Informative line

# Finding Prices Involving Discounts

• We come across the term "discount" many times when we go for shopping.
• It is an offer to the customers to pay less than the original price.
• Discount is the amount of money which is taken away or not to be paid from the original price.
• It is usually in the form of a percent.
• The word "off" is also used for the discount.

Example: $$10\text{%}$$ discount on clothing.

• The discount offer is a marketing strategy that promotes business.
• It induces the customers to shop and helps the business to increase its sales and clear the old stock.

To understand it in a better way, consider an example:

Mr. Watson is running a restaurant. It is well known for its burgers. Today, there are a lot of customers as he offers $$5\text{%}$$ discount on the original price of the burgers.

The price of one burger is $$8.00$$. If Jessica buys one burger, what is the amount of discount and how much will she pay for a burger?

To find the answers to these questions, follow the procedure discussed below:

Step 1: Calculate the amount of discount.

Original price of one burger $$=8.00$$

Discount $$=5\text {%}$$

Amount of discount $$=5\text {%}$$ of the original price

$$=5\text {%}$$ of $$8.00$$ ('of' means multiply)

$$=5\text {%}×8.00$$ ( $$5\text {%}=0.05$$)

$$=0.05×8.00$$

$$\begin{array} {r} 8.00 \\ ×0.05 \\ \hline 0.40 \\ \hline \end{array}$$

Amount of discount $$=0.40$$

Step 2: Subtract the amount of discount from the original price.

$$8.00-0.40=7.60$$

Thus, the amount Jessica will pay for a burger $$=7.60$$

#### Ana goes to a bakery to buy a packet of muffins. The bakery is offering a sale of $$10\text {%}$$ off on each packet of muffins. She buys one packet. If the original price of a packet is $$6.50$$, how much money will she pay for it?

A $$6.25$$

B $$7.25$$

C $$5.85$$

D $$4.65$$

×

"Sale of $$10\,\text {%}$$ off on the original price" means there is a discount of $$10\,\text {%}$$.

Original price of a packet of muffins $$=6.50$$

The amount of discount $$=10\,\text {%}$$ of the original price

$$=10\,\text {%}$$ of $$6.50$$ ('of' means multiply)

$$=10\,\text {%}×6.50$$ ( $$10\,\text {%}=0.10$$ )

$$=0.10×6.50$$

$$\begin {array} {r} 6.50 \\ ×0.10 \\ \hline 0.65 \\ \hline \end {array}$$

Discount $$=0.65$$

Ana will pay = Original Price – Amount of discount

$$=6.50-0.65$$

$$=5.85$$

Hence, option (C) is correct.

### Ana goes to a bakery to buy a packet of muffins. The bakery is offering a sale of $$10\text {%}$$ off on each packet of muffins. She buys one packet. If the original price of a packet is $$6.50$$, how much money will she pay for it?

A

$$6.25$$

.

B

$$7.25$$

C

$$5.85$$

D

$$4.65$$

Option C is Correct

# Calculation of Prices Involving Sales Tax

• Sales tax should be calculated on the original price.
• It is the opposite of discount.
• In discount, we subtract an amount of money from the original price while in sales tax, we add an amount of money to the original price.
• In every country, almost each state levies the sales tax.
• If we purchase something, we have to pay an additional amount of money along with the original price as sales tax.
• It is in the form of a percent.

To understand it in a better way, consider an example:

Alex goes to a restaurant to buy a large cheese pizza. The menu shows the price of a pizza $$7.50$$ excluding the sales tax $$4\text{%}$$. What will be the total cost?

For getting the answer, we should first calculate the sales tax.

Original price of a pizza $$=7.50$$

Sales tax on pizza $$=4\text{%}$$ of $$7.50$$ ($$4\text{%}=0.04$$ )

$$=0.04\text{ of }7.50$$ ( 'of' means multiply)

$$=0.04×7.50$$

$$\begin {array} {r} 7.50 \\ ×0.04 \\ \hline 0.3000 \\ \hline \end{array}$$

Sales tax on the pizza $$=0.30$$

Total amount of bill = Original price + Sales tax

$$=7.50+0.30$$

$$=7.80$$

Thus, he pays $$7.80$$ for the pizza.

#### How much would Alex pay for a pair of shoes costing $$35.00$$, if the sales tax is $$5\text{%}$$?

A $$2.40$$

B $$36.75$$

C $$144$$

D $$22$$

×

Given: The original price of the pair of shoes $$=35.00$$

Sales tax = $$5\text{%}$$

Sales tax is calculated on the original price.

$$\therefore$$ Sales tax $$=5\text{% of }35.00$$ ($$5\text{%}=0.05$$)

$$=0.05\text{ of }35.00$$ ( 'of' means multiply)

$$=0.05×35.00$$

$$\begin {array} {r} 35.00 \\ ×0.05 \\ \hline 01.7500 \\ \hline \end {array}$$

$$=1.75$$

Sales tax on shoes $$=1.75$$

Total bill = Original price + Sales tax

$$=35.00+1.75$$

$$=36.75$$

Thus, he will pay $$36.75$$ for the shoes.

Hence, option (B) is correct.

### How much would Alex pay for a pair of shoes costing $$35.00$$, if the sales tax is $$5\text{%}$$?

A

$$2.40$$

.

B

$$36.75$$

C

$$144$$

D

$$22$$

Option B is Correct

# Percent using Double Number line Diagram

• A percent is a part of a whole, where the whole represents $$100$$.
• We can also represent a percent by using double number line diagram.

To understand it in a better way, let us consider an example:

• Casey has $$30$$ chocolates. She eats $$6$$ of them.

"What percent of chocolates did she eat?"

• The solution of the above problem can be determined by using a double number line diagram.
• The situation is represented on a double number line diagram as shown.

Out of $$30$$ chocolates, Casey eats $$6$$ chocolates.

$$\therefore$$ We should divide $$30$$ by $$5$$ to get $$6$$, i.e.

$$30\div5=6$$

Thus, $$100\text{%}$$ also gets divided by $$5$$, i.e.

$$100\text{%}\div5=20\text{%}$$

On observing the number line diagram, we can say that Casey eats $$20\text{%}$$ chocolates.

#### In the following number line diagram, which value is represented by K?

A $$75\text{%}$$

B $$35\text{%}$$

C $$59\text{%}$$

D $$65\text{%}$$

×

By observing the number line diagram, we get to know that K is placed below $$39$$.

To get $$39$$, we should multiply $$13$$ by $$3$$, i.e.

$$13×3=39$$

Thus, $$25\text{%}$$ will also get multiplied by $$3$$, i.e.

$$25\text{%}×3=75\text{%}$$

Thus, the value of $$K$$ is $$75\text{%}$$.

Hence, option (A) is correct.

### In the following number line diagram, which value is represented by K?

A

$$75\text{%}$$

.

B

$$35\text{%}$$

C

$$59\text{%}$$

D

$$65\text{%}$$

Option A is Correct

# Percent using Tape Diagram

• A percent is a part of a whole, where the whole represents $$100$$.
• To understand percent in a visual way, we can represent it through a tape diagram.

Let us consider an example:

There are $$40$$ students who participated in a school event. $$2$$ students won gold medal, $$3$$ students won silver medal and  $$5$$ students won bronze medal.

Now, the question is,

"What percent of the total students got medals?"

We will calculate it using a tape diagram.

Total number of students $$=40$$

Total number of students who got medals $$=2+3+5$$

$$=10$$

$$10$$ out of $$40$$ students got medals.

We can easily find out the percent of the students who got medals using tape diagram method:

Step 1: Draw a tape diagram and divide it into segments.

Here, each section represents $$10$$ students.

Step 2: Consider the whole as $$100\text{%}$$ and divide it into equal number of parts.

We have $$4$$ sections.

$$\therefore$$ $$100\text{%}$$ should also be divided into $$4$$ parts, i.e.

$$\dfrac {100\text{%}}{4}=25\text{%}$$

Here, each section represents $$25\text{%}$$.

Therefore, we can say that $$25\text{%}$$ of $$40\text{ is } 10$$.

Thus, $$25\text{%}$$ of the total students got medals.

$$\because$$ We have $$4$$ sections

$$\therefore$$ $$100\text{%}$$ should also be divided into $$4$$ parts, i.e.,

$$\dfrac {100\text{%}}{4}=25\text{%}$$

Here, each section represents $$25\text{%}$$.

Therefore, we can say that $$25\text{%}$$ of $$40\text{ is } 10$$.

Thus, $$25\text{%}$$ of the total students got medals.

#### The given tape diagram represents $$5$$ rooms in Mrs. Anderson's house. She is brooming the rooms. If she has broomed $$2$$ rooms, what percent of the total rooms did she broom?

A $$25\text{%}$$

B $$40\text{%}$$

C $$32\text{%}$$

D $$20\text{%}$$

×

Given: Total number of rooms $$=5$$

Number of rooms she broomed $$=2$$

She broomed $$2$$ out of $$5$$ rooms.

By using the given tape diagram, we can find out the percent of the total rooms she broomed.

Here, each section represents $$1$$ room.

Thus, it is clear from the tape diagram that $$40\text{%}$$ of $$5$$ is $$2$$.

$$\therefore$$ She broomed $$40\text{%}$$ of the total rooms.

Hence, option (B) is correct.

### The given tape diagram represents $$5$$ rooms in Mrs. Anderson's house. She is brooming the rooms. If she has broomed $$2$$ rooms, what percent of the total rooms did she broom?

A

$$25\text{%}$$

.

B

$$40\text{%}$$

C

$$32\text{%}$$

D

$$20\text{%}$$

Option B is Correct

# Simple Interest

When we borrow an amount of money from a bank or a money lender for a certain period of time, we have to pay back some extra money along with the borrowed amount.

The extra money paid over and above the original amount is called simple interest.

To calculate it, we have to understand the following three terms:

Principal (P):  Amount of money borrowed/ loan amount

Rate (r): Rate of interest per year, generally in percentage form

Time(t): Time for which money is borrowed

Thus, Interest $$(I)=P×r×t$$

$$=P\,r\,t$$

For example:  Mr. Craven wants to buy a car.

He takes a loan of $$10,000.00$$ from a bank. The bank charges $$9\text{%}$$ interest per year for $$10,000.00$$. Here, the question arises,

"How much interest will he pay after $$4$$ years?"

To determine the answer, we should calculate the interest, i.e.

$$I=P\,r\,t$$

where

$$P=$$ Principal

$$r=$$ Rate percent

$$t=$$ Time taken

$$I=$$ Interest

In the above example:

Principal = $$10,000.00$$

Rate = $$9\text{%}$$

Time = 4 years

$$I=10,000.00×9\text{%}×4$$ years ($$\because\;9\text{%}=0.09$$)

$$I=10000×0.09×4$$

$$=40000×0.09$$

$$\begin{array} {r} 40000 \\ ×0.09 \\ \hline 3600.00\\ \hline \end{array}$$

$$I=3,600.00$$

Thus, Mr. Craven will pay $$3,600$$ as interest.

#### Mr. Wilson wants to buy a house. He takes a loan of $$8,500$$ from a bank. If he has to return $$8\text{%}$$ annually to the bank, how much money will he pay as simple interest in $$3$$ years?

A $$2,040.00$$

B $$1,860.00$$

C $$3,160.00$$

D $$1,780.00$$

×

$$I=P\,r\,t$$

where

$$P=$$ Principal

$$r=$$ Rate percent

$$t=$$ Time taken

$$I=$$ Interest

Principal is the amount borrowed.

Principal (P) = $$8,500.00$$

Rate (r) = $$8\text{%}$$ per year

Time (t) = 3 years

Putting values in

$$I=P\,r\,t$$

$$I=8,500.00×8\text{%}×3$$ years ($$\because\;8\text{%}=0.08$$)

$$I=8500×0.08×3$$

$$=25500×0.08$$

$$\begin{array} {r} 25500 \\ ×0.08 \\ \hline 2040.00\\ \hline \end{array}$$

$$I=2,040.00$$

Hence, option (A) is correct.

### Mr. Wilson wants to buy a house. He takes a loan of $$8,500$$ from a bank. If he has to return $$8\text{%}$$ annually to the bank, how much money will he pay as simple interest in $$3$$ years?

A

$$2,040.00$$

.

B

$$1,860.00$$

C

$$3,160.00$$

D

$$1,780.00$$

Option A is Correct

# Tip Estimation

• Tip is some part of the cost of the meal or other charges in a restaurant or in a hotel.
• Tip is given to the server for his services. It is also called gratuity.
• Generally, it is about $$15\text{%}$$ of the total cost of the bill.

Note:  If the percent of the tip is not given in the problem, we can take it as $$15\text{%}$$ of the total bill.

To understand it in a better way, consider an example:

Jessica and Emma went for dinner in a restaurant. The bill for food was $$25.00$$. They gave $$15\text{%}$$ of the bill as tip to the waiter. What was the estimated amount of tip paid by them?

Here,

Total bill $$=25.00$$

Percent of tip $$=15\text{%}$$

Step 1: Tip $$=15\text{%}$$ of the total bill

$$\text{Tip}=15\text{% of }25.00$$

Step 2: First, we find $$10\text{%}$$ and $$20\text{%}$$ of the total bill because $$15\text{%}$$ lies between $$10\text{%}$$ and $$20\text{%}$$ as finding $$10\text{%}$$ and $$20\text{%}$$ of a number is easy.

Thus, $$10\text{% of }25.00=2.50$$

$$20\text{% of }25.00=2×10\text{% of }25.00$$

$$=2×2.50=5.00$$

Step 3:

$$15\text{%}$$ is between $$10\text{%}$$ and $$20\text{%}$$.

$$\therefore$$ The tip lies between $$2.50$$ and $$5.00$$.

Step 4:

So, we take the estimated tip as $$3.75$$.

#### Mr. Harris stayed in a hotel for two nights. While checking out, he gave tip to the room service man which is approximately $$15\text{%}$$ of the bill. If the bill was $$252.30$$, how much estimated money did he pay to the room service man?

A $$35.27$$

B $$67.25$$

C $$24.05$$

D None of these

×

Given:

Total bill $$=252.30$$

Tip $$=15\text{% of } 252.30$$

Here, Tip $$=15\text{% of } 252.30$$

First, we calculate $$10\text{%}$$ of $$252.30$$ by moving the decimal point one place to the left.

$$10\text{% of } 252.30=25.23$$

Now, to calculate $$20\text{% of } 252.30$$, multiply the $$10\text{% of } 252.30$$ by $$2$$.

$$20\text{% of } 252.30=2×10\text{% of } 252.30$$

$$=2×25.23$$

$$=50.46$$

$$15\text{%}$$ lies between $$10\text{%}$$ and $$20\text{%}$$.

$$\therefore$$ The tip is between $$25.23$$ and $$50.46$$ .

In the given options, only option (A) is correct because $$35.27$$ is between $$25.23$$ and $$50.46$$.

Hence, option (A) is correct.

### Mr. Harris stayed in a hotel for two nights. While checking out, he gave tip to the room service man which is approximately $$15\text{%}$$ of the bill. If the bill was $$252.30$$, how much estimated money did he pay to the room service man?

A

$$35.27$$

.

B

$$67.25$$

C

$$24.05$$

D

None of these

Option A is Correct