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

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 b1, b2 are the values of b corresponding to the values a1 > a2 of a respectively, then a1 b1 = a2 b2 or a1 / a2 = b2 / b1

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 ?
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?
(45/150) × 100 = 30%

→ What is 40% of 120 ?
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$$\left(1+\frac{\mathrm{TR}}{100}\right)$$