Percentage and Discount

Objectives:

This lesson introduces you to comparing quantities like :

  • Ratios and Percentages
  • Increase or Decrease Percent
  • Discounts
  • Estimation in percentages
  • Prices related to buying and selling
  • Sales Tax / Value Added Tax

Recalling Ratios and Percentages

Ratio means comparing two quantities. A basket has two types of fruits, say, 20 apples and 5 oranges.

Then, the ratio of the number of oranges to the number of apples =5 :20.

The comparison can be done by using fractions as, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»5«/mn»«mn»20«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«/math».

The number of oranges are «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«/math» th the number of apples.

In terms of ratio, this is 1 : 4, read as, “1 is to 4” or Number of apples to number of oranges «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mo»§nbsp;«/mo»«mfrac»«mn»20«/mn»«mn»5«/mn»«/mfrac»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mfrac»«mn»4«/mn»«mn»1«/mn»«/mfrac»«/math»  which means, the number of apples are 4 times the number of oranges.

This comparison can also be done using percentages. 

Percent means ‘per hundred’ or out of hundred. Percentage is another way of comparing ratios that compares to hundred.

There are 5 oranges out of 25 fruits. So percentage of oranges is

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»5«/mn»«mn»25«/mn»«/mfrac»«mo»§#215;«/mo»«mfrac»«mn»4«/mn»«mn»4«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mn»20«/mn»«mn»100«/mn»«/mfrac»«mo»=«/mo»«mn»20«/mn»«mo»%«/mo»«/math»[Denominator made 100]. 

By unitary method:

Out of 25 fruits, number of oranges are 5. So out of 100 fruits, number of oranges«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mo»§nbsp;«/mo»«mfrac»«mn»5«/mn»«mn»25«/mn»«/mfrac»«mo»§#215;«/mo»«mn»100«/mn»«mo»=«/mo»«mn»20«/mn»«/math»

Since basket contains only apples and oranges, So, percentage of apples + percentage of oranges = 100 or  percentage of apples + 20 = 100 or  percentage of apples = 100 – 20 = 80. Thus the basket has  20%  oranges and  80%  apples.

Finding the Increase or Decrease Percent

A change in a quantity can be positive, which means an increase, or negative, which means a decrease. Such a change can be measured by an increase percent or a decrease percent.

We often come across such information in our daily life as  25% off on marked prices, 10% hike in the price of petrol. Let us consider a few such examples.

Example -1

The price of a scooter was Rs 34,000 last year. It has increased by 20% this year. What is the price now?

Solution:

Amita said that she would first find the increase in the price, which is 20% of Rs 34,000, and then find the new price.

20% of Rs 34000 = Rs «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»20«/mn»«mn»100«/mn»«/mfrac»«mo»§#215;«/mo»«mn»34000«/mn»«/math» = Rs 6800

New price = Old price + Increase = Rs 34,000 + Rs 6,800 = Rs 40,800

Sunita used the unitary method

20%  increase means, Rs 100 increased to Rs 120.

So, Rs 34,000 will increase to?

Increased price «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mfrac»«mn»120«/mn»«mn»100«/mn»«/mfrac»«mo»§#215;«/mo»«mn»34000«/mn»«/math» «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»40«/mn»«mo»,«/mo»«mn»800«/mn»«/math»

Similarly, a percentage decrease in price would imply finding the actual decrease followed by its subtraction the from original price. Suppose in order to increase its sale, the price of scooter was decreased by 5%. Then let us find the price of scooter.

Price of scooter = Rs 34000

Reduction = 5% of Rs 34000«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mfrac»«mn»5«/mn»«mn»100«/mn»«/mfrac»«mo»§#215;«/mo»«mn»34000«/mn»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»1700«/mn»«/math»

New price = Old price – Reduction = Rs 34000 – Rs 1700 = Rs 32300

Finding Discounts

Discount is a reduction given on the Marked Price (MP) of the article.

This is generally given to attract customers to buy goods or to promote sales of the goods. You can find the discount by subtracting its sale price from its marked price. So, Discount = Marked price – Sale price.

Example-2 

An item marked at Rs 840 is sold for Rs 714. What is the discount and discount %?

Solution:

Discount = Marked Price – Sale Price  = Rs 840 – Rs 714 = Rs 126. Since discount is on marked price, we will have to use marked price as the base.

On marked price of Rs 840, the discount is Rs 126. On MP of Rs 100, how much will the discount be?

Discount «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mfrac»«mn»126«/mn»«mn»840«/mn»«/mfrac»«mo»§#215;«/mo»«mn»100«/mn»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mn»15«/mn»«mo»%«/mo»«/math»

Example-3

The list price of a frock is Rs 220. A discount of 20% is announced on sales. What is the amount of discount on it and its sale price.

Solution:

Marked price is same as the list price.20% discount means that on Rs 100 (MP), the discount is Rs 20.

By unitary method, on Re 1 the discount will be Rs «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»20«/mn»«mn»100«/mn»«/mfrac»«/math»

On Rs 220, discount «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mfrac»«mn»20«/mn»«mn»100«/mn»«/mfrac»«mo»§#215;«/mo»«mn»220«/mn»«mo»§nbsp;«/mo»«mo»=«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»44«/mn»«/math»

The sale price = (Rs 220 – Rs 44) or Rs 176

Rehana found the sale price like this —

A discount of 20% means for a MP of Rs 100, discount is Rs 20. Hence the sale price is Rs 80. Using unitary method, when MP is Rs 100, sale price is Rs 80;

When MP is Re 1, sale price is Rs «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»80«/mn»«mn»100«/mn»«/mfrac»«/math»

Hence when MP is Rs 220, sale price  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mfrac»«mn»80«/mn»«mn»100«/mn»«/mfrac»«mo»§#215;«/mo»«mn»220«/mn»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»176«/mn»«/math»

Estimation in percentages

Your bill in a shop is Rs 577.80 and the shopkeeper gives a discount of 15%. Estimate the amount to be paid.

  1. Round off the bill to the nearest tens of Rs 577.80, i.e., to Rs 580.
  2. Find 10% of this, i.e., Rs «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»10«/mn»«mn»100«/mn»«/mfrac»«mo»§#215;«/mo»«mn»580«/mn»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»58«/mn»«/math»
  3. Take half of this, i.e., «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mo»§#215;«/mo»«mn»58«/mn»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»29«/mn»«/math»
  4. Add the amounts in (2) and (3) to get Rs 87.

You could therefore reduce your bill amount by Rs 87 or by about Rs 85, which will be Rs 495 approximately.

Prices Related to Buying and Selling (Profit and Loss)

Sometimes when an article is bought, some additional expenses are made while buying or before selling it.

These expenses have to be included in the cost price. These expenses are sometimes referred to as overhead charges.These may include expenses like amount spent on repairs, labour charges, transportation etc.

Let's find out the method to calculate cost price/selling price, profit %, loss%

Example-4

Sohan bought a second hand refrigerator for Rs 2,500, then spent Rs 500 on its repairs and sold it for Rs 3,300. Find his loss or gain per cent.

Solution:

Cost Price (CP) = Rs 2500 + Rs 500 (overhead expenses are added to give CP) = Rs 3000

Sale Price (SP) = Rs 3300. As SP > CP, he made a profit = Rs 3300 – Rs 3000 = Rs 300

His profit on Rs 3,000, is Rs 300. 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»P«/mi»«mi»r«/mi»«mi»o«/mi»«mi»f«/mi»«mi»i«/mi»«mi»t«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mfrac»«mn»300«/mn»«mn»3000«/mn»«/mfrac»«mo»§#215;«/mo»«mn»100«/mn»«mo»%«/mo»«mo»=«/mo»«mfrac»«mn»30«/mn»«mn»3«/mn»«/mfrac»«mo»%«/mo»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mn»10«/mn»«mo»%«/mo»«/math»;  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»P«/mi»«mo»%«/mo»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mfrac»«mi»p«/mi»«mrow»«mi»c«/mi»«mi»p«/mi»«/mrow»«/mfrac»«mo»§#215;«/mo»«mn»100«/mn»«/math»

Example-5

A shopkeeper purchased 200 bulbs for Rs 10 each. However 5 bulbs were fused and had to be thrown away. The remaining were sold at Rs 12 each. Find the gain or loss %.

Solution

Cost price of 200 bulbs = Rs 200 × 10 = Rs 2000; 5 bulbs were fused. Hence, number of bulbs left = 200 – 5 = 195

These were sold at Rs 12 each. The SP of 195 bulbs = Rs 195 × 12 = Rs 2340

He obviously made a profit (as SP > CP). Profit = Rs 2340 – Rs 2000 = Rs 340

On Rs 2000, the profit is Rs 340. How much profit is made on Rs 100?

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»P«/mi»«mi»r«/mi»«mi»o«/mi»«mi»f«/mi»«mi»i«/mi»«mi»t«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mfrac»«mn»340«/mn»«mn»2000«/mn»«/mfrac»«mo»§#215;«/mo»«mn»100«/mn»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mn»17«/mn»«mo»%«/mo»«/math»

Example-6

Meenu bought two fans for Rs 1200 each. She sold one at a loss of 5% and the other at a profit of 10%. Find the selling price of each. Also find out the total profit or loss.

Solution:

Overall CP of each fan = Rs 1200. One is sold at a loss of 5%. This means if CP is Rs 100, SP is Rs 95.

Therefore, when CP is Rs 1200, then SP «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mfrac»«mn»95«/mn»«mn»100«/mn»«/mfrac»«mo»§#215;«/mo»«mn»1200«/mn»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»1140«/mn»«/math»

Also second fan is sold at a profit of 10%. It means, if CP is Rs 100, SP is Rs 110.

Therefore, when CP is Rs 1200, then SP  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mfrac»«mn»110«/mn»«mn»100«/mn»«/mfrac»«mo»§#215;«/mo»«mn»1200«/mn»«mo»=«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»1320«/mn»«/math»

Total CP = Rs 1200 + Rs 1200 = Rs 2400. Total SP = Rs 1140 + Rs 1320 = Rs 2460

Since total SP > total CP, a profit of Rs (2460 – 2400) or Rs 60 has been made.

Sales Tax / Value Added Tax

ST means Sales Tax, which we pay when we buy items.This sales tax is charged by the government on the sale of an item. It is collected by the shopkeeper from the customer and given to the government. This is, therefore, always on the selling price of an item and is added to the value of the bill. These days however, the prices include the tax known as Value Added Tax (VAT).

The teacher showed the class a bill in which the following heads were written.

Example-7

(Finding Sales Tax) The cost of a pair of roller skates at a shop was Rs 450. The sales tax charged was 5%. Find the bill amount.

Solution:

On Rs 100, the tax paid was Rs 5.

On Rs 450, the tax paid would be  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mfrac»«mn»5«/mn»«mn»100«/mn»«/mfrac»«mo»§#215;«/mo»«mn»450«/mn»«/math»«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mo»§nbsp;«/mo»«mi»R«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mn»22«/mn»«mo».«/mo»«mn»50«/mn»«/math»

Bill amount = Cost of item + Sales tax = Rs 450 + Rs 22.50 = Rs 472.50.

Example-8

(Value Added Tax (VAT)) Waheeda bought an air cooler for Rs 3300 including a tax of 10%. Find the price of the air cooler before VAT was added.

Solution:

The price includes the VAT, i.e., the value added tax. Thus, a 10% VAT means if the price without VAT is Rs 100 then price including VAT is Rs 110. Now, when price including VAT is Rs 110, original price is Rs 100.

Hence when price including tax is Rs 3300, the original price  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mi»R«/mi»«mi»s«/mi»«mo».«/mo»«mo»§nbsp;«/mo»«mfrac»«mn»100«/mn»«mn»110«/mn»«/mfrac»«mo»§#215;«/mo»«mn»3300«/mn»«mo»§nbsp;«/mo»«mo»=«/mo»«mi»R«/mi»«mi»s«/mi»«mo».«/mo»«mo»§nbsp;«/mo»«mn»3000«/mn»«/math»

Let's summarize what we learned 

  • Discount is a reduction given on marked price.

  • Discount = Marked Price – Sale Price.

  • Discount can be calculated when discount percentage is given.

  • Discount = Discount % of Marked Price

  • Additional expenses made after buying an article are included in the cost price and are known

  • as overhead expenses.

  • CP = Buying price + Overhead expenses

  • Sales tax is charged on the sale of an item by the government and is added to the Bill Amount.

  • Sales tax = Tax% of Bill Amount

 

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