 15,0000 at 15% per year as the rate of interest. Find the interest she has to pay at end of 3 years.

Solution

P = ₹ 150000, R = 15%, T = 3 years

 

     

     

Total amount to be paid by Sunita at the end of one year = ₹ 150000 + ₹ 67500

         = ₹ 217500.

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