Students can go through AP Board 7th Class Maths Notes 7th Lesson Ratio and Proportion to understand and remember the concept easily.

## AP Board 7th Class Maths Notes 7th Lesson Ratio and Proportion

→ Compound Ratio of a: b and c: d is ac: bd.

→ a, b are two quantities if a increases, b also increases or a decreases, b also decreases then a and b are said to be in direct proportion.

→ a and b are in direct proportion, then \(\frac{a}{b}\) = k, where k is proportional constant.

→ a, b are two quantities, if ‘a’ increases, ‘b’ decreases or ‘a’ decreases, ‘b’ increases, then ‘a’ and ‘b’ are said to be in inverse proportion.

→ a and b are in inverse proportion, then a × b = k, where k is proportional constant.

→ Sometimes change in one quantitiy depends upon the change in two or more quantities in some proportion, which is called compound proportion.

1% = 1/100 = 0.01 = 1:100

→ Profit = Selling price – Cost price

Loss = Cost price – Selling price

Profit percentage = \(\frac{\text { profit }}{\text { C.P }}\) × 100

→ Loss percentage = \(\frac{\text { loss }}{\text { C.P }}\) × 100

→ Discount always calculated on marked price

→ Discount = Marked price – Selling

→ Simple Interest I = \(\frac{P \times T \times R}{100}\)

→ Ratio: Comparing two quantities of the same kind by virtue of division is called ‘Ratio’ of these two quantities.

- The ratio of ‘a’ and ‘b’ is a ÷ b or \(\frac{a}{b}\) and is denoted by a: b. and read as ‘a’ is to ‘b’.
- In a: b the first term ‘a’ is called antecedent and the second term ‘b’ is called consequent.
- The terms of the ratio should be expressed in same units.
- Generally a ratio is expressed in its simplest form.

Ex: The ratio of 200 ml and 3l is?

200: 3000 = 1: 15

→ Compound ratio: For two or more ratios, if we take antecedent as product of antecedents of the ratios and conse-quent as product of consequents of the ratios, then the ratio thus formed is called mixed or compound ratio.

- When two or more ratios are multiplied term-wise; the ratio thus obtained is called compound ratio.
- The compound ratio of a: b and c: d is ac: bd.
- The compound ratio of a: b, c: d and e: f is ace: bdf and so on.

→ Direct proportion:

Direct proportion is the relationship between two variables whose ratio is equal to a constant value. In other words, the direct proportion is a situation where an increase in one quantity causes a corresponding increase in the other quantity, or a decrease in one quantity results in a decrease in the other quantity.

→ If a and b are in direct proportion, then an increase (decrease) in a causes an increase (decrease) in b at the same rate. This is represented by a ∝ b

\(\frac{a}{b}\) = k or a = kb where k is proportionality constant.

Example:

- If the number of individuals visiting a restaurant increases, earning of the restaurant also increases and vice versa.
- Speed is directly proportional to distance.
- The cost of the fruits or vegetable increases as the weight for the same increase. Indirect proportion: Two quantities a and b are said to be in inverse proportion if an increase in the quantity a, there will be a decrease in the quantity b, and vice-versa. In other words, the product of their corresponding values should remain constant. Sometimes, it is also known as inverse variation.

→ The statement ‘a is inversely proportional to b’ is written as

a ∝ \(\frac{1}{b}\)

ab = k, where k is called proportionality constant.

That is, if ab = k, then a and b are said to vary inversely. In this case, if b_{1}, b_{2} are the values of b corresponding to the values a_{1} > a_{2} of a respectively, then a_{1} b_{1} = a_{2} b_{2} or a_{1} / a_{2} = b_{2} / b_{1}

Example:

- If speed increases, then the time taken to cover the same distance decreases.
- If number of men increases, then time taken to complete a given work decreases.

→ Compound proportion:

The proportion involving two or more quantities is called Compound propor¬tionality. The quantities could be directlly related or inversely related or both.

Ex.: 195 men working 10 hours a day can finish a job in 20 days. How many men are employed,to finish the job in 15 days if they work 13 hours a day ?

Answer:

Let x be the no. of men required

Days | Hours | Men |

20 | 10 | 195 |

15 | 13 | x |

20 × 10 × 195 = 15 × 13 × x

x = \(\frac{20 \times 10 \times 195}{15 \times 13}\)

= 200 men

→ CASE – 1: If quantity 1 and quantity 2 are directly related and quantity 2 and quantity 3 are also directly related, then we use the following rule:

\(\frac{\mathrm{a} \times \mathrm{b}}{\mathrm{c}}=\frac{\mathrm{d} \times \mathrm{e}}{\mathrm{x}}\)

→ CASE – 2: If quantity 1 and quantity 2 are directly related and quantity 2 and quantity 3 are inversely related, then we use the following rule:

\(\frac{b \times c}{a}=\frac{e \times x}{d}\)

→ CASE – 3: If quantity 1 and quantity 2 are inversely related and quantity 2 and quantity 3 are directly related, then we use the following rule:

\(\frac{a \times b}{c}=\frac{d \times e}{x}\)

→ CASE – 4: If quantity 1 and quantity 2 are inversely related and quantity 2 and quantity 3 are also inversely related, then we use the following rule:

a × b × c = d × e × x

→ Application of percentages: A percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate percent of a number, divide the number by whole and multiply by 100. Hence, the percentage means, a part per hundred. The word per cent means per 100. It represented by the symbol %.

- Percentages have no units. Hence it is called a dimensionless/unitless number.
- If we say, 50% of a number, then it means 50 per cent of its whole.

→ Percentage formula: The formula to calculate percentage of a number out of another number is:

Percentage = (Original number/ Another number) × 100

Express the given number as a part of whole or equivalent fraction. Multiply the fraction with 100.

Assign the symbol %.

→ What is the percentage of 45 out of 150?

Answer:

(45/150) × 100 = 30%

→ What is 40% of 120 ?

Answer:

40% of 120

= 40/100 × 120 = 48

→ Cost Price: It is the price at which a product is purchased. It is commonly abbreviated as C.P.

Example: A shopkeeper has bought 1 kg of apples for Rs.100, then C.P of apples is Rs. 100.

→ Selling Price: It is the price at which a product is sold. It is commonly abbre-viated as S.P.

Example:

- A shopkeeper has bought 1 kg of apples for Rs.100. And sold it for Rs. 120 per kg.
- The S.P of apples is Rs.120

→ Profit or gain: If the selling price of a product is more than the cost price, there will be profit in the deal.

Therefore, Profit or Gain = S.P. – C.P

Example: A shopkeeper has bought 1 kg of apples for Rs.100 and sold it for Rs. 120 per kg, then the profit is

S.P. – C.P. = 120 – 100 = Rs. 20

→ Loss: If the selling price of a product is less than the cost price, the seller will incur a loss. Therefore,

Loss = C.P – S.P.

Example: A shopkeeper has bought 1 kg of apples for Rs.100 and sold it for Rs. 80 per kg, then the loss is

C.P. – S.P = 100 – 80 = Rs. 20.

The profit percent or loss percent is always calculated as some percent of cost price.

→ Profit percentage = (Profit/Cost Price) × 100

Example: A shopkeeper has bought 1 kg of apples for Rs. 100 and sold it for Rs. 120 per kg then the profit percent is?

= (Loss / Cost price) × 100

Example: A shopkeeper has bought 1 kg of apples for Rs.100 and sold it for Rs. 80 per kg. then the loss percent is ?

Loss percent = \(\frac{\text { C.P-S.P. }}{\text { C.P. }}\) × 100%

= \(\frac{100-80}{100}\) × 100%

= \(\frac{20}{100}\) × 100% = 20%

→ Marked Price Formula (M.P): The price shown on an item is called its marked price. This is basically labelled by shop-keepers to offer a discount.

→ Discount: Discount is often given as a percent on C.P.

Discount = Marked Price – Selling Price

Discount percent = \(\frac{\text { M.P. }-\text { S.P. }}{\text { M.P. }}\) × 100%

→ Simple Interest: Sum lent or borrowed is called the Principle denoted by P. Time after which a loan is repaid is called the Time Period denoted by T.

The extra amount to be paid on a loan at the end of agreed time period is always expressed as a percent on P and is known as Rate of interest denoted by R%.

So Interest = R% of P for T years

= R × \(\frac{\mathrm{P}}{100}\) × T = \(\frac{\mathrm{P} \times \mathrm{T} \times \mathrm{R}}{100}\)

Amount = Principle + Interest

= P + \(\frac{\mathrm{P} \times \mathrm{T} \times \mathrm{R}}{100}\) = P\(\left(1+\frac{\mathrm{TR}}{100}\right)\)