# Chapter 8 – Comparing Quantities

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

In your daily life, you might have come across instances when you had to compare and measure certain quantities. Comparing quantities is one of the most common activities that we do everyday, maybe in terms of scoring marks, measuring distances, comparing heights, and many more.

The most important and basic fact for comparing quantities is that they should be similar, i.e two unlike quantities can’t be compared. For example, you can’t compare the height of one person with the age of another. Among different available standards, the percentage is one of the common standards for doing this, which we will be focusing on in this article.

#### Use of percentage for comparing quantities

‘Percentage’ is derived from a Latin word ‘ per centum’ which implies per hundred, denoted by the symbol %. In other words, it is a fraction whose denominator is 100. Recalling ratios and percentages,  we can say that both are popular ways of measurement of quantities.

Let us look at an example. When you get 80% marks, it implies, it is equivalent to 80 out of 100, which is 0.80 in decimal or 4:5 ratio. Similarly, you can find out the percentage of each of the atmospheric gases in our environment and so on. It is widely used since it is the easiest way of comparing quantities. We will be seeing other mathematical topics in which this can be implemented, in the next sections.

#### Finding Discounts using percentage

Marketers generally attract customers by offering a discounted price on any product. This implies a reduction in its marked price. The discount can be found by deducting the sale price from the marked price, i.e:

Discount = Marked price – Sale price

and discount% = (discount/ marked price) * 100

Example:

A toy marked at 800 INR was sold at 720 INR, what is the discount % offered?

Discount percentage = 800-720= 80

So, discount % = (80/800)*100 = 10%

#### Using percentage in the calculation of profit and loss

In exams, you get problems that have prices related to buying and selling where you would be asked to find the profit or loss percentage. Let us see an example.

Ram bought a toaster for 2500 INR. Later he had to spend 500 on repairing it and sold it off for 3300. Was it a loss or profit for him and what was the percentage?

Total cost price for Ram= 2500 + 500 = 3000

Sale price = 3300

Since cost price < sale price, it is profit which is equal to 3300-3000= 300

Profit %= (Profit/ cost price) *100

=(300/3000)*100= 10%

Similarly, when the cost price is higher than the sale price, it is termed as a loss.

Loss = cost price – sale price

and loss %= (Loss/cost price) *100

#### Sales Tax using percentage

Government charges tax when we purchase a product, which is termed as Sales  tax.

Example:

A book costs 450 INR. 3% sales tax was imposed on it. So, its sale price will be:

Tax paid =3% of 450 = 13.50

So amount paid = 450 + 13.50 = 463.5 INR.

#### Compound Interest (CI) using percentage

When the interest obtained on an amount is reinvested by adding it to the principal in the following year, instead of paying it out, we get the interest compounded.

Let us find the SI and CI on 20000 INR at 8% for 2 years and see which is higher.

Calculating CI:

It can be calculated by first finding the Simple Interest in the 1st year, which is:

(20000 x 8 x 2)/ 100= 1600

Now compound interest after 2 years = interest on the new principal

and new principal = Initial amount + Interest obtained in 1st year

= 20000+1600= 21600

Now SI for 2nd year= (21600 *2*8)/100= 1728.

So, the amount that will be received at 2nd year end= 21600+1728= 23328

Hence, total interest earned= 1600+1728=3328

#### Calculation of SI

SI= (20000x 8×2)/100= 3200

This shows that CI is more than SI.

Instead of going through this lengthy process of calculating CI, deducing a formula for compound interest will save time. This formula is:

A= P x [ 1+ (R/100)]^n

and CI = A- P

where, A= amount obtained after “n” years

R= rate of interest.

P= principal amount

#### Calculating interest when rate compounded annually or half yearly

When you find a question with interest being compounded half-yearly, then it implies, there will be 2 conversion periods in 1 year, after 6months. So the rate to be used in this case will be half of the annual rate of interest.

Similarly, if the interest is compounded quarterly, then you will have 4 conversion periods in a year and the rate of interest will be 1/4 th of the annual interest rate.

Example:

What will happen to 500 INR at 10% p.a, over one year, when interest is compounded half-yearly?

For 1st 6 months:

Interest= [P x (R/2) ]/ 100

= [500 x 5]/100 = 50

Amount =  500 + 50= 550

For 2nd 6 months:

Interest= (550 x 5 )/100 =27.5

Amount = 550 + 27.5 = 577.5

#### Applications of Compound Interest Formula

i. It can be used for finding the increase and decrease in percent of the population in an area.
ii. The value of a product, when its cost rises or falls in the immediate years.
iii. You can know the bacterial growth when the rate of growth is given.

#### FAQ

What are comparing quantities?

It implies making a comparison between two similar things. In mathematics, it has to be two quantities with the same units.

What are the formulas for comparing quantities?

There are numerous formulae for comparing quantities, of which percentage and ratios are the popular methods. For various topics, the formula varies, such as in the case of profit and loss, sales tax, compound interest, and others.

What is the formula for discount?

Discount on a product is calculated as:

Discount= [(Marked price – Sale price)/ Marked price] * 100

What is the formula of SP?

SP is the sale price of an object and is calculated as:

SP= Cost price + profit

or SP= Cost price-loss

How do you turn a ratio into a percentage?

Let us see this through an example where the given ratio is 4:5.

When converted into a fraction, it is 4/5

This needs to be multiplied by 100 and % has to be used at the end of the answer obtained:

i.e  (4/5) *100 = 80 %

We covered the basics about comparing quantities in this article. But you can refer to the MSVGo app that has a more extensive explanation of concepts with examples. You can also find visual illustrations with certain examples. It is one of the best apps that aims at ensuring a core understanding of concepts that will help you to clearly understand the underlying concept and solve problems more easily and quickly.

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