Comparing Quantities

In this chapter, we'll learn about different ideas and methods that help you compare quantities in various aspects of our daily lives from simple shopping to complex financial transactions. We'll look at things like ratio and proportion, percentage, profit and loss, discount, simple interest, compound interest and understanding data through examples that relate to real-life situations.

Ratio

When you compare two quantities with the same units by dividing them, you create a ratio like a : b or ??/??.
The first number is called the "antecedent" and the second number is called the "consequent".

Ratio in Simplest Form

A ratio (a : b) is said to be in its simplest form if the highest common factor (HCF) of a and b is 1.

To simplify a ratio, you need to divide both the antecedent and the consequent by their HCF.

Example: To find the simplest form of 35 : 56, follow these steps:

Step 1: HCF of 35 and 56 is 7.

Step 2: Now divide both 35 and 56 by 7 ⇒ (35/7) ÷ (56/7) = 5/8

Hence, the simplest form of 35 : 56 is 5 : 8.

Proportion

When two ratios are equal, it is called a proportion.
If a : b = c : d which means the ratio of the first quantity (a) to the second quantity (b) is the same as the ratio of the third quantity (c) to the fourth quantity (d). The symbols '::' or '=' shows this relationship.

The first and fourth terms are called extremes and the second and third term is called means.

Continued Proportion

Three quantities (a, b and c) are in continued proportion if a : b = b : c or ??/?? = ??/??, which can be expressed as:

Word Problems on Proportion

Example 1: What is the value of p if the proportion is p : 4.5 :: 27 : 0.09?          

a) 1.35
b) 13.5
c) 135
d) 1350

Answer: d) 1350

Explanation: In proportion, the product of the extremes (the first and last numbers) is equal to the product of the means (the middle numbers). 

⇒ p × 0.09 = 4.5 × 27
⇒ p = (4.5 × 27) ÷ 0.09
⇒ p = (45 × 27 × 100) ÷ (9 × 10)
⇒ p = 1350

Example 2: Emma went on a seven-day hike and consumed eleven loaves of bread. If he consumes the same quantity of bread every day, how many loaves of bread will she bring on an eight-week hiking trip?

a) 77 loaves
b) 88 loaves
c) 99 loaves
d) 111 loaves

Answer: b) 88 loaves

Explanation: Let x loaves of bread last for an eight-week hiking trip.

In this case,

11 loaves/7 days :: x loaves/8 weeks
⇒ 11 loaves/7 days :: x loaves/56 days [1 week = 7 days]
⇒ 11 ÷ 7 = x ÷ 56
⇒ x = (11 × 56) ÷ 7
⇒ x = 88

Therefore, 88 loaves of bread last for an eight-week hiking trip.

Percent

Percent means per hundred and it is represented by the symbol %.

Example: If there are 100 eggs in a tray and 15 of them are rotten, we say that 15% of the eggs are rotten and 85% of the eggs are good to use.

Percentage

When we talk about a certain percentage, we mean 'that many' hundredths.
Example: 85% is the same as 85 hundredths or 85/100.

Converting Fraction into Percent: To convert a fraction into a per cent, multiply it by 100.

Example: 9/25 becomes 36% when multiplied by 100.
(9 ÷ 25) × 100 = 36%

Converting Ratio into Percent: Convert the ratio into fractions and then multiply by 100.

Example: Ratio 27 : 30 becomes 90% when converted.
Ratio 27 : 30 into fraction = 27 ÷ 30
(27÷30) × 100 = 90%

Converting Decimal into Percent: Multiply the decimal by 100 which is the same as shifting the decimal point two places to the right.

Example: 0.027 becomes 2.7% when multiplied by 100.
0.027 × 100 = (27 ÷ 1000) × 100 = 2.7%   
Hence, decimal point is shifted by two places to the right.

Converting Percent into Fraction: Remove the % sign and divide by 100.

Example: 35% becomes 7/20.
35% = 35 ÷ 100 = 7/20

Converting Percent into Ratio: Remove the % sign and form a ratio by taking the given number as the 1st term and 100 as the 2nd term.

Example: 45% becomes the ratio 9 : 20.
45% = 45 : 100 = 9 : 20

Converting Percent into Decimal: Remove the per cent (%) sign and divide by 100 which is done by shifting the decimal two places to the left.

Example: 93% becomes 0.93.
93% = 93 ÷ 100 = 0.93

Key Formulae related to Percentage:

Profit and Loss

Cost Price (C.P.): The cost price of an article is the price at which it is bought.
Selling Price (S.P.): The selling price of an article is the price at which it is sold.

If the Selling Price is greater than the Cost Price (S.P. > C.P.), then there is a profit or gain.

Key Formulae Related to Profit:

If the Selling Price is less than the Cost Price (S.P. < C.P.), then there is a loss.

Key Formulae Related to Loss:

Discount

The marked price (M.P.) is the initial printed price for an article. It is also known as printed or listed price.
A discount is a decrease in the marked price.

Key Formulae related to Discount:

Simple Interest

The amount of money borrowed or lent for a specific time is known as the principal (P) or sum.
Simple Interest (S.I.) is when the interest on that amount borrowed or lent which is calculated uniformly throughout the entire loan period. It is the extra money you get for having a certain amount of money (P) for a certain time (T) at an annual interest rate (R%).

The sum of principle and simple interest is the amount (A).

Key Formulae related to Simple Interest:

Compound Interest

Compound interest is the interest added each year on the total amount from the previous year. It is the interest you earn on interest.
The total amount (A) is paid on an initial amount of money (P) which is lent for a certain number of years (T) with an annual interest rate (r%). This calculation also considers how many times (n) the interest is compounded per year.

Key Formulae related to Compound Interest:

The amount and compound interest formula can be expressed for different scenarios such as when the interest rate is compounded yearly, half-yearly, quarterly, etc.

The general formula is given by:

The amount and compound interest compounded annually is given by:

Here, the number of times interest is compounded annually n = 1.

Interest Compounded for Different Years

Interest compounded for different years refers to how the interest on an amount of money is calculated over various time periods.

Compound Interest When the Rate is Compounded Half Yearly

When calculating compound interest for a period of half a year, we divide the interest rate by two and multiply the time by two in the general formula.

Compound interest formulae suitable for every six months (half yearly - n = 2) are as follows:

Compound Interest When the Rate is Compounded

If the interest rate is compounded quarterly, we divide the interest rate by four and multiply the time by four in the general formula.

Compound interest formulae suitable for every three months (quarterly - n = 4) are as follows: