Informative line

# Mathematical Modeling on Addition and Subtraction

Many times in day to day life we need to deal with more than one operation. Only addition or subtraction does not help, and we need to make use of a combination of operations in order to solve the problem.

For example:

• Tyler buys one box of chalk costing $$4$$ and one pack of pens costing $$2$$. He pays $$10$$. How much money does he get back?
• In the given example Tyler paid for two items:

one box of chalk $$=4$$

and one pack of pens $$=2$$

• To calculate total cost of two items we will add the cost of both the items.

Total cost $$=4+2=6$$

But, Tyler paid  $$10$$.

• In order to calculate how much money he gets back, we will subtract the total cost of items from the amount he paid.

$$\Rightarrow\;10-6=4$$

• Thus, Tyler gets $$4$$ back.

#### Hayden bought three smartphones for $$85,\;90$$ and $$87$$, including tax. He gave $$300$$ to the shopkeeper. How much money does he get back?

A $$262$$

B $$38$$

C $$300$$

D $$100$$

×

Hayden bought three smartphones for $$85,\;90$$ and $$87$$.

To calculate the total cost of three smartphones, we will add the costs of all three.

Total cost $$=85+90+87=262$$

But, Hayden gave $$300$$.

In order to calculate how much money Hayden gets back, we will subtract the total cost of smartphones from the amount he paid.

$$=300-262$$

$$=38$$

$$\therefore$$ Hayden gets $$38$$ back.

Hence, option (B) is correct.

### Hayden bought three smartphones for $$85,\;90$$ and $$87$$, including tax. He gave $$300$$ to the shopkeeper. How much money does he get back?

A

$$262$$

.

B

$$38$$

C

$$300$$

D

$$100$$

Option B is Correct

# Mathematical Modeling on Addition and Multiplication

Many times in real life problems we need to deal with more than one operations. Only addition or only multiplication would not help. We need to make use of the combination of operations in order to solve the problem.

For example:

• Amy needs to purchase $$8$$ notebooks at $$2$$ each and $$7$$ sketchpads at $$1$$ each. About how much will she spend?
• In the given example cost of one notebook is $$2$$ and cost of one sketchpad is $$1$$.
• We need to find out costs of $$8$$ notebooks and $$7$$ sketchpads.
• In order to calculate the above, we will multiply the cost of one notebook by $$8$$

$$=8×2$$

$$=16$$

and cost of one sketchpad by $$7$$

$$=7×1$$

$$=7$$

• To calculate how much she spends, we will add the total cost of $$8$$ notebooks and $$7$$ sketchpads.

$$=16+7$$

$$=23$$

• Thus, Amy spends $$23$$.

#### Samuel earns $$9$$ per hour for his part-time job. He worked for $$25$$ hours and $$22$$ hours for first and second week respectively. What is the total amount that Samuel earned for the two weeks?

A $$198$$

B $$225$$

C $$413$$

D $$423$$

×

Samuel earns $$9$$ per hour.

First week he worked for $$25$$ hours.

To calculate total money Samuel earned for first week, we will multiply money per hour by hours he worked.

$$=25×9$$

$$=225$$

Second week he worked for $$22$$ hours.

So, we will multiply money per hour by hours he worked.

$$=22×9$$

$$=198$$

Samuel earned $$225$$ in first week and $$198$$ in second week.

In order to calculate total amount of money he earned for working these two weeks, we will add the total money of first week and second week.

$$=225+198=423$$

So, Samuel earned $$\,423$$ for the two weeks.

Hence, option (D) is correct.

### Samuel earns $$9$$ per hour for his part-time job. He worked for $$25$$ hours and $$22$$ hours for first and second week respectively. What is the total amount that Samuel earned for the two weeks?

A

$$198$$

.

B

$$225$$

C

$$413$$

D

$$423$$

Option D is Correct

# Mathematical Modeling on Subtraction and Multiplication

Many times in everyday problems we need to deal with more than one operation. Only subtraction or multiplication does not help. We need to make use of the combination of operations in order to solve the problem.

For example:

• Roger wants to buy a computer that costs $$575$$, including tax. He has been saving $$20$$ each week for the past $$12$$ weeks. How much more money does Roger need to purchase the computer?
• In the given example, each week Roger saved $$20$$. He has been saving for the past $$12$$ weeks. To calculate total amount of money he saved in $$12$$ weeks, we will multiply $$20$$ by $$12$$.

$$20×12$$

$$=240$$

He saved $$240$$ in $$12$$ weeks.

• To calculate how much more money he needs, we will subtract the money he saved from the total cost of the computer.

$$575-240$$

$$=335$$

Thus, Roger needs $$335$$ more to purchase the computer.

#### Look at the table: Dresses  Cost Per Dress Red $$\,50$$ Black $$\,57$$ How much more would a person pay for $$5$$ black dresses than $$5$$ red dresses?

A $$50$$

B $$57$$

C $$35$$

D $$107$$

×

Given:

Cost of a red dress $$=50$$

Cost of a black dress $$=57$$

To calculate, how much more money is required for one black dress than one red dress, we have to subtract the cost of a red dress from the cost of a black dress.

$$57-50=7$$

$$\therefore$$ The person pays $$7$$ more for one black dress than one red dress.

Now, for $$5$$ black dresses, he will pay

$$(5×7)$$$$=35$$ more

A person pays $$35$$ more for $$5$$ black dresses than  $$5$$ red dresses.

Hence, option (C) is correct.

### Look at the table: Dresses  Cost Per Dress Red $$\,50$$ Black $$\,57$$ How much more would a person pay for $$5$$ black dresses than $$5$$ red dresses?

A

$$50$$

.

B

$$57$$

C

$$35$$

D

$$107$$

Option C is Correct

# Mathematical Modeling on Addition and Division

Many times in everyday problems we need to deal with more than one operation. Only addition or division does not help. We need to make use of a combination of operations in order to solve the problem.

For example:

• Three friends are planning the college function. The music band will charge $$180$$. The food will cost $$300$$ and $$110$$ will be needed for decoration. $$295$$ students have signed up for the dance. Out of the following amounts, how much money can they charge from each of the students to cover all the expenses?
• In the given example, to calculate total charges for college function, we will add the charges of band, food and decoration.

$$180+300+110$$ $$=590$$

Total expenditure on the college function is $$590$$.

To calculate how much money can they charge from each student, we will divide the expenditure by the total number of students,

$$590\div295$$$$=2$$

They will charge $$2$$ from each of the students.

#### In summer, Sarah earned $$150$$ in the first week and $$140$$ in the second week. She equally shared the total amount earned by her among her two brothers. How much money does each brother get?

A $$500$$

B $$290$$

C $$145$$

D $$150$$

×

In the given problem:

Amount earned in first week $$=150$$

Amount earned in second week $$=140$$

To calculate total money earned by her, add the amount earned in the first and second weeks.

$$150+140=290$$

Total amount earned by her $$=290$$

Now, we have to find the total amount of money that each brother gets.

We have to divide the total amount by $$2$$, for finding the share of each brother.

$$290\div2=145$$

$$\therefore$$ Each brother gets $$145$$ each.

Hence, option (C) is correct.

### In summer, Sarah earned $$150$$ in the first week and $$140$$ in the second week. She equally shared the total amount earned by her among her two brothers. How much money does each brother get?

A

$$500$$

.

B

$$290$$

C

$$145$$

D

$$150$$

Option C is Correct

# Mathematical Modeling on Subtraction and Division

Many times in everyday problems we need to deal with more than one operations. Only subtraction or only division does not help. We need to make use of a combination of operations in order to solve the problem.

For example:

• Tim earns $$1000$$ in a month. He paid $$100$$ for charity on Christmas eve. If he divided the remaining amount evenly among his three sisters, then how much money does each sister get?
• In the given example, Tim earns $$1000$$ in a month and he paid $$100$$ for charity. To calculate total amount of money left after paying for charity, we will subtract the money paid for charity from $$1000$$.

$$1000-100=900$$

The remaining amount is $$900$$. Thus, to calculate the amount that each sister gets, we need to divide the remaining amount by $$3.$$

$$900\div3=300$$

Tim gives $$300$$ to each sister.

#### Julie works in an aquarium shop. There are $$100$$ fishes, she sold $$20$$ fishes. She needs to divide the remaining fishes among $$5$$ aquariums. What is the number of fishes she puts in each aquarium?

A $$80$$

B $$16$$

C $$10$$

D $$100$$

×

At first, $$100$$ fishes are there in an aquarium shop. Julie sold $$20$$ fishes. To calculate the remaining fishes, we will subtract $$20$$ from $$100$$.

$$100-20$$

$$=80$$

Now, $$80$$ fishes are remaining in the aquarium shop which she needs to put equally in $$5$$ aquariums. To calculate number of fishes in each aquarium we need to divide $$80$$ by $$5.$$

$$80\div5$$

$$=16$$

She puts $$16$$ fishes in each aquarium.

Hence, option (B) is correct.

### Julie works in an aquarium shop. There are $$100$$ fishes, she sold $$20$$ fishes. She needs to divide the remaining fishes among $$5$$ aquariums. What is the number of fishes she puts in each aquarium?

A

$$80$$

.

B

$$16$$

C

$$10$$

D

$$100$$

Option B is Correct

# Mathematical Modeling on Multiplication and Division

Many times in everyday problems we need to deal with more than one operation. Only multiplication or only division does not help, we need to make use of a combination of operations in order to solve the problem.

For example:

• Carl bought $$5$$ rolls of tape to seal boxes. Each roll contains $$30$$ meter of tape. He uses $$2$$ meter of this tape to seal each box. What is the total number of boxes Carl can seal with these $$5$$ rolls of tape?
• In the given example, Carl bought $$5$$ rolls of tape and each roll is of $$30$$ meter. To calculate total meter of $$5$$ rolls of tape, we will multiply $$5$$ rolls of tape by $$30$$.

$$5×30$$

$$=150\;\text{meter}$$

• Carl has $$150$$ meter of tape. He uses $$2$$ meter to seal each box. To calculate the total number of boxes that can be sealed with $$150$$ meter, we will divide $$150$$ meter by $$2$$ meter.

$$150\div2$$

$$=75$$

• Carl can seal $$75$$ boxes with these $$5$$ rolls of tape.

#### Cooper sells tennis balls. The cost of $$5$$ balls is $$\,25$$. Cooper sells $$11$$ balls. How much money Cooper will receive?

A $$55$$

B $$275$$

C $$11$$

D $$25$$

×

To calculate the cost of one tennis ball, we will divide $$25$$ by $$5.$$

$$25\div5$$

$$=5$$

The cost of one tennis ball is $$5.$$ Cooper sells $$11$$ tennis balls. To calculate the total cost of $$11$$ tennis balls, we will multiply the cost of one ball by $$11$$ balls.

$$11×5$$

$$=55$$

Cooper will receive $$55$$ for $$11$$ tennis balls.

Hence, option (A) is correct.

### Cooper sells tennis balls. The cost of $$5$$ balls is $$\,25$$. Cooper sells $$11$$ balls. How much money Cooper will receive?

A

$$55$$

.

B

$$275$$

C

$$11$$

D

$$25$$

Option A is Correct