# System of Equations Word Problems

### Solutions

1John purchased a computer and a TV for \$2,000 and later sold both items for \$2,260.

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?

2What is the area of a rectangle knowing that its perimeter is 16 cm and its base is three times its height?

3A farm has pigs and turkeys, in total there are 58 heads and 168 paws. How many pigs and turkeys are there?

4John 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?

5A 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?

6The value of the digit in the tens column of a two-digit number is twice the value of the digit in the ones 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?

7Two appliances have been purchased for \$3,500. If a 10% discount was applied to the first item and a 8% discount on the second, the total price for both purchases would have been \$3,170. What is the price of each item?

8Find a two-digit number knowing that its digit in the tens column minus 5 is the same digit in the ones column and if the order of the digits is reversed, the number obtained is equal to the first number, minus 27.

9 A company has three mines with ore deposits:

Nickel (%) Copper (%) Iron (%) 1 2 3 2 5 7 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?

10The 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?

11Three types of grain are sold by a farmer: wheat, barley and millet.

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

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

12There 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.

## 1

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

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. price of sale of the computer. price of sale of the TV.    \$800 price of the computer.

\$1,200 price of the TV.

## 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.  6 cm base of the rectangle.

2 cm height of the rectangle.

## 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.    32 number of turkeys.

26 number of pigs.

## 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.   24 John's money.

12 Peter's money.

## 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. men with glasses. women with glasses.     25 number of men.

35 number of women.

## 6

The value of the digit in the tens column of a two-digit number is twice the value of the digit in the ones 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 = 6

Number 63

## 7

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

x price of the 1st.

y price of the 2nd. discount on the 1st. discount on the 2nd.   \$2,500 price of the 1st.

\$1,000 price of the 2nd.

## 8

Find a two-digit number knowing that its digit in the tens column minus 5 is the same digit in the ones 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     Number 41

## 9

A company has three mines with ore deposits:

Nickel (%) Copper (%) Iron (%) 1 2 3 2 5 7 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  ## 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  At birth, children, the father was 35 and 40, respectively.

## 11

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

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

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

x = volume of wheat.

y = Volume of barley.

z = Volume of millet.  Considering that the three variables are natural numbers, and that their sum is 100, the following solutions are obtained:

S1 S2 S3 S4 S5 x 1 4 7 10 13 y 31 24 17 10 3 z 68 72 76 80 84

## 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:    20/90 = 2/9

In the 2nd ingot, the law is:   30/120 = 1/4

In the 3rd ingot, the law is:   40/180 = 2/9

The equation for gold is: Silver

In the 1st ingot, the law is:    30/90 = 1/3

In the 2nd ingot, the law is:    40/120 = 1/3

In the 3rd ingot, the law is:   50/180 = 5/18

The equation for the silver is: Copper

In the 1st ingot, the law is:   40/90 = 4/9

In the 2nd ingot, the law is:   50/120 = 5/12

In the 3rd ingot, the law is:    90/180 = 1/2

The equation for copper is:  x = 45      y = 48      z = 54