Comparing Quantities

Throughout this chapter, we will deal with various concepts and techniques that empower you to make meaningful comparisons between quantities. You will explore topics such as profit and loss, discount, simple interest, compound interest, ratios and the interpretation of data through practical examples.

Profit and Loss

Cost price (C.P.): The amount at which an item is bought by a store owner or manufactured by a producer is known as its cost price.

Selling price (S.P.): The price at which an item is sold is referred to as its selling price.

(1) When S.P. > C.P., the shopkeeper makes a profit and this profit is calculated as Profit = S.P. − C.P.
(2) When S.P. < C.P., the shopkeeper incurs a loss and this loss is calculated as Loss = C.P. − S.P.

Percentage of Profit and Loss:

Note: The percentages of profit and loss are calculated based on the cost price.

Important Formulae

Example: By selling a Macbook for $1752, a boy loses ?th of its selling price. What is the cost of the Macbook?

a) $584
b) $1168
c) $2350
d) $2920

Answer: d) $2920

Explanation: S.P. of a Macbook = $1752

Loss = ? of S.P.
        = ?  × $1752
        = $1168

C.P. = S.P. + Loss
        = $(1752 + 1168)
        = $2920

Discount

Discount is a pricing strategy used to stimulate sales of a product or clear out old inventory. It involves reducing the price of an item from its Marked Price (M.P.). The reduced price at which the item is sold is called the Selling Price (S.P.).

Key formulae for discounts:

When calculating the Cost Price (C.P.) of an item, any additional expenses incurred after purchasing the article known as overhead expenses are included. Thus,

C.P. = Buying Price + Overhead Expenses

In many transactions, the government imposes a Sales Tax which is a percentage of the bill amount and is added to the total bill to be paid.

Sales tax = Tax% of the bill amount

Simple Interest

Simple Interest is the interest that is earned on a specific amount of money denoted as "P" over a given period of time "T" at an annual interest rate of "R%". This interest is calculated based on the principal amount and serves as compensation for the use of that money.

Compound Interest

Compound interest is when you earn interest on both the money you have saved and the interest you earn. Compound interest is the interest you earn on interest.

Amount accumulated on an initial principal amount denoted as "P" over a period of "T or n" years at an annual interest rate of "R%" when interest is compounded annually is expressed as:

Ratio

A ratio is a comparison between two or more quantities of the same kind and in the same units. It illustrates how many times one quantity contains or relates to the other. Ratios are typically denoted using the symbol ":" and they can be represented as "a : b" where 'a' is referred to as the antecedent and 'b' is referred to as the consequent.

In an alternative representation, the ratio of 'a' to 'b' (a : b) can also be expressed as a fraction, where 'a' is the numerator and 'b' is the denominator. This fraction is written as "a/b".

Ratio of two Quantities

1. Ratios are meaningful when the quantities being compared share the same units of measurement.
   Example: Ratio of 500 litres to 2000 litres = ^500L⁄_2000L = ¹⁄4 = 1 : 4
2. When dealing with ratios where the quantities have different units, it is essential to convert them into a common unit. Typically, the larger unit is converted into a smaller one. 
   Example: Ratio of 200 millilitres to 5 litres = ^200mL⁄_5L = ^200mL⁄_{5× 1000 mL} = ¹⁄₂₅ = 1 : 25
3. The sequence of the quantities in a ratio (A : B) holds significance. If you interchange the antecedent and consequent in a ratio, you obtain a distinct ratio
   (B : A).
   Example: Ratio 5 : 7 is not the same as 7 : 5.

Comparison of Ratios

To make a comparison between two ratios, the approach involves transforming each ratio into a fraction and these fractions are compared by ensuring that they share a common denominator.

Proportion

When two ratios are found to be equal, this relationship is termed a proportion. In other words, if the ratios of four quantities denoted as a, b, c and d satisfy the condition where the ratio of a to b equals the ratio of c to d (a : b = c : d), then a, b, c and d are said to be in proportion. This is typically expressed by writing a : b : c : d, where a, b, c and d are referred to as the first, second, third and fourth terms, respectively.

Example: 3 : 5 = 9 : 15 is written as 3 : 5 :: 9 : 15.

The first and fourth terms are termed extremes while the second and third terms are termed means.

The product of the extreme terms is equal to the product of the means. This relationship is denoted as:

a : b = c : d

⇒ a/b = c/d
⇒ ad = bc
⇒ Product of the extreme terms = Product of the means

Three values a, b, and c are considered to be in a continued proportion when the ratio of a to b is equal to the ratio of b to c, which can be expressed as: 

Here, b is the mean proportion between a and c while c is referred to as the third proportion.

Proportions find application in diverse mathematical and practical scenarios, assisting in the resolution of problems that revolve around comparisons, scaling and the interrelationships among quantities.

Example: What will be the ratio between simple interest and compound interest if the sum of $1000 is deposited in a bank at 5% per annum for 2 years?

a) 39 : 40
b) 39 : 41
c) 40 : 41
d) 41 : 40

Answer: c) 40 : 41

Explanation: