Informative line

# Percent Decrease

• Percent decrease refers to the ratio or proportion of the amount of decrease to the original amount/quantity.
• We can find the percent decrease by using the following steps:

Step 1: Find the amount of decrease by subtracting the new amount from the original amount.

Amount of decrease = Original Amount – New Amount

Step 2: Write a fraction in which, take the amount of decrease as the numerator and the original amount as the denominator.

$$\text {Percent Decrease}=\dfrac {\text{Amount of decrease}}{\text {Original Amount}}$$

Step 3: Write an equivalent fraction with $$x$$ as the numerator and $$100$$ as the denominator.

$$\dfrac {\text{Amount of decrease}}{\text{Original amount}}=\dfrac {x}{100}$$

Step 4: Solve the equation using cross multiplication method.

Example: In the month of June, the average temperature of a city was $$32°C$$ while in October, it dropped to $$16°C$$. What is the percent decrease in temperature?

Average temperature in June $$=32°C$$

Average temperature in October $$=16°C$$

• Difference / Decrease in temperature

$$=32°C-16°C$$

$$=16°C$$

• $$\text {Percent Decrease}=\dfrac {\text{Difference in temperature}}{\text {Original temperature}}$$

$$=\dfrac {16°C}{32°C}$$

$$=\dfrac {1}{2}$$

Write an equivalent fraction with $$x$$ as the numerator and $$100$$ as the denominator.

$$\dfrac {\text{Amount of decrease}}{\text{Original amount}}=\dfrac {x}{100}$$

$$\Rightarrow\dfrac {1}{2}=\dfrac {x}{100}$$

Solve by using cross multiplication method:  $$\Rightarrow\;2x=1×100$$

$$\Rightarrow\;2x=100$$

$$\Rightarrow\;x=\dfrac {100}{2}$$

$$\Rightarrow\;x=50$$

Thus, the temperature decreases by $$50\%$$ in October.

#### The original cost of a dress was $$150$$. If it is now on sale for $$120$$, what is the percent decrease in cost?

A $$20\%$$

B $$30\%$$

C $$40\%$$

D $$120\%$$

×

Given:

Original cost of dress $$=150$$

Sale Price of dress $$=120$$

Difference between the amount of original price and sale price

$$=150-120$$

$$=30$$

Amount of decrease in price $$=30$$

Original Price $$=150$$

$$\text {Percent decrease} = \dfrac {\text{Amount of Decrease}}{\text {Original Price}}$$

$$\Rightarrow\text {Percent decrease} = \dfrac{30}{150}$$ or $$\dfrac {1}{5}$$

Converting the fraction into a percent by using proportion.

$$\Rightarrow\dfrac {1}{5}=\dfrac {x}{100}$$

Solve by using the cross multiplication method: $$\Rightarrow5x=1×100$$

$$\Rightarrow5x=100$$

$$\Rightarrow x=\dfrac {100}{5}$$

$$\Rightarrow x=20$$

Thus, the cost of dress decreases by $$20\%$$.

Hence, option (A) is correct.

### The original cost of a dress was $$150$$. If it is now on sale for $$120$$, what is the percent decrease in cost?

A

$$20\%$$

.

B

$$30\%$$

C

$$40\%$$

D

$$120\%$$

Option A is Correct

# Finding New Amount Using Percent of Change

• If percent increase or percent decrease and the original amount are given, then we can find the new amount.
• Find the new amount by using  the following formula:

Amount of Change = Percent of Change × Original Amount

To calculate the new amount:

(i) In case of increase

New Amount = Original Amount + Amount of Change

(ii) In case of decrease

New Amount = Original Amount – Amount of Change

Let's consider an example to understand how to calculate the new amount using percent of change.

Example: Carl got $$30$$ marks in Physics. After revaluation, his marks increased by $$10\%$$, now what are his marks in Physics after revaluation?

Change in Marks = Percent of Change × Original Marks

$$=10\%×30$$

$$=\dfrac {10}{100}×30$$

$$=0.1×30$$

$$=3$$

Since there is an increase, so add the change in marks to the original marks to calculate the revalued marks.

Thus, Carl now scored

$$=30+3=33$$  marks

#### Alex bought a pair of shoes for $$40$$. His friend bought the same pair of shoes at $$20\%$$ less than that of Alex. Calculate the cost price of Alex's friend's shoes.

A $$34$$

B $$32$$

C $$36$$

D $$38$$

×

Given:

Cost of Alex's Shoes $$=40$$

Now, calculating the cost of Alex's friend's shoes.

Amount of Change = Percent of Change × Original Amount

$$=20\%×40$$

Amount of change $$=20\%×40$$

$$=0.20×40$$

$$=8$$

Since there is a decrease, so subtract the amount of change from the price and calculate the cost price of Alex's friend's shoes.

Thus, the cost price of Alex's friend's shoes

$$=40-8$$

$$=32$$

Hence, option (B) is correct.

### Alex bought a pair of shoes for $$40$$. His friend bought the same pair of shoes at $$20\%$$ less than that of Alex. Calculate the cost price of Alex's friend's shoes.

A

$$34$$

.

B

$$32$$

C

$$36$$

D

$$38$$

Option B is Correct

# Finding New Amount using Percent of Change-II

If the percent of change is above $$100\%$$ then we calculate the new amount by using the following formula:
New Amount = Percent of Change × Original Amount

• Let's consider an example to understand how to calculate new amount using the percent of change.

From a sports store, Alex bought a football for $$25$$. His friend bought a similar football from a different store for $$140\%$$ of what Alex paid.

How much money did Alex's friend spend?

New Amount = Percent of Change × Original Amount

$$=140\%×25$$

$$=\dfrac {140}{100}×25$$

$$=1.40×25$$

$$=35$$

Hence, Alex's friend spent $$35$$ on the football.

#### Two friends, Aron and Keith work in the same company. Aron is getting $$3,500$$ per month while Keith earns $$120\%$$ of what Aron earns. Calculate how much Keith earns.

A $$3,620$$

B $$600$$

C $$4,200$$

D $$4,000$$

×

Given:

Total income of Aron $$=3,500$$ per month

Calculating the income of Keith,

New Amount = Percent of Change × Original Amount

$$=120\%×3,500$$

$$=\dfrac {120}{100}×3,500$$

$$=1.20×3,500$$

$$=4,200$$

Thus, the income of Keith is $$4,200$$.

Hence, option (C) is correct.

### Two friends, Aron and Keith work in the same company. Aron is getting $$3,500$$ per month while Keith earns $$120\%$$ of what Aron earns. Calculate how much Keith earns.

A

$$3,620$$

.

B

$$600$$

C

$$4,200$$

D

$$4,000$$

Option C is Correct

# Percent Increase

• Percent increase refers to the ratio of the amount of increase in a quantity to the original amount of quantity.
• We can find the percent increase by using the following steps:

Step 1: Find the amount of increase by subtracting the original amount from the new amount.

Amount of increase = New Amount – Original Amount

Step 2: Write a fraction in which, take the amount of increase as the numerator and the original amount as the denominator.

$$\text {Percent Increase}=\dfrac {\text{Amount of Increase}}{\text{Original Amount}}$$

Step 3: Write an equivalent fraction with $$x$$ as the numerator and $$100$$ as the denominator.

$$\dfrac {\text{Amount of Increase}}{\text{Original amount}}=\dfrac {x}{100}$$

Step 4: Solve the equation using cross multiplication method.

Example:  The cost of each pen increases from $$0.15$$ to $$0.22$$. Find what percent increase is there in the cost of a single pen?

• The original cost of a pen $$=0.15$$

The cost increases to $$0.22$$

• Difference / Increase in cost $$=0.22-0.15$$

$$=0.07$$

•  $$\text {Percent Increase}=\dfrac {\text{Difference in Cost}}{\text{Original Cost}}$$

$$\dfrac {0.07}{0.15}=\dfrac {7}{15}$$

Write an equivalent fraction with $$x$$ as the numerator and $$100$$ as the denominator.

$$\dfrac {\text{Difference in Cost}}{\text{Original Cost}}=\dfrac {x}{100}$$

Solve by using the cross multiplication method:

$$\Rightarrow\;15x=7×100$$

$$\Rightarrow\;15x=700$$

$$\Rightarrow\;x=\dfrac {700}{15}$$

$$\Rightarrow\;x=46.6\%$$

Thus, the cost of a pen increases by $$46.6\%$$.  #### In the first year, Sam saves $$5,100$$. Next year he saves  $$5,406$$. By what percent has Sam's savings increased in the second year?

A $$5\%$$

B $$8\%$$

C $$7\%$$

D $$6\%$$

×

Given:

Total savings in the first year $$=5,100$$

Total savings in the second year $$=5,406$$

Difference between the savings of $$2$$ years

$$=5406-5100$$

$$=306$$

Increase in savings (Amount of Increase) $$=306$$

Savings in the first year (Original Amount) $$=5,100$$

$$\text {Percent Increase}=\dfrac {\text{Amount of increase}}{\text{Original Amount}}$$

$$\Rightarrow\text {Percent Increase}=\dfrac {306}{5,100}$$ or $$\dfrac {51}{850}$$

Now, converting the fraction into a percent by using proportion,

$$\dfrac {51}{850}=\dfrac {x}{100}$$

Solve by using the cross multiplication method: $$\Rightarrow850x=51×100$$

$$\Rightarrow850x=5100$$

$$\Rightarrow x=\dfrac {5100}{850}$$

$$\Rightarrow x=6$$

Thus, the savings have increased by $$6\%$$.

Hence, option (D) is correct.

### In the first year, Sam saves $$5,100$$. Next year he saves  $$5,406$$. By what percent has Sam's savings increased in the second year?

A

$$5\%$$

.

B

$$8\%$$

C

$$7\%$$

D

$$6\%$$

Option D is Correct

# Circle Graphs (Interpretation)

• A circle graph is a way of displaying the data.
• A circle represents the whole i.e. $$100\%$$.
• It is divided into parts known as sections.
• Each section looks like a pie-shaped wedge.
• Each pie-shaped wedge represents a percent of the whole.

Example:

The given circle graph represents the percentage of the population in a state under different age groups.  A percent represents the part of a whole and a circle represents the $$100\%$$.

The given circle graph is divided into $$5$$ parts known as sections. On observing the above circle graph, we have:

Population of $$0-20$$  age group $$=30\%$$

Population of $$20-40$$  age group $$=33\%$$

Population of $$40-60$$  age group $$=24\%$$

Population of $$60-70$$  age group $$=7\%$$

Population of $$70$$ and above  age group $$=6\%$$

We can observe -

• The highest percentage is $$33\%$$  which is the age group of $$20-40$$ years.
• The least percentage is $$6\%$$ which is the age group of $$70$$ and above.
• The percentage of $$40-60$$ age group is $$4$$ times of the percentage of $$70$$ and above, i.e. $$24=4×6$$
• The percentage of $$0-20$$ age group is $$5$$ times of the percentage of $$70$$ and above, i.e. $$30=5×6$$
• The percentage of $$40-60$$ age group is higher than the percentage of $$60-70$$ age group.

#### The circle graph represents the percentage of students having different weights in a class. Which statement is FALSE? (Weight is in pounds)

A The percentage of students belonging to $$68-70$$ pounds weight category is twice the percentage of students belonging to $$66-68$$ pounds weight category.

B The percentage of students belonging to $$72-74$$ pounds weight category is $$13\%$$.

C The percentage of students belonging to $$64-66$$ pounds weight category is half the percentage of students belonging to  $$70-72$$ pounds weight category.

D The percentage of students belonging to $$64-66$$ pounds weight category is $$9\%$$.

×

A percent represents the part of a whole and a circle represents the $$100\%$$.

Option (A):

We can observe the following from the circle graph:

Percentage of students belonging  to $$68-70$$ pounds category$$=32\%$$

Percentage of students belonging to $$66-68$$ pounds category $$=16\%$$

Since, $$32=2×16$$

Hence, option (A) is true.

Option (B):

According to the circle graph,

The percentage of students belonging to $$72-74$$ pounds category is $$13\%$$.

Hence, option (B) is true.

Option (C):

The percentage of students belonging to $$64-66$$ pounds weight category $$=10\%$$

The percentage of students belonging to $$70-72$$ pounds weight category $$=20\%$$

Since, $$10\%=\dfrac {20\%}{2}$$

Hence, option (C) is true.

Option (D):

The percentage of students belonging to $$64-66$$ pounds weight category is $$10\%$$.

Hence, option (D) is FALSE.

### The circle graph represents the percentage of students having different weights in a class. Which statement is FALSE? (Weight is in pounds) A

The percentage of students belonging to $$68-70$$ pounds weight category is twice the percentage of students belonging to $$66-68$$ pounds weight category.

.

B

The percentage of students belonging to $$72-74$$ pounds weight category is $$13\%$$.

C

The percentage of students belonging to $$64-66$$ pounds weight category is half the percentage of students belonging to  $$70-72$$ pounds weight category.

D

The percentage of students belonging to $$64-66$$ pounds weight category is $$9\%$$.

Option D is Correct

# Percent through Circle Graph-II

• A circle graph is a way of displaying data.
• A full circle represents the $$100\text{%}$$.
• It is divided into a number of sections, known as pie shaped wedges.
• Each wedge represents a percent of the whole.

Let us consider the following example:

Here, the circle represents the percentage of students that play different sports, at a school.

The total number of students is $$200$$.

Now the question is....... "How many students play baseball, cricket and basketball?"

We can calculate it by using the concept of "Percent of a number".

Thus, the number of students who play

Baseball = $$50\text {% of }200$$

$$=.50×200=100$$

Cricket = $$20\text {% of }200$$

$$=.20×200=40$$

Basketball = $$30\text {% of }200$$

$$=.30×200=60$$

This means,  $$100$$ students play Baseball, $$40$$ students play Cricket, and $$60$$ students play Basketball.  #### James starts a business of fruit selling. The circle graph represents the percentage of different types of fruits he has for selling. If there are a total of $$600$$ fruits to sell, find how many apples are there?

A $$146$$

B $$132$$

C $$360$$

D $$115$$

×

Total number of fruits $$=600$$

From the circle graph,

Apples $$=22\text {%}$$

Number of apples $$=22\text {% of }\,600$$

$$=.22×600$$

$$=132$$

Hence, option (B) is correct.

### James starts a business of fruit selling. The circle graph represents the percentage of different types of fruits he has for selling. If there are a total of $$600$$ fruits to sell, find how many apples are there? A

$$146$$

.

B

$$132$$

C

$$360$$

D

$$115$$

Option B is Correct