Exercise 1

John purchased a computer and a TV for 2,000 dollars and later sold both items for 2,260 dollars.

How much did each item cost, knowing that John sold the computer for 10% more than the purchase price, and the TV for 15% more?

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Exercise 2

What is the area of a rectangle knowing that its perimeter is 16 cm and its base is three times its height?

Exercise 3

A farm has pigs and turkeys, in total there are 58 heads and 168 paws. How many pigs and turkeys are there?

Exercise 4

John says to Peter, "I have double the amount of money that you have" and Peter replies, "if you give me six dollars we will have the same amount of money". How much money does each have?

Exercise 5

A company employs 60 people. Of this amount, 16% of the men wear glasses and 20% of the women also wear glasses. If the total number of people who wear glasses is 11, how many men and women are there in the company?

Exercise 6

The value of the digit in the tens column of a two-digit number is twice the value of the digit in the one's column, and if you subtract this by number 27, the number obtained is a number with the same digits but in reverse order. What is the number?

Exercise 7

Two appliances have been purchased for 3,500 dollars. If a 10% discount was applied to the first item and an 8% discount on the second, the total price for both purchases would have been 3,170 dollars. What is the price of each item?

Exercise 8

Find a two-digit number knowing that its digit in the tens column minus 5 is the same digit in the one's column and if the order of the digits is reversed, the number obtained is equal to the first number, minus 27.

Exercise 9

A company has three mines with ore deposits:

Nickel (%) Copper (%) Iron (%)
Mine A 1 2 3
Mine B 2 5 7
Mine C 1 3 1

How many tons from each mine should be used to obtain 7 tons of nickel, 18 tonnes of copper, and 16 tons iron?

Exercise 10

The age of a father is twice the sum of the ages of his two sons. Some years ago (exactly the difference of the current ages of children), the father's age was three times the sum of the ages of his sons. After some years (the amount that is equal to the sum of the current age of the children), the sum of the ages of all three will be 150 years. How old was the father when his sons were born?

Exercise 11

Three types of grain are sold by a farmer: wheat, barley, and millet.

Each portion of wheat sells for 4.00 dollars, the barley for 2.00 dollars, and the millet for 0.50 dollars.

If he sells 100 portions in total and receives 100 dollars from the sale, how many portions are sold of each type?

Exercise 12

There are three ingots.

  • The first of 20 grams of gold, 30 grams of silver, and 40 grams of copper.
  • The second of 30 grams of gold, 40 grams of silver, and 50 grams of copper.
  • The third of 40 grams of gold, 50 grams of silver, and 90 grams of copper.

What weight will be taken from each of the previous ingots to form a new ingot of 34 grams of gold, 46 grams of silver, and 67 grams of copper?

 

 

Solution of exercise 1

John purchased a computer and a TV for 2,000 dollars and later sold both items for 2,260 dollars.

How much did each item cost, knowing that John sold the computer for 10% more than the purchase price, and the TV for 15% more?

x price of the computer.

y price of the TV.

x + \frac { 10x }{ 100 } price of sale of the computer.

y + \frac { 15y }{ 100 } price of sale of the TV.

\left\{\begin{matrix} x + y = 2000 \\ x + \frac { 10x }{ 100 } + y + \frac { 15y }{ 100 } = 2260 \end{matrix}\right

\left\{\begin{matrix} x + y = 2000 \rightarrow \times (-110)  \rightarrow \\ 110x + 115y = 226 000 \end{matrix}\right \left\{\begin{matrix} -110x - 110y = -220 000 \\ 110x +115y = 226 000 \end{matrix}\right

If we add both equations, we will be left with 5y = 6000

y = 1200

 

x + y = 2000

x + 1200 = 2000

x = 800

800 dollar price of the computer.

1,200 dollar price of the TV.

 

Solution of exercise 2

What is the area of a rectangle knowing that its perimeter is 16 cm and its base is three times its height?

x base of the rectangle.

y height of the rectangle.

2x + 2y  perimeter.

\left\{\begin{matrix} x = 3y \\ 2x + 2y = 16 \end{matrix}\right

2 . (3y) + 2y = 16

6y + 2y = 16

8y = 16

y = 2

 

x = 3(2)

x = 6

6 cm base of the rectangle.

2 cm height of the rectangle.

 

Solution of exercise 3

A farm has pigs and turkeys, in total there are 58 heads and 168 paws. How many pigs and turkeys are there?

x number of turkeys.

y number of pigs.

\left\{\begin{matrix} x + y = 58 \\ 2x + 4y = 168 \end{matrix}\right

\left\{\begin{matrix} x + y = 58 \rightarrow \\ 2x + 4y = 168 \end{matrix}\right \left\{\begin{matrix} -2x - 2y = -116 \\ 2x + 4y = 168 \end{matrix}\right

2y = 52

y = 26

 

x + 26 = 58

x = 32

32 number of turkeys.

26 number of pigs.

 

Solution of exercise 4

John says to Peter, "I have double the amount of money that you have" and Peter replies, "if you give me six dollars we will have the same amount of money". How much money does each have?

x John's money.

y Peter's money.

\left\{\begin{matrix} x = 2y \\ y + 6 = x - 6 \end{matrix}\right

y + 6 = 2y - 6

y = 12

 

x = 2y

x = 24

24 John's money.

12 Peter's money.

 

Solution of exercise 5

A company employs 60 people. Of this amount, 16% of the men wear glasses and 20% of the women also wear glasses. If the total number of people who wear glasses is 11, how many men and women are there in the company?

x number of men.

y number of women.

\frac { 16x }{ 100 } men with glasses.

\frac { 20y }{ 100 } women with glasses.

\left\{\begin{matrix} x + y = 60 \\ 16x + 20y = 1100 \end{matrix}\right

x = 60 - y

16(60 - y) + 20y = 1100

960 - 16y + 20y = 1100

4y = 140

y = 35

 

x + 35 = 60

x = 25

25 number of men.

35 number of women.

 

Solution of exercise 6

The value of the digit in the tens column of a two-digit number is twice the value of the digit in the one's column, and if you subtract this by number 27, the number obtained is a number with the same digits but in reverse order. What is the number?

x units (ones column)

y tens (tens column)

 

10y + x number

10x + y number reversed

y = 2x

(10y + x) - 27 = 10x + y

10 . 2x + x - 27 = 10x + 2x

20x + x - 12x = 27

x = 3

 

y = 2x

y = 6

The number is 63

 

Solution of exercise 7

Two appliances have been purchased for 3,500 dollars. If a 10% discount was applied to the first item and an 8% discount on the second, the total price for both purchases would have been 3,170 dollars. What is the price of each item?

x price of the 1st.

y price of the 2nd.

x - \frac { 10x }{ 100 } discount on the 1st.

y - \frac { 8y }{ 100 } discount on the 2nd.

\left\{\begin{matrix} x + y = 3500 \\ x - \frac { 10x }{ 100 } + y - \frac { 8y }{ 100 } = 3170 \end{matrix}\right

\left\{\begin{matrix} x + y = 3500 \rightarrow \times (-90) \\ 90x + 92y = 317 000 \end{matrix}\right \left\{\begin{matrix} -90x - 90y = -315000 \\ 90x + 92y = 317000 \end{matrix}\right

2y = 2000

y = 1000

 

x + 1000 = 3500

x = 2500

2,500 dollars price of the 1st.

1,000 dollars price of the 2nd.

 

Solution of exercise 8

Find a two-digit number knowing that its digit in the tens column minus 5 is the same digit in the one's column and if the order of the digits is reversed, the number obtained is equal to the first number, minus 27.

x unit

y ten

10y + x  number

10x + y  number reversed

\left\{\begin{matrix} y = 5 - x \\ 10x + y = 10y + x - 27 \end{matrix}\right

9x - 9y = -27

9x - 9(5 - x) = -27

9x - 45 + 9x = -27

18x = 18

x = 1

 

y = 5 - x

y = 4

Number 41

 

Solution of exercise 9

A company has three mines with ore deposits:

Nickel (%) Copper (%) Iron (%)
Mine A 1 2 3
Mine B 2 5 7
Mine C 1 3 1

How many tons from each mine should be used to obtain 7 tons of nickel, 18 tonnes of copper, and 16 tons iron?

x = tons of mine A.             x=200 t

y = tons of mine B.            y=100 t

z = tons of mine C.              z=300 t

\left\{\begin{matrix} \frac { x }{ 100 } + \frac { 2y }{ 100 } + \frac { z }{ 100 } = 7 \\ \frac { 2x }{ 100 } + \frac { 5y }{ 100 } + \frac { 3z }{ 100 } = 18 \\ \frac { 3x }{ 100 } + \frac { 7y }{ 100 } + \frac { z }{ 100 } = 16 \end{matrix}\right \qquad \qquad \left\{\begin{matrix} x + 2y + z = 700 \\ 2x + 5y + 3z =1800 \\ 3x + 7y + z = 1600 \end{matrix}\right

x = 200 \qquad y = 100 \qquad z = 300

 

Solution of exercise 10

The age of a father is twice the sum of the ages of his two sons. Some years ago (exactly the difference of the current ages of children), the father's age was three times the sum of the ages of his sons. After some years (the amount that is equal to the sum of the current age of the children), the sum of the ages of all three will be 150 years. How old was the father when his sons were born?

x = Current age of the father.

y = Current age of the eldest son.

z = Current Age of youngest son.

Current Relationship:         x = 2(y + z)

It y - z years        x - (y - z) = 3[y - (y - z) + z - (y - z)]

Within y + z:        x + (y + z) + y + (y + z) + z + (y + z) = 150

 

\left\{\begin{matrix} x - 2y - 2z = 0\\ x + 2y - 8z = 0 \\ x + 4y + 4z = 150 \end{matrix}\right

x = 50 \qquad y = 15 \qquad z = 10

At birth, children, the father was 35 and 40, respectively.

 

Solution of exercise 11

Three types of grain are sold by a farmer: wheat, barley, and millet.

Each portion of wheat sells for 4.00 dollars, the barley for 2.00 dollars, and the millet for 0.50 dollars.

If he sells 100 portions in total and receives 100 dollars from the sale, how many portions are sold of each type?

x = volume of wheat.

y = Volume of barley.

z = Volume of millet.

\left\{\begin{matrix} x + y + z = 100 \\ 4x + 2y + 0.5z = 100 \end{matrix}\right \qquad \qquad \left\{\begin{matrix} y + z = 100 - x \\ 4y + z = 200 - 8x \end{matrix}\right

y = \frac { 100 - 7x }{ 3 } \qquad z = \frac { 200 + 4x }{ 3 }

Considering that the three variables are natural numbers, and that their sum is 100, the following solutions are obtained:

<tableclass="t_izq" style="margin-left:5%;">

S1 S2 S3 S4 S5 x1471013y312417103z6872768084

Solution of exercise 12

There are three ingots.

  • The first of 20 grams of gold, 30 grams of silver, and 40 grams of copper.
  • The second of 30 grams of gold, 40 grams of silver, and 50 grams of copper.
  • The third of 40 grams of gold, 50 grams of silver, and 90 grams of copper.

What weight will be taken from each of the previous ingots to form a new ingot of 34 grams of gold, 46 grams of silver, and 67 grams of copper?

 

x = weight of the 1st ingot.

y = weight of the 2nd ingot.

z = weight of the 3rd ingot.

Gold

In the 1st ingot, the law is:   \frac { 20 }{ 90 } = \frac { 2 }{ 9 }

In the 2nd ingot, the law is:   \frac { 30 }{ 120 } = \frac { 1 }{ 4 }

In the 3rd ingot, the law is:   \frac { 40 }{ 180 } = \frac { 2 }{ 9 }

The equation for gold is:

\frac { 2x }{ 9 } + \frac { y }{ 4 } + \frac { 2z }{ 9 } = 34

Silver

In the 1st ingot, the law is:    \frac { 30 }{ 90 } = \frac { 1 }{ 3 }

In the 2nd ingot, the law is:    \frac { 40 }{ 120 } = \frac { 1 }{ 3 }

In the 3rd ingot, the law is:   \frac { 50 }{ 180 } = \frac { 5 }{ 18 }

The equation for the silver is:

\frac { x }{ 3 } + \frac { y }{ 3 } + \frac { 5z }{ 18 } = 46

Copper

In the 1st ingot, the law is:   \frac { 40 }{ 90 } = \frac { 4 }{ 9 }

In the 2nd ingot, the law is:   \frac { 50 }{ 120 } = \frac { 5 }{ 12 }

In the 3rd ingot, the law is:    \frac { 90 }{ 180 } = \frac { 1 }{ 2 }

The equation for copper is:

\frac { 4x }{ 9 } + \frac { 5y }{ 12 } + \frac { z }{ 2 } = 67

\left\{\begin{matrix} 8x + 9y + 8z = 1224 \\ 6x + 6y + 5z = 828 \\ 16x + 15y + 18z = 2412 \end{matrix}\right

x = 45 \qquad y = 48 \qquad z = 54

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.